#### Sample records for accurate numerical solutions

1. Accurate numerical solution of compressible, linear stability equations

NASA Technical Reports Server (NTRS)

Malik, M. R.; Chuang, S.; Hussaini, M. Y.

1982-01-01

The present investigation is concerned with a fourth order accurate finite difference method and its application to the study of the temporal and spatial stability of the three-dimensional compressible boundary layer flow on a swept wing. This method belongs to the class of compact two-point difference schemes discussed by White (1974) and Keller (1974). The method was apparently first used for solving the two-dimensional boundary layer equations. Attention is given to the governing equations, the solution technique, and the search for eigenvalues. A general purpose subroutine is employed for solving a block tridiagonal system of equations. The computer time can be reduced significantly by exploiting the special structure of two matrices.

2. A novel numerical technique to obtain an accurate solution to the Thomas-Fermi equation

Parand, Kourosh; Yousefi, Hossein; Delkhosh, Mehdi; Ghaderi, Amin

2016-07-01

In this paper, a new algorithm based on the fractional order of rational Euler functions (FRE) is introduced to study the Thomas-Fermi (TF) model which is a nonlinear singular ordinary differential equation on a semi-infinite interval. This problem, using the quasilinearization method (QLM), converts to the sequence of linear ordinary differential equations to obtain the solution. For the first time, the rational Euler (RE) and the FRE have been made based on Euler polynomials. In addition, the equation will be solved on a semi-infinite domain without truncating it to a finite domain by taking FRE as basic functions for the collocation method. This method reduces the solution of this problem to the solution of a system of algebraic equations. We demonstrated that the new proposed algorithm is efficient for obtaining the value of y'(0) , y(x) and y'(x) . Comparison with some numerical and analytical solutions shows that the present solution is highly accurate.

3. Spectrally accurate numerical solution of the single-particle Schrödinger equation

Batcho, P. F.

1998-06-01

We have formulated a three-dimensional fully numerical (i.e., chemical basis-set free) method and applied it to the solution of the single-particle Schrödinger equation. The numerical method combines the rapid ``exponential'' convergence rates of spectral methods with the geometric flexibility of finite-element methods and can be viewed as an extension of the spectral element method. Singularities associated with multicenter systems are efficiently integrated by a Duffy transformation and the discrete operator is formulated by a variational statement. The method is applicable to molecular modeling for quantum chemical calculations on polyatomic systems. The complete system is shown to be efficiently inverted by the preconditioned conjugate gradient method and exponential convergence rates in numerical approximations are demonstrated for suitable benchmark problems including the hydrogenlike orbitals of nitrogen.

4. Third-order-accurate numerical methods for efficient, large time-step solutions of mixed linear and nonlinear problems

SciTech Connect

Cobb, J.W.

1995-02-01

There is an increasing need for more accurate numerical methods for large-scale nonlinear magneto-fluid turbulence calculations. These methods should not only increase the current state of the art in terms of accuracy, but should also continue to optimize other desired properties such as simplicity, minimized computation, minimized memory requirements, and robust stability. This includes the ability to stably solve stiff problems with long time-steps. This work discusses a general methodology for deriving higher-order numerical methods. It also discusses how the selection of various choices can affect the desired properties. The explicit discussion focuses on third-order Runge-Kutta methods, including general solutions and five examples. The study investigates the linear numerical analysis of these methods, including their accuracy, general stability, and stiff stability. Additional appendices discuss linear multistep methods, discuss directions for further work, and exhibit numerical analysis results for some other commonly used lower-order methods.

5. An accurate numerical solution to the Saint-Venant-Hirano model for mixed-sediment morphodynamics in rivers

Stecca, Guglielmo; Siviglia, Annunziato; Blom, Astrid

2016-07-01

We present an accurate numerical approximation to the Saint-Venant-Hirano model for mixed-sediment morphodynamics in one space dimension. Our solution procedure originates from the fully-unsteady matrix-vector formulation developed in [54]. The principal part of the problem is solved by an explicit Finite Volume upwind method of the path-conservative type, by which all the variables are updated simultaneously in a coupled fashion. The solution to the principal part is embedded into a splitting procedure for the treatment of frictional source terms. The numerical scheme is extended to second-order accuracy and includes a bookkeeping procedure for handling the evolution of size stratification in the substrate. We develop a concept of balancedness for the vertical mass flux between the substrate and active layer under bed degradation, which prevents the occurrence of non-physical oscillations in the grainsize distribution of the substrate. We suitably modify the numerical scheme to respect this principle. We finally verify the accuracy in our solution to the equations, and its ability to reproduce one-dimensional morphodynamics due to streamwise and vertical sorting, using three test cases. In detail, (i) we empirically assess the balancedness of vertical mass fluxes under degradation; (ii) we study the convergence to the analytical linearised solution for the propagation of infinitesimal-amplitude waves [54], which is here employed for the first time to assess a mixed-sediment model; (iii) we reproduce Ribberink's E8-E9 flume experiment [46].

6. Numerical Asymptotic Solutions Of Differential Equations

NASA Technical Reports Server (NTRS)

Thurston, Gaylen A.

1992-01-01

Numerical algorithms derived and compared with classical analytical methods. In method, expansions replaced with integrals evaluated numerically. Resulting numerical solutions retain linear independence, main advantage of asymptotic solutions.

7. High order accurate solutions of viscous problems

NASA Technical Reports Server (NTRS)

Hayder, M. E.; Turkel, Eli

1993-01-01

We consider a fourth order extension to MacCormack's scheme. The original extension was fourth order only for the inviscid terms but was second order for the viscous terms. We show how to modify the viscous terms so that the scheme is uniformly fourth order in the spatial derivatives. Applications are given to some boundary layer flows. In addition, for applications to shear flows the effect of the outflow boundary conditions are very important. We compare the accuracy of several of these different boundary conditions for both boundary layer and shear flows. Stretching at the outflow usually increases the oscillations in the numerical solution but the addition of a filtered sponge layer (with or without stretching) reduces such oscillations. The oscillations are generated by insufficient resolution of the shear layer. When the shear layer is sufficiently resolved then oscillations are not generated and there is less of a need for a nonreflecting boundary condition.

8. Numerical evolution of multiple black holes with accurate initial data

SciTech Connect

Galaviz, Pablo; Bruegmann, Bernd; Cao Zhoujian

2010-07-15

We present numerical evolutions of three equal-mass black holes using the moving puncture approach. We calculate puncture initial data for three black holes solving the constraint equations by means of a high-order multigrid elliptic solver. Using these initial data, we show the results for three black hole evolutions with sixth-order waveform convergence. We compare results obtained with the BAM and AMSS-NCKU codes with previous results. The approximate analytic solution to the Hamiltonian constraint used in previous simulations of three black holes leads to different dynamics and waveforms. We present some numerical experiments showing the evolution of four black holes and the resulting gravitational waveform.

9. Numerical solution methods for viscoelastic orthotropic materials

NASA Technical Reports Server (NTRS)

Gramoll, K. C.; Dillard, D. A.; Brinson, H. F.

1988-01-01

Numerical solution methods for viscoelastic orthotropic materials, specifically fiber reinforced composite materials, are examined. The methods include classical lamination theory using time increments, direction solution of the Volterra Integral, Zienkiewicz's linear Prony series method, and a new method called Nonlinear Differential Equation Method (NDEM) which uses a nonlinear Prony series. The criteria used for comparison of the various methods include the stability of the solution technique, time step size stability, computer solution time length, and computer memory storage. The Volterra Integral allowed the implementation of higher order solution techniques but had difficulties solving singular and weakly singular compliance function. The Zienkiewicz solution technique, which requires the viscoelastic response to be modeled by a Prony series, works well for linear viscoelastic isotropic materials and small time steps. The new method, NDEM, uses a modified Prony series which allows nonlinear stress effects to be included and can be used with orthotropic nonlinear viscoelastic materials. The NDEM technique is shown to be accurate and stable for both linear and nonlinear conditions with minimal computer time.

10. Accurate deterministic solutions for the classic Boltzmann shock profile

Yue, Yubei

The Boltzmann equation or Boltzmann transport equation is a classical kinetic equation devised by Ludwig Boltzmann in 1872. It is regarded as a fundamental law in rarefied gas dynamics. Rather than using macroscopic quantities such as density, temperature, and pressure to describe the underlying physics, the Boltzmann equation uses a distribution function in phase space to describe the physical system, and all the macroscopic quantities are weighted averages of the distribution function. The information contained in the Boltzmann equation is surprisingly rich, and the Euler and Navier-Stokes equations of fluid dynamics can be derived from it using series expansions. Moreover, the Boltzmann equation can reach regimes far from the capabilities of fluid dynamical equations, such as the realm of rarefied gases---the topic of this thesis. Although the Boltzmann equation is very powerful, it is extremely difficult to solve in most situations. Thus the only hope is to solve it numerically. But soon one finds that even a numerical simulation of the equation is extremely difficult, due to both the complex and high-dimensional integral in the collision operator, and the hyperbolic phase-space advection terms. For this reason, until few years ago most numerical simulations had to rely on Monte Carlo techniques. In this thesis I will present a new and robust numerical scheme to compute direct deterministic solutions of the Boltzmann equation, and I will use it to explore some classical gas-dynamical problems. In particular, I will study in detail one of the most famous and intrinsically nonlinear problems in rarefied gas dynamics, namely the accurate determination of the Boltzmann shock profile for a gas of hard spheres.

11. Numerical solution of under-resolved detonations

Tosatto, Luca; Vigevano, Luigi

2008-02-01

A new fractional-step method is proposed for the numerical solution of high speed reacting flows, where the chemical time scales are often much smaller than the fluid dynamical time scales. When the problem is stiff, because of insufficient spatial/temporal resolution, a well-known spurious numerical phenomenon occurs in standard finite volume schemes: the incorrect calculation of the speed of propagation of discontinuities. The new method is first illustrated considering a one-dimensional scalar hyperbolic advection/reaction equation with stiff source term, which may be considered as a model problem to under-resolved detonations. During the reaction step, the proposed scheme replaces the cell average representation with a two-value reconstruction, which allows us to locate the discontinuity position inside the cell during the computation of the source term. This results in the correct propagation of discontinuities even in the stiff case. The method is proved to be second-order accurate for smooth solutions of scalar equations and is applied successfully to the solution of the one-dimensional reactive Euler equations for Chapman-Jouguet detonations.

12. Accurate description of calcium solvation in concentrated aqueous solutions.

PubMed

Kohagen, Miriam; Mason, Philip E; Jungwirth, Pavel

2014-07-17

Calcium is one of the biologically most important ions; however, its accurate description by classical molecular dynamics simulations is complicated by strong electrostatic and polarization interactions with surroundings due to its divalent nature. Here, we explore the recently suggested approach for effectively accounting for polarization effects via ionic charge rescaling and develop a new and accurate parametrization of the calcium dication. Comparison to neutron scattering and viscosity measurements demonstrates that our model allows for an accurate description of concentrated aqueous calcium chloride solutions. The present model should find broad use in efficient and accurate modeling of calcium in aqueous environments, such as those encountered in biological and technological applications.

13. Spurious Numerical Solutions Of Differential Equations

NASA Technical Reports Server (NTRS)

Lafon, A.; Yee, H. C.

1995-01-01

Paper presents detailed study of spurious steady-state numerical solutions of differential equations that contain nonlinear source terms. Main objectives of this study are (1) to investigate how well numerical steady-state solutions of model nonlinear reaction/convection boundary-value problem mimic true steady-state solutions and (2) to relate findings of this investigation to implications for interpretation of numerical results from computational-fluid-dynamics algorithms and computer codes used to simulate reacting flows.

14. A time-accurate adaptive grid method and the numerical simulation of a shock-vortex interaction

NASA Technical Reports Server (NTRS)

Bockelie, Michael J.; Eiseman, Peter R.

1990-01-01

A time accurate, general purpose, adaptive grid method is developed that is suitable for multidimensional steady and unsteady numerical simulations. The grid point movement is performed in a manner that generates smooth grids which resolve the severe solution gradients and the sharp transitions in the solution gradients. The temporal coupling of the adaptive grid and the PDE solver is performed with a grid prediction correction method that is simple to implement and ensures the time accuracy of the grid. Time accurate solutions of the 2-D Euler equations for an unsteady shock vortex interaction demonstrate the ability of the adaptive method to accurately adapt the grid to multiple solution features.

15. ACCURATE CHEMICAL MASTER EQUATION SOLUTION USING MULTI-FINITE BUFFERS

PubMed Central

Cao, Youfang; Terebus, Anna; Liang, Jie

2016-01-01

The discrete chemical master equation (dCME) provides a fundamental framework for studying stochasticity in mesoscopic networks. Because of the multi-scale nature of many networks where reaction rates have large disparity, directly solving dCMEs is intractable due to the exploding size of the state space. It is important to truncate the state space effectively with quantified errors, so accurate solutions can be computed. It is also important to know if all major probabilistic peaks have been computed. Here we introduce the Accurate CME (ACME) algorithm for obtaining direct solutions to dCMEs. With multi-finite buffers for reducing the state space by O(n!), exact steady-state and time-evolving network probability landscapes can be computed. We further describe a theoretical framework of aggregating microstates into a smaller number of macrostates by decomposing a network into independent aggregated birth and death processes, and give an a priori method for rapidly determining steady-state truncation errors. The maximal sizes of the finite buffers for a given error tolerance can also be pre-computed without costly trial solutions of dCMEs. We show exactly computed probability landscapes of three multi-scale networks, namely, a 6-node toggle switch, 11-node phage-lambda epigenetic circuit, and 16-node MAPK cascade network, the latter two with no known solutions. We also show how probabilities of rare events can be computed from first-passage times, another class of unsolved problems challenging for simulation-based techniques due to large separations in time scales. Overall, the ACME method enables accurate and efficient solutions of the dCME for a large class of networks. PMID:27761104

16. Numerical and approximate solutions for plume rise

Krishnamurthy, Ramesh; Gordon Hall, J.

Numerical and approximate analytical solutions are compared for turbulent plume rise in a crosswind. The numerical solutions were calculated using the plume rise model of Hoult, Fay and Forney (1969, J. Air Pollut. Control Ass.19, 585-590), over a wide range of pertinent parameters. Some wind shear and elevated inversion effects are included. The numerical solutions are seen to agree with the approximate solutions over a fairly wide range of the parameters. For the conditions considered in the study, wind shear effects are seen to be quite small. A limited study was made of the penetration of elevated inversions by plumes. The results indicate the adequacy of a simple criterion proposed by Briggs (1969, AEC Critical Review Series, USAEC Division of Technical Information extension, Oak Ridge, Tennesse).

17. Fast and Accurate Learning When Making Discrete Numerical Estimates.

PubMed

Sanborn, Adam N; Beierholm, Ulrik R

2016-04-01

18. Fast and Accurate Learning When Making Discrete Numerical Estimates.

PubMed

Sanborn, Adam N; Beierholm, Ulrik R

2016-04-01

19. Fast and Accurate Learning When Making Discrete Numerical Estimates

PubMed Central

Sanborn, Adam N.; Beierholm, Ulrik R.

2016-01-01

20. A new benchmark with high accurate solution for hot-cold fluids mixing

Younes, Anis; Fahs, Marwan; Zidane, Ali; Huggenberger, Peter; Zechner, Eric

2015-09-01

A new benchmark is proposed for the verification of buoyancy-driven flow codes. The benchmark deals with mixing hot and cold fluids from the opposite boundaries of an open channel. A high accurate solution is developed using the Fourier-Galerkin (FG) method and compared to the results of an advanced finite element (FE) model. An excellent agreement is observed between the FG and FE solutions for different Reynolds numbers which demonstrates the viability of the solutions in benchmarking buoyancy-driven flow numerical codes.

1. Analytical and Numerical Solution for a Solidifying Liquid Alloy Slab

NASA Technical Reports Server (NTRS)

Antar, B. N.

1983-01-01

Numerical and analytical solutions are presented for the temperature and concentration distributions during the solidification of a binary liquid alloy slab. The slab is taken to be of a finite depth but infinite in the horizontal direction. The solidification process is started by withdrawing a fixed amount of heat from the lower surface of the slab. The upper surface of the slab is subjected to both radiation and convective conditions. The solution gives the concentration and temperature profiles and the interface position as a function of time. Due to the smallness of the mass diffusion coefficient in the solid, the numerical solution method breaks down whenever the ratio of the diffusivities in the solid and the liquid falls below a certain value. An analytical method is developed which gives accurate solution for any value of the diffusivity ratio.

2. PolyPole-1: An accurate numerical algorithm for intra-granular fission gas release

Pizzocri, D.; Rabiti, C.; Luzzi, L.; Barani, T.; Van Uffelen, P.; Pastore, G.

2016-09-01

The transport of fission gas from within the fuel grains to the grain boundaries (intra-granular fission gas release) is a fundamental controlling mechanism of fission gas release and gaseous swelling in nuclear fuel. Hence, accurate numerical solution of the corresponding mathematical problem needs to be included in fission gas behaviour models used in fuel performance codes. Under the assumption of equilibrium between trapping and resolution, the process can be described mathematically by a single diffusion equation for the gas atom concentration in a grain. In this paper, we propose a new numerical algorithm (PolyPole-1) to efficiently solve the fission gas diffusion equation in time-varying conditions. The PolyPole-1 algorithm is based on the analytic modal solution of the diffusion equation for constant conditions, combined with polynomial corrective terms that embody the information on the deviation from constant conditions. The new algorithm is verified by comparing the results to a finite difference solution over a large number of randomly generated operation histories. Furthermore, comparison to state-of-the-art algorithms used in fuel performance codes demonstrates that the accuracy of PolyPole-1 is superior to other algorithms, with similar computational effort. Finally, the concept of PolyPole-1 may be extended to the solution of the general problem of intra-granular fission gas diffusion during non-equilibrium trapping and resolution, which will be the subject of future work.

3. Accurate numerical simulation of short fiber optical parametric amplifiers.

PubMed

Marhic, M E; Rieznik, A A; Kalogerakis, G; Braimiotis, C; Fragnito, H L; Kazovsky, L G

2008-03-17

We improve the accuracy of numerical simulations for short fiber optical parametric amplifiers (OPAs). Instead of using the usual coarse-step method, we adopt a model for birefringence and dispersion which uses fine-step variations of the parameters. We also improve the split-step Fourier method by exactly treating the nonlinear ellipse rotation terms. We find that results obtained this way for two-pump OPAs can be significantly different from those obtained by using the usual coarse-step fiber model, and/or neglecting ellipse rotation terms.

4. Numerical solution of the electron transport equation

Woods, Mark

The electron transport equation has been solved many times for a variety of reasons. The main difficulty in its numerical solution is that it is a very stiff boundary value problem. The most common numerical methods for solving boundary value problems are symmetric collocation methods and shooting methods. Both of these types of methods can only be applied to the electron transport equation if the boundary conditions are altered with unrealistic assumptions because they require too many points to be practical. Further, they result in oscillating and negative solutions, which are physically meaningless for the problem at hand. For these reasons, all numerical methods for this problem to date are a bit unusual because they were designed to try and avoid the problem of extreme stiffness. This dissertation shows that there is no need to introduce spurious boundary conditions or invent other numerical methods for the electron transport equation. Rather, there already exists methods for very stiff boundary value problems within the numerical analysis literature. We demonstrate one such method in which the fast and slow modes of the boundary value problem are essentially decoupled. This allows for an upwind finite difference method to be applied to each mode as is appropriate. This greatly reduces the number of points needed in the mesh, and we demonstrate how this eliminates the need to define new boundary conditions. This method is verified by showing that under certain restrictive assumptions, the electron transport equation has an exact solution that can be written as an integral. We show that the solution from the upwind method agrees with the quadrature evaluation of the exact solution. This serves to verify that the upwind method is properly solving the electron transport equation. Further, it is demonstrated that the output of the upwind method can be used to compute auroral light emissions.

5. Numerical solution of a tunneling equation

SciTech Connect

Wang, C.Y.; Carter, M.D.; Batchelor, D.B.; Jaeger, E.F.

1994-04-01

A numerical method is presented to solve mode conversion equations resulting from the use of radio frequency (rf) waves to heat plasmas. The solutions of the mode conversion equations contain exponentially growing modes, and ordinary numerical techniques give large errors. To avoid the unphysical growing modes, a set of boundary conditions are found, that eliminate the unphysical modes. The mode conversion equations are then solved with the boundary conditions as a standard two-point boundary value problem. A tunneling equation (one of the mode conversion equations without power absorption) is solved as a specific example of this numerical technique although the technique itself is very general and can be easily applied to solve any mode conversion equation. The results from the numerical calculation agree very well with those found from asymptotic analysis.

6. A numerical method for solving systems of linear ordinary differential equations with rapidly oscillating solutions

NASA Technical Reports Server (NTRS)

Bernstein, Ira B.; Brookshaw, Leigh; Fox, Peter A.

1992-01-01

The present numerical method for accurate and efficient solution of systems of linear equations proceeds by numerically developing a set of basis solutions characterized by slowly varying dependent variables. The solutions thus obtained are shown to have a computational overhead largely independent of the small size of the scale length which characterizes the solutions; in many cases, the technique obviates series solutions near singular points, and its known sources of error can be easily controlled without a substantial increase in computational time.

7. Efficient and accurate numerical methods for the Klein-Gordon-Schroedinger equations

SciTech Connect

Bao, Weizhu . E-mail: bao@math.nus.edu.sg; Yang, Li . E-mail: yangli@nus.edu.sg

2007-08-10

In this paper, we present efficient, unconditionally stable and accurate numerical methods for approximations of the Klein-Gordon-Schroedinger (KGS) equations with/without damping terms. The key features of our methods are based on: (i) the application of a time-splitting spectral discretization for a Schroedinger-type equation in KGS (ii) the utilization of Fourier pseudospectral discretization for spatial derivatives in the Klein-Gordon equation in KGS (iii) the adoption of solving the ordinary differential equations (ODEs) in phase space analytically under appropriate chosen transmission conditions between different time intervals or applying Crank-Nicolson/leap-frog for linear/nonlinear terms for time derivatives. The numerical methods are either explicit or implicit but can be solved explicitly, unconditionally stable, and of spectral accuracy in space and second-order accuracy in time. Moreover, they are time reversible and time transverse invariant when there is no damping terms in KGS, conserve (or keep the same decay rate of) the wave energy as that in KGS without (or with a linear) damping term, keep the same dynamics of the mean value of the meson field, and give exact results for the plane-wave solution. Extensive numerical tests are presented to confirm the above properties of our numerical methods for KGS. Finally, the methods are applied to study solitary-wave collisions in one dimension (1D), as well as dynamics of a 2D problem in KGS.

8. Accurate solutions for transonic viscous flow over finite wings

NASA Technical Reports Server (NTRS)

Vatsa, V. N.

1986-01-01

An explicit multistage Runge-Kutta type time-stepping scheme is used for solving the three-dimensional, compressible, thin-layer Navier-Stokes equations. A finite-volume formulation is employed to facilitate treatment of complex grid topologies encountered in three-dimensional calculations. Convergence to steady state is expedited through usage of acceleration techniques. Further numerical efficiency is achieved through vectorization of the computer code. The accuracy of the overall scheme is evaluated by comparing the computed solutions with the experimental data for a finite wing under different test conditions in the transonic regime. A grid refinement study ir conducted to estimate the grid requirements for adequate resolution of salient features of such flows.

9. Numerical methods for finding stationary gravitational solutions

Dias, Óscar J. C.; Santos, Jorge E.; Way, Benson

2016-07-01

The wide applications of higher dimensional gravity and gauge/gravity duality have fuelled the search for new stationary solutions of the Einstein equation (possibly coupled to matter). In this topical review, we explain the mathematical foundations and give a practical guide for the numerical solution of gravitational boundary value problems. We present these methods by way of example: resolving asymptotically flat black rings, singly spinning lumpy black holes in anti-de Sitter (AdS), and the Gregory-Laflamme zero modes of small rotating black holes in AdS{}5× {S}5. We also include several tools and tricks that have been useful throughout the literature.

10. Accurate numerical solutions for elastic-plastic models. [LMFBR

SciTech Connect

Schreyer, H. L.; Kulak, R. F.; Kramer, J. M.

1980-03-01

The accuracy of two integration algorithms is studied for the common engineering condition of a von Mises, isotropic hardening model under plane stress. Errors in stress predictions for given total strain increments are expressed with contour plots of two parameters: an angle in the pi plane and the difference between the exact and computed yield-surface radii. The two methods are the tangent-predictor/radial-return approach and the elastic-predictor/radial-corrector algorithm originally developed by Mendelson. The accuracy of a combined tangent-predictor/radial-corrector algorithm is also investigated.

11. High-order numerical solutions using cubic splines

NASA Technical Reports Server (NTRS)

Rubin, S. G.; Khosla, P. K.

1975-01-01

The cubic spline collocation procedure for the numerical solution of partial differential equations was reformulated so that the accuracy of the second-derivative approximation is improved and parallels that previously obtained for lower derivative terms. The final result is a numerical procedure having overall third-order accuracy for a nonuniform mesh and overall fourth-order accuracy for a uniform mesh. Application of the technique was made to the Burger's equation, to the flow around a linear corner, to the potential flow over a circular cylinder, and to boundary layer problems. The results confirmed the higher-order accuracy of the spline method and suggest that accurate solutions for more practical flow problems can be obtained with relatively coarse nonuniform meshes.

12. Numerical solution of large Lyapunov equations

NASA Technical Reports Server (NTRS)

1989-01-01

A few methods are proposed for solving large Lyapunov equations that arise in control problems. The common case where the right hand side is a small rank matrix is considered. For the single input case, i.e., when the equation considered is of the form AX + XA(sup T) + bb(sup T) = 0, where b is a column vector, the existence of approximate solutions of the form X = VGV(sup T) where V is N x m and G is m x m, with m small is established. The first class of methods proposed is based on the use of numerical quadrature formulas, such as Gauss-Laguerre formulas, applied to the controllability Grammian. The second is based on a projection process of Galerkin type. Numerical experiments are presented to test the effectiveness of these methods for large problems.

13. Numerical analysis of the asymptotic two-point boundary value solution for N-body trajectories.

NASA Technical Reports Server (NTRS)

Lancaster, J. E.; Allemann, R. A.

1972-01-01

Previously published asymptotic solutions for lunar and interplanetary trajectories have been modified and combined to formulate a general analytical boundary value solution applicable to a broad class of trajectory problems. In addition, the earlier first-order solutions have been extended to second-order to determine if improved accuracy is possible. Comparisons between the asymptotic solution and numerical integration for several lunar and interplanetary trajectories show that the asymptotic solution is generally quite accurate. Also, since no iterations are required, a solution to the boundary value problem is obtained in a fraction of the time required for numerically integrated solutions.

14. Comparison between analytical and numerical solution of mathematical drying model

Shahari, N.; Rasmani, K.; Jamil, N.

2016-02-01

Drying is often related to the food industry as a process of shifting heat and mass inside food, which helps in preserving food. Previous research using a mass transfer equation showed that the results were mostly concerned with the comparison between the simulation model and the experimental data. In this paper, the finite difference method was used to solve a mass equation during drying using different kinds of boundary condition, which are equilibrium and convective boundary conditions. The results of these two models provide a comparison between the analytical and the numerical solution. The result shows a close match between the two solution curves. It is concluded that the two proposed models produce an accurate solution to describe the moisture distribution content during the drying process. This analysis indicates that we have confidence in the behaviour of moisture in the numerical simulation. This result demonstrated that a combined analytical and numerical approach prove that the system is behaving physically. Based on this assumption, the model of mass transfer was extended to include the temperature transfer, and the result shows a similar trend to those presented in the simpler case.

15. Numerical integration of asymptotic solutions of ordinary differential equations

NASA Technical Reports Server (NTRS)

Thurston, Gaylen A.

1989-01-01

Classical asymptotic analysis of ordinary differential equations derives approximate solutions that are numerically stable. However, the analysis also leads to tedious expansions in powers of the relevant parameter for a particular problem. The expansions are replaced with integrals that can be evaluated by numerical integration. The resulting numerical solutions retain the linear independence that is the main advantage of asymptotic solutions. Examples, including the Falkner-Skan equation from laminar boundary layer theory, illustrate the method of asymptotic analysis with numerical integration.

16. Accurate solution of the Dirac equation on Lagrange meshes.

PubMed

Baye, Daniel; Filippin, Livio; Godefroid, Michel

2014-04-01

The Lagrange-mesh method is an approximate variational method taking the form of equations on a grid because of the use of a Gauss quadrature approximation. With a basis of Lagrange functions involving associated Laguerre polynomials related to the Gauss quadrature, the method is applied to the Dirac equation. The potential may possess a 1/r singularity. For hydrogenic atoms, numerically exact energies and wave functions are obtained with small numbers n+1 of mesh points, where n is the principal quantum number. Numerically exact mean values of powers -2 to 3 of the radial coordinate r can also be obtained with n+2 mesh points. For the Yukawa potential, a 15-digit agreement with benchmark energies of the literature is obtained with 50 or fewer mesh points.

17. Highly accurate boronimeter assay of concentrated boric acid solutions

SciTech Connect

Ball, R.M. )

1992-01-01

The Random-Walk Boronimeter has successfully been used as an on-line indicator of boric acid concentration in an operating commercial pressurized water reactor. The principle has been adapted for measurement of discrete samples to high accuracy and to concentrations up to 6000 ppm natural boron in light water. Boric acid concentration in an aqueous solution is a necessary measurement in many nuclear power plants, particularly those that use boric acid dissolved in the reactor coolant as a reactivity control system. Other nuclear plants use a high-concentration boric acid solution as a backup shutdown system. Such a shutdown system depends on rapid injection of the solution and frequent surveillance of the fluid to ensure the presence of the neutron absorber. The two methods typically used to measure boric acid are the chemical and the physical methods. The chemical method uses titration to determine the ionic concentration of the BO[sub 3] ions and infers the boron concentration. The physical method uses the attenuation of neutrons by the solution and infers the boron concentration from the neutron absorption properties. This paper describes the Random-Walk Boronimeter configured to measure discrete samples to high accuracy and high concentration.

18. A numerical solution of Duffing's equations including the prediction of jump phenomena

NASA Technical Reports Server (NTRS)

Moyer, E. T., Jr.; Ghasghai-Abdi, E.

1987-01-01

Numerical methodology for the solution of Duffing's differential equation is presented. Algorithms for the prediction of multiple equilibrium solutions and jump phenomena are developed. In addition, a filtering algorithm for producing steady state solutions is presented. The problem of a rigidly clamped circular plate subjected to cosinusoidal pressure loading is solved using the developed algorithms (the plate is assumed to be in the geometrically nonlinear range). The results accurately predict regions of solution multiplicity and jump phenomena.

19. Accurate Critical Stress Intensity Factor Griffith Crack Theory Measurements by Numerical Techniques

PubMed Central

Petersen, Richard C.

2014-01-01

Critical stress intensity factor (KIc) has been an approximation for fracture toughness using only load-cell measurements. However, artificial man-made cracks several orders of magnitude longer and wider than natural flaws have required a correction factor term (Y) that can be up to about 3 times the recorded experimental value [1-3]. In fact, over 30 years ago a National Academy of Sciences advisory board stated that empirical KIc testing was of serious concern and further requested that an accurate bulk fracture toughness method be found [4]. Now that fracture toughness can be calculated accurately by numerical integration from the load/deflection curve as resilience, work of fracture (WOF) and strain energy release (SIc) [5, 6], KIc appears to be unnecessary. However, the large body of previous KIc experimental test results found in the literature offer the opportunity for continued meta analysis with other more practical and accurate fracture toughness results using energy methods and numerical integration. Therefore, KIc is derived from the classical Griffith Crack Theory [6] to include SIc as a more accurate term for strain energy release rate (𝒢Ic), along with crack surface energy (γ), crack length (a), modulus (E), applied stress (σ), Y, crack-tip plastic zone defect region (rp) and yield strength (σys) that can all be determined from load and deflection data. Polymer matrix discontinuous quartz fiber-reinforced composites to accentuate toughness differences were prepared for flexural mechanical testing comprising of 3 mm fibers at different volume percentages from 0-54.0 vol% and at 28.2 vol% with different fiber lengths from 0.0-6.0 mm. Results provided a new correction factor and regression analyses between several numerical integration fracture toughness test methods to support KIc results. Further, bulk KIc accurate experimental values are compared with empirical test results found in literature. Also, several fracture toughness mechanisms

20. Multiresolution strategies for the numerical solution of optimal control problems

Jain, Sachin

There exist many numerical techniques for solving optimal control problems but less work has been done in the field of making these algorithms run faster and more robustly. The main motivation of this work is to solve optimal control problems accurately in a fast and efficient way. Optimal control problems are often characterized by discontinuities or switchings in the control variables. One way of accurately capturing the irregularities in the solution is to use a high resolution (dense) uniform grid. This requires a large amount of computational resources both in terms of CPU time and memory. Hence, in order to accurately capture any irregularities in the solution using a few computational resources, one can refine the mesh locally in the region close to an irregularity instead of refining the mesh uniformly over the whole domain. Therefore, a novel multiresolution scheme for data compression has been designed which is shown to outperform similar data compression schemes. Specifically, we have shown that the proposed approach results in fewer grid points in the grid compared to a common multiresolution data compression scheme. The validity of the proposed mesh refinement algorithm has been verified by solving several challenging initial-boundary value problems for evolution equations in 1D. The examples have demonstrated the stability and robustness of the proposed algorithm. The algorithm adapted dynamically to any existing or emerging irregularities in the solution by automatically allocating more grid points to the region where the solution exhibited sharp features and fewer points to the region where the solution was smooth. Thereby, the computational time and memory usage has been reduced significantly, while maintaining an accuracy equivalent to the one obtained using a fine uniform mesh. Next, a direct multiresolution-based approach for solving trajectory optimization problems is developed. The original optimal control problem is transcribed into a

1. Explicit numerical solutions of a microbial survival model under nonisothermal conditions.

PubMed

Zhu, Si; Chen, Guibing

2016-03-01

Differential equations used to describe the original and modified Geeraerd models were, respectively, simplified into an explicit equation in which the integration of the specific inactivation rate with respect to time was numerically approximated using the Simpson's rule. The explicit numerical solutions were then used to simulate microbial survival curves and fit nonisothermal survival data for identifying model parameters in Microsoft Excel. The results showed that the explicit numerical solutions provided an easy way to accurately simulate microbial survival and estimate model parameters from nonisothermal survival data using the Geeraerd models.

2. Recommendations for Achieving Accurate Numerical Simulation of Tip Clearance Flows in Transonic Compressor Rotors

NASA Technical Reports Server (NTRS)

VanZante, Dale E.; Strazisar, Anthony J.; Wood, Jerry R,; Hathaway, Michael D.; Okiishi, Theodore H.

2000-01-01

The tip clearance flows of transonic compressor rotors are important because they have a significant impact on rotor and stage performance. While numerical simulations of these flows are quite sophisticated. they are seldom verified through rigorous comparisons of numerical and measured data because these kinds of measurements are rare in the detail necessary to be useful in high-speed machines. In this paper we compare measured tip clearance flow details (e.g. trajectory and radial extent) with corresponding data obtained from a numerical simulation. Recommendations for achieving accurate numerical simulation of tip clearance flows are presented based on this comparison. Laser Doppler Velocimeter (LDV) measurements acquired in a transonic compressor rotor, NASA Rotor 35, are used. The tip clearance flow field of this transonic rotor was simulated using a Navier-Stokes turbomachinery solver that incorporates an advanced k-epsilon turbulence model derived for flows that are not in local equilibrium. Comparison between measured and simulated results indicates that simulation accuracy is primarily dependent upon the ability of the numerical code to resolve important details of a wall-bounded shear layer formed by the relative motion between the over-tip leakage flow and the shroud wall. A simple method is presented for determining the strength of this shear layer.

3. The numerical solution of thermoporoelastoplasticity problems

Sivtsev, P. V.; Kolesov, A. E.; Sirditov, I. K.; Stepanov, S. P.

2016-10-01

Before constructing buildings in permafrost areas the careful study of stress-strain state of soils and building foundations must be performed in order to estimate their bearing capacity and stability to avoid issues with maintenance. To determine stress-strain state of frozen soils the numerical modeling of thermoporoelastoplasticity problems is used. The mathematical model of considered problems includes the elasto-plasticity equations and equations of heat and mass transfer with phase transition. The computational algorithm is based on the finite element approximation in space and the finite difference approximation in time. As the model problem we consider the deformation of soil under house weight and heating. Special attention is given to thawing of frozen soils, which can cause additional deformations and lead to loss of stability.

4. Towards numerically accurate many-body perturbation theory: Short-range correlation effects

SciTech Connect

Gulans, Andris

2014-10-28

The example of the uniform electron gas is used for showing that the short-range electron correlation is difficult to handle numerically, while it noticeably contributes to the self-energy. Nonetheless, in condensed-matter applications studied with advanced methods, such as the GW and random-phase approximations, it is common to neglect contributions due to high-momentum (large q) transfers. Then, the short-range correlation is poorly described, which leads to inaccurate correlation energies and quasiparticle spectra. To circumvent this problem, an accurate extrapolation scheme is proposed. It is based on an analytical derivation for the uniform electron gas presented in this paper, and it provides an explanation why accurate GW quasiparticle spectra are easy to obtain for some compounds and very difficult for others.

5. Final Report for "Accurate Numerical Models of the Secondary Electron Yield from Grazing-incidence Collisions".

SciTech Connect

Seth A Veitzer

2008-10-21

Effects of stray electrons are a main factor limiting performance of many accelerators. Because heavy-ion fusion (HIF) accelerators will operate in regimes of higher current and with walls much closer to the beam than accelerators operating today, stray electrons might have a large, detrimental effect on the performance of an HIF accelerator. A primary source of stray electrons is electrons generated when halo ions strike the beam pipe walls. There is some research on these types of secondary electrons for the HIF community to draw upon, but this work is missing one crucial ingredient: the effect of grazing incidence. The overall goal of this project was to develop the numerical tools necessary to accurately model the effect of grazing incidence on the behavior of halo ions in a HIF accelerator, and further, to provide accurate models of heavy ion stopping powers with applications to ICF, WDM, and HEDP experiments.

6. Evolution of midplate hotspot swells: Numerical solutions

NASA Technical Reports Server (NTRS)

Liu, Mian; Chase, Clement G.

1990-01-01

The evolution of midplate hotspot swells on an oceanic plate moving over a hot, upwelling mantle plume is numerically simulated. The plume supplies a Gaussian-shaped thermal perturbation and thermally-induced dynamic support. The lithosphere is treated as a thermal boundary layer with a strongly temperature-dependent viscosity. The two fundamental mechanisms of transferring heat, conduction and convection, during the interaction of the lithosphere with the mantle plume are considered. The transient heat transfer equations, with boundary conditions varying in both time and space, are solved in cylindrical coordinates using the finite difference ADI (alternating direction implicit) method on a 100 x 100 grid. The topography, geoid anomaly, and heat flow anomaly of the Hawaiian swell and the Bermuda rise are used to constrain the models. Results confirm the conclusion of previous works that the Hawaiian swell can not be explained by conductive heating alone, even if extremely high thermal perturbation is allowed. On the other hand, the model of convective thinning predicts successfully the topography, geoid anomaly, and the heat flow anomaly around the Hawaiian islands, as well as the changes in the topography and anomalous heat flow along the Hawaiian volcanic chain.

7. Numerical solution of integral-algebraic equations for multistep methods

Budnikova, O. S.; Bulatov, M. V.

2012-05-01

Systems of Volterra linear integral equations with identically singular matrices in the principal part (called integral-algebraic equations) are examined. Multistep methods for the numerical solution of a selected class of such systems are proposed and justified.

8. Milne, a routine for the numerical solution of Milne's problem

Rawat, Ajay; Mohankumar, N.

2010-11-01

The routine Milne provides accurate numerical values for the classical Milne's problem of neutron transport for the planar one speed and isotropic scattering case. The solution is based on the Case eigen-function formalism. The relevant X functions are evaluated accurately by the Double Exponential quadrature. The calculated quantities are the extrapolation distance and the scalar and the angular fluxes. Also, the H function needed in astrophysical calculations is evaluated as a byproduct. Program summaryProgram title: Milne Catalogue identifier: AEGS_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEGS_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 701 No. of bytes in distributed program, including test data, etc.: 6845 Distribution format: tar.gz Programming language: Fortran 77 Computer: PC under Linux or Windows Operating system: Ubuntu 8.04 (Kernel version 2.6.24-16-generic), Windows-XP Classification: 4.11, 21.1, 21.2 Nature of problem: The X functions are integral expressions. The convergence of these regular and Cauchy Principal Value integrals are impaired by the singularities of the integrand in the complex plane. The DE quadrature scheme tackles these singularities in a robust manner compared to the standard Gauss quadrature. Running time: The test included in the distribution takes a few seconds to run.

9. Alternative formulations of the Laplace transform boundary element (LTBE) numerical method for the solution of diffusion-type equations

SciTech Connect

Moridis, G.

1992-03-01

The Laplace Transform Boundary Element (LTBE) method is a recently introduced numerical method, and has been used for the solution of diffusion-type PDEs. It completely eliminates the time dependency of the problem and the need for time discretization, yielding solutions numerical in space and semi-analytical in time. In LTBE solutions are obtained in the Laplace spare, and are then inverted numerically to yield the solution in time. The Stehfest and the DeHoog formulations of LTBE, based on two different inversion algorithms, are investigated. Both formulations produce comparable, extremely accurate solutions.

10. Robust numerical solution of the reservoir routing equation

Fiorentini, Marcello; Orlandini, Stefano

2013-09-01

The robustness of numerical methods for the solution of the reservoir routing equation is evaluated. The methods considered in this study are: (1) the Laurenson-Pilgrim method, (2) the fourth-order Runge-Kutta method, and (3) the fixed order Cash-Karp method. Method (1) is unable to handle nonmonotonic outflow rating curves. Method (2) is found to fail under critical conditions occurring, especially at the end of inflow recession limbs, when large time steps (greater than 12 min in this application) are used. Method (3) is computationally intensive and it does not solve the limitations of method (2). The limitations of method (2) can be efficiently overcome by reducing the time step in the critical phases of the simulation so as to ensure that water level remains inside the domains of the storage function and the outflow rating curve. The incorporation of a simple backstepping procedure implementing this control into the method (2) yields a robust and accurate reservoir routing method that can be safely used in distributed time-continuous catchment models.

11. Numerical Methodology for Coupled Time-Accurate Simulations of Primary and Secondary Flowpaths in Gas Turbines

NASA Technical Reports Server (NTRS)

Przekwas, A. J.; Athavale, M. M.; Hendricks, R. C.; Steinetz, B. M.

2006-01-01

Detailed information of the flow-fields in the secondary flowpaths and their interaction with the primary flows in gas turbine engines is necessary for successful designs with optimized secondary flow streams. Present work is focused on the development of a simulation methodology for coupled time-accurate solutions of the two flowpaths. The secondary flowstream is treated using SCISEAL, an unstructured adaptive Cartesian grid code developed for secondary flows and seals, while the mainpath flow is solved using TURBO, a density based code with capability of resolving rotor-stator interaction in multi-stage machines. An interface is being tested that links the two codes at the rim seal to allow data exchange between the two codes for parallel, coupled execution. A description of the coupling methodology and the current status of the interface development is presented. Representative steady-state solutions of the secondary flow in the UTRC HP Rig disc cavity are also presented.

12. Numerical study of the Kerr solution in rotating coordinates

Bai, S.; Izquierdo, G.; Klein, C.

2016-06-01

The Kerr solution in coordinates corotating with the horizon is studied as a testbed for a spacetime with a helical Killing vector in the Ernst picture. The solution is numerically constructed by solving the Ernst equation with a spectral method and a Newton iteration. We discuss convergence of the iteration for several initial iterates and different values of the Kerr parameters.

13. Numerical Solution of Natural Convection in Eccentric Annuli

SciTech Connect

Pepper, D.W.

2001-09-18

The governing equations for transient natural convection in eccentric annular space are solved with two high-order accurate numerical algorithms. The equation set is transformed into bipolar coordinates and split into two one-dimensional equations: finite elements are used in the direction normal to the cylinder surfaces; the pseudospectral technique is used in the azimuthal direction. This report discusses those equations.

14. Recommendations for accurate numerical blood flow simulations of stented intracranial aneurysms.

PubMed

Janiga, Gábor; Berg, Philipp; Beuing, Oliver; Neugebauer, Mathias; Gasteiger, Rocco; Preim, Bernhard; Rose, Georg; Skalej, Martin; Thévenin, Dominique

2013-06-01

The number of scientific publications dealing with stented intracranial aneurysms is rapidly increasing. Powerful computational facilities are now available; an accurate computational modeling of hemodynamics in patient-specific configurations is, however, still being sought. Furthermore, there is still no general agreement on the quantities that should be computed and on the most adequate analysis for intervention support. In this article, the accurate representation of patient geometry is first discussed, involving successive improvements. Concerning the second step, the mesh required for the numerical simulation is especially challenging when deploying a stent with very fine wire structures. Third, the description of the fluid properties is a major challenge. Finally, a founded quantitative analysis of the simulation results is obviously needed to support interventional decisions. In the present work, an attempt has been made to review the most important steps for a high-quality computational fluid dynamics computation of virtually stented intracranial aneurysms. In consequence, this leads to concrete recommendations, whereby the obtained results are not discussed for their medical relevance but for the evaluation of their quality. This investigation might hopefully be helpful for further studies considering stent deployment in patient-specific geometries, in particular regarding the generation of the most appropriate computational model. PMID:23729530

15. Fast and Accurate Prediction of Numerical Relativity Waveforms from Binary Black Hole Coalescences Using Surrogate Models.

PubMed

Blackman, Jonathan; Field, Scott E; Galley, Chad R; Szilágyi, Béla; Scheel, Mark A; Tiglio, Manuel; Hemberger, Daniel A

2015-09-18

Simulating a binary black hole coalescence by solving Einstein's equations is computationally expensive, requiring days to months of supercomputing time. Using reduced order modeling techniques, we construct an accurate surrogate model, which is evaluated in a millisecond to a second, for numerical relativity (NR) waveforms from nonspinning binary black hole coalescences with mass ratios in [1, 10] and durations corresponding to about 15 orbits before merger. We assess the model's uncertainty and show that our modeling strategy predicts NR waveforms not used for the surrogate's training with errors nearly as small as the numerical error of the NR code. Our model includes all spherical-harmonic _{-2}Y_{ℓm} waveform modes resolved by the NR code up to ℓ=8. We compare our surrogate model to effective one body waveforms from 50M_{⊙} to 300M_{⊙} for advanced LIGO detectors and find that the surrogate is always more faithful (by at least an order of magnitude in most cases).

16. Keeping the edge: an accurate numerical method to solve the stream power law

Campforts, B.; Govers, G.

2015-12-01

Bedrock rivers set the base level of surrounding hill slopes and mediate the dynamic interplay between mountain building and denudation. The propensity of rivers to preserve pulses of increased tectonic uplift also allows to reconstruct long term uplift histories from longitudinal river profiles. An accurate reconstruction of river profile development at different timescales is therefore essential. Long term river development is typically modeled by means of the stream power law. Under specific conditions this equation can be solved analytically but numerical Finite Difference Methods (FDMs) are most frequently used. Nonetheless, FDMs suffer from numerical smearing, especially at knickpoint zones which are key to understand transient landscapes. Here, we solve the stream power law by means of a Finite Volume Method (FVM) which is Total Variation Diminishing (TVD). Total volume methods are designed to simulate sharp discontinuities making them very suitable to model river incision. In contrast to FDMs, the TVD_FVM is well capable of preserving knickpoints as illustrated for the fast propagating Niagara falls. Moreover, we show that the TVD_FVM performs much better when reconstructing uplift at timescales exceeding 100 Myr, using Eastern Australia as an example. Finally, uncertainty associated with parameter calibration is dramatically reduced when the TVD_FVM is applied. Therefore, the use of a TVD_FVM to understand long term landscape evolution is an important addition to the toolbox at the disposition of geomorphologists.

17. Fast and Accurate Prediction of Numerical Relativity Waveforms from Binary Black Hole Coalescences Using Surrogate Models.

PubMed

Blackman, Jonathan; Field, Scott E; Galley, Chad R; Szilágyi, Béla; Scheel, Mark A; Tiglio, Manuel; Hemberger, Daniel A

2015-09-18

Simulating a binary black hole coalescence by solving Einstein's equations is computationally expensive, requiring days to months of supercomputing time. Using reduced order modeling techniques, we construct an accurate surrogate model, which is evaluated in a millisecond to a second, for numerical relativity (NR) waveforms from nonspinning binary black hole coalescences with mass ratios in [1, 10] and durations corresponding to about 15 orbits before merger. We assess the model's uncertainty and show that our modeling strategy predicts NR waveforms not used for the surrogate's training with errors nearly as small as the numerical error of the NR code. Our model includes all spherical-harmonic _{-2}Y_{ℓm} waveform modes resolved by the NR code up to ℓ=8. We compare our surrogate model to effective one body waveforms from 50M_{⊙} to 300M_{⊙} for advanced LIGO detectors and find that the surrogate is always more faithful (by at least an order of magnitude in most cases). PMID:26430979

18. Numerical solutions of telegraph equations with the Dirichlet boundary condition

Ashyralyev, Allaberen; Turkcan, Kadriye Tuba; Koksal, Mehmet Emir

2016-08-01

In this study, the Cauchy problem for telegraph equations in a Hilbert space is considered. Stability estimates for the solution of this problem are presented. The third order of accuracy difference scheme is constructed for approximate solutions of the problem. Stability estimates for the solution of this difference scheme are established. As a test problem to support theoretical results, one-dimensional telegraph equation with the Dirichlet boundary condition is considered. Numerical solutions of this equation are obtained by first, second and third order of accuracy difference schemes.

19. Towards an accurate understanding of UHMWPE visco-dynamic behaviour for numerical modelling of implants.

PubMed

Quinci, Federico; Dressler, Matthew; Strickland, Anthony M; Limbert, Georges

2014-04-01

Considerable progress has been made in understanding implant wear and developing numerical models to predict wear for new orthopaedic devices. However any model of wear could be improved through a more accurate representation of the biomaterial mechanics, including time-varying dynamic and inelastic behaviour such as viscosity and plastic deformation. In particular, most computational models of wear of UHMWPE implement a time-invariant version of Archard's law that links the volume of worn material to the contact pressure between the metal implant and the polymeric tibial insert. During in-vivo conditions, however, the contact area is a time-varying quantity and is therefore dependent upon the dynamic deformation response of the material. From this observation one can conclude that creep deformations of UHMWPE may be very important to consider when conducting computational wear analyses, in stark contrast to what can be found in the literature. In this study, different numerical modelling techniques are compared with experimental creep testing on a unicondylar knee replacement system in a physiologically representative context. Linear elastic, plastic and time-varying visco-dynamic models are benchmarked using literature data to predict contact deformations, pressures and areas. The aim of this study is to elucidate the contributions of viscoelastic and plastic effects on these surface quantities. It is concluded that creep deformations have a significant effect on the contact pressure measured (experiment) and calculated (computational models) at the surface of the UHMWPE unicondylar insert. The use of a purely elastoplastic constitutive model for UHMWPE lead to compressive deformations of the insert which are much smaller than those predicted by a creep-capturing viscoelastic model (and those measured experimentally). This shows again the importance of including creep behaviour into a constitutive model in order to predict the right level of surface deformation

20. Towards an accurate understanding of UHMWPE visco-dynamic behaviour for numerical modelling of implants.

PubMed

Quinci, Federico; Dressler, Matthew; Strickland, Anthony M; Limbert, Georges

2014-04-01

Considerable progress has been made in understanding implant wear and developing numerical models to predict wear for new orthopaedic devices. However any model of wear could be improved through a more accurate representation of the biomaterial mechanics, including time-varying dynamic and inelastic behaviour such as viscosity and plastic deformation. In particular, most computational models of wear of UHMWPE implement a time-invariant version of Archard's law that links the volume of worn material to the contact pressure between the metal implant and the polymeric tibial insert. During in-vivo conditions, however, the contact area is a time-varying quantity and is therefore dependent upon the dynamic deformation response of the material. From this observation one can conclude that creep deformations of UHMWPE may be very important to consider when conducting computational wear analyses, in stark contrast to what can be found in the literature. In this study, different numerical modelling techniques are compared with experimental creep testing on a unicondylar knee replacement system in a physiologically representative context. Linear elastic, plastic and time-varying visco-dynamic models are benchmarked using literature data to predict contact deformations, pressures and areas. The aim of this study is to elucidate the contributions of viscoelastic and plastic effects on these surface quantities. It is concluded that creep deformations have a significant effect on the contact pressure measured (experiment) and calculated (computational models) at the surface of the UHMWPE unicondylar insert. The use of a purely elastoplastic constitutive model for UHMWPE lead to compressive deformations of the insert which are much smaller than those predicted by a creep-capturing viscoelastic model (and those measured experimentally). This shows again the importance of including creep behaviour into a constitutive model in order to predict the right level of surface deformation

1. Some recent advances in the numerical solution of differential equations

D'Ambrosio, Raffaele

2016-06-01

The purpose of the talk is the presentation of some recent advances in the numerical solution of differential equations, with special emphasis to reaction-diffusion problems, Hamiltonian problems and ordinary differential equations with discontinuous right-hand side. As a special case, in this short paper we focus on the solution of reaction-diffusion problems by means of special purpose numerical methods particularly adapted to the problem: indeed, following a problem oriented approach, we propose a modified method of lines based on the employ of finite differences shaped on the qualitative behavior of the solutions. Constructive issues and a brief analysis are presented, together with some numerical experiments showing the effectiveness of the approach and a comparison with existing solvers.

2. Numerical solution of a microbial growth model applied to dynamic environments.

PubMed

Zhu, Si; Chen, Guibing

2015-05-01

The Baranyi and Roberts model is one of the most frequently used microbial growth models. It has been successfully applied to numerous studies of various microorganisms in different food products. Under dynamic conditions, the model is implicitly formulated as a set of two coupled differential equations which could be numerically solved using the Runge-Kutta method. In this study, a simplified numerical solution of the coupled differential equations was derived and used to simulate microbial growth under dynamic conditions in Microsoft Excel. As expected, the results obtained were the same as those from solving the coupled differential equations using a MATLAB Solver. In addition, model parameters were accurately identified by fitting the numerical solution to simulated growth curves under dynamic (time-varying) temperature conditions using the Microsoft Excel Solver.

3. Implicit numerical integration for periodic solutions of autonomous nonlinear systems

NASA Technical Reports Server (NTRS)

Thurston, G. A.

1982-01-01

A change of variables that stabilizes numerical computations for periodic solutions of autonomous systems is derived. Computation of the period is decoupled from the rest of the problem for conservative systems of any order and for any second-order system. Numerical results are included for a second-order conservative system under a suddenly applied constant load. Near the critical load for the system, a small increment in load amplitude results in a large increase in amplitude of the response.

4. Numerical solutions for heat flow in adhesive lap joints

NASA Technical Reports Server (NTRS)

Howell, P. A.; Winfree, William P.

1992-01-01

The present formulation for the modeling of heat transfer in thin, adhesively bonded lap joints precludes difficulties associated with large aspect ratio grids required by standard FEM formulations. This quasi-static formulation also reduces the problem dimensionality (by one), thereby minimizing computational requirements. The solutions obtained are found to be in good agreement with both analytical solutions and solutions from standard FEM programs. The approach is noted to yield a more accurate representation of heat-flux changes between layers due to a disbond.

5. Numerical solution to systems of delay integrodifferential algebraic equations

Dmitriev, S. S.; Kuznetsov, E. B.

2008-03-01

The numerical solution of the initial value problem for a system of delay integrodifferential algebraic equations is examined in the framework of the parametric continuation method. Necessary and sufficient conditions are obtained for transforming this problem to the best argument, which is the arc length along the integral curve of the problem. The efficiency of the transformation is demonstrated using test examples.

6. Highly Accurate Beam Torsion Solutions Using the p-Version Finite Element Method

NASA Technical Reports Server (NTRS)

Smith, James P.

1996-01-01

A new treatment of the classical beam torsion boundary value problem is applied. Using the p-version finite element method with shape functions based on Legendre polynomials, torsion solutions for generic cross-sections comprised of isotropic materials are developed. Element shape functions for quadrilateral and triangular elements are discussed, and numerical examples are provided.

7. Optimality conditions for the numerical solution of optimization problems with PDE constraints :

SciTech Connect

Aguilo Valentin, Miguel Alejandro; Ridzal, Denis

2014-03-01

A theoretical framework for the numerical solution of partial di erential equation (PDE) constrained optimization problems is presented in this report. This theoretical framework embodies the fundamental infrastructure required to e ciently implement and solve this class of problems. Detail derivations of the optimality conditions required to accurately solve several parameter identi cation and optimal control problems are also provided in this report. This will allow the reader to further understand how the theoretical abstraction presented in this report translates to the application.

8. Numerical solution-space analysis of satisfiability problems

Mann, Alexander; Hartmann, A. K.

2010-11-01

The solution-space structure of the three-satisfiability problem (3-SAT) is studied as a function of the control parameter α (ratio of the number of clauses to the number of variables) using numerical simulations. For this purpose one has to sample the solution space with uniform weight. It is shown here that standard stochastic local-search (SLS) algorithms like average satisfiability (ASAT) exhibit a sampling bias, as does “Metropolis-coupled Markov chain Monte Carlo” (MCMCMC) (also known as “parallel tempering”) when run for feasible times. Nevertheless, unbiased samples of solutions can be obtained using the “ballistic-networking approach,” which is introduced here. It is a generalization of “ballistic search” methods and yields also a cluster structure of the solution space. As application, solutions of 3-SAT instances are generated using ASAT plus ballistic networking. The numerical results are compatible with a previous analytical prediction of a simple solution-space structure for small values of α and a transition to a clustered phase at αc≈3.86 , where the solution space breaks up into several non-negligible clusters. Furthermore, in the thermodynamic limit there are, even for α=4.25 close to the SAT-UNSAT transition αs≈4.267 , always clusters without any frozen variables. This may explain why some SLS algorithms are able to solve very large 3-SAT instances close to the SAT-UNSAT transition.

9. Numerical solution-space analysis of satisfiability problems.

PubMed

Mann, Alexander; Hartmann, A K

2010-11-01

The solution-space structure of the three-satisfiability problem (3-SAT) is studied as a function of the control parameter α (ratio of the number of clauses to the number of variables) using numerical simulations. For this purpose one has to sample the solution space with uniform weight. It is shown here that standard stochastic local-search (SLS) algorithms like average satisfiability (ASAT) exhibit a sampling bias, as does "Metropolis-coupled Markov chain Monte Carlo" (MCMCMC) (also known as "parallel tempering") when run for feasible times. Nevertheless, unbiased samples of solutions can be obtained using the "ballistic-networking approach," which is introduced here. It is a generalization of "ballistic search" methods and yields also a cluster structure of the solution space. As application, solutions of 3-SAT instances are generated using ASAT plus ballistic networking. The numerical results are compatible with a previous analytical prediction of a simple solution-space structure for small values of α and a transition to a clustered phase at α(c)≈3.86 , where the solution space breaks up into several non-negligible clusters. Furthermore, in the thermodynamic limit there are, even for α=4.25 close to the SAT-UNSAT transition α(s)≈4.267 , always clusters without any frozen variables. This may explain why some SLS algorithms are able to solve very large 3-SAT instances close to the SAT-UNSAT transition. PMID:21230614

10. Orbital Advection by Interpolation: A Fast and Accurate Numerical Scheme for Super-Fast MHD Flows

SciTech Connect

Johnson, B M; Guan, X; Gammie, F

2008-04-11

In numerical models of thin astrophysical disks that use an Eulerian scheme, gas orbits supersonically through a fixed grid. As a result the timestep is sharply limited by the Courant condition. Also, because the mean flow speed with respect to the grid varies with position, the truncation error varies systematically with position. For hydrodynamic (unmagnetized) disks an algorithm called FARGO has been developed that advects the gas along its mean orbit using a separate interpolation substep. This relaxes the constraint imposed by the Courant condition, which now depends only on the peculiar velocity of the gas, and results in a truncation error that is more nearly independent of position. This paper describes a FARGO-like algorithm suitable for evolving magnetized disks. Our method is second order accurate on a smooth flow and preserves {del} {center_dot} B = 0 to machine precision. The main restriction is that B must be discretized on a staggered mesh. We give a detailed description of an implementation of the code and demonstrate that it produces the expected results on linear and nonlinear problems. We also point out how the scheme might be generalized to make the integration of other supersonic/super-fast flows more efficient. Although our scheme reduces the variation of truncation error with position, it does not eliminate it. We show that the residual position dependence leads to characteristic radial variations in the density over long integrations.

11. On the accurate long-time solution of the wave equation in exterior domains: Asymptotic expansions and corrected boundary conditions

NASA Technical Reports Server (NTRS)

Hagstrom, Thomas; Hariharan, S. I.; Maccamy, R. C.

1993-01-01

We consider the solution of scattering problems for the wave equation using approximate boundary conditions at artificial boundaries. These conditions are explicitly viewed as approximations to an exact boundary condition satisfied by the solution on the unbounded domain. We study the short and long term behavior of the error. It is provided that, in two space dimensions, no local in time, constant coefficient boundary operator can lead to accurate results uniformly in time for the class of problems we consider. A variable coefficient operator is developed which attains better accuracy (uniformly in time) than is possible with constant coefficient approximations. The theory is illustrated by numerical examples. We also analyze the proposed boundary conditions using energy methods, leading to asymptotically correct error bounds.

12. Numerical solution of boundary-integral equations for molecular electrostatics.

SciTech Connect

Bardhan, J.; Mathematics and Computer Science; Rush Univ.

2009-03-07

Numerous molecular processes, such as ion permeation through channel proteins, are governed by relatively small changes in energetics. As a result, theoretical investigations of these processes require accurate numerical methods. In the present paper, we evaluate the accuracy of two approaches to simulating boundary-integral equations for continuum models of the electrostatics of solvation. The analysis emphasizes boundary-element method simulations of the integral-equation formulation known as the apparent-surface-charge (ASC) method or polarizable-continuum model (PCM). In many numerical implementations of the ASC/PCM model, one forces the integral equation to be satisfied exactly at a set of discrete points on the boundary. We demonstrate in this paper that this approach to discretization, known as point collocation, is significantly less accurate than an alternative approach known as qualocation. Furthermore, the qualocation method offers this improvement in accuracy without increasing simulation time. Numerical examples demonstrate that electrostatic part of the solvation free energy, when calculated using the collocation and qualocation methods, can differ significantly; for a polypeptide, the answers can differ by as much as 10 kcal/mol (approximately 4% of the total electrostatic contribution to solvation). The applicability of the qualocation discretization to other integral-equation formulations is also discussed, and two equivalences between integral-equation methods are derived.

13. ASYMPTOTICALLY OPTIMAL HIGH-ORDER ACCURATE ALGORITHMS FOR THE SOLUTION OF CERTAIN ELLIPTIC PDEs

SciTech Connect

Leonid Kunyansky, PhD

2008-11-26

The main goal of the project, "Asymptotically Optimal, High-Order Accurate Algorithms for the Solution of Certain Elliptic PDE's" (DE-FG02-03ER25577) was to develop fast, high-order algorithms for the solution of scattering problems and spectral problems of photonic crystals theory. The results we obtained lie in three areas: (1) asymptotically fast, high-order algorithms for the solution of eigenvalue problems of photonics, (2) fast, high-order algorithms for the solution of acoustic and electromagnetic scattering problems in the inhomogeneous media, and (3) inversion formulas and fast algorithms for the inverse source problem for the acoustic wave equation, with applications to thermo- and opto- acoustic tomography.

14. Stability of Inviscid Flow over Airfoils Admitting Multiple Numerical Solutions

Liu, Ya; Xiong, Juntao; Liu, Feng; Luo, Shijun

2012-11-01

Multiple numerical solutions at the same flight condition are found of inviscid transonic flow over certain airfoils (Jameson et al., AIAA 2011-3509) within some Mach number range. Both symmetric and asymmetric solutions exist for a symmetric airfoil at zero angle of attack. Global linear stability analysis of the multiple solutions is conducted. Linear perturbation equations of the Euler equations around a steady-state solution are formed and discretized numerically. An eigenvalue problem is then constructed using the modal analysis approach. Only a small portion of the eigen spectrum is needed and thus can be found efficiently by using Arnoldi's algorithm. The least stable or unstable mode corresponds to the eigenvalue with the largest real part. Analysis of the NACA 0012 airfoil indicates stability of symmetric solutions of the Euler equations at conditions where buffet is found from unsteady Navier-Stokes equations. Euler solutions of the same airfoil but modified to include the displacement thickness of the boundary layer computed from the Navier-Stokes equations, however, exhibit instability based on the present linear stability analysis. Graduate Student.

15. Multi-stencils fast marching methods: a highly accurate solution to the eikonal equation on cartesian domains.

PubMed

Hassouna, M Sabry; Farag, A A

2007-09-01

A wide range of computer vision applications require an accurate solution of a particular Hamilton- Jacobi (HJ) equation, known as the Eikonal equation. In this paper, we propose an improved version of the fast marching method (FMM) that is highly accurate for both 2D and 3D Cartesian domains. The new method is called multi-stencils fast marching (MSFM), which computes the solution at each grid point by solving the Eikonal equation along several stencils and then picks the solution that satisfies the upwind condition. The stencils are centered at each grid point and cover its entire nearest neighbors. In 2D space, 2 stencils cover the 8-neighbors of the point, while in 3D space, 6 stencils cover its 26-neighbors. For those stencils that are not aligned with the natural coordinate system, the Eikonal equation is derived using directional derivatives and then solved using higher order finite difference schemes. The accuracy of the proposed method over the state-of-the-art FMM-based techniques has been demonstrated through comprehensive numerical experiments.

16. Numerical fluid solutions for nonlocal electron transport in hot plasmas: Equivalent diffusion versus nonlocal source

SciTech Connect

Colombant, Denis Manheimer, Wallace

2010-06-01

Flux limitation and preheat are important processes in electron transport occurring in laser produced plasmas. The proper calculation of both of these has been a subject receiving much attention over the entire lifetime of the laser fusion project. Where nonlocal transport (instead of simple single flux limit) has been modeled, it has always been with what we denote the equivalent diffusion solution, namely treating the transport as only a diffusion process. We introduce here a new approach called the nonlocal source solution and show it is numerically viable for laser produced plasmas. It turns out that the equivalent diffusion solution generally underestimates preheat. Furthermore, the advance of the temperature front, and especially the preheat, can be held up by artificial 'thermal barriers'. The nonlocal source method of solution, on the other hand more accurately describes preheat and can stably calculate the solution for the temperature even if the heat flux is up the gradient.

17. Wiener Chaos expansions and numerical solutions of randomly forced equations of fluid mechanics

SciTech Connect

Hou, Thomas Y. . E-mail: hou@ama.caltech.edu; Luo Wuan; Rozovskii, Boris; Zhou Haomin

2006-08-10

In this paper, we propose a numerical method based on Wiener Chaos expansion and apply it to solve the stochastic Burgers and Navier-Stokes equations driven by Brownian motion. The main advantage of the Wiener Chaos approach is that it allows for the separation of random and deterministic effects in a rigorous and effective manner. The separation principle effectively reduces a stochastic equation to its associated propagator, a system of deterministic equations for the coefficients of the Wiener Chaos expansion. Simple formulas for statistical moments of the stochastic solution are presented. These formulas only involve the solutions of the propagator. We demonstrate that for short time solutions the numerical methods based on the Wiener Chaos expansion are more efficient and accurate than those based on the Monte Carlo simulations.

18. An accurate two-phase approximate solution to the acute viral infection model

SciTech Connect

Perelson, Alan S

2009-01-01

During an acute viral infection, virus levels rise, reach a peak and then decline. Data and numerical solutions suggest the growth and decay phases are linear on a log scale. While viral dynamic models are typically nonlinear with analytical solutions difficult to obtain, the exponential nature of the solutions suggests approximations can be found. We derive a two-phase approximate solution to the target cell limited influenza model and illustrate the accuracy using data and previously established parameter values of six patients infected with influenza A. For one patient, the subsequent fall in virus concentration was not consistent with our predictions during the decay phase and an alternate approximation is derived. We find expressions for the rate and length of initial viral growth in terms of the parameters, the extent each parameter is involved in viral peaks, and the single parameter responsible for virus decay. We discuss applications of this analysis in antiviral treatments and investigating host and virus heterogeneities.

19. Numerical time-dependent solutions of the Schrödinger equation with piecewise continuous potentials

van Dijk, Wytse

2016-06-01

We consider accurate numerical solutions of the one-dimensional time-dependent Schrödinger equation when the potential is piecewise continuous. Spatial step sizes are defined for each of the regions between the discontinuities and a matching condition at the boundaries of the regions is employed. The Numerov method for spatial integration is particularly appropriate to this approach. By employing Padé approximants for the time-evolution operator, we obtain solutions with significantly improved precision without increased CPU time. This approach is also appropriate for adaptive changes in spatial step size even when there is no discontinuity of the potential.

20. The comparative study on analytical solutions and numerical solutions of displacement in transversely isotropic rock mass

Zhang, Zhizeng; Zhao, Zhao; Li, Yongtao

2016-06-01

This paper attempts to verify the correctness of the analytical displacement solution in transversely isotropic rock mass, and to determine the scope of its application. The analytical displacement solution of a circular tunnel in transversely isotropic rock mass was derived firstly. The analytical solution was compared with the numerical solution, which was carried out by FLAC3D software. The results show that the expression of the analytical displacement solution is correct, and the allowable engineering range is that the dip angle is less than 15 degrees.

1. A Collocation Method for Numerical Solutions of Coupled Burgers' Equations

Mittal, R. C.; Tripathi, A.

2014-09-01

In this paper, we propose a collocation-based numerical scheme to obtain approximate solutions of coupled Burgers' equations. The scheme employs collocation of modified cubic B-spline functions. We have used modified cubic B-spline functions for unknown dependent variables u, v, and their derivatives w.r.t. space variable x. Collocation forms of the partial differential equations result in systems of first-order ordinary differential equations (ODEs). In this scheme, we did not use any transformation or linearization method to handle nonlinearity. The obtained system of ODEs has been solved by strong stability preserving the Runge-Kutta method. The proposed scheme needs less storage space and execution time. The test problems considered in the literature have been discussed to demonstrate the strength and utility of the proposed scheme. The computed numerical solutions are in good agreement with the exact solutions and competent with those available in earlier studies. The scheme is simple as well as easy to implement. The scheme provides approximate solutions not only at the grid points, but also at any point in the solution range.

2. Numerical solution for option pricing with stochastic volatility model

Mariani, Andi; Nugrahani, Endar H.; Lesmana, Donny C.

2016-01-01

The option pricing equations derived from stochatic volatility models in finance are often cast in the form of nonlinear partial differential equations. To solve the equations, we used the upwind finite difference scheme for the spatial discretisation and a fully implicit time-stepping scheme. The result of this scheme is a matrix system in the form of an M-Matrix and we proof that the approximate solution converges to the viscosity solution to the equation by showing that the scheme is monotone, consistent and stable. Numerical experiments are implemented to show that the behavior and the order of convergence of upwind finite difference method.

3. Nanowire Conductivity: A Numerical Solution of the Boltzmann Equation

Sundaram, Venkat; Mizel, Ari

2002-03-01

The conduction properties of large nanowires (diameter 100nm), are calculated by a direct numerical solution of the Boltzmann equation. Nanowires are modelled by distinguishing a bulk-like interior region from a surface region of finite width and enhanced scattering. Of particular interest is the investigation of a negative longitudinal magnetoresistance in such a model, a result expected from kinetic-theoretical considerations. These calculations can be performed both using the relaxation time approximation and the collision integral, with a view to relating model parameters in either approach to experimental results. Such a direct numerical solution offers the advantage of significant freedom in incorporating nanowire characteristics and conditions such as the equilibrium electron density and temperature profile, or the presence of defects and impurities.

4. Numerical Solution of a Model Equation of Price Formation

Chernogorova, T.; Vulkov, L.

2009-10-01

The paper [2] is devoted to the effect of reconciling the classical Black-Sholes theory of option pricing and hedging with various phenomena observed in the markets such as the influence of trading and hedging on the dynamics of an asset. Here we will discuss the numerical solution of initial boundary-value problems to a model equation of the theory. The lack of regularity in the solution as a result from Dirac delta coefficient reduces the accuracy in the numerical computations. First, we apply the finite volume method to discretize the differential problem. Second, we implement a technique of local regularization introduced by A-K. Tornberg and B. Engquist [7] for handling this equation. We derived the numerical regularization process into two steps: the Dirac delta function is regularized and then the regularized differential equation is discretized by difference schemes. Using the discrete maximum principle a priori bounds are obtained for the difference equations that imply stability and convergence of difference schemes for the problem under consideration. Numerical experiments are discussed.

5. Accurate numerical forward model for optimal retracking of SIRAL2 SAR echoes over open ocean

Phalippou, L.; Demeestere, F.

2011-12-01

The SAR mode of SIRAL-2 on board Cryosat-2 has been designed to measure primarily sea-ice and continental ice (Wingham et al. 2005). In 2005, K. Raney (KR, 2005) pointed out the improvements brought by SAR altimeter for open ocean. KR results were mostly based on 'rule of thumb' considerations on speckle noise reduction due to the higher PRF and to speckle decorrelation after SAR processing. In 2007, Phalippou and Enjolras (PE,2007) provided the theoretical background for optimal retracking of SAR echoes over ocean with a focus on the forward modelling of the power-waveforms. The accuracies of geophysical parameters (range, significant wave heights, and backscattering coefficient) retrieved from SAR altimeter data were derived accounting for SAR echo shape and speckle noise accurate modelling. The step forward to optimal retracking using numerical forward model (NFM) was also pointed out. NFM of the power waveform avoids analytical approximation, a warranty to minimise the geophysical dependent biases in the retrieval. NFM have been used for many years, in operational meteorology in particular, for retrieving temperature and humidity profiles from IR and microwave radiometers as the radiative transfer function is complex (Eyre, 1989). So far this technique was not used in the field of ocean conventional altimetry as analytical models (e.g. Brown's model for instance) were found to give sufficient accuracy. However, although NFM seems desirable even for conventional nadir altimetry, it becomes inevitable if one wish to process SAR altimeter data as the transfer function is too complex to be approximated by a simple analytical function. This was clearly demonstrated in PE 2007. The paper describes the background to SAR data retracking over open ocean. Since PE 2007 improvements have been brought to the forward model and it is shown that the altimeter on-ground and in flight characterisation (e.g antenna pattern range impulse response, azimuth impulse response

6. Physical and Numerical Model Studies of Cross-flow Turbines Towards Accurate Parameterization in Array Simulations

Wosnik, M.; Bachant, P.

2014-12-01

Cross-flow turbines, often referred to as vertical-axis turbines, show potential for success in marine hydrokinetic (MHK) and wind energy applications, ranging from small- to utility-scale installations in tidal/ocean currents and offshore wind. As turbine designs mature, the research focus is shifting from individual devices to the optimization of turbine arrays. It would be expensive and time-consuming to conduct physical model studies of large arrays at large model scales (to achieve sufficiently high Reynolds numbers), and hence numerical techniques are generally better suited to explore the array design parameter space. However, since the computing power available today is not sufficient to conduct simulations of the flow in and around large arrays of turbines with fully resolved turbine geometries (e.g., grid resolution into the viscous sublayer on turbine blades), the turbines' interaction with the energy resource (water current or wind) needs to be parameterized, or modeled. Models used today--a common model is the actuator disk concept--are not able to predict the unique wake structure generated by cross-flow turbines. This wake structure has been shown to create "constructive" interference in some cases, improving turbine performance in array configurations, in contrast with axial-flow, or horizontal axis devices. Towards a more accurate parameterization of cross-flow turbines, an extensive experimental study was carried out using a high-resolution turbine test bed with wake measurement capability in a large cross-section tow tank. The experimental results were then "interpolated" using high-fidelity Navier--Stokes simulations, to gain insight into the turbine's near-wake. The study was designed to achieve sufficiently high Reynolds numbers for the results to be Reynolds number independent with respect to turbine performance and wake statistics, such that they can be reliably extrapolated to full scale and used for model validation. The end product of

7. An Accurate Solution to the Lotka-Volterra Equations by Modified Homotopy Perturbation Method

Chowdhury, M. S. H.; Rahman, M. M.

In this paper, we suggest a method to solve the multispecies Lotka-Voltera equations. The suggested method, which we call modified homotopy perturbation method, can be considered as an extension of the homotopy perturbation method (HPM) which is very efficient in solving a varety of differential and algebraic equations. The HPM is modified in order to obtain the approximate solutions of Lotka-Voltera equation response in a sequence of time intervals. In particular, the example of two species is considered. The accuracy of this method is examined by comparison with the numerical solution of the Runge-Kutta-Verner method. The results prove that the modified HPM is a powerful tool for the solution of nonlinear equations.

8. A new algorithm for generating highly accurate benchmark solutions to transport test problems

SciTech Connect

Azmy, Y.Y.

1997-06-01

We present a new algorithm for solving the neutron transport equation in its discrete-variable form. The new algorithm is based on computing the full matrix relating the scalar flux spatial moments in all cells to the fixed neutron source spatial moments, foregoing the need to compute the angular flux spatial moments, and thereby eliminating the need for sweeping the spatial mesh in each discrete-angular direction. The matrix equation is solved exactly in test cases, producing a solution vector that is free from iteration convergence error, and subject only to truncation and roundoff errors. Our algorithm is designed to provide method developers with a quick and simple solution scheme to test their new methods on difficult test problems without the need to develop sophisticated solution techniques, e.g. acceleration, before establishing the worthiness of their innovation. We demonstrate the utility of the new algorithm by applying it to the Arbitrarily High Order Transport Nodal (AHOT-N) method, and using it to solve two of Burre`s Suite of Test Problems (BSTP). Our results provide highly accurate benchmark solutions, that can be distributed electronically and used to verify the pointwise accuracy of other solution methods and algorithms.

9. Sensitivity of the solution of the Elder problem to density, velocity and numerical perturbations

Park, Chan-Hee; Aral, Mustafa M.

2007-06-01

In this paper the Elder problem is studied with the purpose of evaluating the inherent instabilities associated with the numerical solution of this problem. Our focus is first on the question of the existence of a unique numerical solution for this problem, and second on the grid density and fluid density requirements necessary for a unique numerical solution. In particular we have investigated the instability issues associated with the numerical solution of the Elder problem from the following perspectives: (i) physical instability issues associated with density differences; (ii) sensitivity of the numerical solution to idealization irregularities; and, (iii) the importance of a precise velocity field calculation and the association of this process with the grid density levels that is necessary to solve the Elder problem accurately. In the study discussed here we have used a finite element Galerkin model we have developed for solving density-dependent flow and transport problems, which will be identified as TechFlow. In our study, the numerical results of Frolkovič and de Schepper [Frolkovič, P. and H. de Schepper, 2001. Numerical modeling of convection dominated transport coupled with density-driven flow in porous media, Adv. Water Resour., 24, 63-72.] were replicated using the grid density employed in their work. We were also successful in duplicating the same result with a less dense grid but with more computational effort based on a global velocity estimation process we have adopted. Our results indicate that the global velocity estimation approach recommended by Yeh [Yeh, G.-T., 1981. On the computation of Darcian velocity and mass balance in finite element modelling of groundwater flow, Water Resour. Res., 17(5), 1529-1534.] allows the use of less dense grids while obtaining the same accuracy that can be achieved with denser grids. We have also observed that the regularity of the elements in the discretization of the solution domain does make a difference

10. Numerical solution of flame sheet problems with and without multigrid methods

NASA Technical Reports Server (NTRS)

Douglas, Craig C.; Ern, Alexandre

1993-01-01

Flame sheet problems are on the natural route to the numerical solution of multidimensional flames, which, in turn, are important in many engineering applications. In order to model the structure of flames more accurately, we use the vorticity-velocity formulation of the fluid flow equations, as opposed to the streamfunction-vorticity approach. The numerical solution of the resulting nonlinear coupled elliptic partial differential equations involves a pseudo transient process and a steady state Newton iteration. Rather than working with dimensionless variables, we introduce scale factors that can yield significant savings in the execution time. In this context, we also investigate the applicability and performance of several multigrid methods, focusing on nonlinear damped Newton multigrid, using either one way or correction schemes.

11. Numerical solution techniques for unsteady transonic aerodynamics problems

NASA Technical Reports Server (NTRS)

Ballhaus, W. F.; Bridgeman, J. O.

1980-01-01

Basic concepts of finite difference solution techniques for unsteady transonic flows are presented. The hierarchy of mathematical forumulations that approximate the Navier-Stokes equations are reviewed. The basic concepts involved in constructing numerical algorthms to solve these formulations are given. Semi-implicit and implicit schemes are constructed and analyzed. The discussion focuses primarily on techniques for solving the low frequency transonic small disturbance equation. This is the simplest formulation that contains the essence of inviscid unsteady transonic flow physics. The low frequency formulation is emphasized here because codes based on this theory can be run in minutes of processor time on currently available computers. Furthermore, numerical techniques involved in solving this simple formulation also apply to the more complicated formulations. Extensions to these formulations are briefly described. An indication of the present capability for solving unsteady transonic flows is provided. Important areas of future research for the advancement of computational unsteady transonic aerodynamics are described.

12. An algorithm for the numerical solution of linear differential games

SciTech Connect

Polovinkin, E S; Ivanov, G E; Balashov, M V; Konstantinov, R V; Khorev, A V

2001-10-31

A numerical algorithm for the construction of stable Krasovskii bridges, Pontryagin alternating sets, and also of piecewise program strategies solving two-person linear differential (pursuit or evasion) games on a fixed time interval is developed on the basis of a general theory. The aim of the first player (the pursuer) is to hit a prescribed target (terminal) set by the phase vector of the control system at the prescribed time. The aim of the second player (the evader) is the opposite. A description of numerical algorithms used in the solution of differential games of the type under consideration is presented and estimates of the errors resulting from the approximation of the game sets by polyhedra are presented.

13. NUMERICAL SOLUTION FOR THE POTENTIAL AND DENSITY PROFILE OF A THERMAL EQUILIBRIUM SHEET BEAM

SciTech Connect

Bazouin, Steven M. Lund, Guillaume; Bazouin, Guillaume

2011-04-01

In a recent paper, S. M. Lund, A. Friedman, and G. Bazouin, Sheet beam model for intense space-charge: with application to Debye screening and the distribution of particle oscillation frequencies in a thermal equilibrium beam, in press, Phys. Rev. Special Topics - Accel. and Beams (2011), a 1D sheet beam model was extensively analyzed. In this complementary paper, we present details of a numerical procedure developed to construct the self-consistent electrostatic potential and density profile of a thermal equilibrium sheet beam distribution. This procedure effectively circumvents pathologies which can prevent use of standard numerical integration techniques when space-charge intensity is high. The procedure employs transformations and is straightforward to implement with standard numerical methods and produces accurate solutions which can be applied to thermal equilibria with arbitrarily strong space-charge intensity up to the applied focusing limit.

14. NUMERICAL SOLUTION FOR THE POTENTIAL AND DENSITY PROFILE OF A THERMAL EQUILIBRIUM SHEET BEAM

SciTech Connect

Lund, S M; Bazouin, G

2011-03-29

In a recent paper, S. M. Lund, A. Friedman, and G. Bazouin, Sheet beam model for intense space-charge: with application to Debye screening and the distribution of particle oscillation frequencies in a thermal equilibrium beam, in press, Phys. Rev. Special Topics - Accel. and Beams (2011), a 1D sheet beam model was extensively analyzed. In this complementary paper, we present details of a numerical procedure developed to construct the self-consistent electrostatic potential and density profile of a thermal equilibrium sheet beam distribution. This procedure effectively circumvents pathologies which can prevent use of standard numerical integration techniques when space-charge intensity is high. The procedure employs transformations and is straightforward to implement with standard numerical methods and produces accurate solutions which can be applied to thermal equilibria with arbitrarily strong space-charge intensity up to the applied focusing limit.

15. Numerical solution of a semilinear elliptic equation via difference scheme

2016-08-01

We consider the Bitsadze-Samarskii type nonlocal boundary value problem { -d/2v (t ) d t2 +B v (t ) =h (t ,v (t ) ) ,0 solution of problem (1), we use the first order of accuracy difference scheme. The numerical results are computed by MATLAB.

16. Numerical solution of fractionally damped beam by homotopy perturbation method

Behera, Diptiranjan; Chakraverty, Snehashish

2013-06-01

This paper investigates the numerical solution of a viscoelastic continuous beam whose damping behaviours are defined in term of fractional derivatives of arbitrary order. The Homotopy Perturbation Method (HPM) is used to obtain the dynamic response. Unit step function response is considered for the analysis. The obtained results are depicted in various plots. From the results obtained it is interesting to note that by increasing the order of the fractional derivative the beam suffers less oscillation. Similar observations have also been made by keeping the order of the fractional derivative constant and varying the damping ratios. Comparisons are made with the analytic solutions obtained by Zu-feng and Xiao-yan [Appl. Math. Mech. 28, 219 (2007)] to show the effectiveness and validation of this method.

17. Numerical Solutions of Supersonic and Hypersonic Laminar Compression Corner Flows

NASA Technical Reports Server (NTRS)

Hung, C. M.; MacCormack, R. W.

1976-01-01

An efficient time-splitting, second-order accurate, numerical scheme is used to solve the complete Navier-Stokes equations for supersonic and hypersonic laminar flow over a two-dimensional compression corner. A fine, exponentially stretched mesh spacing is used in the region near the wall for resolving the viscous layer. Good agreement is obtained between the present computed results and experimental measurement for a Mach number of 14.1 and a Reynolds number of 1.04 x 10(exp 5) with wedge angles of 15 deg, 18 deg, and 24 deg. The details of the pressure variation across the boundary layer are given, and a correlation between the leading edge shock and the peaks in surface pressure and heat transfer is observed.

18. Numerical solutions of the complete Navier-Stokes equations

NASA Technical Reports Server (NTRS)

Hassan, H. A.

1988-01-01

The physical phenomena within supersonic flows that sustain chemical reactions are investigated. An earlier study to develop accurate physical models for supersonic reacting flowfields focused on 2-D laminar shear layers. The objective is to examine the mixing and subsequent combustion within turbulent reacting shear layers. To conduct this study, a computer program has been written to solve the axisymmetric Reynolds averaged Navier-Stokes equations. The numerical method uses a cell-centered finite volume approach and a Runge Kutta time stepping scheme. The Reynolds averaged equations are closed using the eddy viscosity concept. Several zero-equation models have been tested by making calculations for an H2-air nonreacting coaxial jet flow. Comparisons made with experimental data show that Cohen's eddy viscosity model provides best agreement. The finite rate chemistry model used in the study of 2-D laminar shear layers is incorporated into the computer program and data is compared from a recent experiment performed at NASA Langley.

19. On the very accurate numerical evaluation of the Generalized Fermi-Dirac Integrals

Mohankumar, N.; Natarajan, A.

2016-10-01

We indicate a new and a very accurate algorithm for the evaluation of the Generalized Fermi-Dirac Integral with a relative error less than 10-20. The method involves Double Exponential, Trapezoidal and Gauss-Legendre quadratures. For the residue correction of the Gauss-Legendre scheme, a simple and precise continued fraction algorithm is used.

20. Variable time-stepping in the pathwise numerical solution of the chemical Langevin equation.

PubMed

Ilie, Silvana

2012-12-21

Stochastic modeling is essential for an accurate description of the biochemical network dynamics at the level of a single cell. Biochemically reacting systems often evolve on multiple time-scales, thus their stochastic mathematical models manifest stiffness. Stochastic models which, in addition, are stiff and computationally very challenging, therefore the need for developing effective and accurate numerical methods for approximating their solution. An important stochastic model of well-stirred biochemical systems is the chemical Langevin Equation. The chemical Langevin equation is a system of stochastic differential equation with multidimensional non-commutative noise. This model is valid in the regime of large molecular populations, far from the thermodynamic limit. In this paper, we propose a variable time-stepping strategy for the numerical solution of a general chemical Langevin equation, which applies for any level of randomness in the system. Our variable stepsize method allows arbitrary values of the time-step. Numerical results on several models arising in applications show significant improvement in accuracy and efficiency of the proposed adaptive scheme over the existing methods, the strategies based on halving/doubling of the stepsize and the fixed step-size ones.

1. A numerical solution for the diffusion equation in hydrogeologic systems

USGS Publications Warehouse

Ishii, A.L.; Healy, R.W.; Striegl, R.G.

1989-01-01

The documentation of a computer code for the numerical solution of the linear diffusion equation in one or two dimensions in Cartesian or cylindrical coordinates is presented. Applications of the program include molecular diffusion, heat conduction, and fluid flow in confined systems. The flow media may be anisotropic and heterogeneous. The model is formulated by replacing the continuous linear diffusion equation by discrete finite-difference approximations at each node in a block-centered grid. The resulting matrix equation is solved by the method of preconditioned conjugate gradients. The conjugate gradient method does not require the estimation of iteration parameters and is guaranteed convergent in the absence of rounding error. The matrixes are preconditioned to decrease the steps to convergence. The model allows the specification of any number of boundary conditions for any number of stress periods, and the output of a summary table for selected nodes showing flux and the concentration of the flux quantity for each time step. The model is written in a modular format for ease of modification. The model was verified by comparison of numerical and analytical solutions for cases of molecular diffusion, two-dimensional heat transfer, and axisymmetric radial saturated fluid flow. Application of the model to a hypothetical two-dimensional field situation of gas diffusion in the unsaturated zone is demonstrated. The input and output files are included as a check on program installation. The definition of variables, input requirements, flow chart, and program listing are included in the attachments. (USGS)

2. A modulation equations approach for numerically solving the moving soliton and radiation solutions of NLS

Soffer, Avy; Zhao, Xiaofei

2016-04-01

Based on our previous work for solving the nonlinear Schrödinger equation with multichannel dynamics that is given by a localized standing wave and radiation, in this work we deal with the multichannel solution which consists of a moving soliton and radiation. We apply the modulation theory to give a system of ODEs coupled to the radiation term for describing the solution, which is valid for all times. The modulation equations are solved accurately by the proposed numerical method. The soliton and radiation are captured separately in the computation, and they are solved on the translated domain that is moving with them. Thus for a fixed finite physical domain in the lab frame, the multichannel solution can pass through the boundary naturally, which cannot be done by imposing any existing boundary conditions. We comment on the differences of this method from the collective coordinates.

3. Numerical solution of High-kappa model of superconductivity

SciTech Connect

Karamikhova, R.

1996-12-31

We present formulation and finite element approximations of High-kappa model of superconductivity which is valid in the high {kappa}, high magnetic field setting and accounts for applied magnetic field and current. Major part of this work deals with steady-state and dynamic computational experiments which illustrate our theoretical results numerically. In our experiments we use Galerkin discretization in space along with Backward-Euler and Crank-Nicolson schemes in time. We show that for moderate values of {kappa}, steady states of the model system, computed using the High-kappa model, are virtually identical with results computed using the full Ginzburg-Landau (G-L) equations. We illustrate numerically optimal rates of convergence in space and time for the L{sup 2} and H{sup 1} norms of the error in the High-kappa solution. Finally, our numerical approximations demonstrate some well-known experimentally observed properties of high-temperature superconductors, such as appearance of vortices, effects of increasing the applied magnetic field and the sample size, and the effect of applied constant current.

4. Numerical solution of optimal control problems using multiple-interval integral Gegenbauer pseudospectral methods

Tang, Xiaojun

2016-04-01

The main purpose of this work is to provide multiple-interval integral Gegenbauer pseudospectral methods for solving optimal control problems. The latest developed single-interval integral Gauss/(flipped Radau) pseudospectral methods can be viewed as special cases of the proposed methods. We present an exact and efficient approach to compute the mesh pseudospectral integration matrices for the Gegenbauer-Gauss and flipped Gegenbauer-Gauss-Radau points. Numerical results on benchmark optimal control problems confirm the ability of the proposed methods to obtain highly accurate solutions.

5. A critical comparison of the numerical solution of the 1D filtered Vlasov-Poisson equation

Viñas, A. F.; Klimas, A. J.

2003-04-01

We present a comparison of the numerical solution of the filtered Vlasov-Poisson system of equations using the Fourier-Fourier and the Flux-Balance-MacCormack methods in the electrostatic, non-relativistic case. We show that the splitting method combined with the Flux-Balance-MacCormack scheme provides an efficient and accurate scheme for integrating the filtered Vlasov-Poisson system in their self-consistent field. Finally we present various typical problems of interest in plasma physics research which can be studied with the scheme presented here.

6. AN ACCURATE AND EFFICIENT ALGORITHM FOR NUMERICAL SIMULATION OF CONDUCTION-TYPE PROBLEMS. (R824801)

EPA Science Inventory

## Abstract

A modification of the finite analytic numerical method for conduction-type (diffusion) problems is presented. The finite analytic discretization scheme is derived by means of the Fourier series expansion for the most general case of nonuniform grid and variabl...

7. Numerical solution of three-dimensional magnetic differential equations

SciTech Connect

Reiman, A.H.; Greenside, H.S.

1987-02-01

A computer code is described that solves differential equations of the form B . del f = h for a single-valued solution f, given a toroidal three-dimensional divergence-free field B and a single-valued function h. The code uses a new algorithm that Fourier decomposes a given function in a set of flux coordinates in which the field lines are straight. The algorithm automatically adjusts the required integration lengths to compensate for proximity to low order rational surfaces. Applying this algorithm to the Cartesian coordinates defines a transformation to magnetic coordinates, in which the magnetic differential equation can be accurately solved. Our method is illustrated by calculating the Pfirsch-Schlueter currents for a stellarator.

8. Numerical Computation of a Continuous-thrust State Transition Matrix Incorporating Accurate Hardware and Ephemeris Models

NASA Technical Reports Server (NTRS)

Ellison, Donald; Conway, Bruce; Englander, Jacob

2015-01-01

A significant body of work exists showing that providing a nonlinear programming (NLP) solver with expressions for the problem constraint gradient substantially increases the speed of program execution and can also improve the robustness of convergence, especially for local optimizers. Calculation of these derivatives is often accomplished through the computation of spacecraft's state transition matrix (STM). If the two-body gravitational model is employed as is often done in the context of preliminary design, closed form expressions for these derivatives may be provided. If a high fidelity dynamics model, that might include perturbing forces such as the gravitational effect from multiple third bodies and solar radiation pressure is used then these STM's must be computed numerically. We present a method for the power hardward model and a full ephemeris model. An adaptive-step embedded eight order Dormand-Prince numerical integrator is discussed and a method for the computation of the time of flight derivatives in this framework is presented. The use of these numerically calculated derivatieves offer a substantial improvement over finite differencing in the context of a global optimizer. Specifically the inclusion of these STM's into the low thrust missiondesign tool chain in use at NASA Goddard Spaceflight Center allows for an increased preliminary mission design cadence.

9. An Efficient Numerical Solution to the Stochastic Collection Equation.

Tzivion (Tzitzvashvili), Shalva; Feingold, Graham; Levin, Zev

1987-11-01

A new, accurate, efficient method for solving the stochastic collection equation (SCE) is proposed. The SCE is converted to a set of moment equations in categories using a new analytical form of Bleck&s approach. The equations are written in a form amenable to solution and to a category-by-category analysis of drop formation and removal. This method is unique in that closure of the equations is achieved using an expression relating high-order moments to any two lower order moments, thereby restricting the need for approximation of the category distribution function only to integrals over incomplete categories. Moments in categories are then expressed in terms of complete moments with the aid of linear or cubic polynomials. The method is checked for the case of the constant kernel and a linear polynomial kernel. Results show that excellent approximation to the analytical solutions for these kernels are obtained. This is achieved without the use of weighting functions and with modest computing time requirements. The method conserves two or more moments of the spectrum (as required) and successfully alleviates the artificial enhancement of the collection process which is a feature of many schemes.

10. Numerical solution of nonlinear heat problem with moving boundary

AL-Mannai, Mona; Khabeev, Nail

2012-01-01

Two phase gas-liquid flow in pipes is widely spread in space applications: bubble flows appear in cryogenic components transport through fuel/oxidant supply lines. Another important application is based on the fact that in liquid flows with small bubbles a close contact between the two phases occurs resulting in high rates of transfer between them. The compactness of a system makes it ideally suited to serve as a space-based two-phase bio-reactor which forms an important unit in environmental control and life support system deployed onboard. A numerical method was developed for solving a nonlinear problem of thermal interaction between a spherical gas bubble and surrounding liquid. The system of equations for describing this interaction was formulated. It includes ordinary and nonlinear partial differential equations. The problem was solved using finite-difference technique by dividing the system into spherical layers inside the bubble and employing the new variable which "freezes" the moving boundary of the bubble. A numerical solution is obtained for the problem of radial bubble motion induced by a sudden pressure change in the liquid—a situation which corresponds to the behavior of bubbles beyond a shock wave front when the latter enters a bubble curtain.

11. Assessment of an efficient numerical solution of the 1D Richards' equation on bare soil

Varado, N.; Braud, I.; Ross, P. J.; Haverkamp, R.

2006-05-01

A new numerical scheme has been proposed by Ross [Ross, P.J., 2003. Modeling soil water and solute transport—fast, simplified numerical solutions. Agronomy Journal 95, 1352-1361] to solve the 1D Richards' equation [Richards, L.A., 1931. Capillary conduction of liquids through porous medium. Physics 1, 318-333]. This non-iterative solution uses the description of soil properties proposed by Brooks and Corey [Brooks, R.H., Corey, A.T., 1964. Hydraulic properties of porous media. Colorado State University, Fort Collins]. It allows the derivation of an analytical expression for the Kirchhoff potential used in the calculation of water fluxes. The degree of saturation is used as the dependent variable when the soil is unsaturated and the Kirchhoff potential is used in case of saturation. A space and time discretisation scheme leads to a tridiagonal set of linear equations that is solved non-iteratively. We propose in this paper an extensive test of this numerical method, evaluated only on a single case by Ross. The tests are conducted in two steps. First, the solution is assessed against two analytical solutions. The first one [Basha, H.A., 1999. Multidimensional linearized nonsteady infiltration with prescribed boundary conditions at the soil surface. Water Resources Research 35(1), 75-93] provides the water content profile when simplified soil characteristics such as the exponential law of Gardner [Gardner, W.R., 1958. Some steady-state solutions of the unsaturated moisture flow equations with application to evaporation from a water table. Soil Science 85, 228-232] are used. The Ross solution is compared to this solution on eight different soils that were fitted to this law. Analytical solution with the Brooks and Corey models is not available at the moment for the moisture profile but some exist for cumulative infiltration. Therefore, the second analytical solution, used in this study, is the one developed by Parlange et al. [Parlange, J.-Y., Haverkamp, R., Touma, J

12. TRANSPORT OF REACTING SOLUTES SUBJECT TO A MOVING DISSOLUTION BOUNDARY: NUMERICAL METHODS AND SOLUTIONS.

USGS Publications Warehouse

Willis, Catherine; Rubin, Jacob

1987-01-01

In this paper we consider examples of chemistry-affected transport processes in porous media. A moving boundary problem which arises during transport with precipitation-dissolution reactions is solved by three different numerical methods. Two of these methods (one explicit and one implicit) are based on an integral formulation of mass balance and lead to an approximation of a weak solution. These methods are compared to a front-tracking scheme. Although the two approaches are conceptually different, the numerical solutions showed good agreement. As the ratio of dispersion to convection decreases, the methods based on the integral formulation become computationally more efficient. Specific reactions were modeled to examine the dependence of the system on the physical and chemical parameters.

13. Towards more accurate numerical modeling of impedance based high frequency harmonic vibration

Lim, Yee Yan; Kiong Soh, Chee

2014-03-01

The application of smart materials in various fields of engineering has recently become increasingly popular. For instance, the high frequency based electromechanical impedance (EMI) technique employing smart piezoelectric materials is found to be versatile in structural health monitoring (SHM). Thus far, considerable efforts have been made to study and improve the technique. Various theoretical models of the EMI technique have been proposed in an attempt to better understand its behavior. So far, the three-dimensional (3D) coupled field finite element (FE) model has proved to be the most accurate. However, large discrepancies between the results of the FE model and experimental tests, especially in terms of the slope and magnitude of the admittance signatures, continue to exist and are yet to be resolved. This paper presents a series of parametric studies using the 3D coupled field finite element method (FEM) on all properties of materials involved in the lead zirconate titanate (PZT) structure interaction of the EMI technique, to investigate their effect on the admittance signatures acquired. FE model updating is then performed by adjusting the parameters to match the experimental results. One of the main reasons for the lower accuracy, especially in terms of magnitude and slope, of previous FE models is the difficulty in determining the damping related coefficients and the stiffness of the bonding layer. In this study, using the hysteretic damping model in place of Rayleigh damping, which is used by most researchers in this field, and updated bonding stiffness, an improved and more accurate FE model is achieved. The results of this paper are expected to be useful for future study of the subject area in terms of research and application, such as modeling, design and optimization.

14. Numerical Solution of the k-Eigenvalue Problem

Hamilton, Steven Paul

2011-12-01

Obtaining solutions to the k-eigenvalue form of the radiation transport equation is an important topic in the design and analysis of nuclear reactors. Although this has been an area of active interest in the nuclear engineering community for several decades, to date no truly satisfactory solution strategies exist. In general, existing techniques are either slow to converge for difficult problems or suffer from stability and robustness issues that can cause solvers to diverge for some problems. This work provides a comparison between a variety of methods and introduces a new strategy based on the Davidson method that has been used in other fields for many years but never for this problem. The Davidson method offers an alternative to the nested iteration structure inherent to standard approaches and allows expensive linear solvers to be replaced by a potentially cheap preconditioner. To fill the role of this preconditioner, a strategy based on a multigrid treatment of the energy variable is developed. Numerical experiments using the 2-D NEWT transport package are presented, demonstrating the effectiveness of the proposed strategy.

15. Numerical solution of Boltzmann equation using discrete velocity grids

Vedula, Prakash

2015-11-01

An importance sampling based approach for numerical solution of the (single species) Boltzmann equation using discrete velocity grids is proposed. This approach involves a stochastic method for evaluation of the collision integral based on sampling of depleting/replenishing collisions and is designed to preserve important symmetries of the collision operator, including collision invariants. The underlying particle distribution function is represented as a collection of delta functions with associated weights that are non-negative. A key feature in the construction of the proposed method is that it ensures that the weights associated with the distribution function remain non-negative during collisional relaxation, thereby satisfying an important realizability condition. Performance of the proposed approach will be studied using test problems involving spatially homogeneous collisional relaxation flow and microchannel flows. Results obtained from the proposed method will be compared with those obtained from the (deterministic) collisional Lattice Boltzmann Method (cLBM) and the traditional direct simulation Monte Carlo (DSMC) method for solution of Boltzmann equation. Extension of the proposed method using discrete velocity grids for multicomponent mixtures will also be discussed.

16. On a numerical solution of the plastic buckling problem of structures

NASA Technical Reports Server (NTRS)

Gupta, K. K.

1978-01-01

An automated digital computer procedure is presented for the accurate and efficient solution of the plastic buckling problem of structures. This is achieved by a Sturm sequence method employing a bisection strategy, which eliminates the need for having to solve the buckling eigenvalue problem at each incremental (decremental) loading stage that is associated with the usual solution techniques. The plastic buckling mode shape is determined by a simple inverse iteration process, once the buckling load has been established. Numerical results are presented for plate problems with various edge conditions. The resulting computer program written in FORTRAN V for the JPL UNIVAC 1108 machine proves to be most economical in comparison with other existing methods of such analysis.

17. Automatically high accurate and efficient photomask defects management solution for advanced lithography manufacture

Zhu, Jun; Chen, Lijun; Ma, Lantao; Li, Dejian; Jiang, Wei; Pan, Lihong; Shen, Huiting; Jia, Hongmin; Hsiang, Chingyun; Cheng, Guojie; Ling, Li; Chen, Shijie; Wang, Jun; Liao, Wenkui; Zhang, Gary

2014-04-01

Defect review is a time consuming job. Human error makes result inconsistent. The defects located on don't care area would not hurt the yield and no need to review them such as defects on dark area. However, critical area defects can impact yield dramatically and need more attention to review them such as defects on clear area. With decrease in integrated circuit dimensions, mask defects are always thousands detected during inspection even more. Traditional manual or simple classification approaches are unable to meet efficient and accuracy requirement. This paper focuses on automatic defect management and classification solution using image output of Lasertec inspection equipment and Anchor pattern centric image process technology. The number of mask defect found during an inspection is always in the range of thousands or even more. This system can handle large number defects with quick and accurate defect classification result. Our experiment includes Die to Die and Single Die modes. The classification accuracy can reach 87.4% and 93.3%. No critical or printable defects are missing in our test cases. The missing classification defects are 0.25% and 0.24% in Die to Die mode and Single Die mode. This kind of missing rate is encouraging and acceptable to apply on production line. The result can be output and reloaded back to inspection machine to have further review. This step helps users to validate some unsure defects with clear and magnification images when captured images can't provide enough information to make judgment. This system effectively reduces expensive inline defect review time. As a fully inline automated defect management solution, the system could be compatible with current inspection approach and integrated with optical simulation even scoring function and guide wafer level defect inspection.

18. Fast and accurate numerical method for predicting gas chromatography retention time.

PubMed

Claumann, Carlos Alberto; Wüst Zibetti, André; Bolzan, Ariovaldo; Machado, Ricardo A F; Pinto, Leonel Teixeira

2015-08-01

Predictive modeling for gas chromatography compound retention depends on the retention factor (ki) and on the flow of the mobile phase. Thus, different approaches for determining an analyte ki in column chromatography have been developed. The main one is based on the thermodynamic properties of the component and on the characteristics of the stationary phase. These models can be used to estimate the parameters and to optimize the programming of temperatures, in gas chromatography, for the separation of compounds. Different authors have proposed the use of numerical methods for solving these models, but these methods demand greater computational time. Hence, a new method for solving the predictive modeling of analyte retention time is presented. This algorithm is an alternative to traditional methods because it transforms its attainments into root determination problems within defined intervals. The proposed approach allows for tr calculation, with accuracy determined by the user of the methods, and significant reductions in computational time; it can also be used to evaluate the performance of other prediction methods.

19. The use of experimental bending tests to more accurate numerical description of TBC damage process

2016-04-01

Thermal barrier coatings (TBCs) have been extensively used in aircraft engines to protect critical engine parts such as blades and combustion chambers, which are exposed to high temperatures and corrosive environment. The blades of turbine engines are additionally exposed to high mechanical loads. These loads are created by the high rotational speed of the rotor (30 000 rot/min), causing the tensile and bending stresses. Therefore, experimental testing of coated samples is necessary in order to determine strength properties of TBCs. Beam samples with dimensions 50×10×2 mm were used in those studies. The TBC system consisted of 150 μm thick bond coat (NiCoCrAlY) and 300 μm thick top coat (YSZ) made by APS (air plasma spray) process. Samples were tested by three-point bending test with various loads. After bending tests, the samples were subjected to microscopic observation to determine the quantity of cracks and their depth. The above mentioned results were used to build numerical model and calibrate material data in Abaqus program. Brittle cracking damage model was applied for the TBC layer, which allows to remove elements after reaching criterion. Surface based cohesive behavior was used to model the delamination which may occur at the boundary between bond coat and top coat.

20. Primitive numerical simulation of circular Couette flow - Carrousel wind tunnel nonturbulent solutions

NASA Technical Reports Server (NTRS)

Hasiuk, Jan; Hindman, Richard; Iversen, James

1988-01-01

The azimuthal-invariant, three-dimensional cylindrical, incompressible Navier-Stokes equations are solved to steady state for a finite-length, physically realistic model. The numerical method relies on an alternating-direction implicit scheme that is formally second-order accurate in space and first-order accurate in time. The equations are linearized and uncoupled by evaluating variable coefficients at the previous time iteration. Wall grid clustering is provided by a Roberts transformation in radial and axial directions. A vorticity-velocity formulation is found to be preferable to a vorticity-streamfunction approach. Subject to no-slip, Dirichlet boundary conditions, except for the inner cylinder rotation velocity (impulsive start-up) and zero-flow initial conditions, nonturbulent solutions are obtained for sub- and supercritical Reynolds numbers of 100 to 400 for a finite geometry where R(outer)/R(inner) = 1.5, H/R(inner) = 0.73, and H/Delta-R = 1.5. An axially-stretched model solution is shown to asymptotically approach the one-dimensional analytic Couette solution at the cylinder midheight. Flowfield change from laminar to Taylor-vortex flow is discussed as a function of Reynolds number. Three-dimensional velocities, vorticity, and streamfunction are presented via two-dimensional graphs and three-dimensional surface and contour plots.

1. State space truncation with quantified errors for accurate solutions to discrete Chemical Master Equation

PubMed Central

Cao, Youfang; Terebus, Anna; Liang, Jie

2016-01-01

truncation and error analysis methods developed here can be used to ensure accurate direct solutions to the dCME for a large number of stochastic networks. PMID:27105653

2. State Space Truncation with Quantified Errors for Accurate Solutions to Discrete Chemical Master Equation.

PubMed

Cao, Youfang; Terebus, Anna; Liang, Jie

2016-04-01

truncation and error analysis methods developed here can be used to ensure accurate direct solutions to the dCME for a large number of stochastic networks.

3. Numerical solutions of the semiclassical Boltzmann ellipsoidal-statistical kinetic model equation

PubMed Central

Yang, Jaw-Yen; Yan, Chin-Yuan; Huang, Juan-Chen; Li, Zhihui

2014-01-01

Computations of rarefied gas dynamical flows governed by the semiclassical Boltzmann ellipsoidal-statistical (ES) kinetic model equation using an accurate numerical method are presented. The semiclassical ES model was derived through the maximum entropy principle and conserves not only the mass, momentum and energy, but also contains additional higher order moments that differ from the standard quantum distributions. A different decoding procedure to obtain the necessary parameters for determining the ES distribution is also devised. The numerical method in phase space combines the discrete-ordinate method in momentum space and the high-resolution shock capturing method in physical space. Numerical solutions of two-dimensional Riemann problems for two configurations covering various degrees of rarefaction are presented and various contours of the quantities unique to this new model are illustrated. When the relaxation time becomes very small, the main flow features a display similar to that of ideal quantum gas dynamics, and the present solutions are found to be consistent with existing calculations for classical gas. The effect of a parameter that permits an adjustable Prandtl number in the flow is also studied. PMID:25104904

4. Numerical solutions of the semiclassical Boltzmann ellipsoidal-statistical kinetic model equation.

PubMed

Yang, Jaw-Yen; Yan, Chin-Yuan; Huang, Juan-Chen; Li, Zhihui

2014-08-01

Computations of rarefied gas dynamical flows governed by the semiclassical Boltzmann ellipsoidal-statistical (ES) kinetic model equation using an accurate numerical method are presented. The semiclassical ES model was derived through the maximum entropy principle and conserves not only the mass, momentum and energy, but also contains additional higher order moments that differ from the standard quantum distributions. A different decoding procedure to obtain the necessary parameters for determining the ES distribution is also devised. The numerical method in phase space combines the discrete-ordinate method in momentum space and the high-resolution shock capturing method in physical space. Numerical solutions of two-dimensional Riemann problems for two configurations covering various degrees of rarefaction are presented and various contours of the quantities unique to this new model are illustrated. When the relaxation time becomes very small, the main flow features a display similar to that of ideal quantum gas dynamics, and the present solutions are found to be consistent with existing calculations for classical gas. The effect of a parameter that permits an adjustable Prandtl number in the flow is also studied. PMID:25104904

5. Sedimentation analysis of noninteracting and self-associating solutes using numerical solutions to the Lamm equation.

PubMed Central

Schuck, P

1998-01-01

The potential of using the Lamm equation in the analysis of hydrodynamic shape and gross conformation of proteins and reversibly formed protein complexes from analytical ultracentrifugation data was investigated. An efficient numerical solution of the Lamm equation for noninteracting and rapidly self-associating proteins by using combined finite-element and moving grid techniques is described. It has been implemented for noninteracting solutes and monomer-dimer and monomer-trimer equilibria. To predict its utility, the error surface of a nonlinear regression of simulated sedimentation profiles was explored. Error contour maps were calculated for conventional independent and global analyses of experiments with noninteracting solutes and with monomer-dimer systems at different solution column heights, loading concentrations, and centrifugal fields. It was found that the rotor speed is the major determinant for the shape of the error surface, and that global analysis of different experiments can allow substantially improved characterization of the solutes. We suggest that the global analysis of the approach to equilibrium in a short-column sedimentation equilibrium experiment followed by a high-speed short-column sedimentation velocity experiment can result in sedimentation and diffusion coefficients of very high statistical accuracy. In addition, in the case of a protein in rapid monomer-dimer equilibrium, this configuration was found to reveal the most precise estimate of the association constant. PMID:9726952

6. A fast numerical solution of scattering by a cylinder: Spectral method for the boundary integral equations

NASA Technical Reports Server (NTRS)

Hu, Fang Q.

1994-01-01

It is known that the exact analytic solutions of wave scattering by a circular cylinder, when they exist, are not in a closed form but in infinite series which converges slowly for high frequency waves. In this paper, we present a fast number solution for the scattering problem in which the boundary integral equations, reformulated from the Helmholtz equation, are solved using a Fourier spectral method. It is shown that the special geometry considered here allows the implementation of the spectral method to be simple and very efficient. The present method differs from previous approaches in that the singularities of the integral kernels are removed and dealt with accurately. The proposed method preserves the spectral accuracy and is shown to have an exponential rate of convergence. Aspects of efficient implementation using FFT are discussed. Moreover, the boundary integral equations of combined single and double-layer representation are used in the present paper. This ensures the uniqueness of the numerical solution for the scattering problem at all frequencies. Although a strongly singular kernel is encountered for the Neumann boundary conditions, we show that the hypersingularity can be handled easily in the spectral method. Numerical examples that demonstrate the validity of the method are also presented.

7. Albedos of homogeneous semi-infinite canopies - Comparison of two-stream analytic and numerical solutions

NASA Technical Reports Server (NTRS)

Dickinson, Robert E.; Sellers, Piers J.; Kimes, Daniel S.

1987-01-01

The albedo of plant canopies is treated as a problem in radiative transfer. Albedos calcualted from an iterative multistream numerical model are compared with those calculated with an analytic two-stream solution. With the assumption of a randomly homogeneous distribution of leaf positions and orientations and isotropic scattering by individual leaves, the single-scattering albedo of the canopy can be found analytically. This single-scattering solution is incorporated into the two-stream solution and used to benchmark the multistream numerical model in the single-scattering limit. Relative errors so established in the multistream model are O(0.3 percent) or less. The two-stream model is also found to be remarkably accurate, with the error in multiply scattered radiation O(5 percent) or less, corresponding to absolute errors in visible albedo of less than 0.001 and near-infrared albedo of less than or equal to 0.01. Thus the two-stream model should be adequate for many purposes, such as climate modeling, provided the assumptions of homogeneous canopy and isotropic scattering are not too unrealistic.

8. Numerical solution of acoustic scattering by finite perforated elastic plates

Cavalieri, A. V. G.; Wolf, W. R.; Jaworski, J. W.

2016-04-01

We present a numerical method to compute the acoustic field scattered by finite perforated elastic plates. A boundary element method is developed to solve the Helmholtz equation subjected to boundary conditions related to the plate vibration. These boundary conditions are recast in terms of the vibration modes of the plate and its porosity, which enables a direct solution procedure. A parametric study is performed for a two-dimensional problem whereby a cantilevered perforated elastic plate scatters sound from a point quadrupole near the free edge. Both elasticity and porosity tend to diminish the scattered sound, in agreement with previous work considering semi-infinite plates. Finite elastic plates are shown to reduce acoustic scattering when excited at high Helmholtz numbers k0 based on the plate length. However, at low k0, finite elastic plates produce only modest reductions or, in cases related to structural resonance, an increase to the scattered sound level relative to the rigid case. Porosity, on the other hand, is shown to be more effective in reducing the radiated sound for low k0. The combined beneficial effects of elasticity and porosity are shown to be effective in reducing the scattered sound for a broader range of k0 for perforated elastic plates.

9. Numerical solution of periodic vortical flows about a thin airfoil

NASA Technical Reports Server (NTRS)

Scott, James R.; Atassi, Hafiz M.

1989-01-01

A numerical method is developed for computing periodic, three-dimensional, vortical flows around isolated airfoils. The unsteady velocity is split into a vortical component which is a known function of the upstream flow conditions and the Lagrangian coordinates of the mean flow, and an irrotational field whose potential satisfies a nonconstant-coefficient, inhomogeneous, convective wave equation. Solutions for thin airfoils at zero degrees incidence to the mean flow are presented in this paper. Using an elliptic coordinate transformation, the computational domain is transformed into a rectangle. The Sommerfeld radiation condition is applied to the unsteady pressure on the grid line corresponding to the far field boundary. The results are compared with a Possio solver, and it is shown that for maximum accuracy the grid should depend on both the Mach number and reduced frequency. Finally, in order to assess the range of validity of the classical thin airfoil approximation, results for airfoils with zero thickness are compared with results for airfoils with small thickness.

10. Elastic turbulence in Taylor-Couette Flow of Dilute Polymeric Solutions: A Direct Numerical Simulation Study

Liu, Nansheng; Khomami, Bamin

2011-11-01

Despite tremendous progress in development of numerical techniques and constitutive theories for polymeric fluids in the past decade, Direct Numerical Simulation (DNS) of elastic turbulence has posed tremendous challenges to researchers engaged in developing first principles models and simulations that can accurately and robustly predict the dynamical behavior of polymeric flows. In this presentation, we report the first DNS of elastic turbulence in the Taylor-Couette (TC) flow. Specifically, our computations with prototypical constitutive equations for dilute polymeric solutions, such as the FENE-P model are capable of reproducing the essential features of the experimentally observed elastic turbulence in TC flow of this class of fluids, namely, randomly fluctuating fluid motion excited in a broad range of spatial and temporal scales, and a significant increase of the flow resistance. Moreover, the experimentally measured Power Spectral Density of radial velocity fluctuations, i.e., two contiguous regions of power-law decay, -1.1 at lower frequencies and -2.2 at high-frequencies is accurately computed. We would like to thank NSF through grant CBET-0755269 and NSFC through grant NO. 10972211 for supporting of this work.

11. Efficient numerical solution of acoustic scattering from doubly-periodic arrays of axisymmetric objects

Liu, Yuxiang; Barnett, Alex H.

2016-11-01

We present a high-order accurate boundary-based solver for three-dimensional (3D) frequency-domain scattering from a doubly-periodic grating of smooth axisymmetric sound-hard or transmission obstacles. We build the one-obstacle solution operator using separation into P azimuthal modes via the FFT, the method of fundamental solutions (with N proxy points lying on a curve), and dense direct least-squares solves; the effort is O (N3 P) with a small constant. Periodizing then combines fast multipole summation of nearest neighbors with an auxiliary global Helmholtz basis expansion to represent the distant contributions, and enforcing quasiperiodicity and radiation conditions on the unit cell walls. Eliminating the auxiliary coefficients, and preconditioning with the one-obstacle solution operator, leaves a well-conditioned square linear system that is solved iteratively. The solution time per incident wave is then O (NP) at fixed frequency. Our scheme avoids singular quadratures, periodic Green's functions, and lattice sums, and its convergence rate is unaffected by resonances within obstacles. We include numerical examples such as scattering from a grating of period 13 λ × 13 λ comprising highly-resonant sound-hard "cups" each needing NP = 64800 surface unknowns, to 10-digit accuracy, in half an hour on a desktop.

12. A robust and accurate numerical method for transcritical turbulent flows at supercritical pressure with an arbitrary equation of state

Kawai, Soshi; Terashima, Hiroshi; Negishi, Hideyo

2015-11-01

This paper addresses issues in high-fidelity numerical simulations of transcritical turbulent flows at supercritical pressure. The proposed strategy builds on a tabulated look-up table method based on REFPROP database for an accurate estimation of non-linear behaviors of thermodynamic and fluid transport properties at the transcritical conditions. Based on the look-up table method we propose a numerical method that satisfies high-order spatial accuracy, spurious-oscillation-free property, and capability of capturing the abrupt variation in thermodynamic properties across the transcritical contact surface. The method introduces artificial mass diffusivity to the continuity and momentum equations in a physically-consistent manner in order to capture the steep transcritical thermodynamic variations robustly while maintaining spurious-oscillation-free property in the velocity field. The pressure evolution equation is derived from the full compressible Navier-Stokes equations and solved instead of solving the total energy equation to achieve the spurious pressure oscillation free property with an arbitrary equation of state including the present look-up table method. Flow problems with and without physical diffusion are employed for the numerical tests to validate the robustness, accuracy, and consistency of the proposed approach.

13. An adaptive grid method for computing time accurate solutions on structured grids

NASA Technical Reports Server (NTRS)

Bockelie, Michael J.; Smith, Robert E.; Eiseman, Peter R.

1991-01-01

The solution method consists of three parts: a grid movement scheme; an unsteady Euler equation solver; and a temporal coupling routine that links the dynamic grid to the Euler solver. The grid movement scheme is an algebraic method containing grid controls that generate a smooth grid that resolves the severe solution gradients and the sharp transitions in the solution gradients. The temporal coupling is performed with a grid prediction correction procedure that is simple to implement and provides a grid that does not lag the solution in time. The adaptive solution method is tested by computing the unsteady inviscid solutions for a one dimensional shock tube and a two dimensional shock vortex iteraction.

14. Scaling laws and accurate small-amplitude stationary solution for the motion of a planar vortex filament in the Cartesian form of the local induction approximation.

PubMed

Van Gorder, Robert A

2013-04-01

We provide a formulation of the local induction approximation (LIA) for the motion of a vortex filament in the Cartesian reference frame (the extrinsic coordinate system) which allows for scaling of the reference coordinate. For general monotone scalings of the reference coordinate, we derive an equation for the planar solution to the derivative nonlinear Schrödinger equation governing the LIA. We proceed to solve this equation perturbatively in small amplitude through an application of multiple-scales analysis, which allows for accurate computation of the period of the planar vortex filament. The perturbation result is shown to agree strongly with numerical simulations, and we also relate this solution back to the solution obtained in the arclength reference frame (the intrinsic coordinate system). Finally, we discuss nonmonotone coordinate scalings and their application for finding self-intersections of vortex filaments. These self-intersecting vortex filaments are likely unstable and collapse into other structures or dissipate completely.

15. A GENERAL MASS-CONSERVATIVE NUMERICAL SOLUTION FOR THE UNSATURATED FLOW EQUATION

EPA Science Inventory

Numerical approximations based on different forms of the governing partial differential equation can lead to significantly different results for unsaturated flow problems. Numerical solution based on the standard h-based form of Richards equation generally yields poor results, ch...

16. Numerical solution of the stochastic parabolic equation with the dependent operator coefficient

SciTech Connect

Ashyralyev, Allaberen; Okur, Ulker

2015-09-18

In the present paper, a single step implicit difference scheme for the numerical solution of the stochastic parabolic equation with the dependent operator coefficient is presented. Theorem on convergence estimates for the solution of this difference scheme is established. In applications, this abstract result permits us to obtain the convergence estimates for the solution of difference schemes for the numerical solution of initial boundary value problems for parabolic equations. The theoretical statements for the solution of this difference scheme are supported by the results of numerical experiments.

17. Asynchronous and corrected-asynchronous numerical solutions of parabolic PDES on MIMD multiprocessors

NASA Technical Reports Server (NTRS)

Amitai, Dganit; Averbuch, Amir; Itzikowitz, Samuel; Turkel, Eli

1991-01-01

A major problem in achieving significant speed-up on parallel machines is the overhead involved with synchronizing the concurrent process. Removing the synchronization constraint has the potential of speeding up the computation. The authors present asynchronous (AS) and corrected-asynchronous (CA) finite difference schemes for the multi-dimensional heat equation. Although the discussion concentrates on the Euler scheme for the solution of the heat equation, it has the potential for being extended to other schemes and other parabolic partial differential equations (PDEs). These schemes are analyzed and implemented on the shared memory multi-user Sequent Balance machine. Numerical results for one and two dimensional problems are presented. It is shown experimentally that the synchronization penalty can be about 50 percent of run time: in most cases, the asynchronous scheme runs twice as fast as the parallel synchronous scheme. In general, the efficiency of the parallel schemes increases with processor load, with the time level, and with the problem dimension. The efficiency of the AS may reach 90 percent and over, but it provides accurate results only for steady-state values. The CA, on the other hand, is less efficient, but provides more accurate results for intermediate (non steady-state) values.

18. Collocation Method for Numerical Solution of Coupled Nonlinear Schroedinger Equation

SciTech Connect

Ismail, M. S.

2010-09-30

The coupled nonlinear Schroedinger equation models several interesting physical phenomena presents a model equation for optical fiber with linear birefringence. In this paper we use collocation method to solve this equation, we test this method for stability and accuracy. Numerical tests using single soliton and interaction of three solitons are used to test the resulting scheme.

19. Exact expressions and accurate approximations for the dependences of radius and index of refraction of solutions of inorganic solutes on relative humidity

SciTech Connect

Lewis, E.R.; Schwartz, S.

2010-03-15

Light scattering by aerosols plays an important role in Earth’s radiative balance, and quantification of this phenomenon is important in understanding and accounting for anthropogenic influences on Earth’s climate. Light scattering by an aerosol particle is determined by its radius and index of refraction, and for aerosol particles that are hygroscopic, both of these quantities vary with relative humidity RH. Here exact expressions are derived for the dependences of the radius ratio (relative to the volume-equivalent dry radius) and index of refraction on RH for aqueous solutions of single solutes. Both of these quantities depend on the apparent molal volume of the solute in solution and on the practical osmotic coefficient of the solution, which in turn depend on concentration and thus implicitly on RH. Simple but accurate approximations are also presented for the RH dependences of both radius ratio and index of refraction for several atmospherically important inorganic solutes over the entire range of RH values for which these substances can exist as solution drops. For all substances considered, the radius ratio is accurate to within a few percent, and the index of refraction to within ~0.02, over this range of RH. Such parameterizations will be useful in radiation transfer models and climate models.

20. Higher-order approximation of contaminant transport equation for turbulent channel flows based on centre manifolds and its numerical solution

Ngo-Cong, D.; Mohammed, F. J.; Strunin, D. V.; Skvortsov, A. T.; Mai-Duy, N.; Tran-Cong, T.

2015-06-01

The contaminant transport process governed by the advection-diffusion equation plays an important role in modelling industrial and environmental flows. In this article, our aim is to accurately reduce the 2-D advection-diffusion equation governing the dispersion of a contaminant in a turbulent open channel flow to its 1-D approximation. The 1-D model helps to quickly estimate the horizontal size of contaminant clouds based on the values of the model coefficients. We derive these coefficients analytically and investigate numerically the model convergence. The derivation is based on the centre manifold theory to obtain successively more accurate approximations in a consistent manner. Two types of the average velocity profile are considered: the classical logarithmic profile and the power profile. We further develop the one-dimensional integrated radial basis function network method as a numerical approach to obtain the numerical solutions to both the original 2-D equation and the approximate 1-D equations. We compare the solutions of the original models with their centre-manifold approximations at very large Reynolds numbers. The numerical results obtained from the approximate 1-D models are in good agreement with those of the original 2-D model for both the logarithmic and power velocity profiles.

1. Numerical solutions of the compressible 3-D boundary-layer equations for aerospace configurations with emphasis on LFC

NASA Technical Reports Server (NTRS)

Harris, Julius E.; Iyer, Venkit; Radwan, Samir

1987-01-01

The application of stability theory in Laminar Flow Control (LFC) research requires that density and velocity profiles be specified throughout the viscous flow field of interest. These profile values must be as numerically accurate as possible and free of any numerically induced oscillations. Guidelines for the present research project are presented: develop an efficient and accurate procedure for solving the 3-D boundary layer equation for aerospace configurations; develop an interface program to couple selected 3-D inviscid programs that span the subsonic to hypersonic Mach number range; and document and release software to the LFC community. The interface program was found to be a dependable approach for developing a user friendly procedure for generating the boundary-layer grid and transforming an inviscid solution from a relatively coarse grid to a sufficiently fine boundary-layer grid. The boundary-layer program was shown to be fourth-order accurate in the direction normal to the wall boundary and second-order accurate in planes parallel to the boundary. The fourth-order accuracy allows accurate calculations with as few as one-fifth the number of grid points required for conventional second-order schemes.

2. Numerical solution of control problems governed by nonlinear differential equations

SciTech Connect

Heinkenschloss, M.

1994-12-31

In this presentation the author investigates an iterative method for the solution of optimal control problems. These problems are formulated as constrained optimization problems with constraints arising from the state equation and in the form of bound constraints on the control. The method for the solution of these problems uses the special structure of the problem arising from the bound constraint and the state equation. It is derived from SQP methods and projected Newton methods and combines the advantages of both methods. The bound constraint is satisfied by all iterates using a projection, the nonlinear state equation is satisfied in the limit. Only a linearized state equation has to be solved in every iteration. The solution of the linearized problems are done using multilevel methods and GMRES.

3. Toward Accurate Modeling of the Effect of Ion-Pair Formation on Solute Redox Potential.

PubMed

2016-09-13

A scheme to model the dependence of a solute redox potential on the supporting electrolyte is proposed, and the results are compared to experimental observations and other reported theoretical models. An improved agreement with experiment is exhibited if the effect of the supporting electrolyte on the redox potential is modeled through a concentration change induced via ion pair formation with the salt, rather than by only considering the direct impact on the redox potential of the solute itself. To exemplify the approach, the scheme is applied to the concentration-dependent redox potential of select molecules proposed for nonaqueous flow batteries. However, the methodology is general and enables rational computational electrolyte design through tuning of the operating window of electrochemical systems by shifting the redox potential of its solutes; including potentially both salts as well as redox active molecules. PMID:27500744

4. Low Reynolds number numerical solutions of chaotic flow

NASA Technical Reports Server (NTRS)

Pulliam, Thomas H.

1989-01-01

Numerical computations of two-dimensional flow past an airfoil at low Mach number, large angle of attack, and low Reynolds number are reported which show a sequence of flow states leading from single-period vortex shedding to chaos via the period-doubling mechanism. Analysis of the flow in terms of phase diagrams, Poincare sections, and flowfield variables are used to substantiate these results. The critical Reynolds number for the period-doubling bifurcations is shown to be sensitive to mesh refinement and the influence of large amounts of numerical dissipation. In extreme cases, large amounts of added dissipation can delay or completely eliminate the chaotic response. The effect of artificial dissipation at these low Reynolds numbers is to produce a new effective Reynolds number for the computations.

5. Numerical Solution of Incompressible Navier-Stokes Equations Using a Fractional-Step Approach

NASA Technical Reports Server (NTRS)

Kiris, Cetin; Kwak, Dochan

1999-01-01

A fractional step method for the solution of steady and unsteady incompressible Navier-Stokes equations is outlined. The method is based on a finite volume formulation and uses the pressure in the cell center and the mass fluxes across the faces of each cell as dependent variables. Implicit treatment of convective and viscous terms in the momentum equations enables the numerical stability restrictions to be relaxed. The linearization error in the implicit solution of momentum equations is reduced by using three subiterations in order to achieve second order temporal accuracy for time-accurate calculations. In spatial discretizations of the momentum equations, a high-order (3rd and 5th) flux-difference splitting for the convective terms and a second-order central difference for the viscous terms are used. The resulting algebraic equations are solved with a line-relaxation scheme which allows the use of large time step. A four color ZEBRA scheme is employed after the line-relaxation procedure in the solution of the Poisson equation for pressure. This procedure is applied to a Couette flow problem using a distorted computational grid to show that the method minimizes grid effects. Additional benchmark cases include the unsteady laminar flow over a circular cylinder for Reynolds Numbers of 200, and a 3-D, steady, turbulent wingtip vortex wake propagation study. The solution algorithm does a very good job in resolving the vortex core when 5th-order upwind differencing and a modified production term in the Baldwin-Barth one-equation turbulence model are used with adequate grid resolution.

6. Petrophysical corner - a numerical solution for CNL environmental corrections

SciTech Connect

Tsay, F.S.; Dindoruk, B. )

1990-04-01

This paper suggests numerical methods for converting a complicated Compensated Neutron Log correction nomograph into systems of simple polynomials for computer programming. Basic programs are presented that can provide a base for writing a complete computer-based CNL environmental corrections program, providing that an additional main program is written to include the inputs and outputs for the necessary environmental conditions and computer results as well as proper structural linkage of the subroutines.

7. An Integrated Numerical Hydrodynamic Shallow Flow-Solute Transport Model for Urban Area

Alias, N. A.; Mohd Sidek, L.

2016-03-01

The rapidly changing on land profiles in the some urban areas in Malaysia led to the increasing of flood risk. Extensive developments on densely populated area and urbanization worsen the flood scenario. An early warning system is really important and the popular method is by numerically simulating the river and flood flows. There are lots of two-dimensional (2D) flood model predicting the flood level but in some circumstances, still it is difficult to resolve the river reach in a 2D manner. A systematic early warning system requires a precisely prediction of flow depth. Hence a reliable one-dimensional (1D) model that provides accurate description of the flow is essential. Research also aims to resolve some of raised issues such as the fate of pollutant in river reach by developing the integrated hydrodynamic shallow flow-solute transport model. Presented in this paper are results on flow prediction for Sungai Penchala and the convection-diffusion of solute transports simulated by the developed model.

8. Numerical solution of two dimensional time fractional-order biological population model

Prakash, Amit; Kumar, Manoj

2016-06-01

In this work, we provide an approximate solution of a parabolic fractional degenerate problem emerging in a spatial diffusion of biological population model using a fractional variational iteration method (FVIM). Four test illustrations are used to show the proficiency and accuracy of the projected scheme. Comparisons between exact solutions and numerical solutions are presented for different values of fractional order α.

9. Numerical solutions of the complete Navier-Stokes equations

NASA Technical Reports Server (NTRS)

Hassan, H. A.

1993-01-01

The objective of this study is to compare the use of assumed pdf (probability density function) approaches for modeling supersonic turbulent reacting flowfields with the more elaborate approach where the pdf evolution equation is solved. Assumed pdf approaches for averaging the chemical source terms require modest increases in CPU time typically of the order of 20 percent above treating the source terms as 'laminar.' However, it is difficult to assume a form for these pdf's a priori that correctly mimics the behavior of the actual pdf governing the flow. Solving the evolution equation for the pdf is a theoretically sound approach, but because of the large dimensionality of this function, its solution requires a Monte Carlo method which is computationally expensive and slow to coverage. Preliminary results show both pdf approaches to yield similar solutions for the mean flow variables.

10. Numerical solution of compressible viscous flows at high Reynolds numbers

NASA Technical Reports Server (NTRS)

Maccormack, R. W.

1981-01-01

A new numerical method which was used to reduce the computation time required in fluid dynamics to solve the Navier-Stokes equations at flight Reynolds numbers is described. The method is the implicit analogue of the explicit finite different method. It uses this as its first stage, while the second stage removes the restrictive stability condition by recasting the difference equations in an implicit form. The resulting matrix equations to be solved are either upper or lower block bidiagonal equations. The new method makes it possible and practical to calculate many important three dimensional, high Reynolds number flow fields on computers.

11. Numerical solution of optimal design for axisymmetrical cooling canal

Salač, Petr; Dvořák, Václav

2013-12-01

In this article we investigate the problem of shape optimization of the cooling cavity of the plunger used in the forming process in the glass industry. The paper deals with finding the algorithm for optimization of the shape of inner canal of plunger in such a way, that its outward surface, which realizes cooling of moulded piece, is attained constant set temperature. The results of the numerical optimization to three required target temperatures 700, 750 and 800°C of the surface Γ1 together with the distribution of temperatures on the interface Γ1 between the body and the heat source before and after the optimization process are presented.

12. Solution of simple numerical problems using spreadsheet programs

Riggi, F.

1986-11-01

Spreadsheet programs are now extensively used for the analysis of business problems. A spreadsheet program reproduces the structure of a large page with columns and rows. The intersection of a column and a row defines a cell, each cell being identified by a column letter and a row number, starting at the upper left. The screen is a window over this matrix, whose dimensions depend on the machine's resources. Unlike that which occurs with paper spreadsheets, the elements (cells) of such a matrix structure can hold not only labels or numerical values but also mathematical formulae relating them to other matrix elements.

13. The accurate solution of Poisson's equation by expansion in Chebyshev polynomials

NASA Technical Reports Server (NTRS)

Haidvogel, D. B.; Zang, T.

1979-01-01

A Chebyshev expansion technique is applied to Poisson's equation on a square with homogeneous Dirichlet boundary conditions. The spectral equations are solved in two ways - by alternating direction and by matrix diagonalization methods. Solutions are sought to both oscillatory and mildly singular problems. The accuracy and efficiency of the Chebyshev approach compare favorably with those of standard second- and fourth-order finite-difference methods.

14. PDE-based Morphology for Matrix Fields: Numerical Solution Schemes

Burgeth, Bernhard; Breuß, Michael; Didas, Stephan; Weickert, Joachim

Tensor fields are important in digital imaging and computer vision. Hence there is a demand for morphological operations to perform e.g. shape analysis, segmentation or enhancement procedures. Recently, fundamental morphological concepts have been transferred to the setting of fields of symmetric positive definite matrices, which are symmetric rank two tensors. This has been achieved by a matrix-valued extension of the nonlinear morphological partial differential equations (PDEs) for dilation and erosion known for grey scale images. Having these two basic operations at our disposal, more advanced morphological operators such as top hats or morphological derivatives for matrix fields with symmetric, positive semidefinite matrices can be constructed. The approach realises a proper coupling of the matrix channels rather than treating them independently. However, from the algorithmic side the usual scalar morphological PDEs are transport equations that require special upwind-schemes or novel high-accuracy predictor-corrector approaches for their adequate numerical treatment. In this chapter we propose the non-trivial extension of these schemes to the matrix-valued setting by exploiting the special algebraic structure available for symmetric matrices. Furthermore we compare the performance and juxtapose the results of these novel matrix-valued high-resolution-type (HRT) numerical schemes by considering top hats and morphological derivatives applied to artificial and real world data sets.

15. Three-dimensional convection in horizontal cylinders - Numerical solutions and comparison with experimental and analytical results

NASA Technical Reports Server (NTRS)

Smutek, C.; Bontoux, P.; Roux, B.; Schiroky, G. H.; Hurford, A. C.

1985-01-01

The results of a three-dimensional numerical simulation of Boussinesq free convection in a horizontal differentially heated cylinder are presented. The computation was based on a Samarskii-Andreyev scheme (described by Leong, 1981) and a false-transient advancement in time, with vorticity, velocity, and temperature as dependent variables. Solutions for velocity and temperature distributions were obtained for Rayleigh numbers (based on the radius) Ra = 74-18,700, thus covering the core- and boundary-layer-driven regimes. Numerical solutions are compared with asymptotic analytical solutions and experimental data. The numerical results well represent the complex three-dimensional flows found experimentally.

16. Numerical solutions of ICRF fields in axisymmetric mirrors

SciTech Connect

Phillips, M.W.

1985-07-01

The results of a new numerical code called GARFIELD (Grumman Aerospace Rf Field code) that calculates ICRF Fields in axisymmetric mirror geometry (such as the central cell of a tandem mirror or an RF test stand) are presented. The code solves the electromagnetic wave equation using a cold plasma dispersion relation with a small collision frequency to simulate absorption. The purpose of the calculation is to examine how ICRF wave structure and propagation is effected by the axial variation of the magnetic field in a mirror for various antenna designs. In the code the wave equation is solved in flux coordinates using a finite element method. This should allow more complex dielectric tensors to be modeled in the future. The resulting matrix is solved iteratively, to maximize the allowable size of the spatial grid. Results for a typical antenna array in a simple mirror will be shown.

17. Analysis of bacterial migration; 1: Numerical solution of balance equation

SciTech Connect

Frymier, P.D.; Ford, R.M.; Cummings, P.T. . Dept. of Chemical Engineering)

1994-04-01

Chemotaxis describes the ability of motile bacteria to bias their motion in the direction of increasing gradients of chemicals, usually energy sources, known as attractants. In experimental studies of the migration of chemotactic bacteria, 1-D phenomenological cell balance equations have been used to quantitatively analyze experimental observations. While attractive for their simplicity and the ease of solution, they are limited in the strict mathematical sense to the situation in which individual bacteria are confined to motion in one dimension and respond to attractant gradients in one dimension only. Recently, Ford and Cummings (1992) reduced the general 3-D cell balance equation of Alt (1980) to obtain an equation describing the migration of a bacterial population in response to a 1-D attractant gradient. Solutions of this equation for single gradients of attractants are compared to those of 1-D balance equations, results from cellular dynamics simulations, and experimental data from the authors' laboratory for E. coli responding to [alpha]-methylaspartate. The authors also investigate two aspects of the experimentally derived expression for the tumbling probability: the effect of different models for the down-gradient swimming behavior of the bacteria and the validity of ignoring the temporal derivative of the attractant concentration.

18. Numerical solution of differential equations by artificial neural networks

NASA Technical Reports Server (NTRS)

1995-01-01

Conventionally programmed digital computers can process numbers with great speed and precision, but do not easily recognize patterns or imprecise or contradictory data. Instead of being programmed in the conventional sense, artificial neural networks (ANN's) are capable of self-learning through exposure to repeated examples. However, the training of an ANN can be a time consuming and unpredictable process. A general method is being developed by the author to mate the adaptability of the ANN with the speed and precision of the digital computer. This method has been successful in building feedforward networks that can approximate functions and their partial derivatives from examples in a single iteration. The general method also allows the formation of feedforward networks that can approximate the solution to nonlinear ordinary and partial differential equations to desired accuracy without the need of examples. It is believed that continued research will produce artificial neural networks that can be used with confidence in practical scientific computing and engineering applications.

19. High-Order Accurate Solutions to the Helmholtz Equation in the Presence of Boundary Singularities

Britt, Darrell Steven, Jr.

Problems of time-harmonic wave propagation arise in important fields of study such as geological surveying, radar detection/evasion, and aircraft design. These often involve highfrequency waves, which demand high-order methods to mitigate the dispersion error. We propose a high-order method for computing solutions to the variable-coefficient inhomogeneous Helmholtz equation in two dimensions on domains bounded by piecewise smooth curves of arbitrary shape with a finite number of boundary singularities at known locations. We utilize compact finite difference (FD) schemes on regular structured grids to achieve highorder accuracy due to their efficiency and simplicity, as well as the capability to approximate variable-coefficient differential operators. In this work, a 4th-order compact FD scheme for the variable-coefficient Helmholtz equation on a Cartesian grid in 2D is derived and tested. The well known limitation of finite differences is that they lose accuracy when the boundary curve does not coincide with the discretization grid, which is a severe restriction on the geometry of the computational domain. Therefore, the algorithm presented in this work combines high-order FD schemes with the method of difference potentials (DP), which retains the efficiency of FD while allowing for boundary shapes that are not aligned with the grid without sacrificing the accuracy of the FD scheme. Additionally, the theory of DP allows for the universal treatment of the boundary conditions. One of the significant contributions of this work is the development of an implementation that accommodates general boundary conditions (BCs). In particular, Robin BCs with discontinuous coefficients are studied, for which we introduce a piecewise parameterization of the boundary curve. Problems with discontinuities in the boundary data itself are also studied. We observe that the design convergence rate suffers whenever the solution loses regularity due to the boundary conditions. This is

20. Discontinuous steady-state analytical solutions of the Boussinesq equation and their numerical representation by MODFLOW.

PubMed

Zaidel, Jacob

2013-01-01

Known analytical solutions of groundwater flow equations are routinely used for verification of computer codes. However, these analytical solutions (e.g., the Dupuit solution for the steady-state unconfined unidirectional flow in a uniform aquifer with a flat bottom) represent smooth and continuous water table configurations, simulating which does not pose any significant problems for the numerical groundwater flow models, like MODFLOW. One of the most challenging numerical cases for MODFLOW arises from drying-rewetting problems often associated with abrupt changes in the elevations of impervious base of a thin unconfined aquifer. Numerical solutions of groundwater flow equations cannot be rigorously verified for such cases due to the lack of corresponding exact analytical solutions. Analytical solutions of the steady-state Boussinesq equation, associated with the discontinuous water table configurations over a stairway impervious base, are presented in this article. Conditions resulting in such configurations are analyzed and discussed. These solutions appear to be well suited for testing and verification of computer codes. Numerical solutions, obtained by the latest versions of MODFLOW (MODFLOW-2005 and MODFLOW-NWT), are compared with the presented discontinuous analytical solutions. It is shown that standard MODFLOW-2005 code (as well as MODFLOW-2000 and older versions) has significant convergence problems simulating such cases. The problems manifest themselves either in a total convergence failure or erroneous results. Alternatively, MODFLOW-NWT, providing a good match to the presented discontinuous analytical solutions, appears to be a more reliable and appropriate code for simulating abrupt changes in water table elevations.

1. Design and Construction Solutions in the Accurate Realization of NCSX Magnetic Fields

SciTech Connect

Heitzenroeder, P.; Dudek, Lawrence E.; Brooks, Arthur W.; Viola, Michael E.; Brown, Thomas; Neilson, George H.; Zarnstorff, Michael C.; Rej, Donald; Cole,Michael J.; Freudenberg, Kevin D.; Harris J. H.; McGinnis, Gary

2008-09-29

The National Compact Stellarator Experiment, NCSX, is being constructed at the Princeton Plasma Physics Laboratory (PPPL) in partnership with the Oak Ridge national Laboratory. The goal of NCSX is to provide the understanding necessary to develop an attractive, disruption free, steady state compact stellaratorbased reactor design. This paper describes the recently revised designs of the critical interfaces between the modular coils, the construction solutions developed to meet assembly tolerances, and the recently revised trim coil system that provides the required compensation to correct for the “as built” conditions and to allow flexibility in the disposition of as-built conditions. In May, 2008, the sponsor decided to terminate the NCSX project due to growth in the project’s cost and schedule estimates. However significant technical challenges in design and construction were overcome, greatly reducing the risk in the remaining work to complete the project.

2. Measurements of accurate x-ray scattering data of protein solutions using small stationary sample cells

SciTech Connect

Hong Xinguo; Hao Quan

2009-01-15

In this paper, we report a method of precise in situ x-ray scattering measurements on protein solutions using small stationary sample cells. Although reduction in the radiation damage induced by intense synchrotron radiation sources is indispensable for the correct interpretation of scattering data, there is still a lack of effective methods to overcome radiation-induced aggregation and extract scattering profiles free from chemical or structural damage. It is found that radiation-induced aggregation mainly begins on the surface of the sample cell and grows along the beam path; the diameter of the damaged region is comparable to the x-ray beam size. Radiation-induced aggregation can be effectively avoided by using a two-dimensional scan (2D mode), with an interval as small as 1.5 times the beam size, at low temperature (e.g., 4 deg. C). A radiation sensitive protein, bovine hemoglobin, was used to test the method. A standard deviation of less than 5% in the small angle region was observed from a series of nine spectra recorded in 2D mode, in contrast to the intensity variation seen using the conventional stationary technique, which can exceed 100%. Wide-angle x-ray scattering data were collected at a standard macromolecular diffraction station using the same data collection protocol and showed a good signal/noise ratio (better than the reported data on the same protein using a flow cell). The results indicate that this method is an effective approach for obtaining precise measurements of protein solution scattering.

3. Finite-difference scheme for the numerical solution of the Schroedinger equation

NASA Technical Reports Server (NTRS)

1992-01-01

A finite-difference scheme for numerical integration of the Schroedinger equation is constructed. Asymptotically (r goes to infinity), the method gives the exact solution correct to terms of order r exp -2.

4. Analytical Solutions Involving Shock Waves for Testing Debris Avalanche Numerical Models

Mungkasi, Sudi; Roberts, Stephen Gwyn

2012-10-01

Analytical solutions to debris avalanche problems involving shock waves are derived. The debris avalanche problems are described in two different coordinate systems, namely, the standard Cartesian and topography-linked coordinate systems. The analytical solutions can then be used to test debris avalanche numerical models. In this article, finite volume methods are applied as the numerical models. We compare the performance of the finite volume method with reconstruction of the conserved quantities based on stage, height, and velocity to that of the conserved quantities based on stage, height, and momentum for solving the debris avalanche problems involving shock waves. The numerical solutions agree with the analytical solution. In addition, both reconstructions lead to similar numerical results. This article is an extension of the work of Mangeney et al. (Pure Appl Geophys 157(6-8):1081-1096, 2000).

5. A numerical inversion of a the Laplace transform solution to radial dispersion in a porous medium.

USGS Publications Warehouse

Moench, A.F.; Ogata, A.

1981-01-01

A special form of the numerical inversion of the Laplace transform described by Stehfest (1970) is applied to the transformed solution of dispersion in a radial flow system in a porous medium. The inversion is extremely simple to use because the weighting coefficients depend only on the number of terms used in the computation and not upon the transform solution as required by most numerical inversion techniques.-from Authors

6. Some notes on the numerical solution of shear-lag and mathematically related problems

NASA Technical Reports Server (NTRS)

Kuhn, Paul

1939-01-01

The analysis of box beams with shear deformation of the flanges can be reduced to the solution of a differential equation. The same equation is met in other problems of stress analysis. No analytical solutions of this equation can be given for practical cases, and numerical methods of evaluation must be used. Available methods are briefly discussed. Two numerical examples show the application of the step-by-step method of integration to shear-lag problems.

7. Numerical solution of hybrid fuzzy differential equations using improved predictor-corrector method

Kim, Hyunsoo; Sakthivel, Rathinasamy

2012-10-01

The hybrid fuzzy differential equations have a wide range of applications in science and engineering. This paper considers numerical solution for hybrid fuzzy differential equations. The improved predictor-corrector method is adapted and modified for solving the hybrid fuzzy differential equations. The proposed algorithm is illustrated by numerical examples and the results obtained using the scheme presented here agree well with the analytical solutions. The computer symbolic systems such as Maple and Mathematica allow us to perform complicated calculations of algorithm.

8. Numerical solution of a coupled pair of elliptic equations from solid state electronics

NASA Technical Reports Server (NTRS)

Phillips, T. N.

1983-01-01

Iterative methods are considered for the solution of a coupled pair of second order elliptic partial differential equations which arise in the field of solid state electronics. A finite difference scheme is used which retains the conservative form of the differential equations. Numerical solutions are obtained in two ways, by multigrid and dynamic alternating direction implicit methods. Numerical results are presented which show the multigrid method to be an efficient way of solving this problem.

9. A solution accurate, efficient and stable unsplit staggered mesh scheme for three dimensional magnetohydrodynamics

Lee, Dongwook

2013-06-01

In this paper, we extend the unsplit staggered mesh scheme (USM) for 2D magnetohydrodynamics (MHD) [D. Lee, A.E. Deane, An unsplit staggered mesh scheme for multidimensional magnetohydrodynamics, J. Comput. Phys. 228 (2009) 952-975] to a full 3D MHD scheme. The scheme is a finite-volume Godunov method consisting of a constrained transport (CT) method and an efficient and accurate single-step, directionally unsplit multidimensional data reconstruction-evolution algorithm, which extends Colella's original 2D corner transport upwind (CTU) method [P. Colella, Multidimensional upwind methods for hyperbolic conservation laws, J. Comput. Phys. 87 (1990) 446-466]. We present two types of data reconstruction-evolution algorithms for 3D: (1) a reduced CTU scheme and (2) a full CTU scheme. The reduced 3D CTU scheme is a variant of a simple 3D extension of Collela's 2D CTU method and is considered as a direct extension from the 2D USM scheme. The full 3D CTU scheme is our primary 3D solver which includes all multidimensional cross-derivative terms for stability. The latter method is logically analogous to the 3D unsplit CTU method by Saltzman [J. Saltzman, An unsplit 3D upwind method for hyperbolic conservation laws, J. Comput. Phys. 115 (1994) 153-168]. The major novelties in our algorithms are twofold. First, we extend the reduced CTU scheme to the full CTU scheme which is able to run with CFL numbers close to unity. Both methods utilize the transverse update technique developed in the 2D USM algorithm to account for transverse fluxes without solving intermediate Riemann problems, which in turn gives cost-effective 3D methods by reducing the total number of Riemann solves. The proposed algorithms are simple and efficient especially when including multidimensional MHD terms that maintain in-plane magnetic field dynamics. Second, we introduce a new CT scheme that makes use of proper upwind information in taking averages of electric fields. Our 3D USM schemes can be easily

10. An iterative transformation procedure for numerical solution of flutter and similar characteristics-value problems

NASA Technical Reports Server (NTRS)

Gossard, Myron L

1952-01-01

An iterative transformation procedure suggested by H. Wielandt for numerical solution of flutter and similar characteristic-value problems is presented. Application of this procedure to ordinary natural-vibration problems and to flutter problems is shown by numerical examples. Comparisons of computed results with experimental values and with results obtained by other methods of analysis are made.

11. Numerical solution of 2D-vector tomography problem using the method of approximate inverse

Svetov, Ivan; Maltseva, Svetlana; Polyakova, Anna

2016-08-01

We propose a numerical solution of reconstruction problem of a two-dimensional vector field in a unit disk from the known values of the longitudinal and transverse ray transforms. The algorithm is based on the method of approximate inverse. Numerical simulations confirm that the proposed method yields good results of reconstruction of vector fields.

12. Numerical solutions of reaction-diffusion equations: Application to neural and cardiac models

Ji, Yanyan Claire; Fenton, Flavio H.

2016-08-01

We describe the implementation of the explicit Euler, Crank-Nicolson, and implicit alternating direction methods for solving partial differential equations and apply these methods to obtain numerical solutions of three excitable-media models used to study neurons and cardiomyocyte dynamics. We discuss the implementation, accuracy, speed, and stability of these numerical methods.

13. A numerical solution to the cattaneo-mindlin problem for viscoelastic materials

Spinu, S.; Cerlinca, D.

2016-08-01

The problem of the frictional mechanical contact with slip and stick, also referred to as the Cattaneo-Mindlin problem, is an important topic in engineering, with applications in the modeling of particle-flow simulations or in the study of contact between rough surfaces. In the frame of Linear Theory of Elasticity, accurate description of the slip-stick contact can only be achieved numerically, due to mutual interaction between normal and shear contact tractions. Additional difficulties arise when considering a viscoelastic constitutive law, as the mechanical response of the contacting materials depends explicitly on time. To overcome this obstacle, an existing algorithm for the purely elastic slip-stick contact is coupled with a semi-analytical method for viscoelastic displacement computation. The main advantage of this approach is that the contact model can be divided in subunits having the same structure as that of the purely elastic frictionless contact model, for which a well-established solution is readily available. In each time step, the contact solver assesses the contact area, the pressure distribution, the stick area and the shear tractions that satisfy the contact compatibility conditions and the static force equilibrium in both normal and tangential directions. A temporal discretization of the simulation windows assures that the memory effect, specific to both viscoelasticity and friction as a path-dependent processes, is properly replicated.

14. Numerical solutions of matrix Riccati equations for radiative transfer in a plane-parallel geometry

Chang, Hung-Wen; Wu, Tso-Lun

1997-01-01

In this paper, we conduct numerical experiments with matrix Riccati equations (MREs) which describe the reflection ( R) and transmission ( T) matrices of the specific intensities in a layer containing randomly distributed scattering particles. The theoretical formulation of MREs is discussed in our previous paper where we show that R and T for a thick layer can be efficiently computed by successively doubling R and T matrices for a thin layer (with small optical thickness 0959-7174/7/1/010/img1). We can compute 0959-7174/7/1/010/img2 and 0959-7174/7/1/010/img3 very accurately using either a fourth-order Runge - Kutta scheme or the fourth-order iterative solution. The differences between these results and those computed by the eigenmode expansion technique (EMET) are very small (< 0.1%). Although the MRE formulation cannot be extended to handle the inhomogeneous term (source term) in the differential equation, we show that the force term can be reformulated as an equivalent boundary condition which is consistent with MRE methods. MRE methods offer an alternative way of solving plane-parallel radiative transport problems. For large problems that do not fit into computer memory, the MRE method provides a significant reduction in computer memory and computational time.

15. Code and Solution Verification of 3D Numerical Modeling of Flow in the Gust Erosion Chamber

Yuen, A.; Bombardelli, F. A.

2014-12-01

Erosion microcosms are devices commonly used to investigate the erosion and transport characteristics of sediments at the bed of rivers, lakes, or estuaries. In order to understand the results these devices provide, the bed shear stress and flow field need to be accurately described. In this research, the UMCES Gust Erosion Microcosm System (U-GEMS) is numerically modeled using Finite Volume Method. The primary aims are to simulate the bed shear stress distribution at the surface of the sediment core/bottom of the microcosm, and to validate the U-GEMS produces uniform bed shear stress at the bottom of the microcosm. The mathematical model equations are solved by on a Cartesian non-uniform grid. Multiple numerical runs were developed with different input conditions and configurations. Prior to developing the U-GEMS model, the General Moving Objects (GMO) model and different momentum algorithms in the code were verified. Code verification of these solvers was done via simulating the flow inside the top wall driven square cavity on different mesh sizes to obtain order of convergence. The GMO model was used to simulate the top wall in the top wall driven square cavity as well as the rotating disk in the U-GEMS. Components simulated with the GMO model were rigid bodies that could have any type of motion. In addition cross-verification was conducted as results were compared with numerical results by Ghia et al. (1982), and good agreement was found. Next, CFD results were validated by simulating the flow within the conventional microcosm system without suction and injection. Good agreement was found when the experimental results by Khalili et al. (2008) were compared. After the ability of the CFD solver was proved through the above code verification steps. The model was utilized to simulate the U-GEMS. The solution was verified via classic mesh convergence study on four consecutive mesh sizes, in addition to that Grid Convergence Index (GCI) was calculated and based on

16. Numerical solution of Q evolution equations for fragmentation functions

Hirai, M.; Kumano, S.

2012-04-01

bytes Classification: 11.5 Nature of problem: This program solves time-like DGLAP Q evolution equations with or without next-to-leading order αs effects for fragmentation functions. The evolved functions can be calculated for Dgh, Duh, Dubarh, Ddh, Ddbarh, Dsh, Dsbarh, Dch, Dcbarh, Dbh and Dbbarh of a hadron h. Solution method: The DGLAP integro-differential equations are solved by the Euler method for the differentiation of ln Q and the Gauss-Legendre method for the x integral as explained in Section 4 of the manuscript. Restrictions: This program is used for calculating Q evolution of fragmentation functions in the leading order or in the next-to-leading order of αs. Q evolution equations are the time-like DGLAP equations. The double precision arithmetic is used. The renormalization scheme is the modified minimal subtraction scheme (MSbar). A user provides initial fragmentation functions as the subroutines FF_INI and HQFF in the end of the distributed code FF_DGLAP.f. In FF_DGLAP.f, the subroutines are given by taking the HKNS07 (2) functions as an example of the initial functions. Then, the user inputs kinematical parameters in the file setup.ini as explained in Section 5.2 of the manuscript. Running time: A few seconds on HP DL360G5-DC-X5160.

17. New numerical methods for open-loop and feedback solutions to dynamic optimization problems

The topic of the first part of this research is trajectory optimization of dynamical systems via computational swarm intelligence. Particle swarm optimization is a nature-inspired heuristic search method that relies on a group of potential solutions to explore the fitness landscape. Conceptually, each particle in the swarm uses its own memory as well as the knowledge accumulated by the entire swarm to iteratively converge on an optimal or near-optimal solution. It is relatively straightforward to implement and unlike gradient-based solvers, does not require an initial guess or continuity in the problem definition. Although particle swarm optimization has been successfully employed in solving static optimization problems, its application in dynamic optimization, as posed in optimal control theory, is still relatively new. In the first half of this thesis particle swarm optimization is used to generate near-optimal solutions to several nontrivial trajectory optimization problems including thrust programming for minimum fuel, multi-burn spacecraft orbit transfer, and computing minimum-time rest-to-rest trajectories for a robotic manipulator. A distinct feature of the particle swarm optimization implementation in this work is the runtime selection of the optimal solution structure. Optimal trajectories are generated by solving instances of constrained nonlinear mixed-integer programming problems with the swarming technique. For each solved optimal programming problem, the particle swarm optimization result is compared with a nearly exact solution found via a direct method using nonlinear programming. Numerical experiments indicate that swarm search can locate solutions to very great accuracy. The second half of this research develops a new extremal-field approach for synthesizing nearly optimal feedback controllers for optimal control and two-player pursuit-evasion games described by general nonlinear differential equations. A notable revelation from this development

18. Polarized bidirectional reflectance of optically thick sparse particulate layers: An efficient numerically exact radiative-transfer solution

Mishchenko, Michael I.; Dlugach, Janna M.; Chowdhary, Jacek; Zakharova, Nadezhda T.

2015-05-01

We describe a simple yet efficient numerical algorithm for computing polarized bidirectional reflectance of an optically thick (semi-infinite), macroscopically flat layer composed of statistically isotropic and mirror symmetric random particles. The spatial distribution of the particles is assumed to be sparse, random, and statistically uniform. The 4×4 Stokes reflection matrix is calculated by iterating the Ambartsumian's vector nonlinear integral equation. The result is a numerically exact solution of the vector radiative transfer equation and as such fully satisfies the energy conservation law and the fundamental reciprocity relation. Since this technique bypasses the computation of the internal radiation field, it is very fast and highly accurate. The FORTRAN implementation of the technique is publicly available on the World Wide Web at

19. Symmetry-plane model of 3D Euler flows: Mapping to regular systems and numerical solutions of blowup

Mulungye, Rachel M.; Lucas, Dan; Bustamante, Miguel D.

2014-11-01

We introduce a family of 2D models describing the dynamics on the so-called symmetry plane of the full 3D Euler fluid equations. These models depend on a free real parameter and can be solved analytically. For selected representative values of the free parameter, we apply the method introduced in [M.D. Bustamante, Physica D: Nonlinear Phenom. 240, 1092 (2011)] to map the fluid equations bijectively to globally regular systems. By comparing the analytical solutions with the results of numerical simulations, we establish that the numerical simulations of the mapped regular systems are far more accurate than the numerical simulations of the original systems, at the same spatial resolution and CPU time. In particular, the numerical integrations of the mapped regular systems produce robust estimates for the growth exponent and singularity time of the main blowup quantity (vorticity stretching rate), converging well to the analytically-predicted values even beyond the time at which the flow becomes under-resolved (i.e. the reliability time). In contrast, direct numerical integrations of the original systems develop unstable oscillations near the reliability time. We discuss the reasons for this improvement in accuracy, and explain how to extend the analysis to the full 3D case. Supported under the programme for Research in Third Level Institutions (PRTLI) Cycle 5 and co-funded by the European Regional Development Fund.

20. Numerical Investigations of Vadose Zone Transport of Saturated Sodium Thiosulfate Solutions

White, M. D.; Ward, A. L.

2001-12-01

Compared with water, hypersaline liquid wastes ([NaNO3] > 10 N) from the reduction-oxidation (REDOX) process at the Hanford site have elevated viscosity (μ > 1.2 cP), density (ρ > 1.4 gm/cm3), and surface tension (σ > 100 dyn/cm). Such liquids have infiltrated into the vadose zone at Hanford from leaking underground storage tanks. The migration behavior of saturated or hypersaline salt solutions through unsaturated soils is largely unknown. Laboratory tests with tank-waste simulants suggest that the elevated density, viscosity, and surface tension properties of these liquids can influence the wetting front behavior, altering its shape and migration rate. Conditions under which these mechanisms are active in the field and the extent to which they contribute to transport through the vadose zone are largely unknown, making it impossible to accurately predict the post-leak distribution of these fluids in the field. To investigate the effects of fluid properties on subsurface migration of hypersaline saline solutions, numerical simulations were conducted of a field-scale, tank-leak experiment. The field experiments consisted of five 4000-L injections, at a depth of 5 m, of saturated sodium thiosulfate brine (used as a surrogate for REDOX type wastes) over a 5-week period, followed by three 4000-L injections of Columbia River water. Pre-test modeling of river water injections at this Hanford field site predicted significant lateral spreading of the moisture plume and were confirmed by geophysical logging. A series of three-dimensional, multifluid (i.e., aqueous and gas phases) numerical simulations were conducted that systematically considered the effects of elevated density, viscosity, and surface tension, and reduced vapor pressure on vadose-zone transport. Hydrologic properties were determined from cores collected at the field site and calibrated using river-water injection experiments. Isothermal conditions were assumed for the simulations, however, the effects of

1. Complete numerical solution of the diffusion equation of random genetic drift.

PubMed

Zhao, Lei; Yue, Xingye; Waxman, David

2013-08-01

A numerical method is presented to solve the diffusion equation for the random genetic drift that occurs at a single unlinked locus with two alleles. The method was designed to conserve probability, and the resulting numerical solution represents a probability distribution whose total probability is unity. We describe solutions of the diffusion equation whose total probability is unity as complete. Thus the numerical method introduced in this work produces complete solutions, and such solutions have the property that whenever fixation and loss can occur, they are automatically included within the solution. This feature demonstrates that the diffusion approximation can describe not only internal allele frequencies, but also the boundary frequencies zero and one. The numerical approach presented here constitutes a single inclusive framework from which to perform calculations for random genetic drift. It has a straightforward implementation, allowing it to be applied to a wide variety of problems, including those with time-dependent parameters, such as changing population sizes. As tests and illustrations of the numerical method, it is used to determine: (i) the probability density and time-dependent probability of fixation for a neutral locus in a population of constant size; (ii) the probability of fixation in the presence of selection; and (iii) the probability of fixation in the presence of selection and demographic change, the latter in the form of a changing population size.

2. Numerical solution of a class of integral equations arising in two-dimensional aerodynamics

NASA Technical Reports Server (NTRS)

Fromme, J.; Golberg, M. A.

1978-01-01

We consider the numerical solution of a class of integral equations arising in the determination of the compressible flow about a thin airfoil in a ventilated wind tunnel. The integral equations are of the first kind with kernels having a Cauchy singularity. Using appropriately chosen Hilbert spaces, it is shown that the kernel gives rise to a mapping which is the sum of a unitary operator and a compact operator. This allows the problem to be studied in terms of an equivalent integral equation of the second kind. A convergent numerical algorithm for its solution is derived by using Galerkin's method. It is shown that this algorithm is numerically equivalent to Bland's collocation method, which is then used as the method of computation. Extensive numerical calculations are presented establishing the validity of the theory.

3. Numerical solution of random singular integral equation appearing in crack problems

NASA Technical Reports Server (NTRS)

Sambandham, M.; Srivatsan, T. S.; Bharucha-Reid, A. T.

1986-01-01

The solution of several elasticity problems, and particularly crack problems, can be reduced to the solution of one-dimensional singular integral equations with a Cauchy-type kernel or to a system of uncoupled singular integral equations. Here a method for the numerical solution of random singular integral equations of Cauchy type is presented. The solution technique involves a Chebyshev series approximation, the coefficients of which are the solutions of a system of random linear equations. This method is applied to the problem of periodic array of straight cracks inside an infinite isotropic elastic medium and subjected to a nonuniform pressure distribution along the crack edges. The statistical properties of the random solution are evaluated numerically, and the random solution is used to determine the values of the stress-intensity factors at the crack tips. The error, expressed as the difference between the mean of the random solution and the deterministic solution, is established. Values of stress-intensity factors at the crack tip for different random input functions are presented.

4. Numerical solution of the incompressible Navier-Stokes equations. Ph.D. Thesis - Stanford Univ., Mar. 1989

NASA Technical Reports Server (NTRS)

Rogers, Stuart E.

1990-01-01

The current work is initiated in an effort to obtain an efficient, accurate, and robust algorithm for the numerical solution of the incompressible Navier-Stokes equations in two- and three-dimensional generalized curvilinear coordinates for both steady-state and time-dependent flow problems. This is accomplished with the use of the method of artificial compressibility and a high-order flux-difference splitting technique for the differencing of the convective terms. Time accuracy is obtained in the numerical solutions by subiterating the equations in psuedo-time for each physical time step. The system of equations is solved with a line-relaxation scheme which allows the use of very large pseudo-time steps leading to fast convergence for steady-state problems as well as for the subiterations of time-dependent problems. Numerous laminar test flow problems are computed and presented with a comparison against analytically known solutions or experimental results. These include the flow in a driven cavity, the flow over a backward-facing step, the steady and unsteady flow over a circular cylinder, flow over an oscillating plate, flow through a one-dimensional inviscid channel with oscillating back pressure, the steady-state flow through a square duct with a 90 degree bend, and the flow through an artificial heart configuration with moving boundaries. An adequate comparison with the analytical or experimental results is obtained in all cases. Numerical comparisons of the upwind differencing with central differencing plus artificial dissipation indicates that the upwind differencing provides a much more robust algorithm, which requires significantly less computing time. The time-dependent problems require on the order of 10 to 20 subiterations, indicating that the elliptical nature of the problem does require a substantial amount of computing effort.

5. Geometric invariants for initial data sets: analysis, exact solutions, computer algebra, numerics

Valiente Kroon, Juan A.

2011-09-01

A personal perspective on the interaction of analytical, numerical and computer algebra methods in classical Relativity is given. This discussion is inspired by the problem of the construction of invariants that characterise key solutions to the Einstein field equations. It is claimed that this kind of ideas will be or importance in the analysis of dynamical black hole spacetimes by either analytical or numerical methods.

6. Numerical solution of differential-algebraic equations using the spline collocation-variation method

Bulatov, M. V.; Rakhvalov, N. P.; Solovarova, L. S.

2013-03-01

Numerical methods for solving initial value problems for differential-algebraic equations are proposed. The approximate solution is represented as a continuous vector spline whose coefficients are found using the collocation conditions stated for a subgrid with the number of collocation points less than the degree of the spline and the minimality condition for the norm of this spline in the corresponding spaces. Numerical results for some model problems are presented.

7. Numerical solution of inviscid and viscous laminar and turbulent flow around the airfoil

Slouka, Martin; Kozel, Karel

2016-03-01

This work deals with the 2D numerical solution of inviscid compressible flow and viscous compressible laminar and turbulent flow around the profile. In a case of turbulent flow algebraic Baldwin-Lomax model is used and compared with Wilcox k-omega model. Calculations are done for NACA 0012 and RAE 2822 airfoil profile for the different angles of upstream flow. Numerical results are compared and discussed with experimental data.

8. Dynamics analysis of electrodynamic satellite tethers. Equations of motion and numerical solution algorithms for the tether

NASA Technical Reports Server (NTRS)

Nacozy, P. E.

1984-01-01

The equations of motion are developed for a perfectly flexible, inelastic tether with a satellite at its extremity. The tether is attached to a space vehicle in orbit. The tether is allowed to possess electrical conductivity. A numerical solution algorithm to provide the motion of the tether and satellite system is presented. The resulting differential equations can be solved by various existing standard numerical integration computer programs. The resulting differential equations allow the introduction of approximations that can lead to analytical, approximate general solutions. The differential equations allow more dynamical insight of the motion.

9. Numerical solution of a coupled pair of elliptic equations from solid state electronics

NASA Technical Reports Server (NTRS)

Phillips, T. N.

1984-01-01

Iterative methods are considered for the solution of a coupled pair of second order elliptic partial differential equations which arise in the field of solid state electronics. A finite difference scheme is used which retains the conservative form of the differential equations. Numerical solutions are obtained in two ways, by multigrid and dynamic alternating direction implicit methods. Numerical results are presented which show the multigrid method to be an efficient way of solving this problem. Previously announced in STAR as N83-30109

10. Numerical solution of the one-dimensional fractional convection diffusion equations based on Chebyshev operational matrix.

PubMed

Xie, Jiaquan; Huang, Qingxue; Yang, Xia

2016-01-01

In this paper, we are concerned with nonlinear one-dimensional fractional convection diffusion equations. An effective approach based on Chebyshev operational matrix is constructed to obtain the numerical solution of fractional convection diffusion equations with variable coefficients. The principal characteristic of the approach is the new orthogonal functions based on Chebyshev polynomials to the fractional calculus. The corresponding fractional differential operational matrix is derived. Then the matrix with the Tau method is utilized to transform the solution of this problem into the solution of a system of linear algebraic equations. By solving the linear algebraic equations, the numerical solution is obtained. The approach is tested via examples. It is shown that the proposed algorithm yields better results. Finally, error analysis shows that the algorithm is convergent.

11. Numerical Simulations of STOVL Hot Gas Ingestion in Ground Proximity Using a Multigrid Solution Procedure

NASA Technical Reports Server (NTRS)

Wang, Gang

2003-01-01

A multi grid solution procedure for the numerical simulation of turbulent flows in complex geometries has been developed. A Full Multigrid-Full Approximation Scheme (FMG-FAS) is incorporated into the continuity and momentum equations, while the scalars are decoupled from the multi grid V-cycle. A standard kappa-Epsilon turbulence model with wall functions has been used to close the governing equations. The numerical solution is accomplished by solving for the Cartesian velocity components either with a traditional grid staggering arrangement or with a multiple velocity grid staggering arrangement. The two solution methodologies are evaluated for relative computational efficiency. The solution procedure with traditional staggering arrangement is subsequently applied to calculate the flow and temperature fields around a model Short Take-off and Vertical Landing (STOVL) aircraft hovering in ground proximity.

12. Numerical solution of the one-dimensional fractional convection diffusion equations based on Chebyshev operational matrix.

PubMed

Xie, Jiaquan; Huang, Qingxue; Yang, Xia

2016-01-01

In this paper, we are concerned with nonlinear one-dimensional fractional convection diffusion equations. An effective approach based on Chebyshev operational matrix is constructed to obtain the numerical solution of fractional convection diffusion equations with variable coefficients. The principal characteristic of the approach is the new orthogonal functions based on Chebyshev polynomials to the fractional calculus. The corresponding fractional differential operational matrix is derived. Then the matrix with the Tau method is utilized to transform the solution of this problem into the solution of a system of linear algebraic equations. By solving the linear algebraic equations, the numerical solution is obtained. The approach is tested via examples. It is shown that the proposed algorithm yields better results. Finally, error analysis shows that the algorithm is convergent. PMID:27504247

13. Numerical parameter constraints for accurate PIC-DSMC simulation of breakdown from arc initiation to stable arcs

Moore, Christopher; Hopkins, Matthew; Moore, Stan; Boerner, Jeremiah; Cartwright, Keith

2015-09-01

Simulation of breakdown is important for understanding and designing a variety of applications such as mitigating undesirable discharge events. Such simulations need to be accurate through early time arc initiation to late time stable arc behavior. Here we examine constraints on the timestep and mesh size required for arc simulations using the particle-in-cell (PIC) method with direct simulation Monte Carlo (DMSC) collisions. Accurate simulation of electron avalanche across a fixed voltage drop and constant neutral density (reduced field of 1000 Td) was found to require a timestep ~ 1/100 of the mean time between collisions and a mesh size ~ 1/25 the mean free path. These constraints are much smaller than the typical PIC-DSMC requirements for timestep and mesh size. Both constraints are related to the fact that charged particles are accelerated by the external field. Thus gradients in the electron energy distribution function can exist at scales smaller than the mean free path and these must be resolved by the mesh size for accurate collision rates. Additionally, the timestep must be small enough that the particle energy change due to the fields be small in order to capture gradients in the cross sections versus energy. Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. DOE's National Nuclear Security Administration under Contract DE-AC04-94AL85000.

14. Two Different Methods for Numerical Solution of the Modified Burgers' Equation

PubMed Central

Karakoç, Seydi Battal Gazi; Başhan, Ali; Geyikli, Turabi

2014-01-01

A numerical solution of the modified Burgers' equation (MBE) is obtained by using quartic B-spline subdomain finite element method (SFEM) over which the nonlinear term is locally linearized and using quartic B-spline differential quadrature (QBDQM) method. The accuracy and efficiency of the methods are discussed by computing L 2 and L ∞ error norms. Comparisons are made with those of some earlier papers. The obtained numerical results show that the methods are effective numerical schemes to solve the MBE. A linear stability analysis, based on the von Neumann scheme, shows the SFEM is unconditionally stable. A rate of convergence analysis is also given for the DQM. PMID:25162064

15. Numerical method and FORTRAN program for the solution of an axisymmetric electrostatic collector design problem

NASA Technical Reports Server (NTRS)

Reese, O. W.

1972-01-01

The numerical calculation is described of the steady-state flow of electrons in an axisymmetric, spherical, electrostatic collector for a range of boundary conditions. The trajectory equations of motion are solved alternately with Poisson's equation for the potential field until convergence is achieved. A direct (noniterative) numerical technique is used to obtain the solution to Poisson's equation. Space charge effects are included for initial current densities as large as 100 A/sq cm. Ways of dealing successfully with the difficulties associated with these high densities are discussed. A description of the mathematical model, a discussion of numerical techniques, results from two typical runs, and the FORTRAN computer program are included.

16. Error propagation in the numerical solutions of the differential equations of orbital mechanics

NASA Technical Reports Server (NTRS)

Bond, V. R.

1982-01-01

The relationship between the eigenvalues of the linearized differential equations of orbital mechanics and the stability characteristics of numerical methods is presented. It is shown that the Cowell, Encke, and Encke formulation with an independent variable related to the eccentric anomaly all have a real positive eigenvalue when linearized about the initial conditions. The real positive eigenvalue causes an amplification of the error of the solution when used in conjunction with a numerical integration method. In contrast an element formulation has zero eigenvalues and is numerically stable.

17. Use of Green's functions in the numerical solution of two-point boundary value problems

NASA Technical Reports Server (NTRS)

Gallaher, L. J.; Perlin, I. E.

1974-01-01

This study investigates the use of Green's functions in the numerical solution of the two-point boundary value problem. The first part deals with the role of the Green's function in solving both linear and nonlinear second order ordinary differential equations with boundary conditions and systems of such equations. The second part describes procedures for numerical construction of Green's functions and considers briefly the conditions for their existence. Finally, there is a description of some numerical experiments using nonlinear problems for which the known existence, uniqueness or convergence theorems do not apply. Examples here include some problems in finding rendezvous orbits of the restricted three body system.

18. Equilibrium points, stability and numerical solutions of fractional-order predator-prey and rabies models

Ahmed, E.; El-Sayed, A. M. A.; El-Saka, H. A. A.

2007-01-01

In this paper we are concerned with the fractional-order predator-prey model and the fractional-order rabies model. Existence and uniqueness of solutions are proved. The stability of equilibrium points are studied. Numerical solutions of these models are given. An example is given where the equilibrium point is a centre for the integer order system but locally asymptotically stable for its fractional-order counterpart.

19. Numerical Solution of the Variational Data Assimilation Problem Using Satellite Data

Agoshkov, V. I.; Lebedv, S. A.; Parmuzin, E. I.

2010-12-01

The problem of variational assimilation of satellite observational data on the ocean surface temperature is formulated and numerically investigated in order to reconstruct surface heat fluxes with the use of the global three-dimensional model of ocean hydrothermodynamics developed at the Institute of Numerical Mathematics, Russian Academy of Sciences (INM RAS), and observational data on the ocean surface temperature over the year 2004. The algorithms of the numerical solution to the problem are elaborated and substantiated, and the data assimilation block is developed and incorporated into the global three-dimensional model. Numerical experiments are carried out with the use of the Indian Ocean water area as an example. Numerical experiments confirm the theoretical conclusions obtained and demonstrate the expediency of combining the model with a block of assimilating operational observational data on the surface temperature.

20. Do inverse ecosystem models accurately reconstruct plankton trophic flows? Comparing two solution methods using field data from the California Current

Stukel, Michael R.; Landry, Michael R.; Ohman, Mark D.; Goericke, Ralf; Samo, Ty; Benitez-Nelson, Claudia R.

2012-03-01

Despite the increasing use of linear inverse modeling techniques to elucidate fluxes in undersampled marine ecosystems, the accuracy with which they estimate food web flows has not been resolved. New Markov Chain Monte Carlo (MCMC) solution methods have also called into question the biases of the commonly used L2 minimum norm (L 2MN) solution technique. Here, we test the abilities of MCMC and L 2MN methods to recover field-measured ecosystem rates that are sequentially excluded from the model input. For data, we use experimental measurements from process cruises of the California Current Ecosystem (CCE-LTER) Program that include rate estimates of phytoplankton and bacterial production, micro- and mesozooplankton grazing, and carbon export from eight study sites varying from rich coastal upwelling to offshore oligotrophic conditions. Both the MCMC and L 2MN methods predicted well-constrained rates of protozoan and mesozooplankton grazing with reasonable accuracy, but the MCMC method overestimated primary production. The MCMC method more accurately predicted the poorly constrained rate of vertical carbon export than the L 2MN method, which consistently overestimated export. Results involving DOC and bacterial production were equivocal. Overall, when primary production is provided as model input, the MCMC method gives a robust depiction of ecosystem processes. Uncertainty in inverse ecosystem models is large and arises primarily from solution under-determinacy. We thus suggest that experimental programs focusing on food web fluxes expand the range of experimental measurements to include the nature and fate of detrital pools, which play large roles in the model.

1. Accurate Iterative Analysis Solution of theKapchinskij-Vladimirskij Equations for the Case of a Matched Beam

SciTech Connect

Anderson, O.A.

2007-01-31

The well-known Kapchinskij-Vladimirskij (KV) equations are difficult to solve in general, but the problem is simplified for the matched-beam case with sufficient symmetry. They show that the interdependence of the two KV equations is eliminated, so that only one needs to be solved--a great simplification. They present an iterative method of solution which can potentially yield any desired level of accuracy. The lowest level, the well-known smooth approximation, yields simple, explicit results with good accuracy for weak or moderate focusing fields. The next level improves the accuracy for high fields; they previously showed [Part. Accel. 52, 133 (1996)] how to maintain a simple explicit format for the results. That paper used expansion in a small parameter to obtain results of second-level accuracy. The present paper, using straightforward iteration, obtains equations of first, second, and third levels of accuracy. For a periodic lattice with beam matched to lattice, they use the lattice and beam parameters as input and solve for phase advances and envelope functions. They find excellent agreement with numerical solutions over a wide range of beam emittances and intensities.

2. Accurate iterative analytic solution of theKapchinskij-Vladimirskij equations for the case of a matched beam

SciTech Connect

Anderson, Oscar A.

2006-08-06

The well-known Kapchinskij-Vladimirskij (KV) equations are difficult to solve in general, but the problem is simplified for the matched-beam case with sufficient symmetry. They show that the interdependence of the two KV equations is eliminated, so that only one needs to be solved--a great simplification. They present an iterative method of solution which can potentially yield any desired level of accuracy. The lowest level, the well-known smooth approximation, yields simple, explicit results with good accuracy for weak or moderate focusing fields. The next level improves the accuracy for high fields; they previously showed how to maintain a simple explicit format for the results. That paper used expansion in a small parameter to obtain the second level. The present paper, using straightforward iteration, obtains equations of first, second, and third levels of accuracy. For a periodic lattice with beam matched to lattice, they use the lattice and beam parameters as input and solve for phase advances and envelope waveforms. They find excellent agreement with numerical solutions over a wide range of beam emittances and intensities.

3. Alfvén wave collisions, the fundamental building block of plasma turbulence. II. Numerical solution

SciTech Connect

Nielson, K. D.; Howes, G. G.; Dorland, W.

2013-07-15

This paper presents the numerical verification of an asymptotic analytical solution for the nonlinear interaction between counterpropagating Alfvén waves, the fundamental building block of astrophysical plasma turbulence. The analytical solution, derived in the weak turbulence limit using the equations of incompressible MHD, is compared to a nonlinear gyrokinetic simulation of an Alfvén wave collision. The agreement between these methods signifies that the incompressible solution satisfactorily describes the essential dynamics of the nonlinear energy transfer, even under the weakly collisional plasma conditions relevant to many astrophysical environments.

4. A numerical method for finding sign-changing solutions of superlinear Dirichlet problems

SciTech Connect

Neuberger, J.M.

1996-12-31

In a recent result it was shown via a variational argument that a class of superlinear elliptic boundary value problems has at least three nontrivial solutions, a pair of one sign and one which sign changes exactly once. These three and all other nontrivial solutions are saddle points of an action functional, and are characterized as local minima of that functional restricted to a codimension one submanifold of the Hilbert space H-0-1-2, or an appropriate higher codimension subset of that manifold. In this paper, we present a numerical Sobolev steepest descent algorithm for finding these three solutions.

5. Numerical solution of Maxwell's Wave Equation in an Axisymmetric Curved Space-Time

Otu, Joseph

2002-03-01

Using an exact Weyl metric with a quadrupole moment, a wave equation valid in flat space-time was derived from the covariant Maxwell's equations in curved space-time. The curvature effects are shown to represent a medium with an effective but variable index of refraction. Numerical solution of the wave equation is studied. The solution shows that the wave amplitude is unchanged as the wave propagates into a stronger gravitational field. However, the frequency changes in a way that is consistent with the analytic solution^1, and as predicted by the gravitational red-shift phenomenon. ^1 J. O. Otu, Bull. of the APS, Vol. 44, No. 1, 676(1999).

6. Numerical study of wave effects on groundwater flow and solute transport in a laboratory beach.

PubMed

Geng, Xiaolong; Boufadel, Michel C; Xia, Yuqiang; Li, Hailong; Zhao, Lin; Jackson, Nancy L; Miller, Richard S

2014-09-01

A numerical study was undertaken to investigate the effects of waves on groundwater flow and associated inland-released solute transport based on tracer experiments in a laboratory beach. The MARUN model was used to simulate the density-dependent groundwater flow and subsurface solute transport in the saturated and unsaturated regions of the beach subjected to waves. The Computational Fluid Dynamics (CFD) software, Fluent, was used to simulate waves, which were the seaward boundary condition for MARUN. A no-wave case was also simulated for comparison. Simulation results matched the observed water table and concentration at numerous locations. The results revealed that waves generated seawater-groundwater circulations in the swash and surf zones of the beach, which induced a large seawater-groundwater exchange across the beach face. In comparison to the no-wave case, waves significantly increased the residence time and spreading of inland-applied solutes in the beach. Waves also altered solute pathways and shifted the solute discharge zone further seaward. Residence Time Maps (RTM) revealed that the wave-induced residence time of the inland-applied solutes was largest near the solute exit zone to the sea. Sensitivity analyses suggested that the change in the permeability in the beach altered solute transport properties in a nonlinear way. Due to the slow movement of solutes in the unsaturated zone, the mass of the solute in the unsaturated zone, which reached up to 10% of the total mass in some cases, constituted a continuous slow release of solutes to the saturated zone of the beach. This means of control was not addressed in prior studies.

7. A Comparison of Numerical and Analytical Radiative-Transfer Solutions for Plane Albedo in Natural Waters

EPA Science Inventory

Several numerical and analytical solutions of the radiative transfer equation (RTE) for plane albedo were compared for solar light reflection by sea water. The study incorporated the simplest case, that being a semi-infinite one-dimensional plane-parallel absorbing and scattering...

8. A Comparison of Numerical and Analytical Radiative-Transfer Solutions for Plane Albedo of Natural Waters

EPA Science Inventory

Three numerical algorithms were compared to provide a solution of a radiative transfer equation (RTE) for plane albedo (hemispherical reflectance) in semi-infinite one-dimensional plane-parallel layer. Algorithms were based on the invariant imbedding method and two different var...

9. A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations

NASA Technical Reports Server (NTRS)

Diethelm, Kai; Ford, Neville J.; Freed, Alan D.; Gray, Hugh R. (Technical Monitor)

2002-01-01

We discuss an Adams-type predictor-corrector method for the numerical solution of fractional differential equations. The method may be used both for linear and for nonlinear problems, and it may be extended to multi-term equations (involving more than one differential operator) too.

10. Numerical Solution to Generalized Burgers'-Fisher Equation Using Exp-Function Method Hybridized with Heuristic Computation

PubMed Central

Malik, Suheel Abdullah; Qureshi, Ijaz Mansoor; Amir, Muhammad; Malik, Aqdas Naveed; Haq, Ihsanul

2015-01-01

In this paper, a new heuristic scheme for the approximate solution of the generalized Burgers'-Fisher equation is proposed. The scheme is based on the hybridization of Exp-function method with nature inspired algorithm. The given nonlinear partial differential equation (NPDE) through substitution is converted into a nonlinear ordinary differential equation (NODE). The travelling wave solution is approximated by the Exp-function method with unknown parameters. The unknown parameters are estimated by transforming the NODE into an equivalent global error minimization problem by using a fitness function. The popular genetic algorithm (GA) is used to solve the minimization problem, and to achieve the unknown parameters. The proposed scheme is successfully implemented to solve the generalized Burgers'-Fisher equation. The comparison of numerical results with the exact solutions, and the solutions obtained using some traditional methods, including adomian decomposition method (ADM), homotopy perturbation method (HPM), and optimal homotopy asymptotic method (OHAM), show that the suggested scheme is fairly accurate and viable for solving such problems. PMID:25811858

11. A Penalty Method for the Numerical Solution of Hamilton-Jacobi-Bellman (HJB) Equations in Finance

Witte, J. H.; Reisinger, C.

2010-09-01

We present a simple and easy to implement method for the numerical solution of a rather general class of Hamilton-Jacobi-Bellman (HJB) equations. In many cases, the considered problems have only a viscosity solution, to which, fortunately, many intuitive (e.g. finite difference based) discretisations can be shown to converge. However, especially when using fully implicit time stepping schemes with their desireable stability properties, one is still faced with the considerable task of solving the resulting nonlinear discrete system. In this paper, we introduce a penalty method which approximates the nonlinear discrete system to an order of O(1/ρ), where ρ>0 is the penalty parameter, and we show that an iterative scheme can be used to solve the penalised discrete problem in finitely many steps. We include a number of examples from mathematical finance for which the described approach yields a rigorous numerical scheme and present numerical results.

12. Numerical solution of stochastic differential equations with Poisson and Lévy white noise

Grigoriu, M.

2009-08-01

A fixed time step method is developed for integrating stochastic differential equations (SDE’s) with Poisson white noise (PWN) and Lévy white noise (LWN). The method for integrating SDE’s with PWN has the same structure as that proposed by Kim [Phys. Rev. E 76, 011109 (2007)], but is established by using different arguments. The integration of SDE’s with LWN is based on a representation of Lévy processes by sums of scaled Brownian motions and compound Poisson processes. It is shown that the numerical solutions of SDE’s with PWN and LWN converge weakly to the exact solutions of these equations, so that they can be used to estimate not only marginal properties but also distributions of functionals of the exact solutions. Numerical examples are used to demonstrate the applications and the accuracy of the proposed integration algorithms.

13. Numerical solution of stochastic differential equations with Poisson and Lévy white noise.

PubMed

Grigoriu, M

2009-08-01

A fixed time step method is developed for integrating stochastic differential equations (SDE's) with Poisson white noise (PWN) and Lévy white noise (LWN). The method for integrating SDE's with PWN has the same structure as that proposed by Kim [Phys. Rev. E 76, 011109 (2007)], but is established by using different arguments. The integration of SDE's with LWN is based on a representation of Lévy processes by sums of scaled Brownian motions and compound Poisson processes. It is shown that the numerical solutions of SDE's with PWN and LWN converge weakly to the exact solutions of these equations, so that they can be used to estimate not only marginal properties but also distributions of functionals of the exact solutions. Numerical examples are used to demonstrate the applications and the accuracy of the proposed integration algorithms.

14. Numerical solution of the full potential equation using a chimera grid approach

NASA Technical Reports Server (NTRS)

Holst, Terry L.

1995-01-01

A numerical scheme utilizing a chimera zonal grid approach for solving the full potential equation in two spatial dimensions is described. Within each grid zone a fully-implicit approximate factorization scheme is used to advance the solution one interaction. This is followed by the explicit advance of all common zonal grid boundaries using a bilinear interpolation of the velocity potential. The presentation is highlighted with numerical results simulating the flow about a two-dimensional, nonlifting, circular cylinder. For this problem, the flow domain is divided into two parts: an inner portion covered by a polar grid and an outer portion covered by a Cartesian grid. Both incompressible and compressible (transonic) flow solutions are included. Comparisons made with an analytic solution as well as single grid results indicate that the chimera zonal grid approach is a viable technique for solving the full potential equation.

15. Numerical Solution of a Plane Jet Impingement on an Infinite Flat Surface

Arora, S.; Irfan, Nagma

2015-03-01

In this paper numerical solution of the unsteady plane incompressible viscous jet impinging on to an infinite flat surface are presented for Re=450. In the present study, all calculations have been done by using Dufort Frankel scheme and over relaxation scheme. Result and graphs have been obtained by using MATLAB programming. The obtained results explain the flow of water after exhaling from nozzle and the streamlines and vorticity of flow ofwater after striking with flat infinite surface. The solutions obtained by proposed method indicate that this approach is easy to implement and computationally very attractive and the results of our investigation are in qualitative agreement with those available in the literature [1, 9]. This method is capable of greatly reducing the size of calculations while still maintaining high accuracy of the numerical solution.

16. Stealth surface modification of surface-enhanced Raman scattering substrates for sensitive and accurate detection in protein solutions.

PubMed

Sun, Fang; Ella-Menye, Jean-Rene; Galvan, Daniel David; Bai, Tao; Hung, Hsiang-Chieh; Chou, Ying-Nien; Zhang, Peng; Jiang, Shaoyi; Yu, Qiuming

2015-03-24

Reliable surface-enhanced Raman scattering (SERS) based biosensing in complex media is impeded by nonspecific protein adsorptions. Because of the near-field effect of SERS, it is challenging to modify SERS-active substrates using conventional nonfouling materials without introducing interference from their SERS signals. Herein, we report a stealth surface modification strategy for sensitive, specific and accurate detection of fructose in protein solutions using SERS by forming a mixed self-assembled monolayer (SAM). The SAM consists of a short zwitterionic thiol, N,N-dimethyl-cysteamine-carboxybetaine (CBT), and a fructose probe 4-mercaptophenylboronic acid (4-MPBA). The specifically designed and synthesized CBT not only resists protein fouling effectively, but also has very weak Raman activity compared to 4-MPBA. Thus, the CBT SAM provides a stealth surface modification to SERS-active substrates. The surface compositions of mixed SAMs were investigated using X-ray photoelectron spectroscopy (XPS) and SERS, and their nonfouling properties were studied with a surface plasmon resonance (SPR) biosensor. The mixed SAM with a surface composition of 94% CBT demonstrated a very low bovine serum albumin (BSA) adsorption (∼3 ng/cm(2)), and moreover, only the 4-MPBA signal appeared in the SERS spectrum. With the use of this surface-modified SERS-active substrate, quantification of fructose over clinically relevant concentrations (0.01-1 mM) was achieved. Partial least-squares regression (PLS) analysis showed that the detection sensitivity and accuracy were maintained for the measurements in 1 mg/mL BSA solutions. This stealth surface modification strategy provides a novel route to introduce nonfouling property to SERS-active substrates for SERS biosensing in complex media.

17. An efficient exponential time integration method for the numerical solution of the shallow water equations on the sphere

Gaudreault, Stéphane; Pudykiewicz, Janusz A.

2016-10-01

The exponential propagation methods were applied in the past for accurate integration of the shallow water equations on the sphere. Despite obvious advantages related to the exact solution of the linear part of the system, their use for the solution of practical problems in geophysics has been limited because efficiency of the traditional algorithm for evaluating the exponential of Jacobian matrix is inadequate. In order to circumvent this limitation, we modify the existing scheme by using the Incomplete Orthogonalization Method instead of the Arnoldi iteration. We also propose a simple strategy to determine the initial size of the Krylov space using information from previous time instants. This strategy is ideally suited for the integration of fluid equations where the structure of the system Jacobian does not change rapidly between the subsequent time steps. A series of standard numerical tests performed with the shallow water model on a geodesic icosahedral grid shows that the new scheme achieves efficiency comparable to the semi-implicit methods. This fact, combined with the accuracy and the mass conservation of the exponential propagation scheme, makes the presented method a good candidate for solving many practical problems, including numerical weather prediction.

18. Numerical solution of the Navier-Stokes equations for high Reynolds number incompressible turbulent flow. M.S. Thesis

NASA Technical Reports Server (NTRS)

Thompson, D. S.

1980-01-01

The full Navier-Stokes equations for incompressible turbulent flow must be solved to accurately represent all flow phenomena which occur in a high Reynolds number incompressible flow. A two layer algebraic eddy viscosity turbulence model is used to represent the Reynolds stress in the primitive variable formulation. The development of the boundary-fitted coordinate systems makes the numerical solution of these equations feasible for arbitrarily shaped bodies. The nondimensional time averaged Navier-Stokes equations, including the turbulence mode, are represented by finite difference approximations in the transformed plane. The resulting coupled system of nonlinear algebraic equations is solved using a point successive over relaxation iteration. The test case considered was a NACA 64A010 airfoil section at an angle of attack of two degrees and a Reynolds number of 2,000,000.

19. The space-time solution element method: A new numerical approach for the Navier-Stokes equations

NASA Technical Reports Server (NTRS)

Scott, James R.; Chang, Sin-Chung

1995-01-01

This paper is one of a series of papers describing the development of a new numerical method for the Navier-Stokes equations. Unlike conventional numerical methods, the current method concentrates on the discrete simulation of both the integral and differential forms of the Navier-Stokes equations. Conservation of mass, momentum, and energy in space-time is explicitly provided for through a rigorous enforcement of both the integral and differential forms of the governing conservation laws. Using local polynomial expansions to represent the discrete primitive variables on each cell, fluxes at cell interfaces are evaluated and balanced using exact functional expressions. No interpolation or flux limiters are required. Because of the generality of the current method, it applies equally to the steady and unsteady Navier-Stokes equations. In this paper, we generalize and extend the authors' 2-D, steady state implicit scheme. A general closure methodology is presented so that all terms up through a given order in the local expansions may be retained. The scheme is also extended to nonorthogonal Cartesian grids. Numerous flow fields are computed and results are compared with known solutions. The high accuracy of the scheme is demonstrated through its ability to accurately resolve developing boundary layers on coarse grids. Finally, we discuss applications of the current method to the unsteady Navier-Stokes equations.

20. Numerical simulation of pharyngeal airflow applied to obstructive sleep apnea: effect of the nasal cavity in anatomically accurate airway models.

PubMed

Cisonni, Julien; Lucey, Anthony D; King, Andrew J C; Islam, Syed Mohammed Shamsul; Lewis, Richard; Goonewardene, Mithran S

2015-11-01

Repetitive brief episodes of soft-tissue collapse within the upper airway during sleep characterize obstructive sleep apnea (OSA), an extremely common and disabling disorder. Failure to maintain the patency of the upper airway is caused by the combination of sleep-related loss of compensatory dilator muscle activity and aerodynamic forces promoting closure. The prediction of soft-tissue movement in patient-specific airway 3D mechanical models is emerging as a useful contribution to clinical understanding and decision making. Such modeling requires reliable estimations of the pharyngeal wall pressure forces. While nasal obstruction has been recognized as a risk factor for OSA, the need to include the nasal cavity in upper-airway models for OSA studies requires consideration, as it is most often omitted because of its complex shape. A quantitative analysis of the flow conditions generated by the nasal cavity and the sinuses during inspiration upstream of the pharynx is presented. Results show that adequate velocity boundary conditions and simple artificial extensions of the flow domain can reproduce the essential effects of the nasal cavity on the pharyngeal flow field. Therefore, the overall complexity and computational cost of accurate flow predictions can be reduced.

1. Precise and accurate measurement of U and Th isotopes via ICP-MS using a single solution

Mertz-Kraus, R.; Sharp, W. D.; Ludwig, K. R.

2012-04-01

, allowing the sample's 238U/235U ratio to be measured. In step 3, we monitor peak-tails at half-mass positions (229.5, 231.5, 234.5) and on mass 237 while aspirating sample solution. Tail measurement requires a distinct cup configuration to maintain 238U in the cups; however, no sample is consumed during automated cup reconfiguration. We monitor the accuracy of 234U/238U ratios using CRM 145, which gives a weighted mean atom ratio of (5.2846 ± 0.0029) - 10-5 (all errors 2σ), consistent with published and reference values. The reproducibility of 230Th/238U ratios is monitored using the Schwartzwalder Mine secular-equilibrium standard (SM). We detect no bias in 230Th/238U or 234U/238U ratios measured for SM at beam intensities ranging over a factor of four, consistent with accurate correction for IC yields. Aladdin's cave coral (AC-1) was analyzed to check our ICP-MS method (and the preceding purification by ion exchange) on a carbonate and yields a mean age of 125.43 ± 0.38 ka, in agreement with published values. We are currently applying the method to corals, speleothems, pedogenic coatings, and tufas.

2. Stochastic coalescence in finite systems: an algorithm for the numerical solution of the multivariate master equation.

Alfonso, Lester; Zamora, Jose; Cruz, Pedro

2015-04-01

The stochastic approach to coagulation considers the coalescence process going in a system of a finite number of particles enclosed in a finite volume. Within this approach, the full description of the system can be obtained from the solution of the multivariate master equation, which models the evolution of the probability distribution of the state vector for the number of particles of a given mass. Unfortunately, due to its complexity, only limited results were obtained for certain type of kernels and monodisperse initial conditions. In this work, a novel numerical algorithm for the solution of the multivariate master equation for stochastic coalescence that works for any type of kernels and initial conditions is introduced. The performance of the method was checked by comparing the numerically calculated particle mass spectrum with analytical solutions obtained for the constant and sum kernels, with an excellent correspondence between the analytical and numerical solutions. In order to increase the speedup of the algorithm, software parallelization techniques with OpenMP standard were used, along with an implementation in order to take advantage of new accelerator technologies. Simulations results show an important speedup of the parallelized algorithms. This study was funded by a grant from Consejo Nacional de Ciencia y Tecnologia de Mexico SEP-CONACYT CB-131879. The authors also thanks LUFAC® Computacion SA de CV for CPU time and all the support provided.

3. Numerical and approximate solution of the high Reynolds number small separation problem

NASA Technical Reports Server (NTRS)

Davis, R. T.

1976-01-01

Several possible methods of solving the small separation problem at high Reynolds number are investigated. In addition to using analytical methods, there are several numerical approaches which are used. High Reynolds number laminar two dimensional problems are used for simplicity. A brief discussion is given of the finite difference methods since these methods are discussed in detail. Most of the emphasis is placed on developing an approximate integral method. As a model problem the supersonic compression ramp problem is chosen since several numerical solutions along with experimental data are available. The techniques discussed are modified and applied to other similar type wall geometries.

4. A fifth order implicit method for the numerical solution of differential-algebraic equations

Skvortsov, L. M.

2015-06-01

An implicit two-step Runge-Kutta method of fifth order is proposed for the numerical solution of differential and differential-algebraic equations. The location of nodes in this method makes it possible to estimate the values of higher derivatives at the initial and terminal points of an integration step. Consequently, the proposed method can be regarded as a finite-difference analog of the Obrechkoff method. Numerical results, some of which are presented in this paper, show that our method preserves its order while solving stiff equations and equations of indices two and three. This is the main advantage of the proposed method as compared with the available ones.

5. A numerical solution of a singular boundary value problem arising in boundary layer theory.

PubMed

Hu, Jiancheng

2016-01-01

In this paper, a second-order nonlinear singular boundary value problem is presented, which is equivalent to the well-known Falkner-Skan equation. And the one-dimensional third-order boundary value problem on interval [Formula: see text] is equivalently transformed into a second-order boundary value problem on finite interval [Formula: see text]. The finite difference method is utilized to solve the singular boundary value problem, in which the amount of computational effort is significantly less than the other numerical methods. The numerical solutions obtained by the finite difference method are in agreement with those obtained by previous authors. PMID:27026894

6. Numerical solutions for steady thermal convection from a concentrated source in a porous medium

SciTech Connect

Hickox, C.E.; Watts, H.A.

1980-06-01

Solutions for the steady, axisymmetric velocity and temperature fields associated with a point source of thermal energy in a fluid-saturated porous medium are obtained numerically through use of similarity transformations. The two cases considered are those of a point source located on the lower, insulated boundary of a semi-infinite region and a point source embedded in an infinite region. Numerical results are presented from which complete descriptions of the velocity and temperature fields can be constructed for Rayleigh numbers ranging from 10/sup -3/ to 10/sup 2/.

7. Numerical Modeling Tools for the Prediction of Solution Migration Applicable to Mining Site

SciTech Connect

Martell, M.; Vaughn, P.

1999-01-06

Mining has always had an important influence on cultures and traditions of communities around the globe and throughout history. Today, because mining legislation places heavy emphasis on environmental protection, there is great interest in having a comprehensive understanding of ancient mining and mining sites. Multi-disciplinary approaches (i.e., Pb isotopes as tracers) are being used to explore the distribution of metals in natural environments. Another successful approach is to model solution migration numerically. A proven method to simulate solution migration in natural rock salt has been applied to project through time for 10,000 years the system performance and solution concentrations surrounding a proposed nuclear waste repository. This capability is readily adaptable to simulate solution migration around mining.

8. Impact of 3D root uptake on solute transport: a numerical study

Schröder, N.; Javaux, M.; Vanderborght, J.; Steffen, B.; Vereecken, H.

2011-12-01

Plant transpiration is an important component of the hydrological cycle. Through root water uptake, plants do not only affect the 3D soil water flow velocity distribution, but also solute movement in soil. This numerical study aims at investigating how solute fate is impacted by root uptake using the 3D biophysical model R-SWMS (Javaux et al., 2008). This model solves the Richards equation in 3D in the soil and the flow equation within the plant root xylem vessels. Furthermore, for solute transport simulations, the 3D particle tracker PARTRACE (Bechtold et al., 2011) was used. . We generated 3D virtual steady-state breakthrough curves (BTC) experiments in soils with transpiring plants. The averaged BTCs were then fitted with a 1D numerical flow model under steady-state conditions to obtain apparent CDE parameters. Two types of root architecture, a fibrous and a taprooted structure, were compared in virtual 3D experiments. The solute uptake type or the transpiration rate were also modified and we analyzed how these parameters affected apparent disperisivity and velocity profiles. Our simulation results show, that both, apparent velocity and dispersivity length are affected by water and solute root uptake. In addition, under high exclusion processes (slight or no active uptake), solute accumulates around roots and generates a long tailing to the breakthrough curves, which cannot be reproduced by 1D models that simulate root water uptake with solute exclusion. This observation may have an important impact on how to model pollutant mass transfer to groundwater at larger scales. Javaux, M., T. Schröder, J. Vanderborght, and H. Vereecken. 2008. Use of a three-dimensional detailed modeling approach for predicting root water uptake. Vadose Zone J. 7:1079-1088.doi: 10.2136/vzj2007.0115. Bechtold, M., S. Haber-Pohlmeier, J. Vanderborght, A. Pohlmeier, P.A. Ferre, and H. Vereecken. 2011. Near-surface solute redistribution during evaporation. Submitted to Geophys. Res. Lett

9. A numerical procedure to compute the stabilising solution of game theoretic Riccati equations of stochastic control

Dragan, Vasile; Ivanov, Ivan

2011-04-01

In this article, the problem of the numerical computation of the stabilising solution of the game theoretic algebraic Riccati equation is investigated. The Riccati equation under consideration occurs in connection with the solution of the H ∞ control problem for a class of stochastic systems affected by state-dependent and control-dependent white noise and subjected to Markovian jumping. The stabilising solution of the considered game theoretic Riccati equation is obtained as a limit of a sequence of approximations constructed based on stabilising solutions of a sequence of algebraic Riccati equations of stochastic control with definite sign of the quadratic part. The proposed algorithm extends to this general framework the method proposed in Lanzon, Feng, Anderson, and Rotkowitz (Lanzon, A., Feng, Y., Anderson, B.D.O., and Rotkowitz, M. (2008), 'Computing the Positive Stabilizing Solution to Algebraic Riccati Equations with an Indefinite Quadratic Term Viaa Recursive Method,' IEEE Transactions on Automatic Control, 53, pp. 2280-2291). In the proof of the convergence of the proposed algorithm different concepts associated the generalised Lyapunov operators as stability, stabilisability and detectability are widely involved. The efficiency of the proposed algorithm is demonstrated by several numerical experiments.

10. Numerical solution of convection-diffusion problems in irregular domains mapped onto a circle

Asako, Yutaka; Nakamura, Hiroshi; Faghri, Mohammad; Asaba, Makoto

1991-01-01

A coordinate transformation methodology has been developed for convection-diffusion problems with an arbitrary solution domain. An algebraic coordinate transformation is used that maps the solution domain onto a circle. The transformed conservation equations are discretized by a control-volume finite difference technique. Sample computations are performed for fully developed flow and heat transfer in a polygonal duct, and for natural convection in a square cavity, to validate the present methodology. The numerical results obtained compared reasonably well, even in the extreme case of a rectangular domain mapped onto a circle.

11. A comparison of numerical and analytical solution of the creeping flame spread over thermally thin material

NASA Technical Reports Server (NTRS)

Bhattacharjee, Subrata

1993-01-01

The present numerical solution for the de Ris (1969) problem of flame-spread over thin condensed fuel in an opposed-flow environment is obtained by reformulating the problem in terms of four nondimensional parameters, while retaining all assumptions of the original theory. While the de Ris theory sees the location of the leading edge and the eigenlocation of the onset of evaporation as identical, this analysis treats the leading edge as part of the solution; the location of the flame leading edge is in this way established to be upstream of the eigenlocation, with significant consequences for the spread rate formula.

12. Numerical solution of two-dimensional integral-algebraic systems using Legendre functions

Nemati, S.; Lima, P.; Ordokhani, Y.

2012-09-01

We consider a method for computing approximate solutions to systems of two-dimensional Volterra integral equations. The approximate solution is sought in the form of a linear combination of two-variable shifted Legendre functions. The operational matrices technique is used to reduce the problem to a system of linear algebraic equations. Some numerical tests have been carried out and the results show that this method has a good performance, even in the case when the system matrix is singular in the whole considered domain.

13. Numerical solution of the time-dependent compressible Navier-Stokes equations in inlet regions

NASA Technical Reports Server (NTRS)

Olson, L. E.; Mcgowan, P. R.; Maccormack, R. W.

1974-01-01

The results of a study to determine the effects of compressibility on the viscous flow through channels that have straight, parallel walls are presented. Two channel configurations are considered, the flow between two semi-infinite flat plates with uniform flow prescribed at the inlet plane and a cascade of semi-infinite flat plates with uniform flow introduced upstream. The flow field is modeled by using the time dependent, compressible Navier-Stokes equations. Time dependent solutions are obtained by using an explicit finite difference technique which advances the pressure on near field subsonic boundaries such that accurate steady state solutions are obtained. Steady state results at Reynolds number 20 and 150 are presented for Mach numbers between 0.09 and 0.36 and compared with the incompressible solutions of previous studies.

14. Numerical solution of the Riemann problem for two-dimensional gas dynamics

SciTech Connect

Schulz-Rinne, C.W. ); Collins, J.P. ); Glaz, H.M. )

1993-11-01

The Riemann problem for two-dimensional gas dynamics with isentropic or polytropic gas is considered. The initial data is constant in each quadrant and chosen so that only a rarefaction wave, shock wave, or slip line connects two neighboring constant initial states. With this restriction sixteen (respectively, fifteen) genuinely different wave combinations for isentropic (respectively, polytropic) gas exist. For each configuration the numerical solution is analyzed and illustrated by contour plots. Additionally, the required relations for the initial data and the symmetry properties of the solutions are given. The chosen calculations correspond closely to the cases studied by T. Zhang and Y. Zheng so that the analytical theory can be directly compared to the numerical study.

15. Numerical solution of shock and ramp compression for general material properties

SciTech Connect

Swift, D C

2009-01-28

A general formulation was developed to represent material models for applications in dynamic loading. Numerical methods were devised to calculate response to shock and ramp compression, and ramp decompression, generalizing previous solutions for scalar equations of state. The numerical methods were found to be flexible and robust, and matched analytic results to a high accuracy. The basic ramp and shock solution methods were coupled to solve for composite deformation paths, such as shock-induced impacts, and shock interactions with a planar interface between different materials. These calculations capture much of the physics of typical material dynamics experiments, without requiring spatially-resolving simulations. Example calculations were made of loading histories in metals, illustrating the effects of plastic work on the temperatures induced in quasi-isentropic and shock-release experiments, and the effect of a phase transition.

16. Numerical Solutions of the Nonlinear Fractional-Order Brusselator System by Bernstein Polynomials

PubMed Central

Khan, Rahmat Ali; Tajadodi, Haleh; Johnston, Sarah Jane

2014-01-01

In this paper we propose the Bernstein polynomials to achieve the numerical solutions of nonlinear fractional-order chaotic system known by fractional-order Brusselator system. We use operational matrices of fractional integration and multiplication of Bernstein polynomials, which turns the nonlinear fractional-order Brusselator system to a system of algebraic equations. Two illustrative examples are given in order to demonstrate the accuracy and simplicity of the proposed techniques. PMID:25485293

17. Numerical solutions of the nonlinear fractional-order brusselator system by Bernstein polynomials.

PubMed

Khan, Hasib; Jafari, Hossein; Khan, Rahmat Ali; Tajadodi, Haleh; Johnston, Sarah Jane

2014-01-01

In this paper we propose the Bernstein polynomials to achieve the numerical solutions of nonlinear fractional-order chaotic system known by fractional-order Brusselator system. We use operational matrices of fractional integration and multiplication of Bernstein polynomials, which turns the nonlinear fractional-order Brusselator system to a system of algebraic equations. Two illustrative examples are given in order to demonstrate the accuracy and simplicity of the proposed techniques. PMID:25485293

18. A note on a corrector formula for the numerical solution of ordinary differential equations

NASA Technical Reports Server (NTRS)

Chien, Y.-C.; Agrawal, K. M.

1979-01-01

A new corrector formula for predictor-corrector methods for numerical solutions of ordinary differential equations is presented. Two considerations for choosing corrector formulas are given: (1) the coefficient in the error term and (2) its stability properties. The graph of the roots of an equation plotted against its stability region, of different values, is presented along with the tables that correspond to various corrector equations, including Hamming's and Milne and Reynolds'.

19. Numerical solution of fluid flow and heat tranfer problems with surface radiation

NASA Technical Reports Server (NTRS)

Ahuja, S.; Bhatia, K.

1995-01-01

This paper presents a numerical scheme, based on the finite element method, to solve strongly coupled fluid flow and heat transfer problems. The surface radiation effect for gray, diffuse and isothermal surfaces is considered. A procedure for obtaining the view factors between the radiating surfaces is discussed. The overall solution strategy is verified by comparing the available results with those obtained using this approach. An analysis of a thermosyphon is undertaken and the effect of considering the surface radiation is clearly explained.

20. A numerical solution algorithm and its application to studies of pulsed light fields propagation

Banakh, V. A.; Gerasimova, L. O.; Smalikho, I. N.; Falits, A. V.

2016-08-01

A new method for studies of pulsed laser beams propagation in a turbulent atmosphere was proposed. The algorithm of numerical simulation is based on the solution of wave parabolic equation for complex spectral amplitude of wave field using method of splitting into physical factors. Examples of the use of the algorithm in the case the propagation pulsed Laguerre-Gaussian beams of femtosecond duration in the turbulence atmosphere has been shown.

1. A comment on the importance of numerical evaluation of analytic solutions involving approximations.

PubMed

Overall, J E; Starbuck, R R; Doyle, S R

1994-07-01

An analytic solution proposed by Senn (1) for removing the effects of covariate imbalance in controlled clinical trials was subjected to Monte Carlo evaluation. For practical applications of his derivation, Senn proposed substitution of sample statistics for parameters of the bivariate normal model. Unfortunately, that substitution produces severe distortion in the size of tests of significance for treatment effects when covariate imbalance is present. Numerical verification of proposed substitutions into analytic models is recommended as a prudent approach. PMID:7951276

2. Solution of population balance equations in applications with fine particles: Mathematical modeling and numerical schemes

Nguyen, T. T.; Laurent, F.; Fox, R. O.; Massot, M.

2016-11-01

The accurate description and robust simulation, at relatively low cost, of global quantities (e.g. number density or volume fraction) as well as the size distribution of a population of fine particles in a carrier fluid is still a major challenge for many applications. For this purpose, two types of methods are investigated for solving the population balance equation with aggregation, continuous particle size change (growth and size reduction), and nucleation: the extended quadrature method of moments (EQMOM) based on the work of Yuan et al. [52] and a hybrid method (TSM) between the sectional and moment methods, considering two moments per section based on the work of Laurent et al. [30]. For both methods, the closure employs a continuous reconstruction of the number density function of the particles from its moments, thus allowing evaluation of all the unclosed terms in the moment equations, including the negative flux due to the disappearance of particles. Here, new robust and efficient algorithms are developed for this reconstruction step and two kinds of reconstruction are tested for each method. Moreover, robust and accurate numerical methods are developed, ensuring the realizability of the moments. The robustness is ensured with efficient and tractable algorithms despite the numerous couplings and various algebraic constraints thanks to a tailored overall strategy. EQMOM and TSM are compared to a sectional method for various simple but relevant test cases, showing their ability to describe accurately the fine-particle population with a much lower number of variables. These results demonstrate the efficiency of the modeling and numerical choices, and their potential for the simulation of real-world applications.

3. Numerical solution of the nonlinear Schrödinger equation using smoothed-particle hydrodynamics

Mocz, Philip; Succi, Sauro

2015-05-01

We formulate a smoothed-particle hydrodynamics numerical method, traditionally used for the Euler equations for fluid dynamics in the context of astrophysical simulations, to solve the nonlinear Schrödinger equation in the Madelung formulation. The probability density of the wave function is discretized into moving particles, whose properties are smoothed by a kernel function. The traditional fluid pressure is replaced by a quantum pressure tensor, for which a robust discretization is found. We demonstrate our numerical method on a variety of numerical test problems involving the simple harmonic oscillator, soliton-soliton collision, Bose-Einstein condensates, collapsing singularities, and dark matter halos governed by the Gross-Pitaevskii-Poisson equation. Our method is conservative, applicable to unbounded domains, and is automatically adaptive in its resolution, making it well suited to study problems with collapsing solutions.

4. Numerical method for the solution of large systems of differential equations of the boundary layer type

NASA Technical Reports Server (NTRS)

Green, M. J.; Nachtsheim, P. R.

1972-01-01

A numerical method for the solution of large systems of nonlinear differential equations of the boundary-layer type is described. The method is a modification of the technique for satisfying asymptotic boundary conditions. The present method employs inverse interpolation instead of the Newton method to adjust the initial conditions of the related initial-value problem. This eliminates the so-called perturbation equations. The elimination of the perturbation equations not only reduces the user's preliminary work in the application of the method, but also reduces the number of time-consuming initial-value problems to be numerically solved at each iteration. For further ease of application, the solution of the overdetermined system for the unknown initial conditions is obtained automatically by applying Golub's linear least-squares algorithm. The relative ease of application of the proposed numerical method increases directly as the order of the differential-equation system increases. Hence, the method is especially attractive for the solution of large-order systems. After the method is described, it is applied to a fifth-order problem from boundary-layer theory.

5. Dynamics of the solar tachocline - III. Numerical solutions of the Gough and McIntyre model

Acevedo-Arreguin, L. A.; Garaud, P.; Wood, T. S.

2013-09-01

We present the first numerical simulations of the solar interior to exhibit a tachocline consistent with the Gough and McIntyre model. We find non-linear, axisymmetric, steady-state numerical solutions in which: (1) a large-scale primordial field is confined within the radiation zone by downwelling meridional flows that are gyroscopically pumped in the convection zone; (2) the radiation zone is in almost uniform rotation, with a rotation rate consistent with observations; (3) the bulk of the high-latitude tachocline is in thermal-wind balance, and the Lorentz force plays a negligible role in its dynamics; (4) the interaction between the field and the flows takes place within a very thin magnetic boundary layer, the tachopause, located at the bottom of the tachocline. We show that the thickness of the high-latitude tachocline scales with the amplitude of the meridional flows exactly as predicted by Gough and McIntyre. We also determine the parameter range in which such solutions can be obtained, and provide a simple explanation for the failure of previous numerical attempts at reproducing the Gough and McIntyre model. Finally, we discuss the implications of our findings for future numerical models of the solar interior, and for future observations of the Sun and other stars.

6. The numerical solution of the boundary inverse problem for a parabolic equation

Vasil'ev, V. V.; Vasilyeva, M. V.; Kardashevsky, A. M.

2016-10-01

Boundary inverse problems occupy an important place among the inverse problems of mathematical physics. They are connected with the problems of diagnosis, when additional measurements on one of the borders or inside the computational domain are necessary to restore the boundary regime in the other border, inaccessible to direct measurements. The boundary inverse problems belong to a class of conditionally correct problems, and therefore, their numerical solution requires the development of special computational algorithms. The paper deals with the solution of the boundary inverse problem for one-dimensional second-order parabolic equations, consisting in the restoration of boundary regime according to measurements inside the computational domain. For the numerical solution of the inverse problem it is proposed to use an analogue of a computational algorithm, proposed and developed to meet the challenges of identification of the right side of the parabolic equations in the works P.N.Vabishchevich and his students based on a special decomposition of solving the problem at each temporal layer. We present and discuss the results of a computational experiment conducted on model problems with quasi-solutions, including with random errors in the input data.

7. Grad-Shafranov Reconstruction: Overview and Improvement of the Numerical Solution Used in Space Physics

González, A. Ojeda; Domingues, M. O.; Mendes, O.; Kaibara, M. K.; Prestes, A.

2015-10-01

The Grad-Shafranov equation is a Poisson's equation, i.e., a partial differential equation of elliptic type. The problem is depending on the initial condition and can be treated as a Cauchy problem. Although it is ill-posed or ill-conditioned, it can be integrated numerically. In the integration of the GS equation, singularities with large values of the potential arise after a certain number of integration steps away from the original data line, and a filter should be used. The Grad-Shafranov reconstruction (GSR) technique was developed from 1996 to 2000 for recovering two-dimensional structures in the magnetopause in an ideal MHD formulation. Other works have used the GSR techniques to study magnetic flux ropes in the solar wind and in the magnetotail from a single spacecraft dataset; posteriorly, it was extended to treat measurements from multiple satellites. From Vlasov equation, it is possible to arrive at the GS-equation in function of the normalized vector potential. A general solution is obtained using complex variable theory. A specific solution was chosen as benchmark case to solve numerically the GS equation. We propose some changes in the resolution scheme of the GS equation to improve the solution. The result of each method is compared with the solution proposed by Hau and Sonnerup (J. Geophys. Res. 104(A4), 6899-6917 (1999)). The main improvement found in the GS resolution was the need to filter B x values at each y value.

8. A numerical model for the solution of the Shallow Water equations in composite channels with movable bed

minatti, L.

2013-12-01

A finite volume model solving the shallow water equations coupled with the sediments continuity equation in composite channels with irregular geometry is presented. The model is essentially 1D but can handle composite cross-sections in which bedload transport is considered to occur inside the main channel only. This assumption is coherent with the observed behavior of rivers on short time scales where main channel areas exhibit more relevant morphological variations than overbanks. Furthermore, such a model allows a more precise prediction of thalweg elevation and cross section shape variations than fully 1D models where bedload transport is considered to occur uniformly over the entire cross section. The coupling of the equations describing water and sediments dynamics results in a hyperbolic non-conservative system that cannot be solved numerically with the use of a conservative scheme. Therefore, a path-conservative scheme, based on the approach proposed by Pares and Castro (2004) has been devised in order to account for the coupling with the sediments continuity equation and for the concurrent presence of bottom elevation and breadth variations of the cross section. In order to correctly compute numerical fluxes related to bedload transport in main channel areas, a special treatment of the equations is employed in the model. The resulting scheme is well balanced and fully coupled and can accurately model abrupt time variations of flow and bedload transport conditions in wide rivers, characterized by the presence of overbank areas that are less active than the main channel. The accuracy of the model has been first tested in fixed bed conditions by solving problems with a known analytical solution: in these tests the model proved to be able to handle shocks and supercritical flow conditions properly(see Fig. 01). A practical application of the model to the Ombrone river, southern Tuscany (Italy) is shown. The river has shown relevant morphological changes during

9. Analytical and Numerical Solutions of Generalized Fokker-Planck Equations - Final Report

SciTech Connect

Prinja, Anil K.

2000-12-31

The overall goal of this project was to develop advanced theoretical and numerical techniques to quantitatively describe the spreading of a collimated beam of charged particles in space, in angle, and in energy, as a result of small deflection, small energy transfer Coulomb collisions with the target nuclei and electrons. Such beams arise in several applications of great interest in nuclear engineering, and include electron and ion radiotherapy, ion beam modification of materials, accelerator transmutation of waste, and accelerator production of tritium, to name some important candidates. These applications present unique and difficult modeling challenges, but from the outset are amenable to the language of ''transport theory'', which is very familiar to nuclear engineers and considerably less-so to physicists and material scientists. Thus, our approach has been to adopt a fundamental description based on transport equations, but the forward peakedness associated with charged particle interactions precludes a direct application of solution methods developed for neutral particle transport. Unique problem formulations and solution techniques are necessary to describe the transport and interaction of charged particles. In particular, we have developed the Generalized Fokker-Planck (GFP) approach to describe the angular and radial spreading of a collimated beam and a renormalized transport model to describe the energy-loss straggling of an initially monoenergetic distribution. Both analytic and numerical solutions have been investigated and in particular novel finite element numerical methods have been developed. In the first phase of the project, asymptotic methods were used to develop closed form solutions to the GFP equation for different orders of expansion, and was described in a previous progress report. In this final report we present a detailed description of (i) a novel energy straggling model based on a Fokker-Planck approximation but which is adapted for a

10. Numerical Solution of Multi-Dimensional Hyperbolic Conservation Laws on Unstructured Meshes

NASA Technical Reports Server (NTRS)

Barth, Timothy J.; Kwak, Dochan (Technical Monitor)

1995-01-01

The lecture material will discuss the application of one-dimensional approximate Riemann solutions and high order accurate data reconstruction as building blocks for solving multi-dimensional hyperbolic equations. This building block procedure is well-documented in the nationally available literature. The relevant stability and convergence theory using positive operator analysis will also be presented. All participants in the minisymposium will be asked to solve one or more generic test problems so that a critical comparison of accuracy can be made among differing approaches.

11. Solution of stochastic media transport problems using a numerical quadrature-based method

SciTech Connect

Pautz, S. D.; Franke, B. C.; Prinja, A. K.; Olson, A. J.

2013-07-01

We present a new conceptual framework for analyzing transport problems in random media. We decompose such problems into stratified subproblems according to the number of material pseudo-interfaces within realizations. For a given subproblem we assign pseudo-interface locations in each realization according to product quadrature rules, which allows us to deterministically generate a fixed number of realizations. Quadrature integration of the solutions of these realizations thus approximately solves each subproblem; the weighted superposition of solutions of the subproblems approximately solves the general stochastic media transport problem. We revisit some benchmark problems to determine the accuracy and efficiency of this approach in comparison to randomly generated realizations. We find that this method is very accurate and fast when the number of pseudo-interfaces in a problem is generally low, but that these advantages quickly degrade as the number of pseudo-interfaces increases. (authors)

12. Numerical solution of the Penna model of biological aging with age-modified mutation rate

Magdoń-Maksymowicz, M. S.; Maksymowicz, A. Z.

2009-06-01

In this paper we present results of numerical calculation of the Penna bit-string model of biological aging, modified for the case of a -dependent mutation rate m(a) , where a is the parent’s age. The mutation rate m(a) is the probability per bit of an extra bad mutation introduced in offspring inherited genome. We assume that m(a) increases with age a . As compared with the reference case of the standard Penna model based on a constant mutation rate m , the dynamics of the population growth shows distinct changes in age distribution of the population. Here we concentrate on mortality q(a) , a fraction of items eliminated from the population when we go from age (a) to (a+1) in simulated transition from time (t) to next time (t+1) . The experimentally observed q(a) dependence essentially follows the Gompertz exponential law for a above the minimum reproduction age. Deviation from the Gompertz law is however observed for the very old items, close to the maximal age. This effect may also result from an increase in mutation rate m with age a discussed in this paper. The numerical calculations are based on analytical solution of the Penna model, presented in a series of papers by Coe [J. B. Coe, Y. Mao, and M. E. Cates, Phys. Rev. Lett. 89, 288103 (2002)]. Results of the numerical calculations are supported by the data obtained from computer simulation based on the solution by Coe

13. Numerical study of solute transport in shallow beach aquifers subjected to waves and tides

2015-02-01

A numerical study was conducted to investigate the fate of solute in a laboratory beach in response to waves and tides. A new temporal upscaling approach labeled "net inflow" was introduced to address impacts of waves on solute transport within beaches. Numerical simulations using a computational fluid dynamic model were used as boundary conditions for the two-dimensional variably saturated flow and solute transport model MARUN. The modeling approach was validated against experimental data of solute transport due to waves and tides. Exchange fluxes across the beach face and subsurface solute transport (e.g., trajectory, movement speed, and residence time) were quantified. Simulation results revealed that waves increased the exchange fluxes, and engendered a wider exchange flux zone along the beach surface. Compared to tide-only forcing, waves superimposed on tide caused the plume to be deeper into the beach, and to migrate more seaward. The infiltration into the beach was found to be directly proportional to the general hydraulic gradient in the beach and inversely proportional to the matrix retention (or capillary) capacity. The simulations showed that a higher inland water table would attenuate wave-caused seawater infiltration, which might impact beach geochemical processes (e.g., nutrient recycle and redox condition), especially at low tide zone. The concept of biochemical residence time maps (BRTM) was introduced to account for the net effect of limiting concentration of chemicals on biochemical reactions. It was found that waves shifted the BRTMs downward and seaward in the beach, and subsequently they engendered different biochemical conditions within the beach.

14. A meshless method using radial basis functions for numerical solution of the two-dimensional KdV-Burgers equation

Zabihi, F.; Saffarian, M.

2016-07-01

The aim of this article is to obtain the numerical solution of the two-dimensional KdV-Burgers equation. We construct the solution by using a different approach, that is based on using collocation points. The solution is based on using the thin plate splines radial basis function, which builds an approximated solution with discretizing the time and the space to small steps. We use a predictor-corrector scheme to avoid solving the nonlinear system. The results of numerical experiments are compared with analytical solutions to confirm the accuracy and efficiency of the presented scheme.

15. Comparison between numerical simulation and experimental measurement of solute segregation during directional solidification

Stelian, Carmen; Duffar, Thierry; Nicoara, Irina

2003-07-01

The effect of Bridgman furnace configuration on the temperature field, melt convection and the solute distribution in the resulting crystal are experimentally and numerically analyzed for the semiconductor diluted alloy solidification. The governing equations of the heat and mass transfer are solved by using the finite element method with help of the commercial software FIDAP ®. Two different solidification experiments of Ga 1- xIn xSb ( x=0.01 and 0.04) are simulated in order to compare the numerical results for thermal, velocity and solute fields. The central objective of the work is to give the conditions for which a more uniform distribution of the solute in the crystal can be obtained. It is found that crystals obtained in conditions of a strong convective regime in the vicinity of the solid-liquid interface are more homogeneous radially and on a significant length than the crystals for which solidification occurred in a quasi-diffusive regime. The results, in terms of axial and radial segregation, are compared to experimental chemical analysis.

16. A brief description of a new numerical framework for solving conservation laws: The method of space-time conservation element and solution element

NASA Technical Reports Server (NTRS)

Chang, Sin-Chung; To, Wai-Ming

1992-01-01

A new numerical method for solving conservation laws is being developed. It differs substantially from the well established methods, i.e., finite difference, finite volume, finite element, and spectral methods, in both concept and methodology. It is much simpler than a typical high resolution method. No flux limiter or any technique related to characteristics is involved. No artificial viscosity or smoothing is introduced, and no moving mesh is used. Yet this method is capable of generating highly accurate shock tube solutions. The slight numerical overshoot and/or oscillations generated can be removed if a simple averaging formula initially used is replaced by a weighted formula. This modification has little effect on other parts of the solution. Because of its simplicity, generalization of this new method for multi-dimensional problems is straightforward.

17. Numerical solution of wave equations for the stability of the inner cometo-sheath

NASA Technical Reports Server (NTRS)

Srivastava, Krishna M.; Tsurutani, Bruce T.; Goldstein, Bruce E.

1993-01-01

Numerical solution of the MHD wave equations for stability of the cometary sheath determined by the balance between the inward Lorentz body force and the outward ion-neutral drag force is obtained by using a two-point boundary value method. The eigenvalues and the eigenfunctions are obtained numerically by treating the cometary inner sheath as a layer of finite thickness, bounded by the contact surface, i.e., the diamagnetic cavity boundary. The magnetic field structure discovered in the ionosphere of Comets Halley and Giacobini-Zinner is found to be unstable. The effects of finite plasma pressure, dissociative recombination, and mass loading due to photoionization are found to be stabilizing but are unable to quench the instability completely. It is also found that the higher the neutral production rate the lesser is the growth rate for the instability.

18. Numerical solution of non-isothermal non-adiabatic flow of real gases in pipelines

Bermúdez, Alfredo; López, Xián; Vázquez-Cendón, M. Elena

2016-10-01

A finite volume scheme for the numerical solution of a mathematical model for non-isothermal non-adiabatic compressible flow of a real gas in a pipeline is introduced. In order to make an upwind discretization of the flux, the Q-scheme of van Leer is used. Unlike standard Euler equations, the model takes into account wall friction, variable height and heat transfer between the pipe and the environment. Since all these terms are sources, in order to get a well-balanced scheme they are discretized by making a similar upwinding to the one in the flux term. The performance of the overall method has been shown for some usual numerical tests. The final goal, which is beyond the scope of this paper, is to consider a network including several pipelines connected at junctions, as those employed for natural gas transport.

19. CSR Fields: Direct Numerical Solution of the Maxwell___s Equation

SciTech Connect

Novokhatski, A.; /SLAC

2011-06-22

We discuss the properties of the coherent electromagnetic fields of a very short, ultra-relativistic bunch in a rectangular vacuum chamber inside a bending magnet. The analysis is based on the results of a direct numerical solution of Maxwell's equations together with Newton's equations. We use a new dispersion-free time-domain algorithm which employs a more efficient use of finite element mesh techniques and hence produces self-consistent and stable solutions for very short bunches. We investigate the fine structure of the CSR fields including coherent edge radiation. This approach should be useful in the study of existing and future concepts of particle accelerators and ultrafast coherent light sources. The coherent synchrotron radiation (CSR) fields have a strong action on the beam dynamics of very short bunches, which are moving in the bends of all kinds of magnetic elements. They are responsible for additional energy loss and energy spread; micro bunching and beam emittance growth. These fields may bound the efficiency of damping rings, electron-positron colliders and ultrafast coherent light sources, where high peak currents and very short bunches are envisioned. This is relevant to most high-brightness beam applications. On the other hand these fields together with transition radiation fields can be used for beam diagnostics or even as a powerful resource of THz radiation. A history of the study of CSR and a good collection of references can be found in [1]. Electromagnetic theory suggests several methods on how to calculate CSR fields. The most popular method is to use Lienard-Wiechert potentials. Other approach is to solve numerically the approximate equations, which are a Schrodinger type equation. These numerical methods are described in [2]. We suggest that a direct solution of Maxwell's equations together with Newton's equations can describe the detailed structure of the CSR fields [3].

20. Numerical simulation of fracture permeability evolution due to reactive transport and pressure solution processes

Watanabe, N.; Sun, Y.; Taron, J.; Shao, H.; Kolditz, O.

2013-12-01

Modeling fracture permeability evolution is of great interest in various geotechnical applications including underground waste repositories, carbon capture and storage, and engineered geothermal systems where fractures dominate transport behaviors. In this study, a numerical model is presented to simulate fracture permeability evolution due to reactive transport and pressure solution processes in single fractures. The model was developed within the international benchmarking project for radioactive waste disposals, DECOVALEX 2015 (Task C1). The model combines bulk behavior in pore spaces with intergranular process at asperity contacts. Hydraulic flow and reactive transport including mineral dissolution and precipitation in fracture pore space are simulated using the Galerkin finite element method. A pressure solution model developed by Taron and Elsworth (2010 JGR) is applied to simulating stress-enhanced dissolution, solute exchange with pore space, and volume removal at grain contacts. Fracture aperture and contact area ratio are updated as a result of the pore-space reaction and intergranular dissolution. In order to increase robustness and time step size, relevant processes are monolithically coupled with the simulations. The model is implemented in a scientific open-source project OpenGeoSys (www.opengeosys.org) for numerical simulation of thermo-hydro-mechanical/chemical processes in porous and fractured media. Numerical results are compared to previous experiment performed by Yasuhara et al. (2006) on flow through fractures in the Arkansas novaculite sample. The novaculite is approximated as pure quartz aggregates. Only with fitted quartz dissolution rate constants and solubility is the current model capable of reproducing observed hydraulic aperture reduction and aqueous silicate concentrations. Future work will examine reaction parameters and further validate the model against experimental results.

1. Improved treatment of asthenosphere flow and melting in 2D numerical solutions for continental rifting: embedded vs nested modeling approaches.

de Monserrat, Albert; Morgan, Jason P.; Taramón, Jorge M.; Hall, Robert

2016-04-01

This work focuses on improving current 2D numerical approaches to modeling the boundary conditions associated with computing accurate deformation and melting associated with continental rifting. Recent models primarily use far-field boundary conditions that have been used for decades with little assessment of their effects on asthenospheric flow beneath the rifting region. All are extremely oversimplified. All are likely to significantly shape the pattern of asthenospheric flow beneath the stretching lithosphere which is associated with pressure-release melting and rift volcanism. The choice of boundary conditions may lead to different predictions of asthenospheric flow and melting associated with lithospheric stretching and breakup. We also find that they may affect the mode of crustal stretching. Here we discuss a suite of numerical experiments using a Lagrangian formulation, that compare these choices to likely more realistic boundary condition choices like the analytical solution for flow associated with two diverging plates stretching over a finite-width region. We also compare embedded and nested meshes with a high-resolution 2-D region within a cartesian 'whole mantle cross-section' box. Our initial results imply that the choice of far-field boundary conditions does indeed significantly influence predicted melting distributions and melt volumes associated with continental breakup. For calculations including asthenospheric melting, the 'finite width plate spreading' and embedded rifting boundary condition treatments lead to significantly smaller BC-influenced signals when using high-resolution calculation regions of order ~1000 km wide and 600 km deep within a lower resolution box of the order of >5000 km wide and 2800 km. We recommend their use when models are attempting to resolve the effects of asthenosphere flow and melting. We also discuss several examples of typical numerical 'artifacts' related to 'edge convection' at the sides of the stretching region

2. Comptonization in Ultra-Strong Magnetic Fields: Numerical Solution to the Radiative Transfer Problem

NASA Technical Reports Server (NTRS)

Ceccobello, C.; Farinelli, R.; Titarchuk, L.

2014-01-01

We consider the radiative transfer problem in a plane-parallel slab of thermal electrons in the presence of an ultra-strong magnetic field (B approximately greater than B(sub c) approx. = 4.4 x 10(exp 13) G). Under these conditions, the magnetic field behaves like a birefringent medium for the propagating photons, and the electromagnetic radiation is split into two polarization modes, ordinary and extraordinary, that have different cross-sections. When the optical depth of the slab is large, the ordinary-mode photons are strongly Comptonized and the photon field is dominated by an isotropic component. Aims. The radiative transfer problem in strong magnetic fields presents many mathematical issues and analytical or numerical solutions can be obtained only under some given approximations. We investigate this problem both from the analytical and numerical point of view, provide a test of the previous analytical estimates, and extend these results with numerical techniques. Methods. We consider here the case of low temperature black-body photons propagating in a sub-relativistic temperature plasma, which allows us to deal with a semi-Fokker-Planck approximation of the radiative transfer equation. The problem can then be treated with the variable separation method, and we use a numerical technique to find solutions to the eigenvalue problem in the case of a singular kernel of the space operator. The singularity of the space kernel is the result of the strong angular dependence of the electron cross-section in the presence of a strong magnetic field. Results. We provide the numerical solution obtained for eigenvalues and eigenfunctions of the space operator, and the emerging Comptonization spectrum of the ordinary-mode photons for any eigenvalue of the space equation and for energies significantly lesser than the cyclotron energy, which is on the order of MeV for the intensity of the magnetic field here considered. Conclusions. We derived the specific intensity of the

3. On the efficient and reliable numerical solution of rate-and-state friction problems

Pipping, Elias; Kornhuber, Ralf; Rosenau, Matthias; Oncken, Onno

2016-03-01

We present a mathematically consistent numerical algorithm for the simulation of earthquake rupture with rate-and-state friction. Its main features are adaptive time stepping, a novel algebraic solution algorithm involving nonlinear multigrid and a fixed point iteration for the rate-and-state decoupling. The algorithm is applied to a laboratory scale subduction zone which allows us to compare our simulations with experimental results. Using physical parameters from the experiment, we find a good fit of recurrence time of slip events as well as their rupture width and peak slip. Computations in 3-D confirm efficiency and robustness of our algorithm.

4. A high-resolution numerical technique for inviscid gas-dynamic problems with weak solutions

NASA Technical Reports Server (NTRS)

Yee, H. C.; Warming, R. F.; Harten, A.

1982-01-01

The shock resolution of Harten's (1982) second-order explicit method for one-dimensional hyperbolic conservation laws is investigated for a two-dimensional gas-dynamic problem. The possible extension to a high resolution implicit method for both one- and two-dimensional problems is also investigated. Applications of Harten's method to the quasi-one-dimensional nozzle problem with two nozzle shapes (divergent and convergent-divergent) and the two-dimensional shock-reflection problem resulted in high shock resolution steady-state numerical solutions.

5. Numerical solution of continuous-time mean-variance portfolio selection with nonlinear constraints

Yan, Wei; Li, Shurong

2010-03-01

An investment problem is considered with dynamic mean-variance (M-V) portfolio criterion under discontinuous prices described by jump-diffusion processes. Some investment strategies are restricted in the study. This M-V portfolio with restrictions can lead to a stochastic optimal control model. The corresponding stochastic Hamilton-Jacobi-Bellman equation of the problem with linear and nonlinear constraints is derived. Numerical algorithms are presented for finding the optimal solution in this article. Finally, a computational experiment is to illustrate the proposed methods by comparing with M-V portfolio problem which does not have any constraints.

6. Human-computer interfaces applied to numerical solution of the Plateau problem

Elias Fabris, Antonio; Soares Bandeira, Ivana; Ramos Batista, Valério

2015-09-01

In this work we present a code in Matlab to solve the Problem of Plateau numerically, and the code will include human-computer interface. The Problem of Plateau has applications in areas of knowledge like, for instance, Computer Graphics. The solution method will be the same one of the Surface Evolver, but the difference will be a complete graphical interface with the user. This will enable us to implement other kinds of interface like ocular mouse, voice, touch, etc. To date, Evolver does not include any graphical interface, which restricts its use by the scientific community. Specially, its use is practically impossible for most of the Physically Challenged People.

7. WATSFAR: numerical simulation of soil WATer and Solute fluxes using a FAst and Robust method

Crevoisier, David; Voltz, Marc

2013-04-01

To simulate the evolution of hydro- and agro-systems, numerous spatialised models are based on a multi-local approach and improvement of simulation accuracy by data-assimilation techniques are now used in many application field. The latest acquisition techniques provide a large amount of experimental data, which increase the efficiency of parameters estimation and inverse modelling approaches. In turn simulations are often run on large temporal and spatial domains which requires a large number of model runs. Eventually, despite the regular increase in computing capacities, the development of fast and robust methods describing the evolution of saturated-unsaturated soil water and solute fluxes is still a challenge. Ross (2003, Agron J; 95:1352-1361) proposed a method, solving 1D Richards' and convection-diffusion equation, that fulfil these characteristics. The method is based on a non iterative approach which reduces the numerical divergence risks and allows the use of coarser spatial and temporal discretisations, while assuring a satisfying accuracy of the results. Crevoisier et al. (2009, Adv Wat Res; 32:936-947) proposed some technical improvements and validated this method on a wider range of agro- pedo- climatic situations. In this poster, we present the simulation code WATSFAR which generalises the Ross method to other mathematical representations of soil water retention curve (i.e. standard and modified van Genuchten model) and includes a dual permeability context (preferential fluxes) for both water and solute transfers. The situations tested are those known to be the less favourable when using standard numerical methods: fine textured and extremely dry soils, intense rainfall and solute fluxes, soils near saturation, ... The results of WATSFAR have been compared with the standard finite element model Hydrus. The analysis of these comparisons highlights two main advantages for WATSFAR, i) robustness: even on fine textured soil or high water and solute

8. Direct numerical solution of the transonic perturbation integral equation for lifting and nonlifting airfoils

NASA Technical Reports Server (NTRS)

Nixon, D.

1978-01-01

The linear transonic perturbation integral equation previously derived for nonlifting airfoils is formulated for lifting cases. In order to treat shock wave motions, a strained coordinate system is used in which the shock location is invariant. The tangency boundary conditions are either formulated using the thin airfoil approximation or by using the analytic continuation concept. A direct numerical solution to this equation is derived in contrast to the iterative scheme initially used, and results of both lifting and nonlifting examples indicate that the method is satisfactory.

9. Numerical solution of nonlinear partial differential equations of mixed type. [finite difference approximation

NASA Technical Reports Server (NTRS)

Jameson, A.

1976-01-01

A review is presented of some recently developed numerical methods for the solution of nonlinear equations of mixed type. The methods considered use finite difference approximations to the differential equation. Central difference formulas are employed in the subsonic zone and upwind difference formulas are used in the supersonic zone. The relaxation method for the small disturbance equation is discussed and a description is given of difference schemes for the potential flow equation in quasi-linear form. Attention is also given to difference schemes for the potential flow equation in conservation form, the analysis of relaxation schemes by the time dependent analogy, the accelerated iterative method, and three-dimensional calculations.

10. A numerical solution for thermal tides in the atmosphere of Venus

SciTech Connect

Shen, M.; Zhang, C.Z. )

1990-05-01

The problems which emerge in choosing the basic equations and selecting the atmospheric heating function of the atmosphere during studies of Venus' atmospheric tides are presently approached via the numerical solution of the linear form of the basic equations. An analysis of the results thus obtained indicates that observations of the thermal parameters near the ground are important to the determination of the ground-heating mode of this planet's atmosphere. Pioneer Venus probe data are presently used to construct three heating models; the torque of the atmospheric tides is estimated to be 1.4 x 10 to the 16th J. 37 refs.

11. Numerical and Series Solutions for Stagnation-Point Flow of Nanofluid over an Exponentially Stretching Sheet

PubMed Central

Mustafa, Meraj; Farooq, Muhammad A.; Hayat, Tasawar; Alsaedi, Ahmed

2013-01-01

This investigation is concerned with the stagnation-point flow of nanofluid past an exponentially stretching sheet. The presence of Brownian motion and thermophoretic effects yields a coupled nonlinear boundary-value problem (BVP). Similarity transformations are invoked to reduce the partial differential equations into ordinary ones. Local similarity solutions are obtained by homotopy analysis method (HAM), which enables us to investigate the effects of parameters at a fixed location above the sheet. The numerical solutions are also derived using the built-in solver bvp4c of the software MATLAB. The results indicate that temperature and the thermal boundary layer thickness appreciably increase when the Brownian motion and thermophoresis effects are strengthened. Moreover the nanoparticles volume fraction is found to increase when the thermophoretic effect intensifies. PMID:23671576

12. Numerical and series solutions for stagnation-point flow of nanofluid over an exponentially stretching sheet.

PubMed

Mustafa, Meraj; Farooq, Muhammad A; Hayat, Tasawar; Alsaedi, Ahmed

2013-01-01

This investigation is concerned with the stagnation-point flow of nanofluid past an exponentially stretching sheet. The presence of Brownian motion and thermophoretic effects yields a coupled nonlinear boundary-value problem (BVP). Similarity transformations are invoked to reduce the partial differential equations into ordinary ones. Local similarity solutions are obtained by homotopy analysis method (HAM), which enables us to investigate the effects of parameters at a fixed location above the sheet. The numerical solutions are also derived using the built-in solver bvp4c of the software MATLAB. The results indicate that temperature and the thermal boundary layer thickness appreciably increase when the Brownian motion and thermophoresis effects are strengthened. Moreover the nanoparticles volume fraction is found to increase when the thermophoretic effect intensifies.

13. Numerical solutions for a flow with mixed convection in a vertical geometry

Torczynski, J. R.

The K-12 Aerospace Heat Transfer Committee of the American Society of Mechanical Engineers recently specified a computational benchmark problem involving steady incompressible laminar flow with mixed convection using the Boussinesq approximation in a two-dimensional backstep geometry. FIDAP v6.0 (Fluid Dynamics International) and NEKTON v2.85 (Nektonics, Fluent) are capable of simulating situations with this type of coupled fluid flow and heat transfer. FIDAP uses conventional finite elements and has both steady and transient solvers, whereas NEKTON uses spectral elements with a transient solver (for large problems). Numerical solutions to the benchmark problem are obtained with both of these codes, and grid-refinement studies are performed to verify that grid-independence is achieved. The grid-independent solutions from both codes are found to be in excellent agreement with each other and with results in the archival literature regarding velocity and temperature profiles and the locations of separation and reattachment points.

14. Numerical solution of the problem of optimizing the process of oil displacement by steam

Temirbekov, N. M.; Baigereyev, D. R.

2016-06-01

The paper is devoted to the problem of optimizing the process of steam stimulation on the oil reservoir by controlling the steam pressure on the injection well to achieve preassigned temperature distribution along the reservoir at a given time of development. The relevance of the study of this problem is related to the need to improve methods of heavy oil development, the proportion of which exceeds the reserves of light oils, and it tends to grow. As a mathematical model of oil displacement by steam, three-phase non-isothermal flow equations is considered. The problem of optimal control is formulated, an algorithm for the numerical solution is proposed. As a reference regime, temperature distribution corresponding to the constant pressure of injected steam is accepted. The solution of the optimization problem shows that choosing the steam pressure on the injection well, one can improve the efficiency of steam-stimulation and reduce the pressure of the injected steam.

15. Numerical solutions of three-dimensional MHD flows in strong non-uniform transverse magnetic fields

SciTech Connect

Hua, T.Q.; Walker, J.S.

1988-07-01

Magnetohydrodynamic flows of liquid metals in thin conducting ducts of various geometries in the presence of strong nonuniform transverse magnetic fields are examined. The interaction parameter and Hartmann number are assumed to be large, whereas the magnetic Reynolds number is assumed to be small. Under these assumptions, viscous and inertial effects are confined in very thin boundary layers adjacent to the walls. At walls parallel to the magnetic field lines, as at the side walls of a rectangular duct, the boundary layers (side layers) carry a significant fraction of the volumetric flow rate in the form of high velocity jets. This paper describes the analysis and summarizes the numerical methods for obtaining 3-D solutions (core solutions) for flow parameters outside these layers, without solving explicitly for the layers themselves. 13 refs., 1 fig.

16. A Numerical Algorithm for the Solution of a Phase-Field Model of Polycrystalline Materials

SciTech Connect

Dorr, M R; Fattebert, J; Wickett, M E; Belak, J F; Turchi, P A

2008-12-04

We describe an algorithm for the numerical solution of a phase-field model (PFM) of microstructure evolution in polycrystalline materials. The PFM system of equations includes a local order parameter, a quaternion representation of local orientation and a species composition parameter. The algorithm is based on the implicit integration of a semidiscretization of the PFM system using a backward difference formula (BDF) temporal discretization combined with a Newton-Krylov algorithm to solve the nonlinear system at each time step. The BDF algorithm is combined with a coordinate projection method to maintain quaternion unit length, which is related to an important solution invariant. A key element of the Newton-Krylov algorithm is the selection of a preconditioner to accelerate the convergence of the Generalized Minimum Residual algorithm used to solve the Jacobian linear system in each Newton step. Results are presented for the application of the algorithm to 2D and 3D examples.

17. Numerical solution of the problem of flame propagation by the use of the random element method

NASA Technical Reports Server (NTRS)

Ghoniem, A. F.; Oppenheim, A. K.

1983-01-01

A numerical, grid-free algorithm is presented for one-dimensional reaction-diffusion model of laminar flame propagation in premixed gases. It is based on the random element method we developed for the analysis of diffusional processes. The effect of combustion is taken into account by applying the principle of fractional steps to separate the process of diffusion, modeled by the random walk of computational elements, from the exothermic effects of chemical reaction, monitoring their strength. The validity of the algorithm is demonstrated by application to flame propagation problems for which exact solutions exist. The flame speed evaluated by its use oscillates around the exact value at a relatively small amplitude, while the temperature and species concentration profiles are self-correcting in their convergence to the exact solution. A satisfactory resolution is obtained by the use of quite a small number of computational elements which automatically adjust their distribution of fit sharp gradients.

18. Solution of the main problem of the lunar physical libration by a numerical method

Zagidullin, Arthur; Petrova, Natalia; Nefediev, Yurii

2016-10-01

Series of the lunar programs requires highly accurate ephemeris of the Moon at any given time. In the light of the new requirements on the accuracy the requirements to the lunar physical libration theory increase.At the Kazan University there is the experience of constructing the lunar rotation theory in the analytical approach. Analytical theory is very informative in terms of the interpretation of the observed data, but inferior to the accuracy of numerical theories. The most accurate numerical ephemeris of the Moon is by far the ephemeris DE430 / 431 built in the USA. It takes into account a large number of subtle effects both in external perturbations of the Moon, and in its internal structure. Before the Russian scientists the task is to create its own numerical theory that would be consistent with the American ephemeris. On the other hand, even the practical application of the american ephemeris requires a deep understanding of the principles of their construction and the intelligent application.As the first step, we constructed a theory in the framework of the main problem. Because we compare our theory with the analytical theory of Petrova (1996), all the constants and the theory of orbital motion are taken identical to the analytical theory. The maximum precision, which the model can provide is 0.01 seconds of arc, which is insufficient to meet the accuracy of modern observations, but this model provides the necessary basis for further development.We have constructed the system of the libration equations, for which the numerical integrator was developed. The internal accuracy of the software integrator is several nanoseconds. When compared with the data of Petrova the differences of order of 1 second are observed at the resonant frequencies. The reason, we believe, in the inaccuracy of the analytical theory. We carried out a comparison with the Eroshkin's data [2], which gave satisfactory agreement, and with Rambaux data. In the latter case, as expected

19. On the numerical solution of the cylindrical Poisson equation for isolated self-gravitating systems

Cohl, Howard Saul

This dissertation addresses the need for an accurate and efficient technique which solves the Poisson equation for arbitrarily complex, isolated, self-gravitating fluid systems. Generally speaking, a potential solver is composed of two distinct pieces: a boundary solver and an interior solver. The boundary solver computes the potential, Φ(xB) on a surface which bounds some finite volume of space, V, and contains an isolated mass-density distribution, ρ(x). Given ρ(x) and Φ(xB), the interior solver computes the potential Φ(x) everywhere within V. Herein, we describe the development of a numerical technique which efficiently solves Poisson's equation in cylindrical coordinates on massively parallel computing architectures. First, we report the discovery of a compact cylindrical Green's function (CCGF) expansion and show how the CCGF can be used to efficiently compute the exact numerical representation of Φ(xB). As an analytical representation, the CCGF should prove to be extremely useful wherever one requires the isolated azimuthal modes of a self-gravitating system. We then discuss some mathematical consequences of the CCGF expansion, such as it's applicability to all nine axisymmetric coordinate systems which are R -separable for Laplace's equation. The CCGF expansion, as applied to the spherical coordinate system, leads to a second addition theorem for spherical harmonics. Finally, we present a massively parallel implementation of an interior solver which is based on a data-transpose technique applied to a Fourier-ADI (Alternating Direction Implicit) scheme. The data-transpose technique is a parallelization strategy in which all communication is restricted to global 3D data-transposition operations and all computations are subsequently performed with perfect load balance and zero communication. The potential solver, as implemented here in conjunction with the CCGF expansion, should prove to be an extremely useful tool in a wide variety of astrophysical

20. Symbolic-numeric efficient solution of optimal control problems for multibody systems

Bertolazzi, Enrico; Biral, Francesco; da Lio, Mauro

2006-01-01

This paper presents an efficient symbolic-numerical approach for generating and solving the boundary value problem-differential algebraic equation (BVP-DAE) originating from the variational form of the optimal control problem (OCP). This paper presents the method for the symbolic derivation, by means of symbolic manipulation software (Maple), of the equations of the OCP applied to a generic multibody system. The constrained problem is transformed into a nonconstrained problem, by means of the Lagrange multipliers and penalty functions. From the first variation of the nonconstrained problem a BVP-DAE is obtained, and the finite difference discretization yields a nonlinear systems. For the numerical solution of the nonlinear system a damped Newton scheme is used. The sparse and structured Jacobians is quickly inverted by exploiting the sparsity pattern in the solution strategy. The proposed method is implemented in an object oriented fashion, and coded in C++ language. Efficiency is ensured in core routines by using Lapack and Blas for linear algebra.

1. Numerical solution of an elastic and viscoelastic gravitational models by the finite element method

Arjona Almodóvar, A.; Chacón Rebollo, T.; Gómez Marmol, M.

2014-12-01

Volcanic areas present a lower effective viscosity than usually in the Earth's crust. Both the elastic-gravitational and the viscoelastic-gravitational models allow the computation of gravity, deformation, and gravitational potential changes in order to investigate crustal deformations of Earth (see for instance Battaglia & Segall, 2004; Fernández et al. 1999, 2001; Rundle 1980 and 1983). These models can be represented by a coupled system of linear parabolic (for the elastic deformations), hyperbolic (for the viscoelastic deformations) and elliptic partial differential equations (for gravitational potential changes) (see for instance Arjona et al. 2008 and 2010). The existence and uniqueness of weak solutions for both the elastic-gravitational and viscoelastic-gravitational problem was demonstrated in Arjona et al. (2008 and 2014). The stabilization to solutions of the associated stationary system was proved in Arjona and Díaz (2007). Here we consider the internal source as response to the effect of a pressurized magma reservoir into a multilayered, elastic-gravitational and viscoelastic-gravitational earth model. We introduce the numerical analysis of a simplified steady elastic-gravitational model, solved by means of the finite element method. We also present some numerical tests in realistic situations that confirm the predictions of theoretical order of convergence. Finally, we describe the methodology for both the elastic-gravitational and the viscoelastic-gravitational models using 2D and 3D test examples performed with FreeFEM++.

2. Numerical analysis of water and solute transport in variably-saturated fractured clayey till.

PubMed

Rosenbom, Annette E; Therrien, Rene; Refsgaard, Jens Christian; Jensen, Karsten H; Ernstsen, Vibeke; Klint, Knud Erik S

2009-02-16

This study numerically investigates the influence of initial water content and rain intensities on the preferential migration of two fluorescent tracers, Acid Yellow 7 (AY7) and Sulforhodamine B (SB), through variably-saturated fractured clayey till. The simulations are based on the numerical model HydroGeoSphere, which solves 3D variably-saturated flow and solute transport in discretely-fractured porous media. Using detailed knowledge of the matrix, fracture, and biopore properties, the numerical model is calibrated and validated against experimental high-resolution tracer images/data collected under dry and wet soil conditions and for three different rain events. The model could reproduce reasonably well the observed preferential migration of AY7 and SB through the fractured till, although it did not capture the exact depth of migration and the negligible impact of the dead-end biopores in a near-saturated matrix. A sensitivity analysis suggests fast flow mechanisms and dynamic surface coating in the biopores, and the presence of a plough pan in the till.

3. Numerical solution of transonic wing flows using an Euler/Navier-Stokes zonal approach

NASA Technical Reports Server (NTRS)

Holst, T. L.; Gundy, K. L.; Thomas, S. D.; Chaderjian, N. M.; Flores, J.

1985-01-01

Transonic flow fields about wing geometries are computed using an Euler/Navier-Stokes approach in which the flow field is divided into several zones. The grid zones immediately adjacent to the wing surface are suitably clustered and solved with the Navier-Stokes equations. Grid zones removed from the wing are less finely clustered and are solved with the Euler equations. Wind tunnel wall effects are easily and accurately modeled with the new grid-zoning algorithm because the wind tunnel grid is constructed as an exact subset of the corresponding free-air grid. Solutions are obtained that are in good agreement with experiment, including cases with significant wind tunnel wall effects and shock-induced separation on the upper wing surface.

4. A compositional multiphase model for groundwater contamination by petroleum products. 2. Numerical solution

USGS Publications Warehouse

Baehr, A.L.; Corapcioglu, M.Y.

1987-01-01

In this paper we develop a numerical solution to equations developed in part 1 (M.Y. Corapcioglu and A.L. Baehr, this issue) to predict the fate of an immiscible organic contaminant such as gasoline in the unsaturated zone subsequent to plume establishment. This solution, obtained by using a finite difference scheme and a method of forward projection to evaluate nonlinear coefficients, provides estimates of the flux of solubilized hydrocarbon constituents to groundwater from the portion of a spill which remains trapped in a soil after routine remedial efforts to recover the product have ceased. The procedure was used to solve the one-dimensional (vertical) form of the system of nonlinear partial differential equations defining the transport for each constituent of the product. Additionally, a homogeneous, isothermal soil with constant water content was assumed. An equilibrium assumption partitions the constituents between air, water, adsorbed, and immiscible phases. Free oxygen transport in the soil was also simulated to provide an upper bound estimate of aerobic biodegradation rates. Results are presented for a hypothetical gasoline consisting of eight groups of hydrocarbon constituents. Rates at which hydrocarbon mass is removed from the soil, entering either the atmosphere or groundwater, or is biodegraded are presented. A significant sensitivity to model parameters, particularly the parameters characterizing diffusive vapor transport, was discovered. We conclude that hypocarbon solute composition in groundwater beneath a gasoline contaminated soil would be heavily weighted toward aromatic constituents like benzene, toluene, and xylene.In this paper we develop a numerical solution to equations developed in part 1 (M. Y. Corapcioglu and A. L. Baehr, this issue) to predict the fate of an immiscible organic contaminant such as gasoline in the unsaturated zone subsequent to plume establishment. This solution, obtained by using a finite difference scheme and a method

5. Numerical solutions of Navier-Stokes equations for the structure of a trailing vortex

NASA Technical Reports Server (NTRS)

Jain, A. C.

1977-01-01

The structure and decay of a trailing vortex were analyzed during the numerical solutions of the full Navier-Stokes equations. Unsteady forms of the governing equations were recast in terms of circulation, vorticity, and stream function as dependent variables, and a second upwind finite difference scheme was used to integrate them with prescribed initial and boundary conditions. The boundary conditions at the outer edge and at the outflow section of the trailing vortex were considered. Different models of the flow were postulated, and solutions were obtained describing the development of the flow as integration proceeds in time. A parametric study was undertaken with a view to understanding the various phenomena that may possibly occur in the trailing vortex. Using the Hoffman and Joubert law of circulation at the inflow section, the results of this investigation were compared with experimental data for a Convair 990 wind model and a rectangular wing. With an exponentially decaying law of circulation at the inflow section and an adverse pressure gradient at the outer edge of the trailing vortex, solutions depict vortex bursting through the sudden expansion of the core and/or through the stagnation and consequent reversal of the flow on the axis. It was found that this bursting takes place at lower values of the swirl ratio as the Reynolds number increases.

6. Electron transport and energy degradation in the ionosphere: Evaluation of the numerical solution, comparison with laboratory experiments and auroral observations

NASA Technical Reports Server (NTRS)

Lummerzheim, D.; Lilensten, J.

1994-01-01

Auroral electron transport calculations are a critical part of auroral models. We evaluate a numerical solution to the transport and energy degradation problem. The numerical solution is verified by reproducing simplified problems to which analytic solutions exist, internal self-consistency tests, comparison with laboratory experiments of electron beams penetrating a collision chamber, and by comparison with auroral observations, particularly the emission ratio of the N2 second positive to N2(+) first negative emissions. Our numerical solutions agree with range measurements in collision chambers. The calculated N(2)2P to N2(+)1N emission ratio is independent of the spectral characteristics of the incident electrons, and agrees with the value observed in aurora. Using different sets of energy loss cross sections and different functions to describe the energy distribution of secondary electrons that emerge from ionization collisions, we discuss the uncertainties of the solutions to the electron transport equation resulting from the uncertainties of these input parameters.

7. SOLA-DM: A numerical solution algorithm for transient three-dimensional flows

SciTech Connect

Wilson, T.L.; Nichols, B.D.; Hirt, C.W.; Stein, L.R.

1988-02-01

SOLA-DM is a three-dimensional time-explicit, finite-difference, Eulerian, fluid-dynamics computer code for solving the time-dependent incompressible Navier-Stokes equations. The solution algorithm (SOLA) evolved from the marker-and-cell (MAC) method, and the code is highly vectorized for efficient performance on a Cray computer. The computational domain is discretized by a mesh of parallelepiped cells in either cartesian or cylindrical geometry. The primary hydrodynamic variables for approximating the solution of the momentum equations are cell-face-centered velocity components and cell-centered pressures. Spatial accuracy is selected by the user to be first or second order; the time differencing is first-order accurate. The incompressibility condition results in an elliptic equation for pressure that is solved by a conjugate gradient method. Boundary conditions of five general types may be chosen: free-slip, no-slip, continuative, periodic, and specified pressure. In addition, internal mesh specifications to model obstacles and walls are provided. SOLA-DM also solves the equations for discrete particle dynamics, permitting the transport of marker particles or other solid particles through the fluid to be modeled. 7 refs., 7 figs.

8. A novel stress-accurate FE technology for highly non-linear analysis with incompressibility constraint. Application to the numerical simulation of the FSW process

Chiumenti, M.; Cervera, M.; Agelet de Saracibar, C.; Dialami, N.

2013-05-01

In this work a novel finite element technology based on a three-field mixed formulation is presented. The Variational Multi Scale (VMS) method is used to circumvent the LBB stability condition allowing the use of linear piece-wise interpolations for displacement, stress and pressure fields, respectively. The result is an enhanced stress field approximation which enables for stress-accurate results in nonlinear computational mechanics. The use of an independent nodal variable for the pressure field allows for an adhoc treatment of the incompressibility constraint. This is a mandatory requirement due to the isochoric nature of the plastic strain in metal forming processes. The highly non-linear stress field typically encountered in the Friction Stir Welding (FSW) process is used as an example to show the performance of this new FE technology. The numerical simulation of the FSW process is tackled by means of an Arbitrary-Lagrangian-Eulerian (ALE) formulation. The computational domain is split into three different zones: the work.piece (defined by a rigid visco-plastic behaviour in the Eulerian framework), the pin (within the Lagrangian framework) and finally the stirzone (ALE formulation). A fully coupled thermo-mechanical analysis is introduced showing the heat fluxes generated by the plastic dissipation in the stir-zone (Sheppard rigid-viscoplastic constitutive model) as well as the frictional dissipation at the contact interface (Norton frictional contact model). Finally, tracers have been implemented to show the material flow around the pin allowing a better understanding of the welding mechanism. Numerical results are compared with experimental evidence.

9. Analytical and Numerical Solutions of a Generalized Hyperbolic Non-Newtonian Fluid Flow

Pakdemirli, Mehmet; Sarı, Pınar; Solmaz, Bekir

2010-03-01

The generalized hyperbolic non-Newtonian fluid model first proposed by Al-Zahrani [J. Petroleum Sci. Eng. 17, 211 (1997)] is considered. This model was successfully applied to some drilling fluids with a better performance in relating shear stress and velocity gradient compared to power-law and the Hershel-Bulkley model. Special flow geometries namely pipe flow, parallel plate flow, and flow between two rotating cylinders are treated. For the first two cases, analytical solutions of velocity profiles and discharges in the form of integrals are presented. These quantities are calculated by numerically evaluating the integrals. For the flow between two rotating cylinders, the differential equation is solved by the Runge-Kutta method combined with shooting. For all problems, the power-law approximation of the model is compared with the generalized hyperbolic model, too.

10. Numerical solution of the Navier-Stokes equations for blunt nosed bodies in supersonic flows

NASA Technical Reports Server (NTRS)

Warsi, Z. U. A.; Devarayalu, K.; Thompson, J. F.

1978-01-01

A time dependent, two dimensional Navier-Stokes code employing the method of body fitted coordinate technique was developed for supersonic flows past blunt bodies of arbitrary shapes. The bow shock ahead of the body is obtained as part of the solution, viz., by shock capturing. A first attempt at mesh refinement in the shock region was made by using the forcing function in the coordinate generating equations as a linear function of the density gradients. The technique displaces a few lines from the neighboring region into the shock region. Numerical calculations for Mach numbers 2 and 4.6 and Reynolds numbers from 320 to 10,000 were performed for a circular cylinder with and without a fairing. Results of Mach number 4.6 and Reynolds number 10,000 for an isothermal wall temperature of 556 K are presented in detail.

11. Parallel solution of high-order numerical schemes for solving incompressible flows

NASA Technical Reports Server (NTRS)

Milner, Edward J.; Lin, Avi; Liou, May-Fun; Blech, Richard A.

1993-01-01

A new parallel numerical scheme for solving incompressible steady-state flows is presented. The algorithm uses a finite-difference approach to solving the Navier-Stokes equations. The algorithms are scalable and expandable. They may be used with only two processors or with as many processors as are available. The code is general and expandable. Any size grid may be used. Four processors of the NASA LeRC Hypercluster were used to solve for steady-state flow in a driven square cavity. The Hypercluster was configured in a distributed-memory, hypercube-like architecture. By using a 50-by-50 finite-difference solution grid, an efficiency of 74 percent (a speedup of 2.96) was obtained.

12. Numerical solutions of the Navier-Stokes equations for transonic afterbody flows

NASA Technical Reports Server (NTRS)

Swanson, R. C., Jr.

1980-01-01

The time dependent Navier-Stokes equations in mass averaged variables are solved for transonic flow over axisymmetric boattail plume simulator configurations. Numerical solution of these equations is accomplished with the unsplit explict finite difference algorithm of MacCormack. A grid subcycling procedure and computer code vectorization are used to improve computational efficiency. The two layer algebraic turbulence models of Cebeci-Smith and Baldwin-Lomax are employed for investigating turbulence closure. Two relaxation models based on these baseline models are also considered. Results in the form of surface pressure distribution for three different circular arc boattails at two free stream Mach numbers are compared with experimental data. The pressures in the recirculating flow region for all separated cases are poorly predicted with the baseline turbulence models. Significant improvements in the predictions are usually obtained by using the relaxation models.

13. Incorporation of a boundary condition to numerical solution of POISSON's equation

SciTech Connect

Caspi, S.; Helm, M.; Laslett, L.J.

1988-10-01

Two-dimensional and axially-symmetric problems in electrostatics, magnetostatics or potential fluid flow frequently are solved numerically by means of relaxation techniques -- employing, for example, the finite-difference program POISSON. In many such problems, the ''sources'' (charges or currents, vorticity, and regions of permeable material) lie exclusively within a finite closed boundary curve and the relaxation process, in principle, then can be confined to the region interior to such a boundary -- provided that a suitable boundary condition is imposed on the solution at the boundary. This paper is a review and illustration of a computational method that uses a boundary condition of such a nature as to avoid the inaccuracies and more extensive meshes present when, alternatively, a simple Dirichlet or Neumann boundary condition is specified on a somewhat more remote outer boundary. 2 refs., 5 figs., 1 tab.

14. Numerical investigations of solute transport in bimodal porous media under dynamic boundary conditions

Cremer, Clemens; Neuweiler, Insa; Bechtold, Michel; Vanderborght, Jan

2016-04-01

Quantification of flow and solute transport in the shallow subsurface adjacent to the atmosphere is decisive to prevent groundwater pollution and conserve groundwater quality, to develop successful remediation strategies and to understand nutrient cycling. In nature, due to erratic precipitation-evaporation patterns, soil moisture content and related hydraulic conductivity in the vadose zone are not only variable in space but also in time. Flow directions and flow paths locally change between precipitation and evaporation periods. This makes the identification and description of solute transport processes in the vadose zone a complex problem. Recent studies (Lehmann and Or, 2009; Bechtold et al., 2011a) focused on the investigation of upward transport of solutes during evaporation in heterogeneous soil columns, where heterogeneity was introduced by a sharp vertical material interface between two types of sand. Lateral solute transport through the interface in both (lateral) directions was observed at different depths of the investigated soil columns. Following recent approaches, we conduct two-dimensional numerical simulations in a similar setup which is composed of two sands with a sharp vertical material interface. The investigation is broadened from the sole evaporation to combined precipitation-evaporation cycles in order to quantify transport processes resulting from the combined effects of heterogeneous soil structure and dynamic flow conditions. Simulations are performed with a coupled finite volume and random walk particle tracking algorithm (Ippisch et al., 2006; Bechtold et al., 2011b). By comparing scenarios with cyclic boundary conditions and stationary counterparts with the same net flow rate, we found that duration and intensity of precipitation and evaporation periods potentially have an influence on lateral redistribution of solutes and thus leaching rates. Whether or not dynamic boundary conditions lead to significant deviations in the transport

15. Numerical solutions and error estimations for the space fractional diffusion equation with variable coefficients via Fibonacci collocation method.

PubMed

Bahşı, Ayşe Kurt; Yalçınbaş, Salih

2016-01-01

In this study, the Fibonacci collocation method based on the Fibonacci polynomials are presented to solve for the fractional diffusion equations with variable coefficients. The fractional derivatives are described in the Caputo sense. This method is derived by expanding the approximate solution with Fibonacci polynomials. Using this method of the fractional derivative this equation can be reduced to a set of linear algebraic equations. Also, an error estimation algorithm which is based on the residual functions is presented for this method. The approximate solutions are improved by using this error estimation algorithm. If the exact solution of the problem is not known, the absolute error function of the problems can be approximately computed by using the Fibonacci polynomial solution. By using this error estimation function, we can find improved solutions which are more efficient than direct numerical solutions. Numerical examples, figures, tables are comparisons have been presented to show efficiency and usable of proposed method. PMID:27610294

16. An Evaluation of Solution Algorithms and Numerical Approximation Methods for Modeling an Ion Exchange Process

PubMed Central

Bu, Sunyoung; Huang, Jingfang; Boyer, Treavor H.; Miller, Cass T.

2010-01-01

The focus of this work is on the modeling of an ion exchange process that occurs in drinking water treatment applications. The model formulation consists of a two-scale model in which a set of microscale diffusion equations representing ion exchange resin particles that vary in size and age are coupled through a boundary condition with a macroscopic ordinary differential equation (ODE), which represents the concentration of a species in a well-mixed reactor. We introduce a new age-averaged model (AAM) that averages all ion exchange particle ages for a given size particle to avoid the expensive Monte-Carlo simulation associated with previous modeling applications. We discuss two different numerical schemes to approximate both the original Monte Carlo algorithm and the new AAM for this two-scale problem. The first scheme is based on the finite element formulation in space coupled with an existing backward-difference-formula-based ODE solver in time. The second scheme uses an integral equation based Krylov deferred correction (KDC) method and a fast elliptic solver (FES) for the resulting elliptic equations. Numerical results are presented to validate the new AAM algorithm, which is also shown to be more computationally efficient than the original Monte Carlo algorithm. We also demonstrate that the higher order KDC scheme is more efficient than the traditional finite element solution approach and this advantage becomes increasingly important as the desired accuracy of the solution increases. We also discuss issues of smoothness, which affect the efficiency of the KDC-FES approach, and outline additional algorithmic changes that would further improve the efficiency of these developing methods for a wide range of applications. PMID:20577570

17. An evaluation of solution algorithms and numerical approximation methods for modeling an ion exchange process

Bu, Sunyoung; Huang, Jingfang; Boyer, Treavor H.; Miller, Cass T.

2010-07-01

The focus of this work is on the modeling of an ion exchange process that occurs in drinking water treatment applications. The model formulation consists of a two-scale model in which a set of microscale diffusion equations representing ion exchange resin particles that vary in size and age are coupled through a boundary condition with a macroscopic ordinary differential equation (ODE), which represents the concentration of a species in a well-mixed reactor. We introduce a new age-averaged model (AAM) that averages all ion exchange particle ages for a given size particle to avoid the expensive Monte-Carlo simulation associated with previous modeling applications. We discuss two different numerical schemes to approximate both the original Monte-Carlo algorithm and the new AAM for this two-scale problem. The first scheme is based on the finite element formulation in space coupled with an existing backward difference formula-based ODE solver in time. The second scheme uses an integral equation based Krylov deferred correction (KDC) method and a fast elliptic solver (FES) for the resulting elliptic equations. Numerical results are presented to validate the new AAM algorithm, which is also shown to be more computationally efficient than the original Monte-Carlo algorithm. We also demonstrate that the higher order KDC scheme is more efficient than the traditional finite element solution approach and this advantage becomes increasingly important as the desired accuracy of the solution increases. We also discuss issues of smoothness, which affect the efficiency of the KDC-FES approach, and outline additional algorithmic changes that would further improve the efficiency of these developing methods for a wide range of applications.

18. An Evaluation of Solution Algorithms and Numerical Approximation Methods for Modeling an Ion Exchange Process.

PubMed

Bu, Sunyoung; Huang, Jingfang; Boyer, Treavor H; Miller, Cass T

2010-07-01

The focus of this work is on the modeling of an ion exchange process that occurs in drinking water treatment applications. The model formulation consists of a two-scale model in which a set of microscale diffusion equations representing ion exchange resin particles that vary in size and age are coupled through a boundary condition with a macroscopic ordinary differential equation (ODE), which represents the concentration of a species in a well-mixed reactor. We introduce a new age-averaged model (AAM) that averages all ion exchange particle ages for a given size particle to avoid the expensive Monte-Carlo simulation associated with previous modeling applications. We discuss two different numerical schemes to approximate both the original Monte Carlo algorithm and the new AAM for this two-scale problem. The first scheme is based on the finite element formulation in space coupled with an existing backward-difference-formula-based ODE solver in time. The second scheme uses an integral equation based Krylov deferred correction (KDC) method and a fast elliptic solver (FES) for the resulting elliptic equations. Numerical results are presented to validate the new AAM algorithm, which is also shown to be more computationally efficient than the original Monte Carlo algorithm. We also demonstrate that the higher order KDC scheme is more efficient than the traditional finite element solution approach and this advantage becomes increasingly important as the desired accuracy of the solution increases. We also discuss issues of smoothness, which affect the efficiency of the KDC-FES approach, and outline additional algorithmic changes that would further improve the efficiency of these developing methods for a wide range of applications.

19. Applications of MMPBSA to Membrane Proteins I: Efficient Numerical Solutions of Periodic Poisson-Boltzmann Equation

PubMed Central

Botello-Smith, Wesley M.; Luo, Ray

2016-01-01

Continuum solvent models have been widely used in biomolecular modeling applications. Recently much attention has been given to inclusion of implicit membrane into existing continuum Poisson-Boltzmann solvent models to extend their applications to membrane systems. Inclusion of an implicit membrane complicates numerical solutions of the underlining Poisson-Boltzmann equation due to the dielectric inhomogeneity on the boundary surfaces of a computation grid. This can be alleviated by the use of the periodic boundary condition, a common practice in electrostatic computations in particle simulations. The conjugate gradient and successive over-relaxation methods are relatively straightforward to be adapted to periodic calculations, but their convergence rates are quite low, limiting their applications to free energy simulations that require a large number of conformations to be processed. To accelerate convergence, the Incomplete Cholesky preconditioning and the geometric multi-grid methods have been extended to incorporate periodicity for biomolecular applications. Impressive convergence behaviors were found as in the previous applications of these numerical methods to tested biomolecules and MMPBSA calculations. PMID:26389966

20. A meshfree formulation for the numerical solution of the viscous, compressible Navier-Stokes equations

Gunther, Frank Christian

A meshfree numerical solution procedure consisting of a streamline-upwind Petrov-Galerkin formulation with shock capturing term is presented for the viscous, compressible Navier-Stokes equations in terms of conservation variables. Meshfree methods show similarities to finite elements but result in more general shape functions. Some concepts of multiresolution analysis and multiple scale analysis are formulated in the context of meshfree methods. Special emphasis is put on orthogonality properties against a set of basis functions. A technique of determining and eliminating hidden zero energy modes in wavelet RKPM and similar methods is developed from the reproducing conditions. The effectiveness of SUPG for meshfree formulations is ascertained by numerical experiments. With d'Alembert's principle, a method of imposing general boundary and interface conditions for meshfree methods is introduced. Essential boundary conditions are enforced by orthogonalizing against general constraints. Example computations for viscous, supersonic flows illustrate the viability of the method. The meshfree results compare well to those obtained analytically for changes in flow properties across shock fronts.

1. Numerical solution of a conspicuous consumption model with constant control delay☆

PubMed Central

Huschto, Tony; Feichtinger, Gustav; Hartl, Richard F.; Kort, Peter M.; Sager, Sebastian; Seidl, Andrea

2011-01-01

We derive optimal pricing strategies for conspicuous consumption products in periods of recession. To that end, we formulate and investigate a two-stage economic optimal control problem that takes uncertainty of the recession period length and delay effects of the pricing strategy into account. This non-standard optimal control problem is difficult to solve analytically, and solutions depend on the variable model parameters. Therefore, we use a numerical result-driven approach. We propose a structure-exploiting direct method for optimal control to solve this challenging optimization problem. In particular, we discretize the uncertainties in the model formulation by using scenario trees and target the control delays by introduction of slack control functions. Numerical results illustrate the validity of our approach and show the impact of uncertainties and delay effects on optimal economic strategies. During the recession, delayed optimal prices are higher than the non-delayed ones. In the normal economic period, however, this effect is reversed and optimal prices with a delayed impact are smaller compared to the non-delayed case. PMID:22267871

2. Numerical solution for linear cyclotron and diocotron modes in a nonneutral plasma column

Walsh, Daniel; Dubin, Daniel H. E.

2014-10-01

This poster presents numerical methods for solution of the linearized Vlasov-Poisson (LVP) equation applied to a cylindrical single-species plasma in a uniform magnetic field. The code is used to study z-independent cyclotron and diocotron modes of these plasmas, including kinetic effects. We transform to polar coordinates in both position and velocity space and Fourier expand in both polar angles (i.e. the cyclotron gyro angle and θ). In one approach, we then discretize in the remaining variables r and v (where v is the magnitude of the perpendicular velocity). However, using centered differences the method is unstable to unphysical eigenmodes with rapid variation on the scale of the grid. We remedy this problem by averaging particular terms in the discretized LVP operator over neighboring gridpoints. We also present a stable Galerkin method that expands the r and v dependence in basis functions. We compare the numerical results from both methods to exact analytic results for various modes. Supported by NSF/DOE Partnership Grants PHY-0903877 and DE-SC0002451.

3. Solution of the surface Euler equations for accurate three-dimensional boundary-layer analysis of aerodynamic configurations

NASA Technical Reports Server (NTRS)

Iyer, V.; Harris, J. E.

1987-01-01

The three-dimensional boundary-layer equations in the limit as the normal coordinate tends to infinity are called the surface Euler equations. The present paper describes an accurate method for generating edge conditions for three-dimensional boundary-layer codes using these equations. The inviscid pressure distribution is first interpolated to the boundary-layer grid. The surface Euler equations are then solved with this pressure field and a prescribed set of initial and boundary conditions to yield the velocities along the two surface coordinate directions. Results for typical wing and fuselage geometries are presented. The smoothness and accuracy of the edge conditions obtained are found to be superior to the conventional interpolation procedures.

4. A rescaling algorithm for the numerical solution to the porous medium equation in a two-component domain

Filo, Ján; Hundertmark-Zaušková, Anna

2016-10-01

The aim of this paper is to design a rescaling algorithm for the numerical solution to the system of two porous medium equations defined on two different components of the real line, that are connected by the nonlinear contact condition. The algorithm is based on the self-similarity of solutions on different scales and it presents a space-time adaptable method producing more exact numerical solution in the area of the interface between the components, whereas the number of grid points stays fixed.

5. Numerical solvers to the stabilizing solution of perturbed algebraic Riccati equations in LQ zero-sum games

Ivanov, I. G.; Netov, N. C.; Bogdanova, B. C.

2015-10-01

This paper addresses the problem of solving a generalized algebraic Riccati equation with an indefinite sign of its quadratic term. We extend the approach introduced by Lanzon, Feng, Anderson and Rotkowitz (2008) for solving similar Riccati equations. We numerically investigate two types of iterative methods for computing the stabilizing solution. The first type of iterative methods constructs two matrix sequences, where the sum of them converges to the stabilizing solution. The second type of methods defines one matrix sequence which converges to the stabilizing solution. Computer realizations of the presented methods are numerically tested and compared on the test of family examples. Based on the experiments some conclusions are derived.

6. Asymptotic and numerical solutions for thermally developing flows of Newtonian and non-Newtonian fluids in circular tubes with uniform wall temperature

SciTech Connect

Prusa, J. . Dept. of Mechanical Engineering); Manglik, R.M. . Dept. of Mechanical, Industrial, and Nuclear Engineering)

1994-08-01

Methods that predict heat transfer rates in thermally developing flows, important in engineering design, are often compared with the classical Graetz problem. Surprisingly, numerical solutions to this problem generally do not give accurate results in the entrance region. This inaccuracy stems from the existence of a singularity at the tube inlet. By adopting a fundamental approach based upon singular perturbation theory, the heat transfer process in the tube entrance has been analyzed to bring out the asymptotic boundary layer structure of the generalized problem with non-Newtonian flows. Using a standard finite difference method with only 21 radial nodes, results within 0.3% of the exact solution to the Graetz problem (Newtonian limit of generalized power law fluid flows) are obtained. Compared with previous numerical solutions reported in the literature, these results are an order of magnitude improvement in the accuracy with an order of magnitude decrease in the required number of radial nodes. Also, the number of radial nodes does not have to be increased in the present method to maintain this high level of accuracy as the initial singularity is approached. Solutions for power law, non-Newtonian fluid flows are presented, and generalized correlations are given for predicting Nusselt numbers in both the thermal entrance region and fully developed flows with 0 < n [<=] [infinity].

7. Numerical modeling of the elution peak profiles of retained solutes in supercritical fluid chromatography

SciTech Connect

Kaczmarski, Krzysztof; Guiochon, Georges A

2011-01-01

In supercritical fluid chromatography (SFC), the significant expansion of the mobile phase along the column causes the formation of axial and radial gradients of temperature. Due to these gradients, the mobile phase density, its viscosity, its velocity, its diffusion coefficients, etc. are not constant throughout the column. This results in a nonuniform flow velocity distribution, itself causing a loss of column efficiency in certain cases, even at low flow rates, as they do in HPLC. At high flow rates, an important deformation of the elution profiles of the sample components may occur. The model previously used to account satisfactorily for the retention of an unsorbed solute in SFC is applied to the modeling of the elution peak profiles of retained compounds. The numerical solution of the combined heat and mass balance equations provides the temperature and the pressure profiles inside the column and values of the retention time and the band profiles of retained compounds that are in excellent agreement with independent experimental data for large value of mobile phase reduced density. At low reduced densities, the band profiles can strongly depend on the column axial distribution of porosity.

8. Optimal and Numerical Solutions for an MHD Micropolar Nanofluid between Rotating Horizontal Parallel Plates

PubMed Central

2015-01-01

The present analysis deals with flow and heat transfer aspects of a micropolar nanofluid between two horizontal parallel plates in a rotating system. The governing partial differential equations for momentum, energy, micro rotation and nano-particles concentration are presented. Similarity transformations are utilized to convert the system of partial differential equations into system of ordinary differential equations. The reduced equations are solved analytically with the help of optimal homotopy analysis method (OHAM). Analytical solutions for velocity, temperature, micro-rotation and concentration profiles are expressed graphically against various emerging physical parameters. Physical quantities of interest such as skin friction co-efficient, local heat and local mass fluxes are also computed both analytically and numerically through mid-point integration scheme. It is found that both the solutions are in excellent agreement. Local skin friction coefficient is found to be higher for the case of strong concentration i.e. n=0, as compared to the case of weak concentration n=0.50. Influence of strong and weak concentration on Nusselt and Sherwood number appear to be similar in a quantitative sense. PMID:26046637

9. Numerical solution for Laplace equation with mixed boundary condition for ship problem in the sea

Silalahi, Fitriani Tupa R.; Budhi, Wono Setya; Adytia, Didit; van Groesen, E.

2015-09-01

One interesting phenomena is investigating the movement of ships at the sea. To start with the investigation in modelling of this problem, we will assume that the ship is only a one-dimensional object that is floating on the sea surface. Similarly, we assume that the water flow is uniform in parallel directions to the ship. Therefore, we simply use the two-dimensional Laplace equation in this problem. In the section that describes the surface of sea, Neumann boundary condition is imposed in part related to the ship and the Dirichlet boundary condition for others. Then on the other three boundaries, we imposed the Neumann boundary condition by assuming that the water does not flow on the bottom, and both end. The model is solved by numerical solution using the finite element method. Velocity potential solution on the whole domain is demonstrated as a result of the implementation of the finite element method. In this paper, we initiate an investigation with assuming that the ship is on the water so that the domain of the Laplace equation is rectangular. Then we assume the drift ship. Furthermore, we also study the dependence of width and depth of the domain to the velocity potential.

10. Sliding droplets of Xanthan solutions: A joint experimental and numerical study.

PubMed

Varagnolo, Silvia; Mistura, Giampaolo; Pierno, Matteo; Sbragaglia, Mauro

2015-11-01

We have investigated the sliding of droplets made of solutions of Xanthan, a stiff rodlike polysaccharide exhibiting a non-Newtonian behavior, notably characterized by a shear thinning viscosity accompanied by the emergence of normal stress difference as the polymer concentration is increased. These experimental results are quantitatively compared with those of Newtonian fluids (water). The impact of the non-Newtonian behavior on the sliding process was shown through the relation between the average dimensionless velocity (i.e. the capillary number) and the dimensionless volume forces (i.e. the Bond number). To this aim, it is needed to define operative strategies to compute the capillary number for the shear thinning fluids and compare with the corresponding Newtonian case. The resulting capillary number for the Xanthan solutions scales linearly with the Bond number at small inclinations, as well known for Newtonian fluids, while it shows a plateau as the Bond number is increased. Experimental data were complemented with lattice Boltzmann numerical simulations of sliding droplets, aimed to disentangle the specific contribution of shear thinning and elastic effects on the sliding behavior. In particular the deviation from the linear (Newtonian) trend is more likely attributed to the emergence of normal stresses inside the non-Newtonian droplet. PMID:26614497

11. Optimal and Numerical Solutions for an MHD Micropolar Nanofluid between Rotating Horizontal Parallel Plates.

PubMed

2015-01-01

The present analysis deals with flow and heat transfer aspects of a micropolar nanofluid between two horizontal parallel plates in a rotating system. The governing partial differential equations for momentum, energy, micro rotation and nano-particles concentration are presented. Similarity transformations are utilized to convert the system of partial differential equations into system of ordinary differential equations. The reduced equations are solved analytically with the help of optimal homotopy analysis method (OHAM). Analytical solutions for velocity, temperature, micro-rotation and concentration profiles are expressed graphically against various emerging physical parameters. Physical quantities of interest such as skin friction co-efficient, local heat and local mass fluxes are also computed both analytically and numerically through mid-point integration scheme. It is found that both the solutions are in excellent agreement. Local skin friction coefficient is found to be higher for the case of strong concentration i.e. n=0, as compared to the case of weak concentration n=0.50. Influence of strong and weak concentration on Nusselt and Sherwood number appear to be similar in a quantitative sense. PMID:26046637

12. Analytical and numerical solutions for mass diffusion in a composite cylindrical body

Kumar, A.

1980-12-01

The analytical and numerical solution techniques were investigated to study moisture diffusion problems in cylindrical bodies that are assumed to be composed of a finite number of layers of different materials. A generalized diffusion model for an n-layer cylindrical body with discontinuous moisture content at the interfaces was developed and the formal solutions were obtained. The model is to be used for describing mass transfer rates of any composite body, such as an ear of corn which could be assumed of consisting two different layers: the inner core represents the woody cob and the outer cylinder represents the kernel layer. Data describing the fully exposed drying characteristics of ear corn at high air velocity were obtained under different drying conditions. Ear corns were modeled as homogeneous bodies since composite model did not improve the fit substantially. A computer program using multidimensional optimization technique showed that diffusivity was an exponential function of moisture content and an arrhenius function of temperature of drying air.

13. Compaction of granular materials: numerical simulation of "elastic" compression and pressure solution creep

Bernabe, Y.; Evans, J.

2012-12-01

In a previous work we investigated stress transfer in a pair of grain contacts undergoing pressure solution (PS) creep, showed that stress transfer resulted in a significant decrease in overall strain rate, and concluded that PS creep rates of a randomly packed granular aggregate should be affected by packing evolution and the formation of new contacts during creep. To test these conclusions further, we are numerically simulating the "elastic" hydrostatic compression of a random pack of spheres, using a numerical method similar to that of Cundall and Strack [1979]. We assumed that the spheres were frictionless (i.e., spheres in contact only interacted through normal forces) and that the contact forces obeyed the non-linear Digby [1981] model. In order to determine the PS creep compression of the sphere pack subjected to a constant confining pressure pc, we calculated the thicknesses of the dissolved layers at each individual grain contact during a small time increment and, from these, the overall deformation of the sphere pack. We used an analytical expression discussed in our previous paper and originating from Lehner and Leroy [2004]. During these simulations, we also computed the mean coordination number of the grain contact z, the effective bulk modulus K of the sphere pack and others parameters characterizing the topological and mechanical properties of the sphere assembly. Our results show strong non-linear increase of z and K with pc during "elastic" compression and, with time, during PS creep. The packing rearrangements associated with PS creep produce complex time dependence of the overall deformation ɛ(t). We observed a regular transition from ɛ∝t^3/4 at early times (i.e., less than 0.1 years) and ɛ∝t^1/3 at late times (i.e., more than 1000 years). Cundall, P.A., and O.D.L. Strack (1979), A discrete numerical model for granular assemblies, Geotech., 29, 47-65. Digby, P.J. (1981), The effective elastic moduli of porous rocks, J. Appl. Mech., 48, 803

14. Use of numerically generated body-fitted coordinate systems for solution of the Navier-Stokes equations

NASA Technical Reports Server (NTRS)

Thompson, J. F.; Mastin, C. W.; Thames, F. C.; Shanks, S. P.

1975-01-01

A procedure for numerical solution of the time-dependent, two-dimensional incompressible Navier-Stokes equations that can treat the unsteady laminar flow about bodies of arbitrary shape, such as two-dimensional airfoils, multiple airfoils, and submerged hydrofoils, as naturally as it can deal with the flow about simple bodies. The solution is based on a method of automatic numerical generation of a general curvilinear coordinate system with coordinate lines coincident with all boundaries of a general multiconnected region containing any number of arbitrarily shaped bodies. The curvilinear coordinates are generated as the solution of two elliptical partial differential equations with Dirichlet boundary conditions, one coordinate being specified to be constant on each of the boundaries, and a distribution of the other being specified along the boundaries. The solution compares excellently with the Blasius boundary layer solution for the flow past a semiinfinite flat plate.

15. Supersonic flow of chemically reacting gas-particle mixtures. Volume 1: A theoretical analysis and development of the numerical solution

NASA Technical Reports Server (NTRS)

Penny, M. M.; Smith, S. D.; Anderson, P. G.; Sulyma, P. R.; Pearson, M. L.

1976-01-01

A numerical solution for chemically reacting supersonic gas-particle flows in rocket nozzles and exhaust plumes was described. The gas-particle flow solution is fully coupled in that the effects of particle drag and heat transfer between the gas and particle phases are treated. Gas and particles exchange momentum via the drag exerted on the gas by the particles. Energy is exchanged between the phases via heat transfer (convection and/or radiation). Thermochemistry calculations (chemical equilibrium, frozen or chemical kinetics) were shown to be uncoupled from the flow solution and, as such, can be solved separately. The solution to the set of governing equations is obtained by utilizing the method of characteristics. The equations cast in characteristic form are shown to be formally the same for ideal, frozen, chemical equilibrium and chemical non-equilibrium reacting gas mixtures. The particle distribution is represented in the numerical solution by a finite distribution of particle sizes.

16. Melt-rock reaction in the asthenospheric mantle: Perspectives from high-order accurate numerical simulations in 2D and 3D

Tirupathi, S.; Schiemenz, A. R.; Liang, Y.; Parmentier, E.; Hesthaven, J.

2013-12-01

The style and mode of melt migration in the mantle are important to the interpretation of basalts erupted on the surface. Both grain-scale diffuse porous flow and channelized melt migration have been proposed. To better understand the mechanisms and consequences of melt migration in a heterogeneous mantle, we have undertaken a numerical study of reactive dissolution in an upwelling and viscously deformable mantle where solubility of pyroxene increases upwards. Our setup is similar to that described in [1], except we use a larger domain size in 2D and 3D and a new numerical method. To enable efficient simulations in 3D through parallel computing, we developed a high-order accurate numerical method for the magma dynamics problem using discontinuous Galerkin methods and constructed the problem using the numerical library deal.II [2]. Linear stability analyses of the reactive dissolution problem reveal three dynamically distinct regimes [3] and the simulations reported in this study were run in the stable regime and the unstable wave regime where small perturbations in porosity grows periodically. The wave regime is more relevant to melt migration beneath the mid-ocean ridges but computationally more challenging. Extending the 2D simulations in the stable regime in [1] to 3D using various combinations of sustained perturbations in porosity at the base of the upwelling column (which may result from a viened mantle), we show the geometry and distribution of dunite channel and high-porosity melt channels are highly correlated with inflow perturbation through superposition. Strong nonlinear interactions among compaction, dissolution, and upwelling give rise to porosity waves and high-porosity melt channels in the wave regime. These compaction-dissolution waves have well organized but time-dependent structures in the lower part of the simulation domain. High-porosity melt channels nucleate along nodal lines of the porosity waves, growing downwards. The wavelength scales

17. Understanding the Composition and Reactivity of Au/Cu Electrocatalyst Nanoparticles in Solution Using Highly Accurate Reactive Potentials

Artrith, Nongnuch; Kolpak, Alexie

2014-03-01

The shape, size, and composition of catalyst nanoparticles can have a significant influence on their catalytic activity. Understanding such structure-reactivity relationships is crucial for the optimization of industrial catalysts and the design of novel catalysts with enhanced properties. In this work, we investigate the equilibrium shape and surface structure/composition of Au/Cu nanoparticles in solution, which have recently been shown to be stable and efficient catalysts for CO2 reduction. Using a combination of density functional theory calculations and large-scale Monte-Carlo and molecular dynamics simulations with reactive atomistic potentials, we determine how the nanoparticle shape, surface structure, and surface stoichiometry (i.e., fraction of Au at the surface relative to overall composition), evolve as a function of varying catalytic conditions. We discuss the effects of these changes on the surface electronic structure and binding energies of CO2, H2, and CH3OH. Our results emphasize the important relationships between catalytic environment (e.g., solvent effects), catalyst structure, and catalytic activity. We thank the Schlumberger Foundation Faculty for the Future for financial support. Computing time at XSEDE and NERSC clusters are gratefully acknowledged.

18. The correlation contracted Schrödinger equation: An accurate solution of the G-particle-hole hypervirial

Alcoba, D. R.; Valdemoro, C.; Tel, L. M.; Pérez-Romero, E.

The equation obtained by mapping the matrix representation of the Schrödinger equation with the 2nd-order correlation transition matrix elements into the 2-body space is the so called correlation contracted Schrödinger equation (CCSE) (Alcoba, Phys Rev A 2002, 65, 032519). As shown by Alcoba (Phys Rev A 2002, 65, 032519) the solution of the CCSE coincides with that of the Schrödinger equation. Here the attention is focused in the vanishing hypervirial of the correlation operator (GHV), which can be identified with the anti-Hermitian part of the CCSE. A comparative analysis of the GHV and the anti-Hermitian part of the contracted Schrödinger equation (ACSE) indicates that the former is a stronger stationarity condition than the latter. By applying a Heisenberg-like unitary transformation to the G-particle-hole operator (Valdemoro et al., Phys Rev A 2000, 61, 032507), a good approximation of the expectation value of this operator as well as of the GHV is obtained. The method is illustrated for the case of the Beryllium isoelectronic series as well as for the Li2 and BeH2 molecules. The correlation energies obtained are within 98.80-100.09% of the full-configuration interaction ones. The convergence of these calculations was faster when using the GHV than with the ACSE.

19. Numerical Solution of Inverse Radiative-Conductive Transient Heat Transfer Problem in a Grey Participating Medium

Zmywaczyk, J.; Koniorczyk, P.

2009-08-01

The problem of simultaneous identification of the thermal conductivity Λ(T) and the asymmetry parameter g of the Henyey-Greenstein scattering phase function is under consideration. A one-dimensional configuration in a grey participating medium with respect to silica fibers for which the thermophysical and optical properties are known from the literature is accepted. To find the unknown parameters, it is assumed that the thermal conductivity Λ(T) may be represented in a base of functions {1, T, T 2, . . .,T K } so the inverse problem can be applied to determine a set of coefficients {Λ0, Λ1, . . ., Λ K ; g}. The solution of the inverse problem is based on minimization of the ordinary squared differences between the measured and model temperatures. The measured temperatures are considered known. Temperature responses measured or theoretically generated at several different distances from the heat source along an x axis of the specimen set are known as a result of the numerical solution of the transient coupled heat transfer in a grey participating medium. An implicit finite volume method (FVM) is used for handling the energy equation, while a finite difference method (FDM) is applied to find the sensitivity coefficients with respect to the unknown set of coefficients. There are free parameters in a model, so these parameters are changed during an iteration process used by the fitting procedure. The Levenberg- Marquardt fitting procedure is iteratively searching for best fit of these parameters. The source term in the governing conservation-of-energy equation taking into account absorption, emission, and scattering of radiation is calculated by means of a discrete ordinate method together with an FDM while the scattering phase function approximated by the Henyey-Greenstein function is expanded in a series of Legendre polynomials with coefficients {c l } = (2l + 1)g l . The numerical procedure proposed here also allows consideration of some cases of coupled heat

20. On the formation, growth, and shapes of solution pipes - insights from numerical modeling

Szymczak, Piotr; Tredak, Hanna; Upadhyay, Virat; Kondratiuk, Paweł; Ladd, Anthony J. C.

2015-04-01

Cylindrical, vertical structures called solution pipes are a characteristic feature of epikarst, encountered in different parts of the world, both in relatively cold areas such as England and Poland (where their formation is linked to glacial processes) [1] and in coastal areas in tropical or subtropical climate (Bermuda, Australia, South Africa, Caribbean, Mediterranean) [2,3]. They are invariably associated with weakly cemented, porous limestones and relatively high groundwater fluxes. Many of them develop under the colluvial sandy cover and contain the fill of clayey silt. Although it is widely accepted that they are solutional in origin, the exact mechanism by which the flow becomes focused is still under debate. The hypotheses include the concentration of acidified water around stems and roots of plants, or the presence of pre-existing fractures or steeply dipping bedding planes, which would determine the points of entry for the focused groundwater flows. However, there are field sites where neither of this mechanisms was apparently at play and yet the pipes are formed in large quantities [1]. In this communication we show that the systems of solution pipes can develop spontaneously in nearly uniform matrix due to the reactive-infiltration instability: a homogeneous porous matrix is unstable with respect to small variations in local permeability; regions of high permeability dissolve faster because of enhanced transport of reactants, which leads to increased rippling of the front. This leads to the formation of a system of solution pipes which then advance into the matrix. We study this process numerically, by a combination of 2d- and 3d-simulations, solving the coupled flow and transport equations at the Darcy scale. The relative simplicity of this system (pipes developing in a uniform porous matrix, without any pre-existing structure) makes it very attractive from the modeling standpoint. We quantify the factors which control the pipe diameters and the

1. Shear-driven particle size segregation: Models, analysis, numerical solutions, and experiments

May, Lindsay Bard Hilbert

, we find a layer of small particles below a layer of large particles. We also measure a velocity profile from the Couette cell experimental data, which provides parameters used to derive the solution of the initial boundary value problem. The initial condition for the partial differential equation corresponds to the one dimensional initial configuration of the experiment. We solve two initial boundary value problems, one with a piecewise linear shear rate and one with an exponential shear rate, where the parameters for both cases are derived from the experimental data. In each case, we use the method of characteristics to solve the initial boundary value problem. In both cases, almost all pieces of the solution can be explicitly calculated, and those that cannot are calculated numerically. In the piecewise linear case, there is a material interface across which the characteristic speed jumps; in the exponential case, the characteristics are curved. We compare the model with the exponential shear rate to the experimental data. The model solution is the volume fraction of small particles at time t and location z. We cannot measure the volume fraction locally in the experiment; instead, we map the volume fraction to a theoretical height which we compare to the experimental experimental height data. The height of the sample is an indirect measurement of the amount of mixing or segregation. We conclude that the model captures qualitative features of the experimental data, but there are features of the experiment that the current version of the model does not describe.

2. Type curve and numerical solutions for estimation of Transmissivity and Storage coefficient with variable discharge condition

Zhang, Guowei

2013-01-01

SummaryMost of the coal mines of China are mining under the ground, so the artesian test which is one of the aquifer tests is conducted normally several hundred of meters below the earth surface. And the target aquifer is with very high hydraulic pressure, sometimes more than 3 MPa. Because of the high hydraulic pressure, it is most difficult to control the rate of flow out of the artesian well. Moreover, the velocity of flow out of the well cannot descend rapidly to zero, thus the analytical solution of Jacob and Lohman type curve for the artesian test will not be applicable. It is more reasonable if analyzing this test as a pumping test but with variable discharge. It is considered to rebuild that hydrogeological conceptual model in this paper. This conceptual model is similar with Theis model but with the variable discharge merely. And a general equation for any discharge variability is given. Its application for the linearly decreasing discharge is presented subsequently, and a type curve of this equation with linearly decreasing discharge will be given as well. Then a simple numerical model using FEFLOW will be built to simulate the linearly decreasing discharge from giving several different groups of the values of Transmissivity (T) and Storage coefficient (S). Both of them are much important hydrogeological parameters, and will be evaluated by using the type curve developed for this linear decreasing discharge well. The error between the given values of T and S in FEFLOW and the values of those calculated by matching point are much small. The solution gives really satisfactory values of these hydrogeological parameters.

3. Analytical study and numerical solution of the inverse source problem arising in thermoacoustic tomography

Holman, Benjamin R.

In recent years, revolutionary "hybrid" or "multi-physics" methods of medical imaging have emerged. By combining two or three different types of waves these methods overcome limitations of classical tomography techniques and deliver otherwise unavailable, potentially life-saving diagnostic information. Thermoacoustic (and photoacoustic) tomography is the most developed multi-physics imaging modality. Thermo- and photo- acoustic tomography require reconstructing initial acoustic pressure in a body from time series of pressure measured on a surface surrounding the body. For the classical case of free space wave propagation, various reconstruction techniques are well known. However, some novel measurement schemes place the object of interest between reflecting walls that form a de facto resonant cavity. In this case, known methods cannot be used. In chapter 2 we present a fast iterative reconstruction algorithm for measurements made at the walls of a rectangular reverberant cavity with a constant speed of sound. We prove the convergence of the iterations under a certain sufficient condition, and demonstrate the effectiveness and efficiency of the algorithm in numerical simulations. In chapter 3 we consider the more general problem of an arbitrarily shaped resonant cavity with a non constant speed of sound and present the gradual time reversal method for computing solutions to the inverse source problem. It consists in solving back in time on the interval [0, T] the initial/boundary value problem for the wave equation, with the Dirichlet boundary data multiplied by a smooth cutoff function. If T is sufficiently large one obtains a good approximation to the initial pressure; in the limit of large T such an approximation converges (under certain conditions) to the exact solution.

4. Numerical solution of acoustic response due to hydro/aerodynamic turbulence

Roknaldin, Farzam

In this work, a new methodology has been proposed which determines the acoustic response due to interaction of unsteady hydro/aero-dynamic sources with rigid/flexible structures. This methodology is based on Lighthill's acoustic analogy in which acoustic sources are pre-determined from unsteady flow calculations. The key feature of this methodology is the numerical solution of the acoustic problem. For this purpose, a new variational formulation of Lighthill's acoustic analogy has been developed which can be solved using the finite element method. This enables the true geometry of the structure and acoustically non-compact sources to be considered with relative ease. The feasibility of the approach has been investigated by studying the trailing-edge noise of the Eppler 387 airfoil due to a single quadrupole source, and the noise due to vortices shed from the NACA 0018 airfoil. In both cases the results are compared with analytical solutions that are available for certain limits. As an application to a practical problem, this methodology is used to compute the acoustic signature due to the boundary layer/wake turbulence over and behind the Eppler 387 wing at a cruise condition. Turbulent sources were obtained via Large Eddy Simulation, over an infinite span wing, using an unstructured grid finite element method in conjunction with the Dynamic Smagorinsky subgrid model. For this problem, sufficient numbers of grid points were used to resolve the wall layer. Flow separation, transition and turbulent reattachment were all captured and compared with the experimental data available from other sources. Finally, the acoustic problem is solved to obtain directivity patterns of acoustic pressures. The analysis indicates the importance of both wing geometry and the extent of acoustic sources on directivity.

5. Mechanical Behavior of Salt Caverns: Closed-Form Solutions vs Numerical Computations

Wang, Linlin; Bérest, Pierre; Brouard, Benoît

2015-11-01

Creep closure and structural stability of a cylindrical elongated cavern leached out from a salt formation are discussed. The Norton-Hoff creep law, or "power law", is used to capture the main features of salt rheological behavior. Two failure criteria are considered: (1) shear stresses must not be larger than a certain fraction of the mean stress (dilation criterion); and (2) the effective stress at the cavern wall (actual stress plus cavern fluid pressure) must not be tensile. The case of a brine-filled cavern whose pressure is kept constant is discussed first. It is proved that creep closure reaches a steady state such that stresses in the rock mass remain constant. However, decades are needed to reach such a state. During the transient phase that results from the slow redistribution of stresses in the rock mass, deviatoric stresses decrease at the vicinity of the cavern wall, and onset of dilation is less and less likely. At this point, the case of a rapid brine pressure increase, typical of a tightness test, is considered. It is proved that during such a swift pressure increase, cavern behavior is almost perfectly elastic; there is no risk of dilation onset. However, even when cavern pressure remains significantly smaller than geostatic, the effective stress at cavern wall can become tensile. These results, obtained through numerical computations, are confirmed by closed-form solutions obtained in the case of an idealized perfectly cylindrical cavern; these solutions provide a better insight into the main structural features of the behavior of the cavern.

6. Evaluation of approximate relations for Delta /Q/ using a numerical solution of the Boltzmann equation. [collision integral

NASA Technical Reports Server (NTRS)

Nathenson, M.; Baganoff, D.; Yen, S. M.

1974-01-01

Data obtained from a numerical solution of the Boltzmann equation for shock-wave structure are used to test the accuracy of accepted approximate expressions for the two moments of the collision integral Delta (Q) for general intermolecular potentials in systems with a large translational nonequilibrium. The accuracy of the numerical scheme is established by comparison of the numerical results with exact expressions in the case of Maxwell molecules. They are then used in the case of hard-sphere molecules, which are the furthest-removed inverse power potential from the Maxwell molecule; and the accuracy of the approximate expressions in this domain is gauged. A number of approximate solutions are judged in this manner, and the general advantages of the numerical approach in itself are considered.

7. Computational simulations of the total cavo-pulmonary connection: insights in optimizing numerical solutions.

PubMed

DeGroff, Curt; Birnbaum, Brian; Shandas, Robin; Orlando, Wendy; Hertzberg, Jean

2005-03-01

The Fontan procedure is a palliative surgical technique that is used to treat patients with congenital heart defects that include complex lesions such as those with a hypoplastic ventricle. In vitro, in vivo, and computational models of a set of modifications to the Fontan procedure, called the total cavopulmonary connection (TCPC), have been developed. Using these modeling methods, attempts have been made at finding the most energy efficient TCPC circuit. Computational modeling has distinct advantages to other modeling methods. However, discrepancies have been found in validation studies of TCPC computational models. There is little in the literature available to help explain and correct for such discrepancies. Differences in computational results can occur when choosing between steady flow versus transient flow numerical solvers. In this study transient flow solver results were shown to be more consistent with results from previous TCPC in vitro experiments. Using a transient flow solver we found complex fluctuating flow patterns can exist with steady inflow boundary conditions in computational models of the TCPC. To date such findings have not been reported in the literature. Furthermore, our computational modeling results suggest fluctuating flow patterns as well as the magnitudes of these secondary flow structures diminish if the TCPC offset between vena cavae is increased or if flanged connections are added. An association was found between these modifications and improvements in TCPC circuit flow efficiencies. In summary, development of accurate computational simulations in the validation process is critical to efforts in finding the most efficient TCPC circuits, efforts aimed at potentially improving the long term outcome for Fontan patients. PMID:15642509

8. A homogenized model for solute dispersion in unsaturated double-porosity medium: numerical and experimental applications

Tran Ngoc, T.; Lewandowska, J.; Vauclin, M.; Bertin, H.; Gentier, S.

2009-12-01

The complex processes of water flow and solute transport occurring in subsurface environment have to be well modeled in order to be able to protect the water aquifers against contamination, for security of nuclear waste depositories or CO2 sequestration, in the problem of extraction of geothermal energy. Since natural geological formations are often heterogeneous at different scales, it leads to preferential flow and transport observed in the breakthrough curves which is difficult to model. In such a case the concept of “double-porosity medium” originally introduced by Barenblatt et al. (1960), can be used. In this paper it was applied to a class of heterogeneous media (aggregated soils, fractured porous rocks) in which a strong contrast in the local pore size characteristics is manifested. It was assumed that the interactions/exchanges between the macro- and micro-porosity are responsible for solute spreading in the local non equilibrium conditions and contribute to the non Fickian behaviour. This study presents a macroscopic dispersion model associated with the unsaturated water flow, which was developped using the asymptotic homogenization method. This model consists of two equations describing the processes of solute transfer in the macro- and micro-porosity domains. A coupling between two concentration fields can be seen in the model, which gives an early breakthrough and a long tail effect. In order to enable the two-scale computations, the model was implemented using the commercial code COMSOL Multiphysics®. A particular strategy was proposed to take into account the micro-macro coupling. Finally, a series of experiments of tracer dispersion in a double-porosity physical model was performed under unsaturated steady-state flow conditions. The double-porosity medium presenting the periodic microstructure was composed of a regular assemblage between sintered clayey spheres and a fine sand. The model validation was carried out in two different stages. In

9. Low Mach number analysis of idealized thermoacoustic engines with numerical solution.

PubMed

Hireche, Omar; Weisman, Catherine; Baltean-Carlès, Diana; Le Quéré, Patrick; Bauwens, Luc

2010-12-01

A model of an idealized thermoacoustic engine is formulated, coupling nonlinear flow and heat exchange in the heat exchangers and stack with a simple linear acoustic model of the resonator and load. Correct coupling results in an asymptotically consistent global model, in the small Mach number approximation. A well-resolved numerical solution is obtained for two-dimensional heat exchangers and stack. The model assumes that the heat exchangers and stack are shorter than the overall length by a factor of the order of a representative Mach number. The model is well-suited for simulation of the entire startup process, whereby as a result of some excitation, an initially specified temperature profile in the stack evolves toward a near-steady profile, eventually reaching stationary operation. A validation analysis is presented, together with results showing the early amplitude growth and approach of a stationary regime. Two types of initial excitation are used: Random noise and a small periodic wave. The set of assumptions made leads to a heat-exchanger section that acts as a source of volume but is transparent to pressure and to a local heat-exchanger model characterized by a dynamically incompressible flow to which a locally spatially uniform acoustic pressure fluctuation is superimposed. PMID:21218877

10. A numerical solution of the Navier-Stokes equations for supercritical fluid thermodynamic analysis

NASA Technical Reports Server (NTRS)

Heinmiller, P. J.

1971-01-01

An explicit numerical solution of the compressible Navier-Stokes equations is applied to the thermodynamic analysis of supercritical oxygen in the Apollo cryogenic storage system. The wave character is retained in the conservation equations which are written in the basic fluid variables for a two-dimensional Cartesian coordinate system. Control-volume cells are employed to simplify imposition of boundary conditions and to ensure strict observance of local and global conservation principles. Non-linear real-gas thermodynamic properties responsible for the pressure collapse phenomonon in supercritical fluids are represented by tabular and empirical functions relating pressure and temperature to density and internal energy. Wall boundary conditions are adjusted at one cell face to emit a prescribed mass flowrate. Scaling principles are invoked to achieve acceptable computer execution times for very low Mach number convection problems. Detailed simulations of thermal stratification and fluid mixing occurring under low acceleration in the Apollo 12 supercritical oxygen tank are presented which model the pressure decay associated with de-stratification induced by an ordinary vehicle maneuver and heater cycle operation.

11. Numerical and Analytical Solutions of Hypersonic Interactions Involving Surface Property Discontinuities

NASA Technical Reports Server (NTRS)

Gnoffo, Peter A.; Inger, George R.

1999-01-01

The local viscous-inviscid interaction field generated by a wall temperature jump on a flat plate in supersonic flow and on the windside of a Reusable Launch Vehicle in hypersonic flow is studied in detail by both a Navier-Stokes numerical code and an analytical triple-deck model. Treatment of the rapid heat transfer changes both upstream and downstream of the jump is included. Closed form relationships derived from the triple-deck theory are presented. The analytically predicted pressure and heating variations including upstream influence are found to be in generally good agreement with the Computational Fluid Dynamic (CFD) predictions. These analyses not only clarify the interactive physics involved but also are useful in preliminary design of thermal protection systems and as an insertable module to improve CFD code efficiency when applied to such small-scale interaction problems. The analyses only require conditions at the wall and boundary-layer edge which are easily extracted from a baseline, constant wall temperature, CFD solution.

12. Simulations of emissivity in passive microwave remote sensing with three-dimensional numerical solutions of Maxwell equations and fast algorithm

Zhou, Lin

In the first part of the work, we developed coding for large-scale computation to solve 3-dimensional microwave scattering problem. Maxwell integral equations are solved by using MoM with RWG basis functions in conjunction with fast computation algorithms. The cost-effective solutions of parallel and distributed simulation were implemented on a low cost PC cluster, which consists of 32 processors connected to a fast Ethernet switch. More than a million of surface current unknowns were solved at unprecedented speeds. Accurate simulations of emissivities and bistatic coefficients from ocean and soil were achieved. Exponential correlation function and ocean spectrum are implementd for generating soil and ocean surfaces. They have fine scale features with large rms slope. The results were justified by comparison with numerical results from original code, which is based on pulse basis function, and from analytic methods like SPM, and also with experiments. In the second part of the work, fully polarimetric microwave emissions from wind-generated foam-covered ocean surfaces were investigated. The foam is treated as densely packed air bubbles coated with thin seawater coating. The absorption, scattering and extinction coefficients were calculated by Monte Carlo simulations of solutionsof Maxwell equations of a collection of coated particles. The effects of boundary roughness of ocean surfaces were included by using the second-order small perturbation method (SPM) describing the reflection coefficients between foam and ocean. An empirical wave-number spectrum was used to represent the small-scale wind-generated sea surfaces. The theoretical results of four Stokes brightness temperatures with typical parameters of foam in passive remote sensing at 10.8 GHz, 19.0 GHz and 36.5 GHz were illustrated. The azimuth variations of polarimetric brightness temperature were calculated. Emission with various wind speed and foam layer thickness was studied. The results were also compared

13. Calculation of double-lunar swingby trajectories: Part 2: Numerical solutions in the restricted problem of three bodies

NASA Technical Reports Server (NTRS)

Stalos, S.

1990-01-01

The double-lunar swingby trajectory is a method for maintaining alignment of an Earth satellite's line of apsides with the Sun-Earth line. From a Keplerian point of view, successive close encounters with the Moon cause discrete, instantaneous changes in the satellite's eccentricity and semimajor axis. Numerical solutions to the planar, restricted problem of three bodies as double-lunar swingby trajectories are identified. The method of solution is described and the results compared to the Keplerian formulation.

14. Numerical solution to the problem of variational assimilation of operational observational data on the ocean surface temperature

Agoshkov, V. I.; Lebedev, S. A.; Parmuzin, E. I.

2009-02-01

The problem of variational assimilation of satellite observational data on the ocean surface temperature is formulated and numerically investigated in order to reconstruct surface heat fluxes with the use of the global three-dimensional model of ocean hydrothermodynamics developed at the Institute of Numerical Mathematics, Russian Academy of Sciences (INM RAS), and observational data close to the data actually observed in specified time intervals. The algorithms of the numerical solution to the problem are elaborated and substantiated, and the data assimilation block is developed and incorporated into the global three-dimensional model. Numerical experiments are carried out with the use of the Indian Ocean water area as an example. The data on the ocean surface temperature over the year 2000 are used as observational data. Numerical experiments confirm the theoretical conclusions obtained and demonstrate the expediency of combining the model with a block of assimilating operational observational data on the surface temperature.

15. Different nonideality relationships, different databases and their effects on modeling precipitation from concentrated solutions using numerical speciation codes

SciTech Connect

Brown, L.F.; Ebinger, M.H.

1996-08-01

Four simple precipitation problems are solved to examine the use of numerical equilibrium codes. The study emphasizes concentrated solutions, assumes both ideal and nonideal solutions, and employs different databases and different activity-coefficient relationships. The study uses the EQ3/6 numerical speciation codes. The results show satisfactory material balances and agreement between solubility products calculated from free-energy relationships and those calculated from concentrations and activity coefficients. Precipitates show slightly higher solubilities when the solutions are regarded as nonideal than when considered ideal, agreeing with theory. When a substance may precipitate from a solution dilute in the precipitating substance, a code may or may not predict precipitation, depending on the database or activity-coefficient relationship used. In a problem involving a two-component precipitation, there are only small differences in the precipitate mass and composition between the ideal and nonideal solution calculations. Analysis of this result indicates that this may be a frequent occurrence. An analytical approach is derived for judging whether this phenomenon will occur in any real or postulated precipitation situation. The discussion looks at applications of this approach. In the solutes remaining after the precipitations, there seems to be little consistency in the calculated concentrations and activity coefficients. They do not appear to depend in any coherent manner on the database or activity-coefficient relationship used. These results reinforce warnings in the literature about perfunctory or mechanical use of numerical speciation codes.

16. Numerical solution of the potential due to dipole sources in volume conductors with arbitrary geometry and conductivity.

PubMed

Rosenfeld, M; Tanami, R; Abboud, S

1996-07-01

The integral conservation equation for biological volume conductors with general geometry and arbitrary distribution of electrical conductivity is solved using a finite volume method. An effective conductivity was defined for the boundaries between regions with abrupt change of the conductivity to allow the simultaneous solution of the entire domain although the derivatives are not continuous. The geometrical singularities arising from the spherical topology of the coordinate system are removed using the conservation law. The resulting finite volume solution method is efficient both in central processing unit (CPU) time and memory requirements, allowing the solution of the volume conductor equation using a large number of mesh points (of the order of 10(5)) even on small workstations (like SGI Indigo). It results in very accurate solutions, as several comparisons with analytical solutions of head models reveal. The proposed finite volume method is an attractive alternative to the finite element and boundary element methods that are more common in bioelectric applications.

17. Conceptual and numerical models of solute diffusion around a HLW repository in clay

Samper, Javier; Naves, Acacia; Lu, Chuanhe; Li, Yanmei; Fritz, Bertrand; Clement, Alain

Reactive transport models have been used to simulate solute diffusion, canister corrosion, interactions of the corrosion products with the bentonite and the long-term hydrochemical evolution of porewater composition around radioactive waste repositories. Such models usually rely on simplifications of the geometry and dimensionality of the problem. Detailed three-dimensional flow and transport models, on the other hand, are used which often oversimplify the geochemical reactions. There is a clear need to identify which simplifications and assumptions are admissible. Here we present conceptual and numerical models of radionuclide diffusion and sorption around a HLW repository in clay according to the French reference concept. Models of increasing dimensionality have been performed for: (1) 1D transport perpendicular to the axes of the disposal cells; (2) 1D axisymmetric transport around disposal cells for bounded and unbounded domains; (3) 2D transport through vertical planes; and (4) 1D vertical transport from the disposal cells into the overlying Oxfordian formation. Model results are compared for simulation times up to 10 6 years and for the following radionuclides and tracers: tritium, HTO, which is treated here as an ideal and conservative tracer, 36Cl - which experiences anion exclusion, 133Cs + which sorbs moderately and 238U (IV) which shows a strong sorption capacity. Radionuclides are released into the disposal cell either at a fixed concentration or as an instantaneous unit pulse. Model results indicate that the 1D unbounded model is always acceptable for 238U (IV) and is valid for 133Cs + for t < 10 4 years. It is valid for HTO and 36Cl - only for t < 10 3 years. These conclusions hold true for both release modes. Computed concentrations with the 1D parallel and the 1D axisymmetric models are significantly different. Inasmuch as solute diffusion in a radioactive waste repository is expected to show radial symmetry around the cells, the use of the

18. Accretion-powered Stellar Winds. II. Numerical Solutions for Stellar Wind Torques

Matt, Sean; Pudritz, Ralph E.

2008-05-01

In order to explain the slow rotation observed in a large fraction of accreting pre-main-sequence stars (CTTSs), we explore the role of stellar winds in torquing down the stars. For this mechanism to be effective, the stellar winds need to have relatively high outflow rates, and thus would likely be powered by the accretion process itself. Here, we use numerical magnetohydrodynamical simulations to compute detailed two-dimensional (axisymmetric) stellar wind solutions, in order to determine the spin-down torque on the star. We discuss wind driving mechanisms and then adopt a Parker-like (thermal pressure driven) wind, modified by rotation, magnetic fields, and enhanced mass-loss rate (relative to the Sun). We explore a range of parameters relevant for CTTSs, including variations in the stellar mass, radius, spin rate, surface magnetic field strength, mass-loss rate, and wind acceleration rate. We also consider both dipole and quadrupole magnetic field geometries. Our simulations indicate that the stellar wind torque is of sufficient magnitude to be important for spinning down a "typical" CTTS, for a mass-loss rate of ~10-9 M⊙ yr-1. The winds are wide-angle, self-collimated flows, as expected of magnetic rotator winds with moderately fast rotation. The cases with quadrupolar field produce a much weaker torque than for a dipole with the same surface field strength, demonstrating that magnetic geometry plays a fundamental role in determining the torque. Cases with varying wind acceleration rate show much smaller variations in the torque, suggesting that the details of the wind driving are less important. We use our computed results to fit a semianalytic formula for the effective Alfvén radius in the wind, as well as the torque. This allows for considerable predictive power, and is an improvement over existing approximations.

19. Numerical solution of fractional sub-diffusion and time-fractional diffusion-wave equations via fractional-order Legendre functions

Hooshmandasl, M. R.; Heydari, M. H.; Cattani, C.

2016-08-01

Fractional calculus has been used to model physical and engineering processes that are best described by fractional differential equations. Therefore designing efficient and reliable techniques for the solution of such equations is an important task. In this paper, we propose an efficient and accurate Galerkin method based on the fractional-order Legendre functions (FLFs) for solving the fractional sub-diffusion equation (FSDE) and the time-fractional diffusion-wave equation (FDWE). The time-fractional derivatives for FSDE are described in the Riemann-Liouville sense, while for FDWE are described in the Caputo sense. To this end, we first derive a new operational matrix of fractional integration (OMFI) in the Riemann-Liouville sense for FLFs. Next, we transform the original FSDE into an equivalent problem with fractional derivatives in the Caputo sense. Then the FLFs and their OMFI together with the Galerkin method are used to transform the problems under consideration into the corresponding linear systems of algebraic equations, which can be simply solved to achieve the numerical solutions of the problems. The proposed method is very convenient for solving such kind of problems, since the initial and boundary conditions are taken into account automatically. Furthermore, the efficiency of the proposed method is shown for some concrete examples. The results reveal that the proposed method is very accurate and efficient.

20. Paraxial light distribution in the focal region of a lens: a comparison of several analytical solutions and a numerical result

Wu, Yang; Kelly, Damien P.

2014-12-01

The distribution of the complex field in the focal region of a lens is a classical optical diffraction problem. Today, it remains of significant theoretical importance for understanding the properties of imaging systems. In the paraxial regime, it is possible to find analytical solutions in the neighborhood of the focus, when a plane wave is incident on a focusing lens whose finite extent is limited by a circular aperture. For example, in Born and Wolf's treatment of this problem, two different, but mathematically equivalent analytical solutions, are presented that describe the 3D field distribution using infinite sums of ? and ? type Lommel functions. An alternative solution expresses the distribution in terms of Zernike polynomials, and was presented by Nijboer in 1947. More recently, Cao derived an alternative analytical solution by expanding the Fresnel kernel using a Taylor series expansion. In practical calculations, however, only a finite number of terms from these infinite series expansions is actually used to calculate the distribution in the focal region. In this manuscript, we compare and contrast each of these different solutions to a numerically calculated result, paying particular attention to how quickly each solution converges for a range of different spatial locations behind the focusing lens. We also examine the time taken to calculate each of the analytical solutions. The numerical solution is calculated in a polar coordinate system and is semi-analytic. The integration over the angle is solved analytically, while the radial coordinate is sampled with a sampling interval of ? and then numerically integrated. This produces an infinite set of replicas in the diffraction plane, that are located in circular rings centered at the optical axis and each with radii given by ?, where ? is the replica order. These circular replicas are shown to be fundamentally different from the replicas that arise in a Cartesian coordinate system.

1. Paraxial light distribution in the focal region of a lens: a comparison of several analytical solutions and a numerical result

PubMed Central

Wu, Yang; Kelly, Damien P.

2014-01-01

The distribution of the complex field in the focal region of a lens is a classical optical diffraction problem. Today, it remains of significant theoretical importance for understanding the properties of imaging systems. In the paraxial regime, it is possible to find analytical solutions in the neighborhood of the focus, when a plane wave is incident on a focusing lens whose finite extent is limited by a circular aperture. For example, in Born and Wolf’s treatment of this problem, two different, but mathematically equivalent analytical solutions, are presented that describe the 3D field distribution using infinite sums of Un and Vn type Lommel functions. An alternative solution expresses the distribution in terms of Zernike polynomials, and was presented by Nijboer in 1947. More recently, Cao derived an alternative analytical solution by expanding the Fresnel kernel using a Taylor series expansion. In practical calculations, however, only a finite number of terms from these infinite series expansions is actually used to calculate the distribution in the focal region. In this manuscript, we compare and contrast each of these different solutions to a numerically calculated result, paying particular attention to how quickly each solution converges for a range of different spatial locations behind the focusing lens. We also examine the time taken to calculate each of the analytical solutions. The numerical solution is calculated in a polar coordinate system and is semi-analytic. The integration over the angle is solved analytically, while the radial coordinate is sampled with a sampling interval of Δρ and then numerically integrated. This produces an infinite set of replicas in the diffraction plane, that are located in circular rings centered at the optical axis and each with radii given by 2πm/Δρ, where m is the replica order. These circular replicas are shown to be fundamentally different from the replicas that arise in a Cartesian coordinate system. PMID

2. A block iterative finite element algorithm for numerical solution of the steady-state, compressible Navier-Stokes equations

NASA Technical Reports Server (NTRS)

Cooke, C. H.

1976-01-01

An iterative method for numerically solving the time independent Navier-Stokes equations for viscous compressible flows is presented. The method is based upon partial application of the Gauss-Seidel principle in block form to the systems of nonlinear algebraic equations which arise in construction of finite element (Galerkin) models approximating solutions of fluid dynamic problems. The C deg-cubic element on triangles is employed for function approximation. Computational results for a free shear flow at Re = 1,000 indicate significant achievement of economy in iterative convergence rate over finite element and finite difference models which employ the customary time dependent equations and asymptotic time marching procedure to steady solution. Numerical results are in excellent agreement with those obtained for the same test problem employing time marching finite element and finite difference solution techniques.

3. 3D models of slow motions in the Earth's crust and upper mantle in the source zones of seismically active regions and their comparison with highly accurate observational data: II. Results of numerical calculations

Molodenskii, S. M.; Molodenskii, M. S.; Begitova, T. A.

2016-09-01

In the first part of the paper, a new method was developed for solving the inverse problem of coseismic and postseismic deformations in the real (imperfectly elastic, radially and horizontally heterogeneous, self-gravitating) Earth with hydrostatic initial stresses from highly accurate modern satellite data. The method is based on the decomposition of the sought parameters in the orthogonalized basis. The method was suggested for estimating the ambiguity of the solution of the inverse problem for coseismic and postseismic deformations. For obtaining this estimate, the orthogonal complement is constructed to the n-dimensional space spanned by the system of functional derivatives of the residuals in the system of n observed and model data on the coseismic and postseismic displacements at a variety of sites on the ground surface with small variations in the models. Below, we present the results of the numerical modeling of the elastic displacements of the ground surface, which were based on calculating Green's functions of the real Earth for the plane dislocation surface and different orientations of the displacement vector as described in part I of the paper. The calculations were conducted for the model of a horizontally homogeneous but radially heterogeneous selfgravitating Earth with hydrostatic initial stresses and the mantle rheology described by the Lomnitz logarithmic creep function according to (M. Molodenskii, 2014). We compare our results with the previous numerical calculations (Okado, 1985; 1992) for the simplest model of a perfectly elastic nongravitating homogeneous Earth. It is shown that with the source depths starting from the first hundreds of kilometers and with magnitudes of about 8.0 and higher, the discrepancies significantly exceed the errors of the observations and should therefore be taken into account. We present the examples of the numerical calculations of the creep function of the crust and upper mantle for the coseismic deformations. We

4. Numerical solution for plasticity models using consistency bisection and a transformed-space closest-point return: a nongradient solution method

Homel, Michael A.; Guilkey, James E.; Brannon, Rebecca M.

2015-10-01

A new approach is presented for computing the return in numerical solutions for computational plasticity models that ensures convergence through bisection of the plastic consistency parameter, while using a transformed-space closest-point return based on a geometric search that eliminates the need to compute gradients of the yield function or a consistent tangent operator. Numerical solution of the governing equations for computational plasticity is highly-nontrivial for complex constitutive laws. In particular for geomaterials, a predictive model may account for nonlinear elasticity, shear strength that depends nonlinearly on pressure and Lode angle, and nonlinear evolution models for internal variables such as porosity or pore pressure. Traditional gradient-based integration methods may perform poorly when the hardening laws are highly nonlinear or when the yield function has an ill-defined or cumbersome gradient because of high curvature, vertices, or complicated functional form. The application of this new approach to geomaterial modeling is described, along with verification benchmarks.

5. Numerical Solution of 3D Poisson-Nernst-Planck Equations Coupled with Classical Density Functional Theory for Modeling Ion and Electron Transport in a Confined Environment

SciTech Connect

Meng, Da; Zheng, Bin; Lin, Guang; Sushko, Maria L.

2014-08-29

We have developed efficient numerical algorithms for the solution of 3D steady-state Poisson-Nernst-Planck equations (PNP) with excess chemical potentials described by the classical density functional theory (cDFT). The coupled PNP equations are discretized by finite difference scheme and solved iteratively by Gummel method with relaxation. The Nernst-Planck equations are transformed into Laplace equations through the Slotboom transformation. Algebraic multigrid method is then applied to efficiently solve the Poisson equation and the transformed Nernst-Planck equations. A novel strategy for calculating excess chemical potentials through fast Fourier transforms is proposed which reduces computational complexity from O(N2) to O(NlogN) where N is the number of grid points. Integrals involving Dirac delta function are evaluated directly by coordinate transformation which yields more accurate result compared to applying numerical quadrature to an approximated delta function. Numerical results for ion and electron transport in solid electrolyte for Li ion batteries are shown to be in good agreement with the experimental data and the results from previous studies.

6. Numerical solutions of the macroscopic Maxwell equations for scattering by non-spherical particles: A tutorial review

Kahnert, Michael

2016-07-01

Numerical solution methods for electromagnetic scattering by non-spherical particles comprise a variety of different techniques, which can be traced back to different assumptions and solution strategies applied to the macroscopic Maxwell equations. One can distinguish between time- and frequency-domain methods; further, one can divide numerical techniques into finite-difference methods (which are based on approximating the differential operators), separation-of-variables methods (which are based on expanding the solution in a complete set of functions, thus approximating the fields), and volume integral-equation methods (which are usually solved by discretisation of the target volume and invoking the long-wave approximation in each volume cell). While existing reviews of the topic often tend to have a target audience of program developers and expert users, this tutorial review is intended to accommodate the needs of practitioners as well as novices to the field. The required conciseness is achieved by limiting the presentation to a selection of illustrative methods, and by omitting many technical details that are not essential at a first exposure to the subject. On the other hand, the theoretical basis of numerical methods is explained with little compromises in mathematical rigour; the rationale is that a good grasp of numerical light scattering methods is best achieved by understanding their foundation in Maxwell's theory.

7. Numerical solution to the Bloch equations: paramagnetic solutions under wideband continuous radio frequency irradiation in a pulsed magnetic field

Chen, Wen-Jun; Ma, Hong; Yu, De; Zeng, Xiao-Hu

2016-08-01

A novel nuclear magnetic resonance (NMR) experimental scheme, called wideband continuous wave NMR (WB-CW-NMR), is presented in this article. This experimental scheme has promising applications in pulsed magnetic fields, and can dramatically improve the utilization of the pulsed field. The feasibility of WB-CW-NMR scheme is verified by numerically solving modified Bloch equations. In the numerical simulation, the applied magnetic field is a pulsed magnetic field up to 80 T, and the wideband continuous radio frequency (RF) excitation is a band-limited (0.68–3.40 GHz) white noise. Furthermore, the influences of some experimental parameters, such as relaxation time, applied magnetic field strength and wideband continuous RF power, on the WB-CW-NMR signal are analyzed briefly. Finally, a multi-channel system framework for transmitting and receiving ultra wideband signals is proposed, and the basic requirements of this experimental system are discussed. Meanwhile, the amplitude of the NMR signal, the level of noise and RF interference in WB-CW-NMR experiments are estimated, and a preliminary adaptive cancellation plan is given for detecting WB-CW-NMR signal from large background interference. Supported by National Natural Science Foundation of China (11475067), the Innovative Research Foundation of Huazhong University of Science and Technology (2015 ZDTD017) and the Experimental Apparatus Research Project of Wuhan Pulsed High Magnetic Field Center (2015KF17)

8. Numerical solution to the Bloch equations: paramagnetic solutions under wideband continuous radio frequency irradiation in a pulsed magnetic field

Chen, Wen-Jun; Ma, Hong; Yu, De; Zeng, Xiao-Hu

2016-08-01

A novel nuclear magnetic resonance (NMR) experimental scheme, called wideband continuous wave NMR (WB-CW-NMR), is presented in this article. This experimental scheme has promising applications in pulsed magnetic fields, and can dramatically improve the utilization of the pulsed field. The feasibility of WB-CW-NMR scheme is verified by numerically solving modified Bloch equations. In the numerical simulation, the applied magnetic field is a pulsed magnetic field up to 80 T, and the wideband continuous radio frequency (RF) excitation is a band-limited (0.68-3.40 GHz) white noise. Furthermore, the influences of some experimental parameters, such as relaxation time, applied magnetic field strength and wideband continuous RF power, on the WB-CW-NMR signal are analyzed briefly. Finally, a multi-channel system framework for transmitting and receiving ultra wideband signals is proposed, and the basic requirements of this experimental system are discussed. Meanwhile, the amplitude of the NMR signal, the level of noise and RF interference in WB-CW-NMR experiments are estimated, and a preliminary adaptive cancellation plan is given for detecting WB-CW-NMR signal from large background interference. Supported by National Natural Science Foundation of China (11475067), the Innovative Research Foundation of Huazhong University of Science and Technology (2015 ZDTD017) and the Experimental Apparatus Research Project of Wuhan Pulsed High Magnetic Field Center (2015KF17)

9. Numerical solution of an optimal control problem governed by three-phase non-isothermal flow equations

Temirbekov, Nurlan M.; Baigereyev, Dossan R.

2016-08-01

The paper focuses on the numerical implementation of a model optimal control problem governed by equations of three-phase non-isothermal flow in porous media. The objective is to achieve preassigned temperature distribution along the reservoir at a given time of development by controlling mass flow rate of heat transfer agent on the injection well. The problem of optimal control is formulated, the adjoint problem is presented, and an algorithm for the numerical solution is proposed. Results of computational experiments are presented for a test problem.

10. Water-wave gap solitons: an approximate theory and numerical solutions of the exact equations of motion.

PubMed

Ruban, V P

2008-12-01

It is demonstrated that a standard coupled-mode theory can successfully describe weakly nonlinear gravity water waves in Bragg resonance with a periodic one-dimensional topography. Analytical solutions for gap solitons provided by this theory are in reasonable agreement with numerical simulations of the exact equations of motion for ideal planar potential free-surface flows, even for strongly nonlinear waves. In numerical experiments, self-localized groups of nearly standing water waves can exist up to hundreds of wave periods. Generalizations of the model to the three-dimensional case are also derived. PMID:19256946

11. A note on the numerical solution of the von Karman small disturbance equation

Pilant, M. S.

1985-09-01

In this short note, the von Karman small disturbance equation is derived from the full potential equation of gas dynamics through perturbation methods. Guderley (1962) and Germain (1964) have previously computed exact solutions, in similarity form, for the small disturbance equation. It is shown that these solutions can be computed efficiently by solving a single nonlinear second-order differential equation. The shock and entropy conditions are automatically satisfied, and a one-parameter family of solutions is recovered.

12. Numerical modeling of subsurface radioactive solute transport from waste seepage ponds at the Idaho National Engineering Laboratory

USGS Publications Warehouse

Robertson, John B.

1976-01-01

Aqueous chemical and low-level radioactive effluents have been disposed to seepage ponds since 1952 at the Idaho National Engineering Laboratory. The solutions percolate toward the Snake River Plain aquifer (135 m below) through interlayered basalts and unconsolidated sediments and an extensive zone of ground water perched on a sedimentary layer about 40 m beneath the ponds. A three-segment numerical model was developed to simulate the system, including effects of convection, hydrodynamic dispersion, radioactive decay, and adsorption. Simulated hydraulics and solute migration patterns for all segments agree adequately with the available field data. The model can be used to project subsurface distributions of waste solutes under a variety of assumed conditions for the future. Although chloride and tritium reached the aquifer several years ago, the model analysis suggests that the more easily sorbed solutes, such as cesium-137 and strontium-90, would not reach the aquifer in detectable concentrations within 150 years for the conditions assumed. (Woodard-USGS)

13. Expressions of the fundamental equation of gradient elution and a numerical solution of these equations under any gradient profile.

PubMed

Nikitas, P; Pappa-Louisi, A

2005-09-01

The original work carried out by Freiling and Drake in gradient liquid chromatography is rewritten in the current language of reversed-phase liquid chromatography. This allows for the rigorous derivation of the fundamental equation for gradient elution and the development of two alternative expressions of this equation, one of which is free from the constraint that the holdup time must be constant. In addition, the above derivation results in a very simple numerical solution of the various equations of gradient elution under any gradient profile. The theory was tested using eight catechol-related solutes in mobile phases modified with methanol, acetonitrile, or 2-propanol. It was found to be a satisfactory prediction of solute gradient retention behavior even if we used a simple linear description for the isocratic elution of these solutes. PMID:16131080

14. Numerical solution of the two-dimensional time-dependent incompressible Euler equations

NASA Technical Reports Server (NTRS)

Whitfield, David L.; Taylor, Lafayette K.

1994-01-01

A numerical method is presented for solving the artificial compressibility form of the 2D time-dependent incompressible Euler equations. The approach is based on using an approximate Riemann solver for the cell face numerical flux of a finite volume discretization. Characteristic variable boundary conditions are developed and presented for all boundaries and in-flow out-flow situations. The system of algebraic equations is solved using the discretized Newton-relaxation (DNR) implicit method. Numerical results are presented for both steady and unsteady flow.

15. Solute drag in polycrystalline materials: Derivation and numerical analysis of a variational model for the effect of solute on the motion of boundaries and junctions during coarsening

Wilson, Seth Robert

A mathematical model that results in an expression for the local acceleration of a network of sharp interfaces interacting with an ambient solute field is proposed. This expression comprises a first-order differential equation for the local velocity that, given the appropriate initial conditions, may be used to predict the subsequent time evolution of the system, including non-steady state absorption and desorption of solute. Evolution equations for both interfaces and the junction of interfaces are derived by maximizing a functional approximating the rate at which the local Gibbs free energy density decreases, as a function of the local solute content and the instantaneous velocity. The model has been formulated in three dimensions, and non-equilibrium effects such as grain boundary diffusion, solute gradients, and time-dependant segregation are taken into account. As a consequence of this model, it is shown that both interfaces and the junctions between interfaces obey evolution equations that closely resemble Newton's second law. In particular, the concept of "thrust" in variable-mass systems is shown to have a direct analog in solute-interface interaction. Numerical analysis of the equations that result reveals that a double cusp catastrophe governs the behavior of the solute-interface system, for which trajectories that include hysteresis, slip-stick motion, and jerky motion are all conceivable. The geometry of the cusp catastrophe is quantified, and a number of relations between physical parameters and system behavior are consequently predicted.

16. Numerical solution of an inverse electrocardiography problem for a medium with piecewise constant electrical conductivity

Denisov, A. M.; Zakharov, E. V.; Kalinin, A. V.; Kalinin, V. V.

2010-07-01

A numerical method is proposed for solving an inverse electrocardiography problem for a medium with a piecewise constant electrical conductivity. The method is based on the method of boundary integral equations and Tikhonov regularization.

17. Impact of Energy Slope Averaging Methods on Numerical Solution of 1D Steady Gradually Varied Flow

Artichowicz, Wojciech; Prybytak, Dzmitry

2015-12-01

In this paper, energy slope averaging in the one-dimensional steady gradually varied flow model is considered. For this purpose, different methods of averaging the energy slope between cross-sections are used. The most popular are arithmetic, geometric, harmonic and hydraulic means. However, from the formal viewpoint, the application of different averaging formulas results in different numerical integration formulas. This study examines the basic properties of numerical methods resulting from different types of averaging.

18. Numerical solution of seismic exploration problems in the Arctic region by applying the grid-characteristic method

Petrov, D. I.; Petrov, I. B.; Favorskaya, A. V.; Khokhlov, N. I.

2016-06-01

The goal of this paper is the numerical solution of direct problems concerning hydrocarbon seismic exploration on the Arctic shelf. The task is addressed by solving a complete system of linear elasticity equations and a system of acoustic field equations. Both systems are solved by applying the grid-characteristic method, which takes into account all wave processes in a detailed and physically correct manner and produces a solution near the boundaries and interfaces of the integration domain, including the interface between the acoustic and linear elastic media involved. The seismograms and wave patterns obtained by numerically solving these systems are compared. The effect of ice structures on the resulting wave patterns is examined.

19. Numerical solution of steady-state buoyancy-driven flow of an incompressible fluid with temperature dependent viscosity

SciTech Connect

Abrous, A.; Emery, A.F.

1995-12-31

The steady-state, buoyancy-driven flow of an incompressible fluid with temperature dependent viscosity within a square enclosure is solved numerically and the results are presented. The benchmark problem`s geometrical and mathematical descriptions adopted herein are those specified by the AdHoc Committee of Computational Heat Transfer which is compiling different solutions for this benchmark problem in heat transfer analysis. The objective is to compare solutions computed with several different algorithmic approaches for a problem having a large variation in fluid viscosity, characteristic of modeling turbulent flows with eddy diffusivity concepts. The results of the present analysis are submitted as a contribution to this comparison exercise that has for objective the assessment of the numerical accuracy of modeling the diffusion terms in the conservation equations with variable property.

20. Effect of airfoil (trailing-edge) thickness on the numerical solution of panel methods based on the Dirichlet boundary condition

NASA Technical Reports Server (NTRS)

Yon, Steven; Katz, Joseph; Plotkin, Allen

1992-01-01

The practical limit of airfoil thickness ratio for which acceptable engineering results are obtainable with the Dirichlet boundary-condition-based numerical methods is investigated. This is done by studying the effect of thickness on the calculated pressure distribution near the trailing edge and by comparing the aerodynamic coefficients with available exact solutions. The first objective of this study, owing to the wide use of such computational methods, is to demonstrate the numerical symptoms that occur when the body or wing thickness approaches zero and to increase the awareness of potential users of these methods. Additionally, an effort is made to obtain the practical limits of the trailing-edge thickness where such problems will appear in the flow solution, and to propose some possible cures for very thin airfoils or those with cusped trailing edges.

1. Mesh adaptation on the sphere using optimal transport and the numerical solution of a Monge-Ampère type equation

Weller, Hilary; Browne, Philip; Budd, Chris; Cullen, Mike

2016-03-01

An equation of Monge-Ampère type has, for the first time, been solved numerically on the surface of the sphere in order to generate optimally transported (OT) meshes, equidistributed with respect to a monitor function. Optimal transport generates meshes that keep the same connectivity as the original mesh, making them suitable for r-adaptive simulations, in which the equations of motion can be solved in a moving frame of reference in order to avoid mapping the solution between old and new meshes and to avoid load balancing problems on parallel computers. The semi-implicit solution of the Monge-Ampère type equation involves a new linearisation of the Hessian term, and exponential maps are used to map from old to new meshes on the sphere. The determinant of the Hessian is evaluated as the change in volume between old and new mesh cells, rather than using numerical approximations to the gradients. OT meshes are generated to compare with centroidal Voronoi tessellations on the sphere and are found to have advantages and disadvantages; OT equidistribution is more accurate, the number of iterations to convergence is independent of the mesh size, face skewness is reduced and the connectivity does not change. However anisotropy is higher and the OT meshes are non-orthogonal. It is shown that optimal transport on the sphere leads to meshes that do not tangle. However, tangling can be introduced by numerical errors in calculating the gradient of the mesh potential. Methods for alleviating this problem are explored. Finally, OT meshes are generated using observed precipitation as a monitor function, in order to demonstrate the potential power of the technique.

2. Coherence motion of photoinduced nonadiabatic charge transfer reaction in solution: A numerical study of pump-probe spectroscopy

Sheu, Sheh-Yi; Yang, Dah-Yen

1998-10-01

We study the photoinduced charge separation processes in solution through a pump-probe spectroscopy theory [Dah-Yen Yang and Sheh-Yi Sheu, J. Chem. Phys. 106, 9427 (1997)] numerically. We investigate the detailed mechanism of nonadiabatic transition processes via the transition differential flux analysis. For the harmonic potential surfaces, an electronic coherence motion is observed in the overdamped exothermic activationless and inverted regimes.

3. Numerical Solution for the Effect of Suction or Injection on Flow of Nanofluids Past a Stretching Sheet

2016-06-01

The flow of nanofluids past a stretching sheet has attracted much attention owing to its wide applications in industry and engineering. Numerical solution has been discussed in this article for studying the effect of suction (or injection) on flow of nanofluids past a stretching sheet. The numerical results carried out using Chebyshev collocation method (ChCM). Useful results for temperature profile, concentration profile, reduced Nusselt number, and reduced Sherwood number are discussed in tabular and graphical forms. It was also demonstrated that both temperature and concentration profiles decrease by an increase from injection to suction. Moreover, the numerical results show that the temperature profiles decrease at high values of Prandtl number Pr. Finally, the present results showed that the reduced Nusselt number is a decreasing function, whereas the reduced Sherwood number is an increasing function at fixed values of Prandtl number Pr, Lewis number Le and suction (or injection) parameter s for variation of Brownian motion parameter Nb, and thermophoresis parameter Nt.

4. Solutions of the Two Dimensional Hubbard Model: Benchmarks and Results from a Wide Range of Numerical Algorithms

Leblanc, James

In this talk we present numerical results for ground state and excited state properties (energies, double occupancies, and Matsubara-axis self energies) of the single-orbital Hubbard model on a two-dimensional square lattice. In order to provide an assessment of our ability to compute accurate results in the thermodynamic limit we employ numerous methods including auxiliary field quantum Monte Carlo, bare and bold-line diagrammatic Monte Carlo, method of dual fermions, density matrix embedding theory, density matrix renormalization group, dynamical cluster approximation, diffusion Monte Carlo within a fixed node approximation, unrestricted coupled cluster theory, and multireference projected Hartree-Fock. We illustrate cases where agreement between different methods is obtained in order to establish benchmark results that should be useful in the validation of future results.

5. Numerical solution of the time dependent neutron transport equation by the method of the characteristics

SciTech Connect

Talamo, Alberto

2013-05-01

This study presents three numerical algorithms to solve the time dependent neutron transport equation by the method of the characteristics. The algorithms have been developed taking into account delayed neutrons and they have been implemented into the novel MCART code, which solves the neutron transport equation for two-dimensional geometry and an arbitrary number of energy groups. The MCART code uses regular mesh for the representation of the spatial domain, it models up-scattering, and takes advantage of OPENMP and OPENGL algorithms for parallel computing and plotting, respectively. The code has been benchmarked with the multiplication factor results of a Boiling Water Reactor, with the analytical results for a prompt jump transient in an infinite medium, and with PARTISN and TDTORT results for cross section and source transients. The numerical simulations have shown that only two numerical algorithms are stable for small time steps.

6. Numerical solution of the Boltzmann equation for the collective modes of trapped Fermi gases

SciTech Connect

Lepers, Thomas; Davesne, Dany; Chiacchiera, Silvia; Urban, Michael

2010-08-15

We numerically solve the Boltzmann equation for trapped fermions in the normal phase by using the test-particle method. After discussing a couple of tests in order to estimate the reliability of the method, we apply it to the description of collective modes in a spherical harmonic trap. The numerical results are compared with those obtained previously by taking moments of the Boltzmann equation. We find that the general shape of the response function is very similar in both methods, but the relaxation time obtained from the simulation is significantly longer than that predicted by the method of moments. It is shown that the result of the method of moments can be corrected by including fourth-order moments in addition to the usual second-order ones and that this method agrees very well with our numerical simulations.

7. Study compares methods for the numerical solution of ordinary differential equations

NASA Technical Reports Server (NTRS)

1966-01-01

Study compares the use of five different methods for the computer solution of the restricted three-body problem. It describes the implementation of each method on a burroughs B-5000 computer and in terms of speed and accuracy.

8. Efficient numerical solution of the time fractional diffusion equation by mapping from its Brownian counterpart

Stokes, Peter W.; Philippa, Bronson; Read, Wayne; White, Ronald D.

2015-02-01

The solution of a Caputo time fractional diffusion equation of order 0 < α < 1 is expressed in terms of the solution of a corresponding integer order diffusion equation. We demonstrate a linear time mapping between these solutions that allows for accelerated computation of the solution of the fractional order problem. In the context of an N-point finite difference time discretisation, the mapping allows for an improvement in time computational complexity from O (N2) to O (Nα), given a precomputation of O (N 1 + α ln ⁡ N). The mapping is applied successfully to the least squares fitting of a fractional advection-diffusion model for the current in a time-of-flight experiment, resulting in a computational speed up in the range of one to three orders of magnitude for realistic problem sizes.

9. Alternating irrigation water quality as a method to control solute concentrations and mass fluxes below irrigated fields: A numerical study

Russo, David

2016-05-01

The aim of the present numerical study was to extend the data-driven protocol for the control of soil salinity, to control chloride and nitrate concentrations and mass fluxes below agricultural fields irrigated with treated waste water (TWW). The protocol is based on alternating irrigation water quality between TWW and desalinized water (DSW), guided by solute concentrations at soil depth, zs. Two different schemes, the first requires measurements of soil solution concentrations of chloride and nitrate at zs, while, the second scheme requires only measurements of soil solution EC at zs, were investigated. For this purpose, 3-D numerical simulations of flow and transport were performed for variably saturated, spatially heterogeneous, flow domains located at two different field sites. The sites differ in crop type, irrigation method, and in their lithology; these differences, in turn, considerably affect the performance of the proposed schemes, expressed in terms of their ability to reduce solute concentrations that drained below the root zone. Results of the analyses suggest that the proposed data-driven schemes allow the use of low-quality water for irrigation, while minimizing the consumption of high-quality water to a level, which, for given climate, soil, crop, irrigation method, and water quality, may be determined by the allowable nitrate and chloride concentrations in the groundwater. The results of the present study indicate that with respect to the diminution of groundwater contamination by chloride and nitrate, the more data demanding, first scheme is superior the second scheme.

10. Distributions for negative-feedback-regulated stochastic gene expression: dimension reduction and numerical solution of the chemical master equation.

PubMed

Zeron, Eduardo S; Santillán, Moisés

2010-05-21

In this work we introduce a novel approach to study biochemical noise. It comprises a simplification of the master equation of complex reaction schemes (via an adiabatic approximation) and the numerical solution of the reduced master equation. The accuracy of this procedure is tested by comparing its results with analytic solutions (when available) and with Gillespie stochastic simulations. We further employ our approach to study the stochastic expression of a simple gene network, which is subject to negative feedback regulation at the transcriptional level. Special attention is paid to the influence of negative feedback on the amplitude of intrinsic noise, as well as on the relaxation rate of the system probability distribution function to the steady solution. Our results suggest the existence of an optimal feedback strength that maximizes this relaxation rate.

11. Numerical continuation of solution at a singular point of high codimension for systems of nonlinear algebraic or transcendental equations

Krasnikov, S. D.; Kuznetsov, E. B.

2016-09-01

Numerical continuation of solution through certain singular points of the curve of the set of solutions to a system of nonlinear algebraic or transcendental equations with a parameter is considered. Bifurcation points of codimension two and three are investigated. Algorithms and computer programs are developed that implement the procedure of discrete parametric continuation of the solution and find all branches at simple bifurcation points of codimension two and three. Corresponding theorems are proved, and each algorithm is rigorously justified. A novel algorithm for the estimation of errors of tangential vectors at simple bifurcation points of a finite codimension m is proposed. The operation of the computer programs is demonstrated by test examples, which allows one to estimate their efficiency and confirm the theoretical results.

12. Numerical solutions of linear differential-algebraic equation systems via Hartley series

Ünal, Emrah; Yalçın, Numan; ćelik, Ercan

2014-08-01

In this paper, Hartley series are presented first. Then, the operational matrix of integration together with the product and coefficient matrices are presented. They are used to transform linear differential equation systems to a set of linear algebraic equations. Finally, numerical examples are given.

13. ORDMET: A General Algorithm for Constructing All Numerical Solutions to Ordered Metric Data

ERIC Educational Resources Information Center

McClelland, Gary; Coombs, Clyde H.

1975-01-01

ORDMET is applicable to structures obtained from additive conjoint measurement designs, unfolding theory, general Fechnerian scaling, types of multidimensional scaling, and ordinal multiple regression. A description is obtained of the space containing all possible numerical representations which can satisfy the structure, size, and shape of which…

14. Numerical solution of an extended White-Metzner model for eccentric Taylor-Couette flow

Germann, N.; Dressler, M.; Windhab, E. J.

2011-09-01

In this study, we have developed a new numerical approach to solve differential-type viscoelastic fluid models for a commonly used benchmark problem, namely, the steady Taylor—Couette flow between eccentric cylinders. The proposed numerical approach is special in that the nonlinear system of discretized algebraic flow equations is solved iteratively using a Newton-Krylov method along with an inverse-based incomplete lower-upper preconditioner. The numerical approach has been validated by solving the benchmark problem for the upper-convected Maxwell model at a large Deborah number. Excellent agreement with the numerical data reported in the literature has been found. In addition, a parameter study was performed for an extended White-Metzner model. A large eccentricity ratio was chosen for the cylinder system in order to allow flow recirculation to occur. We detected several interesting phenomena caused by the large eccentricity ratio of the cylinder system and by the viscoelastic nature of the fluid. Encouraged by the results of this study, we intend to investigate other polymeric fluids having a more complex microstructure in an eccentric annular flow field.

15. Numerical solution of the scalar-wave equation for inhomogeneous cylindrical dielectric waveguides.

PubMed

Rose, J W; Mitra, S S

1981-09-01

An initial-value algorithm derived from the Ricatti transformation of the scalar-wave equation is used to find the eigenvalues of inhomogeneous cylindrical dielectric waveguides. The numerical accuracy of the technique is investigated for cladded parabolic and step-index cylindrical refractive-index profiles.

16. Return Trajectory of the SpaceShipTwo Spacecraft--Numerical Solution

ERIC Educational Resources Information Center

Slegr, J.; Kraus, I.

2012-01-01

SpaceShipTwo is a private spaceplane project which is intended for space tourism. Very few details about its construction and flight characteristics are available for the public, but with proper numerical methods some interesting results can be obtained using secondary school mathematics. An exercise about SpaceShipTwo can be used as a…

17. Return trajectory of the SpaceShipTwo spacecraft—numerical solution

Slegr, J.; Kraus, I.

2012-05-01

SpaceShipTwo is a private spaceplane project which is intended for space tourism. Very few details about its construction and flight characteristics are available for the public, but with proper numerical methods some interesting results can be obtained using secondary school mathematics. An exercise about SpaceShipTwo can be used as a motivational factor in physics lessons.

18. Accurate and absolute diffusion measurements of Rhodamine 6G in low-concentration aqueous solutions by the PGSE-WATERGATE sequence

SciTech Connect

Majer, G.; Zick, K.

2015-04-28

A pulsed field gradient spin-echo nuclear magnetic resonance (NMR) sequence with solvent suppression (PGSE-WATERGATE) was applied to accurately measure the diffusion coefficients of Rhodamine 6G (Rh6G) in low-concentration aqueous solutions. Three samples with Rh6G concentrations of C{sub Rh6G} = 1, 4.5, and 25 μM were investigated. The precise determination of the diffusion coefficients in this low-concentration range was made possible by using a cryogenically cooled NMR probe and by the effective solvent suppression of the PGSE-WATERGATE sequence. The present results bridge the gap between diffusion data measured by fluorescence correlation spectroscopy in the single molecule limit and diffusivities obtained by pulsed field gradient NMR (PFG-NMR) without solvent suppression at higher concentrations. To further extend the concentration range, the diffusion coefficient of Rh6G was also measured on a sample with C{sub Rh6G} = 410 μM by PFG-NMR. The overall concentration dependence of the Rh6G diffusion at 25 °C is discussed in terms of dimerization of the Rh6G molecules. The concentration-dependent monomer/dimer proportion is deduced from the diffusion data.

19. Recommendations for numerical solution of reinforced-panel and fuselage-ring problems

NASA Technical Reports Server (NTRS)

Hoff, N J; Libby, Paul A

1949-01-01

Procedures are recommended for solving the equations of equilibrium of reinforced panels and isolated fuselage rings as represented by the external loads and the operations table established according to Southwell's method. From the solution of these equations the stress distribution can be easily determined. The method of systematic relaxations, the matrix-calculus method, and several other methods applicable in special cases are discussed. Definite recommendations are made for obtaining the solution of reinforced-panel problems which are generally designated as shear lag problems. The procedures recommended are demonstrated in the analysis of a number of panels. In the case of fuselage rings it is not possible to make definite recommendations for the solution of the equilibrium equations for all rings and loadings. However, suggestions based on the latest experience are made and demonstrated on several rings.

20. A long-term numerical solution for the insolation quantities of the Earth

Laskar, J.; Robutel, P.; Joutel, F.; Gastineau, M.; Correia, A. C. M.; Levrard, B.

2004-12-01

We present here a new solution for the astronomical computation of the insolation quantities on Earth spanning from -250 Myr to 250 Myr. This solution has been improved with respect to La93 (Laskar et al. \\cite{Laskar1993}) by using a direct integration of the gravitational equations for the orbital motion, and by improving the dissipative contributions, in particular in the evolution of the Earth-Moon System. The orbital solution has been used for the calibration of the Neogene period (Lourens et al. \\cite{Lourens2004}), and is expected to be used for age calibrations of paleoclimatic data over 40 to 50 Myr, eventually over the full Palaeogene period (65 Myr) with caution. Beyond this time span, the chaotic evolution of the orbits prevents a precise determination of the Earth's motion. However, the most regular components of the orbital solution could still be used over a much longer time span, which is why we provide here the solution over 250 Myr. Over this time interval, the most striking feature of the obliquity solution, apart from a secular global increase due to tidal dissipation, is a strong decrease of about 0.38 degree in the next few millions of years, due to the crossing of the s6+g5-g6 resonance (Laskar et al. \\cite{Laskar1993}). For the calibration of the Mesozoic time scale (about 65 to 250 Myr), we propose to use the term of largest amplitude in the eccentricity, related to g2-g5, with a fixed frequency of 3.200''/yr, corresponding to a period of 405 000 yr. The uncertainty of this time scale over 100 Myr should be about 0.1%, and 0.2% over the full Mesozoic era.

1. Application of multiquadric method for numerical solution of elliptic partial differential equations

SciTech Connect

Sharan, M.; Kansa, E.J.; Gupta, S.

1994-01-01

We have used the multiquadric (MQ) approximation scheme for the solution of elliptic partial differential equations with Dirichlet and/or Neumann boundary conditions. The scheme has the advantage to use the data points in arbitrary locations with an arbitrary ordering. Two dimensional Laplace, Poisson and Biharmonic equations describing the various physical processes, have been taken as the test examples. The agreement is found to be very good between the computed and exact solutions. The method also provides an excellent approximation with curve boundary.

2. Numerical solution of the thermal influence of oil well cluster on permafrost

Afanaseva, N. M.; Kolesov, A. E.

2016-10-01

In this work, we study the thermal effects around the oil well cluster on permafrost using numerical modeling. We use the mathematical model of heat transfer with phase transitions. To take into account the arrangement of wells in a cluster, three-dimensional domains with complex geometry are employed, which leads to the use of finite element approximation in space. For time approximation we use fully implicit scheme with linearization of nonlinear coefficients. Numerical implementations are performed using open-source libraries and programs for scientific and engineering computations. To predict the temperature field and formation of thawing area around wells with different sets of input parameters we conduct large-scale computational experiments on the supercomputer of the North-Eastern Federal University.

3. Numerical solutions and laser-Doppler measurements of spin-up

NASA Technical Reports Server (NTRS)

Warn-Varnas, A.; Piacsek, S.; Fowlis, W. W.; Lee, S. M.

1978-01-01

The spin-up flow in a cylinder of homogeneous fluid has been examined both experimentally and numerically. A series of laser-Doppler measurements was made of the zonal flow over a range of Ekman numbers and Rossby numbers at various locations in the interior of the flow. These measurements exceed previous ones in accuracy. The weak inertial modes excited by the impulsive start are detectable. The numerical simulations used the primitive equations in axisymmetric form and employed finite-difference techniques on both constant and variable grids. The number of grid points necessary to resolve the Ekman layers was determined. A thorough comparison of the simulations and the experimental measurements is made which includes the details of the amplitude and frequency of the inertial modes. Agreement to within the experimental tolerance is achieved. Analytical results for conditions identical to those in the experiments are not available but some similar linear and nonlinear theories are also compared with the experiments.

4. Study and numerical solution of a generalized mathematical model of isothermal adsorption

SciTech Connect

Komissarov, Yu.A.; Vetokhin, V.N.; Tsenev, V.A.; Gordeeva, E.L.

1995-06-01

A generalized mathematical model of isothermal adsorption that takes into account mass transfer on the surface of a particle, diffusion in micro- and macropores, and dispersion along the length of the apparatus is considered The parameters {lambda} and {var_phi}{sup 2} determine the dominating effect of any of the mass transfer mechanisms of the adsorption process. A numerical algorithm for solving the generalized adsorption model is suggested.

5. A Numerical Algorithm for Finding Solution of Cross-Coupled Algebraic Riccati Equations

Mukaidani, Hiroaki; Yamamoto, Seiji; Yamamoto, Toru

In this letter, a computational approach for solving cross-coupled algebraic Riccati equations (CAREs) is investigated. The main purpose of this letter is to propose a new algorithm that combines Newton's method with a gradient-based iterative (GI) algorithm for solving CAREs. In particular, it is noteworthy that both a quadratic convergence under an appropriate initial condition and reduction in dimensions for matrix computation are both achieved. A numerical example is provided to demonstrate the efficiency of this proposed algorithm.

6. AN EFFICIENT NUMERICAL METHOD FOR THE SOLUTION OF THE L(2) OPTIMAL MASS TRANSFER PROBLEM.

PubMed

Haber, Eldad; Rehman, Tauseef; Tannenbaum, Allen

2010-01-01

In this paper we present a new computationally efficient numerical scheme for the minimizing flow approach for the computation of the optimal L(2) mass transport mapping. In contrast to the integration of a time dependent partial differential equation proposed in [S. Angenent, S. Haker, and A. Tannenbaum, SIAM J. Math. Anal., 35 (2003), pp. 61-97], we employ in the present work a direct variational method. The efficacy of the approach is demonstrated on both real and synthetic data. PMID:21278828

7. AN EFFICIENT NUMERICAL METHOD FOR THE SOLUTION OF THE L2 OPTIMAL MASS TRANSFER PROBLEM*

PubMed Central

Haber, Eldad; Rehman, Tauseef; Tannenbaum, Allen

2010-01-01

In this paper we present a new computationally efficient numerical scheme for the minimizing flow approach for the computation of the optimal L2 mass transport mapping. In contrast to the integration of a time dependent partial differential equation proposed in [S. Angenent, S. Haker, and A. Tannenbaum, SIAM J. Math. Anal., 35 (2003), pp. 61–97], we employ in the present work a direct variational method. The efficacy of the approach is demonstrated on both real and synthetic data. PMID:21278828

8. Relaxation of hot and massive tracers using numerical solutions of the Boltzmann equation.

PubMed

Khurana, Saheba; Thachuk, Mark

2016-03-14

A numerical method using B-splines is used to solve the linear Boltzmann equation describing the energy relaxation of massive tracer particles moving through a dilute bath gas. The smooth and rough hard sphere and Maxwell molecule models are used with a variety of mass ratios and initial energies to test the capability of the numerical method. Massive tracers are initialized with energies typically found in energy loss experiments in mass spectrometry using biomolecules. The method is also used to examine the applicability of known expressions for the kinetic energy decay from the Fokker-Planck equation for the Rayleigh gas, where we find that results are generally good provided that the initial energy is properly bounded. Otherwise, the energy decay is not constant and a more complex behaviour occurs. The validity of analytical expressions for drag coefficients for spherical particles under specular and diffuse scattering is also tested. We find such expressions are generally good for hard spheres but cannot account, as expected, for the softer repulsive walls of the Maxwell (and real) molecules. Overall, the numerical method performed well even when tracers more than 400 times as massive as the bath were initialized with energies very far from equilibrium. This is a range of applicability beyond many of the standard methods for solving the Boltzmann equation. PMID:26979675

9. Numerical Solution of Problems in Calculus of Variations by Homotopy Perturbation Method

SciTech Connect

Jafari, M. A.; Aminataei, A.

2008-09-01

In this work we use Homotopy Perturbation Method (HPM) to solve differential equations that arise in variational problems. To illustrate the method some examples are provided. The results show the efficiency and accuracy of the HPM. HPM can be considered an alternative method to Adomian decomposition method. Both of these methods can obtain analytic form of the solution in some cases.

10. Numerical Solutions and Structures of Double Quantum Jet Solving by an Upwind Scheme

Lin, San-Yih

2005-11-01

The solutions of a double quantum jet are analyzed by solving the quantum fluid dynamical formulation (QFD) of the Schr"odinger equation. The QFD equations are obtained by expressing the Schr"odinger wave function as =ρ^1/2(iS/)and u=(u,v). In QFD, Q=-ρ-1/2δρ^1/2 is called as quantum potential. An upwind method is developed to solve the QFD equations. The method use a third-order upwind method to discrete convection terms and the central finite difference method to discrete the quantum potential. A fourth-order Runge-Kutta method is used for time marching. Two cases, one-dimensional free particle with external potential and two-dimensional free particle with external potential, are presented to illustrate the accuracy of the QFD solver. The computational results are compared well with the results obtained by solving the Schr"odinger equation. Finally, the QFD solver is applied to solve the solutions of a double quantum jet and to investigate its structures. First, a mathematical formulation is derived to describe the double quantum jet. The jet has the probability density equals 2 and the velocity equals 2 at the inlet of the jet. Then, the solutions are computed by the QFD solver. The structures of the solutions are affected by the strength of the quantum potential. The interesting phenomena of quantum clustering are found.

11. Kmonodium, a Program for the Numerical Solution of the One-Dimensional Schrodinger Equation

ERIC Educational Resources Information Center

Angeli, Celestino; Borini, Stefano; Cimiraglia, Renzo

2005-01-01

A very simple strategy for the solution of the Schrodinger equation of a particle moving in one dimension subjected to a generic potential is presented. This strategy is implemented in a computer program called Kmonodium, which is free and distributed under the General Public License (GPL).

12. Numerical investigation of stability of breather-type solutions of the nonlinear Schrödinger equation

Calini, A.; Schober, C. M.

2013-09-01

In this article we present the results of a broad numerical investigation on the stability of breather-type solutions of the nonlinear Schrödinger (NLS) equation, specifically the one- and two-mode breathers for an unstable plane wave, which are frequently used to model rogue waves. The numerical experiments involve large ensembles of perturbed initial data for six typical random perturbations. Ensemble estimates of the "closeness", A(t), of the perturbed solution to an element of the respective unperturbed family indicate that the only neutrally stable breathers are the ones of maximal dimension, that is: given an unstable background with N unstable modes, the only neutrally stable breathers are the N-dimensional ones (obtained as a superimposition of N simple breathers via iterated Backlund transformations). Conversely, breathers which are not fully saturated are sensitive to noisy environments and are unstable. Interestingly, A(t) is smallest for the coalesced two-mode breather indicating the coalesced case may be the most robust two-mode breather in a laboratory setting. The numerical simulations confirm and provide a realistic realization of the stability behavior established analytically by the authors.

13. Numerical solution of the stationary multicomponent nonlinear Schrödinger equation with a constraint on the angular momentum.

PubMed

Sandin, Patrik; Ögren, Magnus; Gulliksson, Mårten

2016-03-01

We formulate a damped oscillating particle method to solve the stationary nonlinear Schrödinger equation (NLSE). The ground-state solutions are found by a converging damped oscillating evolution equation that can be discretized with symplectic numerical techniques. The method is demonstrated for three different cases: for the single-component NLSE with an attractive self-interaction, for the single-component NLSE with a repulsive self-interaction and a constraint on the angular momentum, and for the two-component NLSE with a constraint on the total angular momentum. We reproduce the so-called yrast curve for the single-component case, described in [A. D. Jackson et al., Europhys. Lett. 95, 30002 (2011)], and produce for the first time an analogous curve for the two-component NLSE. The numerical results are compared with analytic solutions and competing numerical methods. Our method is well suited to handle a large class of equations and can easily be adapted to further constraints and components.

14. Mapping Spatio-Temporal Diffusion inside the Human Brain Using a Numerical Solution of the Diffusion Equation

PubMed Central

Zhan, Wang; Jiang, Li; Loew, Murray; Yang, Yihong

2008-01-01

Diffusion is an important mechanism for molecular transport in living biological tissues. Diffusion magnetic resonance imaging (dMRI) provides a unique probe to examine microscopic structures of the tissues in vivo, but current dMRI techniques usually ignore the spatio-temporal evolution process of the diffusive medium. In the present study, we demonstrate the feasibility to reveal the spatio-temporal diffusion process inside the human brain based on a numerical solution of the diffusion equation. Normal human subjects were scanned with a diffusion tensor imaging (DTI) technique on a 3-Tesla MRI scanner, and the diffusion tensor in each voxel was calculated from the DTI data. The diffusion equation, a partial-derivative description of Fick’s Law for the diffusion process, was discretized into equivalent algebraic equations. A finite-difference method was employed to obtain the numerical solution of the diffusion equation with a Crank-Nicholson iteration scheme to enhance the numerical stability. By specifying boundary and initial conditions, the spatio-temporal evolution of the diffusion process inside the brain can be virtually reconstructed. Our results exhibit similar medium profiles and diffusion coefficients as those of light fluorescence dextrans measured in integrative optical imaging experiments. The proposed method highlights the feasibility to non-invasively estimate the macroscopic diffusive transport time for a molecule in a given region of the brain. PMID:18440744

15. Numerous Numerals.

ERIC Educational Resources Information Center

Henle, James M.

This pamphlet consists of 17 brief chapters, each containing a discussion of a numeration system and a set of problems on the use of that system. The numeration systems used include Egyptian fractions, ordinary continued fractions and variants of that method, and systems using positive and negative bases. The book is informal and addressed to…

16. Multiphase flow experiments, mathematical modeling and numerical simulation of the water - gas - solute movement

Li, Y.; Ma, X.; Su, N.

2013-12-01

The movement of water and solute into and through the vadose zone is, in essence, an issue of immiscible displacement in pore-space network of a soil. Therefore, multiphase flow and transport in porous media, referring to three medium: air, water, and the solute, pose one of the largest unresolved challenges for porous medium fluid seepage. However, this phenomenon has always been largely neglected. It is expected that a reliable analysis model of the multi-phase flow in soil can truly reflect the process of natural movement about the infiltration, which is impossible to be observed directly. In such cases, geophysical applications of the nuclear magnetic resonance (NMR) provides the opportunity to measure the water movements into soils directly over a large scale from tiny pore to regional scale, accordingly enable it available both on the laboratory and on the field. In addition, the NMR provides useful information about the pore space properties. In this study, we proposed both laboratory and field experiments to measure the multi-phase flow parameters, together with optimize the model in computer programming based on the fractional partial differential equations (fPDE). In addition, we establish, for the first time, an infiltration model including solute flowing with water, which has huge influence on agriculture and soil environment pollution. Afterwards, with data collected from experiments, we simulate the model and analyze the spatial variability of parameters. Simulations are also conducted according to the model to evaluate the effects of airflow on water infiltration and other effects such as solute and absorption. It has significant meaning to oxygen irrigation aiming to higher crop yield, and shed more light into the dam slope stability. In summary, our framework is a first-time model added in solute to have a mathematic analysis with the fPDE and more instructive to agriculture activities.

17. Evidence that bisphenol A (BPA) can be accurately measured without contamination in human serum and urine, and that BPA causes numerous hazards from multiple routes of exposure.

PubMed

vom Saal, Frederick S; Welshons, Wade V

2014-12-01

There is extensive evidence that bisphenol A (BPA) is related to a wide range of adverse health effects based on both human and experimental animal studies. However, a number of regulatory agencies have ignored all hazard findings. Reports of high levels of unconjugated (bioactive) serum BPA in dozens of human biomonitoring studies have also been rejected based on the prediction that the findings are due to assay contamination and that virtually all ingested BPA is rapidly converted to inactive metabolites. NIH and industry-sponsored round robin studies have demonstrated that serum BPA can be accurately assayed without contamination, while the FDA lab has acknowledged uncontrolled assay contamination. In reviewing the published BPA biomonitoring data, we find that assay contamination is, in fact, well controlled in most labs, and cannot be used as the basis for discounting evidence that significant and virtually continuous exposure to BPA must be occurring from multiple sources.

18. Evidence that bisphenol A (BPA) can be accurately measured without contamination in human serum and urine, and that BPA causes numerous hazards from multiple routes of exposure

PubMed Central

vom Saal, Frederick S.; Welshons, Wade V.

2016-01-01

There is extensive evidence that bisphenol A (BPA) is related to a wide range of adverse health effects based on both human and experimental animal studies. However, a number of regulatory agencies have ignored all hazard findings. Reports of high levels of unconjugated (bioactive) serum BPA in dozens of human biomonitoring studies have also been rejected based on the prediction that the findings are due to assay contamination and that virtually all ingested BPA is rapidly converted to inactive metabolites. NIH and industry-sponsored round robin studies have demonstrated that serum BPA can be accurately assayed without contamination, while the FDA lab has acknowledged uncontrolled assay contamination. In reviewing the published BPA biomonitoring data, we find that assay contamination is, in fact, well controlled in most labs, and cannot be used as the basis for discounting evidence that significant and virtually continuous exposure to BPA must be occurring from multiple sources. PMID:25304273

19. Practical aspects of spatially high accurate methods

NASA Technical Reports Server (NTRS)

Godfrey, Andrew G.; Mitchell, Curtis R.; Walters, Robert W.

1992-01-01

The computational qualities of high order spatially accurate methods for the finite volume solution of the Euler equations are presented. Two dimensional essentially non-oscillatory (ENO), k-exact, and 'dimension by dimension' ENO reconstruction operators are discussed and compared in terms of reconstruction and solution accuracy, computational cost and oscillatory behavior in supersonic flows with shocks. Inherent steady state convergence difficulties are demonstrated for adaptive stencil algorithms. An exact solution to the heat equation is used to determine reconstruction error, and the computational intensity is reflected in operation counts. Standard MUSCL differencing is included for comparison. Numerical experiments presented include the Ringleb flow for numerical accuracy and a shock reflection problem. A vortex-shock interaction demonstrates the ability of the ENO scheme to excel in simulating unsteady high-frequency flow physics.

20. Numerical solutions of the Navier-Stokes equations for the supersonic laminar flow over a two-dimensional compression corner

NASA Technical Reports Server (NTRS)

Carter, J. E.

1972-01-01

Numerical solutions have been obtained for the supersonic, laminar flow over a two-dimensional compression corner. These solutions were obtained as steady-state solutions to the unsteady Navier-Stokes equations using the finite difference method of Brailovskaya, which has second-order accuracy in the spatial coordinates. Good agreement was obtained between the computed results and wall pressure distributions measured experimentally for Mach numbers of 4 and 6.06, and respective Reynolds numbers, based on free-stream conditions and the distance from the leading edge to the corner. In those calculations, as well as in others, sufficient resolution was obtained to show the streamline pattern in the separation bubble. Upstream boundary conditions to the compression corner flow were provided by numerically solving the unsteady Navier-Stokes equations for the flat plate flow field, beginning at the leading edge. The compression corner flow field was enclosed by a computational boundary with the unknown boundary conditions supplied by extrapolation from internally computed points.

1. The Space-Time Conservation Element and Solution Element Method-A New High-Resolution and Genuinely Multidimensional Paradigm for Solving Conservation Laws. 2; Numerical Simulation of Shock Waves and Contact Discontinuities

NASA Technical Reports Server (NTRS)

Wang, Xiao-Yen; Chow, Chuen-Yen; Chang, Sin-Chung

1998-01-01

Without resorting to special treatment for each individual test case, the 1D and 2D CE/SE shock-capturing schemes described previously (in Part I) are used to simulate flows involving phenomena such as shock waves, contact discontinuities, expansion waves and their interactions. Five 1D and six 2D problems are considered to examine the capability and robustness of these schemes. Despite their simple logical structures and low computational cost (for the 2D CE/SE shock-capturing scheme, the CPU time is about 2 micro-secs per mesh point per marching step on a Cray C90 machine), the numerical results, when compared with experimental data, exact solutions or numerical solutions by other methods, indicate that these schemes can accurately resolve shock and contact discontinuities consistently.

2. Numerical solutions of several reflected shock-wave flow fields with nonequilibrium chemical reactions

NASA Technical Reports Server (NTRS)

Hanson, R. K.; Presley, L. L.; Williams, E. V.

1972-01-01

The method of characteristics for a chemically reacting gas is used in the construction of the time-dependent, one-dimensional flow field resulting from the normal reflection of an incident shock wave at the end wall of a shock tube. Nonequilibrium chemical reactions are allowed behind both the incident and reflected shock waves. All the solutions are evaluated for oxygen, but the results are generally representative of any inviscid, nonconducting, and nonradiating diatomic gas. The solutions clearly show that: (1) both the incident- and reflected-shock chemical relaxation times are important in governing the time to attain steady state thermodynamic properties; and (2) adjacent to the end wall, an excess-entropy layer develops wherein the steady state values of all the thermodynamic variables except pressure differ significantly from their corresponding Rankine-Hugoniot equilibrium values.

3. A fully vectorized numerical solution of the incompressible Navier-Stokes equations. Ph.D. Thesis

NASA Technical Reports Server (NTRS)

Patel, N.

1983-01-01

A vectorizable algorithm is presented for the implicit finite difference solution of the incompressible Navier-Stokes equations in general curvilinear coordinates. The unsteady Reynolds averaged Navier-Stokes equations solved are in two dimension and non-conservative primitive variable form. A two-layer algebraic eddy viscosity turbulence model is used to incorporate the effects of turbulence. Two momentum equations and a Poisson pressure equation, which is obtained by taking the divergence of the momentum equations and satisfying the continuity equation, are solved simultaneously at each time step. An elliptic grid generation approach is used to generate a boundary conforming coordinate system about an airfoil. The governing equations are expressed in terms of the curvilinear coordinates and are solved on a uniform rectangular computational domain. A checkerboard SOR, which can effectively utilize the computer architectural concept of vector processing, is used for iterative solution of the governing equations.

4. The Numerical Solution of the Time-Dependent Nernst-Planck Equations

PubMed Central

Cohen, H.; Cooley, J. W.

1965-01-01

Calculations are reported of the time-dependent Nernst-Planck equations for a thin permeable membrane between electrolytic solutions. Charge neutrality is assumed for the time-dependent case. The response of such a membrane system to step current input is measured in terms of the time and space changes in concentration, electrical potential, and effective conductance. The report also includes discussion of boundary effects that occur when charge neutrality does not hold in the steady-state case. PMID:14268950

5. Influence of karst evolution on solute transport evaluated by process-based numerical modelling

Hubinger, Bernhard; Birk, Steffen

2010-05-01

Karst waters are of major interest in water resources management. Because of their inherent properties karst systems show great vulnerability with regard to contaminants. Karst systems include highly permeable solution conduit networks formed by chemical aggressive water embedded in a fissured matrix. Small initial voids are widened and thus act as preferential passages, where flow is rapid and often turbulent. Water discharging at karst spring originates from different pathways with different residence times. Contaminant transport through conduit pathways is very rapid, whereas flow through the fissured porous matrix is much slower. Thus, on the one hand, pollutants may be rapidly transported and reach high concentrations at the karst spring shortly after their release; on the other hand, the existence of slow flow components may cause the pollution to last for long times. In this work, solute transport properties of karst aquifers are investigated using generic conduit networks of hydraulically connected proto-conduits with initially log-normally distributed apertures in the millimetre range and below. Conduit evolution is modelled by coupling flow, transport, and dissolution processes, whereby single conduits are widened up to the metre range. Thus, different stages of karst evolution can be distinguished. The resulting flow systems provide the basis for modelling advective-dispersive transport of non-reactive solutes through the network of more or less widened (proto-)conduits. The general transport characteristics in karst systems as well as the influence of heterogeneities and structures on solute transport are illustrated for cases of direct injection into the conduit systems at different evolutionary stages. The resulting breakthrough curves typically show several distinct, chronologically shifted peaks with long tailings, which appears to be similar to data from field tracer experiments.

6. Numerical solution of 2D and 3D turbulent internal flow problems

Chen, Naixing; Xu, Yanji

1991-08-01

The paper describes a method for solving numerically two-dimensional or axisymmetric, and three-dimensional turbulent internal flow problems. The method is based on an implicit upwinding relaxation scheme with an arbitrarily shaped conservative control volume. The compressible Reynolds-averaged Navier-Stokes equations are solved with a two-equation turbulence model. All these equations are expressed by using a nonorthogonal curvilinear coordinate system. The method is applied to study the compressible internal flow in modern power installations. It has been observed that predictions for two-dimensional and three-dimensional channels show very good agreement with experimental results.

7. Memory efficient solution of the primitive equations for numerical weather prediction on the CYBER 205

NASA Technical Reports Server (NTRS)

Tuccillo, J. J.

1984-01-01

Numerical Weather Prediction (NWP), for both operational and research purposes, requires only fast computational speed but also large memory. A technique for solving the Primitive Equations for atmospheric motion on the CYBER 205, as implemented in the Mesoscale Atmospheric Simulation System, which is fully vectorized and requires substantially less memory than other techniques such as the Leapfrog or Adams-Bashforth Schemes is discussed. The technique presented uses the Euler-Backard time marching scheme. Also discussed are several techniques for reducing computational time of the model by replacing slow intrinsic routines by faster algorithms which use only hardware vector instructions.

8. Numerical solution of the controlled Duffing oscillator by semi-orthogonal spline wavelets

Lakestani, M.; Razzaghi, M.; Dehghan, M.

2006-09-01

This paper presents a numerical method for solving the controlled Duffing oscillator. The method can be extended to nonlinear calculus of variations and optimal control problems. The method is based upon compactly supported linear semi-orthogonal B-spline wavelets. The differential and integral expressions which arise in the system dynamics, the performance index and the boundary conditions are converted into some algebraic equations which can be solved for the unknown coefficients. Illustrative examples are included to demonstrate the validity and applicability of the technique.

9. Numerical solution for weight reduction model due to health campaigns in Spain

Mohammed, Maha A.; Noor, Noor Fadiya Mohd; Siri, Zailan; Ibrahim, Adriana Irawati Nur

2015-10-01

Transition model between three subpopulations based on Body Mass Index of Valencia community in Spain is considered. No changes in population nutritional habits and public health strategies on weight reduction until 2030 are assumed. The system of ordinary differential equations is solved using Runge-Kutta method of higher order. The numerical results obtained are compared with the predicted values of subpopulation proportion based on statistical estimation in 2013, 2015 and 2030. Relative approximate error is calculated. The consistency of the Runge-Kutta method in solving the model is discussed.

10. Numerical solution of non-Newtonian nanofluid flow over a stretching sheet

Nadeem, S.; Haq, Rizwan Ul; Khan, Z. H.

2014-06-01

The steady flow of a Jeffrey fluid model in the presence of nano particles is studied. Similarity transformation is used to convert the governing partial differential equations to a set of coupled nonlinear ordinary differential equations which are solved numerically. Behavior of emerging parameters is presented graphically and discussed for velocity, temperature and nanoparticles fraction. Variation of the reduced Nusselt and Sherwood number against physical parameters is presented graphically. It was found that reduced Nusselt number is decreasing function and reduced Sherwood number is increasing function of Brownian parameter and thermophoresis parameter.

11. Kondo Impurities in the Kitaev Spin Liquid: Numerical Renormalization Group Solution and Gauge-Flux-Driven Screening.

PubMed

Vojta, Matthias; Mitchell, Andrew K; Zschocke, Fabian

2016-07-15

Kitaev's honeycomb-lattice compass model describes a spin liquid with emergent fractionalized excitations. Here, we study the physics of isolated magnetic impurities coupled to the Kitaev spin-liquid host. We reformulate this Kondo-type problem in terms of a many-state quantum impurity coupled to a multichannel bath of Majorana fermions and present the numerically exact solution using Wilson's numerical renormalization group technique. Quantum phase transitions occur as a function of Kondo coupling and locally applied field. At zero field, the impurity moment is partially screened only when it binds an emergent gauge flux, while otherwise it becomes free at low temperatures. We show how Majorana degrees of freedom determine the fixed-point properties, make contact with Kondo screening in pseudogap Fermi systems, and discuss effects away from the dilute limit. PMID:27472132

12. Kondo Impurities in the Kitaev Spin Liquid: Numerical Renormalization Group Solution and Gauge-Flux-Driven Screening

Vojta, Matthias; Mitchell, Andrew K.; Zschocke, Fabian

2016-07-01

Kitaev's honeycomb-lattice compass model describes a spin liquid with emergent fractionalized excitations. Here, we study the physics of isolated magnetic impurities coupled to the Kitaev spin-liquid host. We reformulate this Kondo-type problem in terms of a many-state quantum impurity coupled to a multichannel bath of Majorana fermions and present the numerically exact solution using Wilson's numerical renormalization group technique. Quantum phase transitions occur as a function of Kondo coupling and locally applied field. At zero field, the impurity moment is partially screened only when it binds an emergent gauge flux, while otherwise it becomes free at low temperatures. We show how Majorana degrees of freedom determine the fixed-point properties, make contact with Kondo screening in pseudogap Fermi systems, and discuss effects away from the dilute limit.

13. Numerical solution of multi-dimensional compressible reactive flow using a parallel wavelet adaptive multi-resolution method

Grenga, Temistocle

The aim of this research is to further develop a dynamically adaptive algorithm based on wavelets that is able to solve efficiently multi-dimensional compressible reactive flow problems. This work demonstrates the great potential for the method to perform direct numerical simulation (DNS) of combustion with detailed chemistry and multi-component diffusion. In particular, it addresses the performance obtained using a massive parallel implementation and demonstrates important savings in memory storage and computational time over conventional methods. In addition, fully-resolved simulations of challenging three dimensional problems involving mixing and combustion processes are performed. These problems are particularly challenging due to their strong multiscale characteristics. For these solutions, it is necessary to combine the advanced numerical techniques applied to modern computational resources.

14. Numerical solution for melting of unfixed rectangular phase-change material under low-gravity environment

SciTech Connect

Asako, Y. . Dept. of Mechanical Engineering); Faghri, M. . Dept. of Mechanical Engineering); Charmchi, M. . Dept. of Mechanical Engineering); Bahrami, P.A. )

1994-02-01

An enthalpy method is employed to solve transport processes associated with melting of an unfixed rectangular phase change material (PCM) in a low-gravitational environment. This method permits the phase-change problems to be solved within fixed numerical grids, hence eliminating the need for coordinate transformation. The PCM, initially at its melting temperature, is placed inside a rectangular enclosure. The lower surface of the container is then exposed to a uniform temperature higher than the PCM melting temperature. The difference in densities of solid and liquid causes a force imbalance on the solid phase exceeds that of the liquid, the solid continually moves downward as melting progresses and hence generates a flow field within the liquid. The problem is formulated as a one-domain problem with the possibility of melting from all the PCM surfaces, and no approximation is made about the liquid film thickness under the melt. The governing equations are discretized by using a control-volume-based finite difference scheme with a new iterative method to correct for the downward solid-phase velocity. This will also speed up the convergence of the numerical procedure. The results are presented in the form of a parametric study of the effects of Archimedes number, Stefan number, Prandtl number, and the geometric parameters on the melt thickness, the downward solid velocity, the elevation of the top surface, and the volume of the solid PCM. They show that in a low-gravitational environment, the melting rate is very slow.

15. Numerical solutions of ideal quantum gas dynamical flows governed by semiclassical ellipsoidal-statistical distribution.

PubMed

Yang, Jaw-Yen; Yan, Chih-Yuan; Diaz, Manuel; Huang, Juan-Chen; Li, Zhihui; Zhang, Hanxin

2014-01-01

The ideal quantum gas dynamics as manifested by the semiclassical ellipsoidal-statistical (ES) equilibrium distribution derived in Wu et al. (Wu et al. 2012 Proc. R. Soc. A 468, 1799-1823 (doi:10.1098/rspa.2011.0673)) is numerically studied for particles of three statistics. This anisotropic ES equilibrium distribution was derived using the maximum entropy principle and conserves the mass, momentum and energy, but differs from the standard Fermi-Dirac or Bose-Einstein distribution. The present numerical method combines the discrete velocity (or momentum) ordinate method in momentum space and the high-resolution shock-capturing method in physical space. A decoding procedure to obtain the necessary parameters for determining the ES distribution is also devised. Computations of two-dimensional Riemann problems are presented, and various contours of the quantities unique to this ES model are illustrated. The main flow features, such as shock waves, expansion waves and slip lines and their complex nonlinear interactions, are depicted and found to be consistent with existing calculations for a classical gas. PMID:24399919

16. Unmitigated numerical solution to the diffraction term in the parabolic nonlinear ultrasound wave equation.

PubMed

Hasani, Mojtaba H; Gharibzadeh, Shahriar; Farjami, Yaghoub; Tavakkoli, Jahan

2013-09-01

Various numerical algorithms have been developed to solve the Khokhlov-Kuznetsov-Zabolotskaya (KZK) parabolic nonlinear wave equation. In this work, a generalized time-domain numerical algorithm is proposed to solve the diffraction term of the KZK equation. This algorithm solves the transverse Laplacian operator of the KZK equation in three-dimensional (3D) Cartesian coordinates using a finite-difference method based on the five-point implicit backward finite difference and the five-point Crank-Nicolson finite difference discretization techniques. This leads to a more uniform discretization of the Laplacian operator which in turn results in fewer calculation gridding nodes without compromising accuracy in the diffraction term. In addition, a new empirical algorithm based on the LU decomposition technique is proposed to solve the system of linear equations obtained from this discretization. The proposed empirical algorithm improves the calculation speed and memory usage, while the order of computational complexity remains linear in calculation of the diffraction term in the KZK equation. For evaluating the accuracy of the proposed algorithm, two previously published algorithms are used as comparison references: the conventional 2D Texas code and its generalization for 3D geometries. The results show that the accuracy/efficiency performance of the proposed algorithm is comparable with the established time-domain methods.

17. A Non-linear Conjugate Gradient Numerical Inverse Solution for the Problem of 3-D Global Electromagnetic Induction

Kelbert, A.; Schultz, A.; Egbert, G.

2006-12-01

We address the non-linear ill-posed inverse problem of reconstructing the global three-dimensional distribution of electrical conductivity in Earth's mantle. The authors have developed a numerical regularized least-squares inverse solution based on the non-linear conjugate gradients approach. We apply this methodology to the most current low-frequency global observatory data set by Fujii &Schultz (2002), that includes c- and d-responses. We obtain 4-8 layer models satisfying the data. We then describe the features common to all these models and discuss the resolution of our method.

18. Numerical solution of supersonic three-dimensional free-mixing flows using the parabolic-elliptic Navier-Stokes equations

NASA Technical Reports Server (NTRS)

Hirsh, R. S.

1976-01-01

A numerical method is presented for solving the parabolic-elliptic Navier-Stokes equations. The solution procedure is applied to three-dimensional supersonic laminar jet flow issuing parallel with a supersonic free stream. A coordinate transformation is introduced which maps the boundaries at infinity into a finite computational domain in order to eliminate difficulties associated with the imposition of free-stream boundary conditions. Results are presented for an approximate circular jet, a square jet, varying aspect ratio rectangular jets, and interacting square jets. The solution behavior varies from axisymmetric to nearly two-dimensional in character. For cases where comparisons of the present results with those obtained from shear layer calculations could be made, agreement was good.

19. Localized numerical impulse solutions in diffuse neural networks modeled by the complex fractional Ginzburg-Landau equation

Mvogo, Alain; Tambue, Antoine; Ben-Bolie, Germain H.; Kofané, Timoléon C.

2016-10-01

We investigate localized wave solutions in a network of Hindmarsh-Rose neural model taking into account the long-range diffusive couplings. We show by a specific analytical technique that the model equations in the infrared limit (wave number k → 0) can be governed by the complex fractional Ginzburg-Landau (CFGL) equation. According to the stiffness of the system, we propose both the semi and the linearly implicit Riesz fractional finite-difference schemes to solve efficiently the CFGL equation. The obtained fractional numerical solutions for the nerve impulse reveal localized short impulse properties. We also show the equivalence between the continuous CFGL and the discrete Hindmarsh-Rose models for relatively large network.

20. Numerical solution for Sakiadis flow of upper-convected Maxwell fluid using Cattaneo-Christov heat flux model

Mushtaq, A.; Abbasbandy, S.; Mustafa, M.; Hayat, T.; Alsaedi, A.

2016-01-01

Present work studies the well-known Sakiadis flow of Maxwell fluid along a moving plate in a calm fluid by considering the Cattaneo-Christov heat flux model. This recently developed model has the tendency to describe the characteristics of relaxation time for heat flux. Some numerical local similarity solutions of the associated problem are computed by two approaches namely (i) the shooting method and (ii) the Keller-box method. The solution is dependent on some interesting parameters which include the viscoelastic fluid parameter β, the dimensionless thermal relaxation time γ and the Prandtl number Pr. Our simulations indicate that variation in the temperature distribution with an increase in local Deborah number γ is non-monotonic. The results for the Fourier's heat conduction law can be obtained as special cases of the present study.

1. Numerical and experimental study of capillary-driven flow of PCR solution in hybrid hydrophobic microfluidic networks.

PubMed

Ramalingam, Naveen; Warkiani, Majid Ebrahimi; Ramalingam, Neevan; Keshavarzi, Gholamreza; Hao-Bing, Liu; Hai-Qing, Thomas Gong

2016-08-01

Capillary-driven microfluidics is essential for development of point-of-care diagnostic micro-devices. Polymerase chain reaction (PCR)-based micro-devices are widely developed and used in such point-of-care settings. It is imperative to characterize the fluid parameters of PCR solution for designing efficient capillary-driven microfluidic networks. Generally, for numeric modelling, the fluid parameters of PCR solution are approximated to that of water. This procedure leads to inaccurate results, which are discrepant to experimental data. This paper describes mathematical modeling and experimental validation of capillary-driven flow inside Poly-(dimethyl) siloxane (PDMS)-glass hybrid micro-channels. Using experimentally measured PCR fluid parameters, the capillary meniscus displacement in PDMS-glass microfluidic ladder network is simulated using computational fluid dynamic (CFD), and experimentally verified to match with the simulated data. PMID:27432321

2. Numerical Simulation of Natural Convection of a Nanofluid in an Inclined Heated Enclosure Using Two-Phase Lattice Boltzmann Method: Accurate Effects of Thermophoresis and Brownian Forces.

PubMed

Ahmed, Mahmoud; Eslamian, Morteza

2015-12-01

Laminar natural convection in differentially heated (β = 0°, where β is the inclination angle), inclined (β = 30° and 60°), and bottom-heated (β = 90°) square enclosures filled with a nanofluid is investigated, using a two-phase lattice Boltzmann simulation approach. The effects of the inclination angle on Nu number and convection heat transfer coefficient are studied. The effects of thermophoresis and Brownian forces which create a relative drift or slip velocity between the particles and the base fluid are included in the simulation. The effect of thermophoresis is considered using an accurate and quantitative formula proposed by the authors. Some of the existing results on natural convection are erroneous due to using wrong thermophoresis models or simply ignoring the effect. Here we show that thermophoresis has a considerable effect on heat transfer augmentation in laminar natural convection. Our non-homogenous modeling approach shows that heat transfer in nanofluids is a function of the inclination angle and Ra number. It also reveals some details of flow behavior which cannot be captured by single-phase models. The minimum heat transfer rate is associated with β = 90° (bottom-heated) and the maximum heat transfer rate occurs in an inclination angle which varies with the Ra number.

3. Numerical Simulation of Natural Convection of a Nanofluid in an Inclined Heated Enclosure Using Two-Phase Lattice Boltzmann Method: Accurate Effects of Thermophoresis and Brownian Forces.

PubMed

Ahmed, Mahmoud; Eslamian, Morteza

2015-12-01

Laminar natural convection in differentially heated (β = 0°, where β is the inclination angle), inclined (β = 30° and 60°), and bottom-heated (β = 90°) square enclosures filled with a nanofluid is investigated, using a two-phase lattice Boltzmann simulation approach. The effects of the inclination angle on Nu number and convection heat transfer coefficient are studied. The effects of thermophoresis and Brownian forces which create a relative drift or slip velocity between the particles and the base fluid are included in the simulation. The effect of thermophoresis is considered using an accurate and quantitative formula proposed by the authors. Some of the existing results on natural convection are erroneous due to using wrong thermophoresis models or simply ignoring the effect. Here we show that thermophoresis has a considerable effect on heat transfer augmentation in laminar natural convection. Our non-homogenous modeling approach shows that heat transfer in nanofluids is a function of the inclination angle and Ra number. It also reveals some details of flow behavior which cannot be captured by single-phase models. The minimum heat transfer rate is associated with β = 90° (bottom-heated) and the maximum heat transfer rate occurs in an inclination angle which varies with the Ra number. PMID:26183389

4. Numerical solution of a three-dimensional cubic cavity flow by using the Boltzmann equation

NASA Technical Reports Server (NTRS)

Hwang, Danny P.

1992-01-01

A three-dimensional cubic cavity flow has been analyzed for diatomic gases by using the Boltzmann equation with the Bhatnagar-Gross-Krook (B-G-K) model. The method of discrete ordinate was applied, and the diffuse reflection boundary condition was assumed. The results, which show a consistent trend toward the Navier-Stokes solution as the Knudson number is reduced, give us confidence to apply the method to a three-dimensional geometry for practical predictions of rarefied-flow characteristics. The CPU time and the main memory required for a three-dimensional geometry using this method seem reasonable.

5. Numerical solution of 3D Navier-Stokes equations with upwind implicit schemes

NASA Technical Reports Server (NTRS)

Marx, Yves P.

1990-01-01

An upwind MUSCL type implicit scheme for the three-dimensional Navier-Stokes equations is presented. Comparison between different approximate Riemann solvers (Roe and Osher) are performed and the influence of the reconstructions schemes on the accuracy of the solution as well as on the convergence of the method is studied. A new limiter is introduced in order to remove the problems usually associated with non-linear upwind schemes. The implementation of a diagonal upwind implicit operator for the three-dimensional Navier-Stokes equations is also discussed. Finally the turbulence modeling is assessed. Good prediction of separated flows are demonstrated if a non-equilibrium turbulence model is used.

6. On substructuring algorithms and solution techniques for the numerical approximation of partial differential equations

NASA Technical Reports Server (NTRS)

Gunzburger, M. D.; Nicolaides, R. A.

1986-01-01

Substructuring methods are in common use in mechanics problems where typically the associated linear systems of algebraic equations are positive definite. Here these methods are extended to problems which lead to nonpositive definite, nonsymmetric matrices. The extension is based on an algorithm which carries out the block Gauss elimination procedure without the need for interchanges even when a pivot matrix is singular. Examples are provided wherein the method is used in connection with finite element solutions of the stationary Stokes equations and the Helmholtz equation, and dual methods for second-order elliptic equations.

7. Numerical solution of the nonlinear Schrödinger equation with wave operator on unbounded domains.

PubMed

Li, Hongwei; Wu, Xiaonan; Zhang, Jiwei

2014-09-01

In this paper, we generalize the unified approach proposed in Zhang et al. [J. Zhang, Z. Xu, and X. Wu, Phys. Rev. E 78, 026709 (2008)] to design the nonlinear local absorbing boundary conditions (LABCs) for the nonlinear Schrödinger equation with wave operator on unbounded domains. In fact, based on the methodology underlying the unified approach, we first split the original equation into two parts-the linear equation and the nonlinear equation-then achieve a one-way operator to approximate the linear equation to make the wave outgoing, and finally combine the one-way operator with the nonlinear equation to achieve the nonlinear LABCs. The stability of the equation with the nonlinear LABCs is also analyzed by introducing some auxiliary variables, and some numerical examples are presented to verify the accuracy and effectiveness of our proposed method.

8. Light-opals interaction modeling by direct numerical solution of Maxwell's equations.

PubMed

Vaccari, Alessandro; Lesina, Antonino Calà; Cristoforetti, Luca; Chiappini, Andrea; Crema, Luigi; Calliari, Lucia; Ramunno, Lora; Berini, Pierre; Ferrari, Maurizio

2014-11-01

This work describes a 3-D Finite-Difference Time-Domain (FDTD) computational approach for the optical characterization of an opal photonic crystal. To fully validate the approach we compare the computed transmittance of a crystal model with the transmittance of an actual crystal sample, as measured over the 400 ÷ 750 nm wavelength range. The opal photonic crystal considered has a face-centered cubic (FCC) lattice structure of spherical particles made of polystyrene (a non-absorptive material with constant relative dielectric permittivity). Light-matter interaction is described by numerically solving Maxwell's equations via a parallelized FDTD code. Periodic boundary conditions (PBCs) at the outer edges of the crystal are used to effectively enforce an infinite lateral extension of the sample. A method to study the propagating Bloch modes inside the crystal bulk is also proposed, which allows the reconstruction of the ω-k dispersion curve for k sweeping discretely the Brillouin zone of the crystal. PMID:25401918

9. Numerical solution of a nonlinear least squares problem in digital breast tomosynthesis

Landi, G.; Loli Piccolomini, E.; Nagy, J. G.

2015-11-01

In digital tomosynthesis imaging, multiple projections of an object are obtained along a small range of different incident angles in order to reconstruct a pseudo-3D representation (i.e., a set of 2D slices) of the object. In this paper we describe some mathematical models for polyenergetic digital breast tomosynthesis image reconstruction that explicitly takes into account various materials composing the object and the polyenergetic nature of the x-ray beam. A polyenergetic model helps to reduce beam hardening artifacts, but the disadvantage is that it requires solving a large-scale nonlinear ill-posed inverse problem. We formulate the image reconstruction process (i.e., the method to solve the ill-posed inverse problem) in a nonlinear least squares framework, and use a Levenberg-Marquardt scheme to solve it. Some implementation details are discussed, and numerical experiments are provided to illustrate the performance of the methods.

10. Numerical solution of differential algebraic equations (DAEs) by mix-multistep method

Rahim, Yong Faezah; Suleiman, Mohamed; Ibrahim, Zarina Bibi

2014-06-01

Differential Algebraic Equations (DAEs) are regarded as stiff Ordinary Differential Equations (ODEs). Therefore they are solved using implicit method such as Backward Differentiation Formula (BDF) type of methods which require the use of Newton iteration which need much computational effort. However, not all of the ODEs in DAE system are stiff. In this paper, we describe a new technique for solving DAE, where the ODEs are treated as non-stiff at the start of the integration and putting the non-stiff ODEs into stiff subsystem should instability occurs. Adams type of method is used to solve the non-stiff part and BDF method for solving the stiff part. This strategy is shown to be competitive in terms of computational effort and accuracy. Numerical experiments are presented to validate its efficiency.

11. Numerical solutions for spin-up from rest in a cylinder

NASA Technical Reports Server (NTRS)

Hyun, J. M.; Leslie, F.; Fowlis, W. W.; Warn-Varnas, A.

1983-01-01

A set of three-dimensional flow-field data for the region around a cylinder impulsively spun-up from rest was derived with a numerical model based on the Navier-Stokes equations. Laser-Doppler anemometer data in the azimuthal direction was employed to test the model predictions, and data was developed for a flowfield with Ekman numbers from 9.18/1,000,000 to 9.18/10,000. The contributions of inviscid and viscous terms were determined as functions of radius and time. It was found that immediately after start-up viscous diffusion is the dominant factor, which is replaced by nonlinear radial advection. The Coriolis force dominates in the later stages of spin-up. The inward radial flow is a maximum near the front, where the vertical velocity is small, but features strong radial gradients, as it does at the edge of the Ekman layer.

12. The numerical solution of ordinary differential equations by the Taylor series method

NASA Technical Reports Server (NTRS)

Silver, A. H.; Sullivan, E.

1973-01-01

A programming implementation of the Taylor series method is presented for solving ordinary differential equations. The compiler is written in PL/1, and the target language is FORTRAN IV. The reduction of a differential system to rational form is described along with the procedures required for automatic numerical integration. The Taylor method is compared with two other methods for a number of differential equations. Algorithms using the Taylor method to find the zeroes of a given differential equation and to evaluate partial derivatives are presented. An annotated listing of the PL/1 program which performs the reduction and code generation is given. Listings of the FORTRAN routines used by the Taylor series method are included along with a compilation of all the recurrence formulas used to generate the Taylor coefficients for non-rational functions.

13. Light-opals interaction modeling by direct numerical solution of Maxwell's equations.

PubMed

Vaccari, Alessandro; Lesina, Antonino Calà; Cristoforetti, Luca; Chiappini, Andrea; Crema, Luigi; Calliari, Lucia; Ramunno, Lora; Berini, Pierre; Ferrari, Maurizio

2014-11-01

This work describes a 3-D Finite-Difference Time-Domain (FDTD) computational approach for the optical characterization of an opal photonic crystal. To fully validate the approach we compare the computed transmittance of a crystal model with the transmittance of an actual crystal sample, as measured over the 400 ÷ 750 nm wavelength range. The opal photonic crystal considered has a face-centered cubic (FCC) lattice structure of spherical particles made of polystyrene (a non-absorptive material with constant relative dielectric permittivity). Light-matter interaction is described by numerically solving Maxwell's equations via a parallelized FDTD code. Periodic boundary conditions (PBCs) at the outer edges of the crystal are used to effectively enforce an infinite lateral extension of the sample. A method to study the propagating Bloch modes inside the crystal bulk is also proposed, which allows the reconstruction of the ω-k dispersion curve for k sweeping discretely the Brillouin zone of the crystal.

14. Numerical solution of the Navier-Stokes equations for super-sonic flows with strong shocks. Ph.D. Thesis

NASA Technical Reports Server (NTRS)

Devarayalu, K.

1978-01-01

The numerical solution of the full Navier-Stokes Equations for viscous flows with high Mach numbers and a strong detached bow shock was obtained. Two dimensional flows around a circular cylinder, and a circular cylinder with an aft-body in the form of a fairing, were considered. The solution of the compressible N.S. equations was accomplished by the method of finite differences. An implicit scheme of solution, the S.O.R., was used with the optimum acceleration parameters determined by trial and error. The tensor notation was used in writing the N-S Equations transformed into general curvilinear coordinates. The equations for the generation of the coordinate system were solved, followed by the solution of the N.S. equations, at the end of a set of given number of time steps. "Wiggles", constituted the one major problem that needed to be overcome. These oscillations give rise to quantities such as negative temperatures, which ultimately caused the computational program to break down. Certain dissipative finite-difference schemes damped these oscillations.

15. Numerical matching of the sheath and presheath solutions for a spherical probe in radial-motion theory

SciTech Connect

Din, Alif; Kuhn, Siegbert

2014-10-15

The theory of positive-ion collection by a probe immersed in a low-pressure plasma was reviewed and extended by Allen, Boyd, and Reynolds [Proc. Phys. Soc. 70, 297 (1957)]. For a given value of the ion current, the boundary values of the matched “nonneutral” or “sheath” solution V{sup ~}{sub nn}{sup (m)}(r; r{sub m}) were obtained from the “quasineutral” or “presheath” solution V{sup ~}{sub qn}(r) by choosing the small potential and electric-field values corresponding to some large “matching radius” r{sub m}. Here, a straightforward but efficient numerical method is presented for systematically determining an optimal value of the matching radius at which the presheath and sheath solutions are joined to yield the “matched” potential profile. Some suitable initial matching radius r{sub m1} is chosen and the related potential and electric-field values of the quasineutral solution are calculated. Using these as boundary conditions, Poisson's equation is integrated to yield the matched nonneutral solution including the corresponding potential at the probe surface. This procedure is repeated for increasing values r{sub m2}, r{sub m3},…. until the resulting potential at the probe surface becomes practically constant. The corresponding value of r{sub m} is taken as the “optimal” matching radius r{sub mo} at which the presheath and sheath solutions are ultimately joined to yield the “optimal” matched potential profile in the entire plasma-probe transition region. It is also shown that the Bohm criterion is inapplicable in the present problem.

16. On Comparison of Series and Numerical Solutions for Flow of Eyring-Powell Fluid with Newtonian Heating And Internal Heat Generation/Absorption

PubMed Central

2015-01-01

In this paper, we have investigated the combined effects of Newtonian heating and internal heat generation/absorption in the two-dimensional flow of Eyring-Powell fluid over a stretching surface. The governing non-linear analysis of partial differential equations is reduced into the ordinary differential equations using similarity transformations. The resulting problems are computed for both series and numerical solutions. Series solution is constructed using homotopy analysis method (HAM) whereas numerical solution is presented by two different techniques namely shooting method and bvp4c. A comparison of homotopy solution with numerical solution is also tabulated. Both solutions are found in an excellent agreement. Dimensionless velocity and temperature profiles are plotted and discussed for various emerging physical parameters. PMID:26402366

17. On Comparison of Series and Numerical Solutions for Flow of Eyring-Powell Fluid with Newtonian Heating And Internal Heat Generation/Absorption.

PubMed

2015-01-01

In this paper, we have investigated the combined effects of Newtonian heating and internal heat generation/absorption in the two-dimensional flow of Eyring-Powell fluid over a stretching surface. The governing non-linear analysis of partial differential equations is reduced into the ordinary differential equations using similarity transformations. The resulting problems are computed for both series and numerical solutions. Series solution is constructed using homotopy analysis method (HAM) whereas numerical solution is presented by two different techniques namely shooting method and bvp4c. A comparison of homotopy solution with numerical solution is also tabulated. Both solutions are found in an excellent agreement. Dimensionless velocity and temperature profiles are plotted and discussed for various emerging physical parameters.

18. On Comparison of Series and Numerical Solutions for Flow of Eyring-Powell Fluid with Newtonian Heating And Internal Heat Generation/Absorption.

PubMed

2015-01-01

In this paper, we have investigated the combined effects of Newtonian heating and internal heat generation/absorption in the two-dimensional flow of Eyring-Powell fluid over a stretching surface. The governing non-linear analysis of partial differential equations is reduced into the ordinary differential equations using similarity transformations. The resulting problems are computed for both series and numerical solutions. Series solution is constructed using homotopy analysis method (HAM) whereas numerical solution is presented by two different techniques namely shooting method and bvp4c. A comparison of homotopy solution with numerical solution is also tabulated. Both solutions are found in an excellent agreement. Dimensionless velocity and temperature profiles are plotted and discussed for various emerging physical parameters. PMID:26402366

19. Solutions of the Two-Dimensional Hubbard Model: Benchmarks and Results from a Wide Range of Numerical Algorithms

LeBlanc, J. P. F.; Antipov, Andrey E.; Becca, Federico; Bulik, Ireneusz W.; Chan, Garnet Kin-Lic; Chung, Chia-Min; Deng, Youjin; Ferrero, Michel; Henderson, Thomas M.; Jiménez-Hoyos, Carlos A.; Kozik, E.; Liu, Xuan-Wen; Millis, Andrew J.; Prokof'ev, N. V.; Qin, Mingpu; Scuseria, Gustavo E.; Shi, Hao; Svistunov, B. V.; Tocchio, Luca F.; Tupitsyn, I. S.; White, Steven R.; Zhang, Shiwei; Zheng, Bo-Xiao; Zhu, Zhenyue; Gull, Emanuel; Simons Collaboration on the Many-Electron Problem

2015-10-01

Numerical results for ground-state and excited-state properties (energies, double occupancies, and Matsubara-axis self-energies) of the single-orbital Hubbard model on a two-dimensional square lattice are presented, in order to provide an assessment of our ability to compute accurate results in the thermodynamic limit. Many methods are employed, including auxiliary-field quantum Monte Carlo, bare and bold-line diagrammatic Monte Carlo, method of dual fermions, density matrix embedding theory, density matrix renormalization group, dynamical cluster approximation, diffusion Monte Carlo within a fixed-node approximation, unrestricted coupled cluster theory, and multireference projected Hartree-Fock methods. Comparison of results obtained by different methods allows for the identification of uncertainties and systematic errors. The importance of extrapolation to converged thermodynamic-limit values is emphasized. Cases where agreement between different methods is obtained establish benchmark results that may be useful in the validation of new approaches and the improvement of existing methods.

20. Evaluation of the Oberbeck-Boussinesq Approximation for the numerical simulation of variable-density flow and solute transport in porous media

Guevara, Carlos; Graf, Thomas

2013-04-01

Subsurface water systems are endangered due to salt water intrusion in coastal aquifers, leachate infiltration from waste disposal sites and salt transport in agricultural sites. This leads to the situation where more dense fluid overlies a less dense fluid creating a density gradient. Under certain conditions this density gradient produces instabilities in form dense plume fingers that move downwards. This free convection increases solute transport over large distances and shorter times. In cases where a significantly larger density gradient exists, the effect of free convection on transport is non-negligible. The assumption of a constant density distribution in space and time is no longer valid. Therefore variable-density flow must be considered. The flow equation and the transport equation govern the numerical modeling of variable-density flow and solute transport. Computer simulation programs mathematically describe variable-density flow using the Oberbeck-Boussinesq Approximation (OBA). Three levels of simplifications can de considered, which are denoted by OB1, OB2 and OB3. OB1 is the usually applied simplification where variable density is taken into account in the hydraulic potential. In OB2 variable density is considered in the flow equation and in OB3 variable density is additionally considered in the transport equation. Using the results from a laboratory-scale experiment of variable-density flow and solute transport (Simmons et al., Transp. Porous Medium, 2002) it is investigated which level of mathematical accuracy is required to represent the physical experiment the most accurate. Differences between the physical and mathematical model are evaluated using qualitative indicators (e.g. mass fluxes, Nusselt number). Results show that OB1 is required for small density gradients and OB3 is required for large density gradients.

1. Evaluating lightning hazards to building environments using explicit numerical solutions of Maxwell's equations

NASA Technical Reports Server (NTRS)

Collier, Richard S.; Mckenna, Paul M.; Perala, Rodney A.

1991-01-01

The objective here is to describe the lightning hazards to buildings and their internal environments using advanced formulations of Maxwell's Equations. The method described is the Three Dimensional Finite Difference Time Domain Solution. It can be used to solve for the lightning interaction with such structures in three dimensions with the inclusion of a considerable amount of detail. Special techniques were developed for including wire, plumbing, and rebar into the model. Some buildings have provisions for lightning protection in the form of air terminals connected to a ground counterpoise system. It is shown that fields and currents within these structures can be significantly high during a lightning strike. Time lapse video presentations were made showing the electric and magnetic field distributions on selected cross sections of the buildings during a simulated lightning strike.

2. Application of implicit numerical techniques to the solution of the three-dimensional diffusion equation

NASA Technical Reports Server (NTRS)

Peltier, Leonard Joel; Biringen, Sedat; Chait, Arnon

1990-01-01

Implicit techniques for calculating three-dimensional, time-dependent heat diffusion in a cube are tested with emphasis on storage efficiency, accuracy, and speed of calculation. For this purpose, a tensor product technique with both Chebyshev collocation and finite differences and a generalized conjugate gradient technique with finite differences are used in conjunction with Crank-Nicolson discretization. An Euler explicit finite difference calculation is performed for use as a benchmark. The implicit techniques are found to be competitive with the Euler explicit method in terms of storage efficiency and speed of calculation and offer advantages both in accuracy and stability. Mesh stretching in the finite difference calculations is shown to markedly improve the accuracy of the solution.

3. Numerical solution of the Dirac equation in Schwarzschild de Sitter spacetime

Lyu, Y.; Gui, Y. X.

2007-02-01

The radial parts of the Dirac equation between the inner and the outer horizon in Schwarzschild-de Sitter geometry are solved. Two limiting cases are concerned. The first case is when the two horizons are far apart and the second case is when the horizons are close to each other. In each case, a 'tangent' approximation is used to replace the modified 'tortoise' coordinate r*, which leads to a simple analytically invertible relation between r* and the radius r. The potential V(r*) is replaced by a collection of step functions in sequence. Then the solutions of the wave equation as well as the reflection and transmission coefficients are computed by a quantum mechanical method.

4. On the numerical solution of a three-dimensional inverse medium scattering problem

Hohage, Thorsten

2001-12-01

We examine the scattering of time-harmonic acoustic waves in inhomogeneous media. The problem is to recover a spatially varying refractive index in a three-dimensional medium from far-field measurements of scattered waves corresponding to incoming waves from all directions. This problem is exponentially ill-posed and of a large scale since a solution of the direct problem corresponds to solving a partial differential equation in R3 for each incident wave. We construct a preconditioner for the conjugate gradient method applied to the normal equation to solve the regularized linearized operator equation in each Newton step. This reduces the number of operator evaluations dramatically compared to standard regularized Newton methods. Our method can also be applied effectively to other exponentially ill-posed problems, for example, in impedance tomography, heat conduction and obstacle scattering. To solve the direct problems, we use an improved fast solver for the Lippmann-Schwinger equation suggested by Vainikko.

5. Numerical study of blow-up and dispersive shocks in solutions to generalized Korteweg-de Vries equations

Klein, C.; Peter, R.

2015-06-01

We present a detailed numerical study of solutions to general Korteweg-de Vries equations with critical and supercritical nonlinearity, both in the context of dispersive shocks and blow-up. We study the stability of solitons and show that they are unstable against being radiated away and blow-up. In the L2 critical case, the blow-up mechanism by Martel, Merle and Raphaël can be numerically identified. In the limit of small dispersion, it is shown that a dispersive shock always appears before an eventual blow-up. In the latter case, always the first soliton to appear will blow up. It is shown that the same type of blow-up as for the perturbations of the soliton can be observed which indicates that the theory by Martel, Merle and Raphaël is also applicable to initial data with a mass much larger than the soliton mass. We study the scaling of the blow-up time t∗ in dependence of the small dispersion parameter ɛ and find an exponential dependence t∗(ɛ) and that there is a minimal blow-up time t0∗ greater than the critical time of the corresponding Hopf solution for ɛ → 0. To study the cases with blow-up in detail, we apply the first dynamic rescaling for generalized Korteweg-de Vries equations. This allows to identify the type of the singularity.

6. Numerical solution of differential equations using multiquadric radial basis functions networks.

PubMed

Mai-Duy, N; Tran-Cong, T

2001-03-01

This paper presents mesh-free procedures for solving linear differential equations (ODEs and elliptic PDEs) based on multiquadric (MQ) radial basis function networks (RBFNs). Based on our study of approximation of function and its derivatives using RBFNs that was reported in an earlier paper (Mai-Duy, N. & Tran-Cong, T. (1999). Approximation of function and its derivatives using radial basis function networks. Neural networks, submitted), new RBFN approximation procedures are developed in this paper for solving DEs, which can also be classified into two types: a direct (DRBFN) and an indirect (IRBFN) RBFN procedure. In the present procedures, the width of the RBFs is the only adjustable parameter according to a(i) = betad(i), where d(i) is the distance from the ith centre to the nearest centre. The IRBFN method is more accurate than the DRBFN one and experience so far shows that beta can be chosen in the range 7 < or = beta 10 for the former. Different combinations of RBF centres and collocation points (uniformly and randomly distributed) are tested on both regularly and irregularly shaped domains. The results for a 1D Poisson's equation show that the DRBFN and the IRBFN procedures achieve a norm of error of at least O(1.0 x 10(-4)) and O(1.0 x 10(-8)), respectively, with a centre density of 50. Similarly, the results for a 2D Poisson's equation show that the DRBFN and the IRBFN procedures achieve a norm of error of at least O(1.0 x 10(-3)) and O(1.0 x10(-6)) respectively, with a centre density of 12 X 12.

7. Public-domain-software solution to data-access problems for numerical modelers

USGS Publications Warehouse

Jenter, Harry; Signell, Richard

1992-01-01

Unidata's network Common Data Form, netCDF, provides users with an efficient set of software for scientific-data-storage, retrieval, and manipulation. The netCDF file format is machine-independent, direct-access, self-describing, and in the public domain, thereby alleviating many problems associated with accessing output from large hydrodynamic models. NetCDF has programming interfaces in both the Fortran and C computer language with an interface to C++ planned for release in the future. NetCDF also has an abstract data type that relieves users from understanding details of the binary file structure; data are written and retrieved by an intuitive, user-supplied name rather than by file position. Users are aided further by Unidata's inclusion of the Common Data Language, CDL, a printable text-equivalent of the contents of a netCDF file. Unidata provides numerous operators and utilities for processing netCDF files. In addition, a number of public-domain and proprietary netCDF utilities from other sources are available at this time or will be available later this year. The U.S. Geological Survey has produced and is producing a number of public-domain netCDF utilities.

8. Ab initio theory of superconductivity in a magnetic field. II. Numerical solution

Linscheid, A.; Sanna, A.; Gross, E. K. U.

2015-07-01

We numerically investigate the spin density functional theory for superconductors (SpinSCDFT) and the approximated exchange-correlation functional, derived and presented in the preceding Paper I [A. Linscheid et al., Phys. Rev. B 92, 024505 (2015), 10.1103/PhysRevB.92.024505]. As a test system, we employ a free-electron gas featuring an exchange splitting, a phononic pairing field, and a Coulomb repulsion. SpinSCDFT results are compared with Sarma, the Bardeen-Cooper-Schrieffer theory, and with an Eliashberg type of approach. We find that the spectrum of the superconducting Kohn-Sham SpinSCDFT system is not in agreement with the true quasiparticle structure. Therefore, starting from the Dyson equation, we derive a scheme that allows to compute the many-body excitations of the superconductor and represents the extension to superconductivity of the G0W0 method in band-structure theory. This superconducting G0W0 method vastly improves the predicted spectra.

9. Numerical solution of the Stratonovich- and Ito–Euler equations: Application to the stochastic piston problem

SciTech Connect

Zhang, Zhongqiang; Yang, Xiu; Lin, Guang; Karniadakis, George Em

2013-03-01

We consider a piston with a velocity perturbed by Brownian motion moving into a straight tube filled with a perfect gas at rest. The shock generated ahead of the piston can be located by solving the one-dimensional Euler equations driven by white noise using the Stratonovich or Ito formulations. We approximate the Brownian motion with its spectral truncation and subsequently apply stochastic collocation using either sparse grid or the quasi-Monte Carlo (QMC) method. In particular, we first transform the Euler equations with an unsteady stochastic boundary into stochastic Euler equations over a fixed domain with a time-dependent stochastic source term. We then solve the transformed equations by splitting them up into two parts, i.e., a ‘deterministic part’ and a ‘stochastic part’. Numerical results verify the Stratonovich–Euler and Ito–Euler models against stochastic perturbation results, and demonstrate the efficiency of sparse grid and QMC for small and large random piston motions, respectively. The variance of shock location of the piston grows cubically in the case of white noise in contrast to colored noise reported in [1], where the variance of shock location grows quadratically with time for short times and linearly for longer times.

10. Numerical solution of three-dimensional unsteady transonic flow over wings including inviscid/viscous interactions

NASA Technical Reports Server (NTRS)

Rizzetta, D. P.; Borland, C. J.

1983-01-01

A numerical procedure is presented for computing the unsteady transonic flow field about three dimensional swept wings undergoing general time dependent motion. The outer inviscid portion of the flow is assumed to be governed by the modified unsteady transonic small disturbance potential equation which is integrated in the time domain by means of an efficient alternating direction implicit approximate factorization algorithm. Gross dominant effects of the shock boundary layer interaction are accounted for by a simple empirically defined model. Viscous flow regions adjacent to the wing surface and in the trailing wake are described by a set of integral equations appropriate for compressible turbulent shear layers. The two dimensional boundary layer equations are applied quasi-statically stripwise across the span. Coupling with the outer inviscid flow is implemented through use of the displacement thickness concept within the limitations of small disturbance theory. Validity of the assumptions underlying the method is established by comparison with experimental data for the flow about a high aspect ratio transport wing having an advanced airfoil section.

11. Efficient technique for the numerical solution of the one-dimensional inverse problem of heat conduction

Blackwell, B. F.

1981-06-01

A very efficient numerical technique has been developed to solve the one-dimensional inverse problem of heat conduction. The Gauss elimination algorithm for solving the tridiagonal system of linear algebraic equations associated with most implicit heat conduction codes is specialized to the inverse problem. When compared to the corresponding direct problem, the upper limit in additional computation time generally does not exceed 27-36%. The technique can be adapted to existing one-dimensional implicit heat conduction codes with minimal effort and applied to difference equations obtained from finite-difference, finite-element, finite control volume, or similar techniques, provided the difference equations are tridiagonal in form. It is also applicable to the nonlinear case in which thermal properties are temperature-dependent and is valid for one-dimensional radial cylindrical and spherical geometries as well as composite bodies. The calculations reported here were done by modifying a one-dimensional implicit (direct) heat conduction code. Program changes consisted of 13 additional lines of FORTRAN coding.

12. Computational time analysis of the numerical solution of 3D electrostatic Poisson's equation

Kamboh, Shakeel Ahmed; Labadin, Jane; Rigit, Andrew Ragai Henri; Ling, Tech Chaw; Amur, Khuda Bux; Chaudhary, Muhammad Tayyab

2015-05-01

3D Poisson's equation is solved numerically to simulate the electric potential in a prototype design of electrohydrodynamic (EHD) ion-drag micropump. Finite difference method (FDM) is employed to discretize the governing equation. The system of linear equations resulting from FDM is solved iteratively by using the sequential Jacobi (SJ) and sequential Gauss-Seidel (SGS) methods, simulation results are also compared to examine the difference between the results. The main objective was to analyze the computational time required by both the methods with respect to different grid sizes and parallelize the Jacobi method to reduce the computational time. In common, the SGS method is faster than the SJ method but the data parallelism of Jacobi method may produce good speedup over SGS method. In this study, the feasibility of using parallel Jacobi (PJ) method is attempted in relation to SGS method. MATLAB Parallel/Distributed computing environment is used and a parallel code for SJ method is implemented. It was found that for small grid size the SGS method remains dominant over SJ method and PJ method while for large grid size both the sequential methods may take nearly too much processing time to converge. Yet, the PJ method reduces computational time to some extent for large grid sizes.

13. Laboratory analog and numerical study of groundwater flow and solute transport in a karst aquifer with conduit and matrix domains

Faulkner, Jonathan; Hu, Bill X.; Kish, Stephen; Hua, Fei

2009-11-01

New mathematical and laboratory methods have been developed for simulating groundwater flow and solute transport in karst aquifers having conduits imbedded in a porous medium, such as limestone. The Stokes equations are used to model the flow in the conduits and the Darcy equation is used for the flow in the matrix. The Beavers-Joseph interface boundary conditions are adopted to describe the flow exchange at the interface boundary between the two domains. A laboratory analog is used to simulate the conduit and matrix domains of a karst aquifer. The conduit domain is located at the bottom of the transparent plexiglas laboratory analog and glass beads occupy the remaining space to represent the matrix domain. Water flows into and out of the two domains separately and each has its own supply and outflow reservoirs. Water and solute are exchanged through an interface between the two domains. Pressure transducers located within the matrix and conduit domains of the analog provide data that is processed and stored in digital format. Dye tracing experiments are recorded using time-lapse imaging. The data and images produced are analyzed by a spatial analysis program. The experiments provide not only hydraulic head distribution but also capture solute front images and mass exchange measurements between the conduit and matrix domains. In the experiment, we measure and record pressures, and quantify flow rates and solute transport. The results present a plausible argument that laboratory analogs can characterize groundwater water flow, solute transport, and mass exchange between the conduit and matrix domains in a karst aquifer. The analog validates the predictions of a numerical model and demonstrates the need of laboratory analogs to provide verification of proposed theories and the calibration of mathematical models.

14. Discretization and its proof for numerical solution of a Cauchy problem for Laplace equation with inaccurately given Cauchy conditions on an inaccurately defined arbitrary surface

Laneev, E. B.; Mouratov, M. N.; Zhidkov, E. P.

2008-05-01

Cauchy problem for the Laplace equation with inaccurately given Cauchy conditions on an inaccurately defined arbitrary surface is considered. Discretization was performed and proved to obtain a numerical solution. An economic algorithm is proposed.

15. Numerical study of a binary Yukawa model in regimes characteristic of globular proteins in solutions

SciTech Connect

Giacometti, Achille; Gazzillo, Domenico; Pastore, Giorgio; Das, Tushar Kanti

2005-03-01

The main goal of this paper is to assess the limits of validity, in the regime of low concentration and strong Coulomb coupling (high molecular charges), of a simple perturbative approximation to the radial distribution functions (RDF's), based upon a low-density expansion of the potential of mean force and proposed to describe protein-protein interactions in a recent small-angle-scattering (SAS) experimental study. A highly simplified Yukawa (screened Coulomb) model of monomers and dimers of a charged globular protein ({beta}-lactoglobulin) in solution is considered. We test the accuracy of the RDF approximation, as a necessary complementary part of the previous experimental investigation, by comparison with the fluid structure predicted by approximate integral equations and exact Monte Carlo (MC) simulations. In the MC calculations, an Ewald construction for Yukawa potentials has been used to take into account the long-range part of the interactions in the weakly screened cases. Our results confirm that the perturbative first-order approximation is valid for this system even at strong Coulomb coupling, provided that the screening is not too weak (i.e., for Debye length smaller than monomer radius). A comparison of the MC results with integral equation calculations shows that both the hypernetted-chain (HNC) and Percus-Yevick closures have a satisfactory behavior under these regimes, with the HNC being superior throughout. The relevance of our findings for interpreting SAS results is also discussed.

16. An Experimenting Field Approach for the Numerical Solution of Multiphase Flow in Porous Media.

PubMed

Salama, Amgad; Sun, Shuyu; Bao, Kai

2016-03-01

In this work, we apply the experimenting pressure field technique to the problem of the flow of two or more immiscible phases in porous media. In this technique, a set of predefined pressure fields are introduced to the governing partial differential equations. This implies that the velocity vector field and the divergence at each cell of the solution mesh can be determined. However, since none of these fields is the true pressure field entailed by the boundary conditions and/or the source terms, the divergence at each cell will not be the correct one. Rather the residue which is the difference between the true divergence and the calculated one is obtained. These fields are designed such that these residuals are used to construct the matrix of coefficients of the pressure equation and the right-hand side. The experimenting pressure fields are generated in the solver routine and are fed to the different routines, which may be called physics routines, which return to the solver the elements of the matrix of coefficients. Therefore, this methodology separates the solver routines from the physics routines and therefore results in simpler, easy to construct, maintain, and update algorithms.

17. Mathematical Models, Analytical Solutions and Numerical Simulations of Self-Assembled Magnetic Colloidal Structures

Piet, David L.

Ferromagnetic microparticles suspended at the interface between immiscible liquids and energized by an external alternating magnetic field show a rich variety of self-assembled structures, from linear snakes to radial asters, elongated wires to spinning chains to less dense clouds of particles called snails. In order to obtain insight into the fundamental physical mechanisms and the overall balance of forces governing self-assembly, we develop a modeling approach based on analytical solutions of the time-averaged Navier-Stokes equations. These analytical expressions for the self-consistent hydrodynamic flows are then employed to modify effective interactions between the particles, which in turn are formulated in terms of the time-averaged quantities. Our method allows effective computational verification of the mechanisms of self-assembly and leads to a testable predictions on the transitions between various self-assembled patterns. In one set of experiments, it was observed that viscosity is the primary driving force that determines whether asters or snakes appear at steady state. In the second set of experiments where hydrodynamics are less critical, the amplitude and frequency of the applied magnetic field determine whether wires, spinners or snails will appear. The ability to better understand what drives self-assembly and how to control which dynamic structures appear is necessary for further development of such structures and their applications.

18. Mosaic-skeleton method as applied to the numerical solution of three-dimensional Dirichlet problems for the Helmholtz equation in integral form

Kashirin, A. A.; Smagin, S. I.; Taltykina, M. Yu.

2016-04-01

Interior and exterior three-dimensional Dirichlet problems for the Helmholtz equation are solved numerically. They are formulated as equivalent boundary Fredholm integral equations of the first kind and are approximated by systems of linear algebraic equations, which are then solved numerically by applying an iteration method. The mosaic-skeleton method is used to speed up the solution procedure.

19. A Cosserat point element (CPE) for the numerical solution of transient large planar motions of elastic-plastic and elastic-viscoplastic beams

Brand, M.; Rubin, M. B.

2013-02-01

The objective of this paper is to develop constitutive equations of a Cosserat point element (CPE) for the numerical solution of transient large planar motions of elastic-plastic and elastic-viscoplastic beams with rigid cross-sections. Specifically, attention is limited to response of a material with constant yield strength. A yield function is proposed which couples the inelastic responses of tension and shear. Another yield function is proposed for bending which depends on a hardening variable that models motion of the elastic-plastic boundary in the beam's cross-section. Evolution equations are proposed for elastic strains and the hardening variable and an overstress-type formulation is used for elastic-viscoplastic response. In contrast, with standard finite element approaches the CPE model needs no integration through the element region. Also, an implicit scheme is developed to integrate the evolution equations without iteration. Examples of transient large motions of beams, which are impulsively loaded, indicate that the CPE produces reasonably accurate response relative results in the literature and full three-dimensional calculations using ABAQUS.

20. LSENS: A General Chemical Kinetics and Sensitivity Analysis Code for homogeneous gas-phase reactions. Part 1: Theory and numerical solution procedures

NASA Technical Reports Server (NTRS)

1994-01-01

LSENS, the Lewis General Chemical Kinetics and Sensitivity Analysis Code, has been developed for solving complex, homogeneous, gas-phase chemical kinetics problems and contains sensitivity analysis for a variety of problems, including nonisothermal situations. This report is part 1 of a series of three reference publications that describe LENS, provide a detailed guide to its usage, and present many example problems. Part 1 derives the governing equations and describes the numerical solution procedures for the types of problems that can be solved. The accuracy and efficiency of LSENS are examined by means of various test problems, and comparisons with other methods and codes are presented. LSENS is a flexible, convenient, accurate, and efficient solver for chemical reaction problems such as static system; steady, one-dimensional, inviscid flow; reaction behind incident shock wave, including boundary layer correction; and perfectly stirred (highly backmixed) reactor. In addition, the chemical equilibrium state can be computed for the following assigned states: temperature and pressure, enthalpy and pressure, temperature and volume, and internal energy and volume. For static problems the code computes the sensitivity coefficients of the dependent variables and their temporal derivatives with respect to the initial values of the dependent variables and/or the three rate coefficient parameters of the chemical reactions.

1. Numerical Solution of the Radiative Transfer Equation: X-Ray Spectral Formation from Cylindrical Accretion onto a Magnetized Neutron Star

NASA Technical Reports Server (NTRS)

Fairnelli, R.; Ceccobello, C.; Romano, P.; Titarchuk, L.

2011-01-01

Predicting the emerging X-ray spectra in several astrophysical objects is of great importance, in particular when the observational data are compared with theoretical models. This requires developing numerical routines for the solution of the radiative transfer equation according to the expected physical conditions of the systems under study. Aims. We have developed an algorithm solving the radiative transfer equation in the Fokker-Planck approximation when both thermal and bulk Comptonization take place. The algorithm is essentially a relaxation method, where stable solutions are obtained when the system has reached its steady-state equilibrium. Methods. We obtained the solution of the radiative transfer equation in the two-dimensional domain defined by the photon energy E and optical depth of the system pi using finite-differences for the partial derivatives, and imposing specific boundary conditions for the solutions. We treated the case of cylindrical accretion onto a magnetized neutron star. Results. We considered a blackbody seed spectrum of photons with exponential distribution across the accretion column and for an accretion where the velocity reaches its maximum at the stellar surface and at the top of the accretion column, respectively. In both cases higher values of the electron temperature and of the optical depth pi produce flatter and harder spectra. Other parameters contributing to the spectral formation are the steepness of the vertical velocity profile, the albedo at the star surface, and the radius of the accretion column. The latter parameter modifies the emerging spectra in a specular way for the two assumed accretion profiles. Conclusions. The algorithm has been implemented in the XPEC package for X-ray fitting and is specifically dedicated to the physical framework of accretion at the polar cap of a neutron star with a high magnetic field (approx > 10(exp 12) G). This latter case is expected to be of typical accreting systems such as X

2. DOUAR: A new three-dimensional creeping flow numerical model for the solution of geological problems

Braun, Jean; Thieulot, Cédric; Fullsack, Philippe; DeKool, Marthijn; Beaumont, Christopher; Huismans, Ritske

2008-12-01

We present a new finite element code for the solution of the Stokes and energy (or heat transport) equations that has been purposely designed to address crustal-scale to mantle-scale flow problems in three dimensions. Although it is based on an Eulerian description of deformation and flow, the code, which we named DOUAR ('Earth' in Breton language), has the ability to track interfaces and, in particular, the free surface, by using a dual representation based on a set of particles placed on the interface and the computation of a level set function on the nodes of the finite element grid, thus ensuring accuracy and efficiency. The code also makes use of a new method to compute the dynamic Delaunay triangulation connecting the particles based on non-Euclidian, curvilinear measure of distance, ensuring that the density of particles remains uniform and/or dynamically adapted to the curvature of the interface. The finite element discretization is based on a non-uniform, yet regular octree division of space within a unit cube that allows efficient adaptation of the finite element discretization, i.e. in regions of strong velocity gradient or high interface curvature. The finite elements are cubes (the leaves of the octree) in which a q1- p0 interpolation scheme is used. Nodal incompatibilities across faces separating elements of differing size are dealt with by introducing linear constraints among nodal degrees of freedom. Discontinuities in material properties across the interfaces are accommodated by the use of a novel method (which we called divFEM) to integrate the finite element equations in which the elemental volume is divided by a local octree to an appropriate depth (resolution). A variety of rheologies have been implemented including linear, non-linear and thermally activated creep and brittle (or plastic) frictional deformation. A simple smoothing operator has been defined to avoid checkerboard oscillations in pressure that tend to develop when using a highly

3. A Nested Genetic Algorithm for the Numerical Solution of Non-Linear Coupled Equations in Water Quality Modeling

García, Hermes A.; Guerrero-Bolaño, Francisco J.; Obregón-Neira, Nelson

2010-05-01

Due to both mathematical tractability and efficiency on computational resources, it is very common to find in the realm of numerical modeling in hydro-engineering that regular linearization techniques have been applied to nonlinear partial differential equations properly obtained in environmental flow studies. Sometimes this simplification is also made along with omission of nonlinear terms involved in such equations which in turn diminishes the performance of any implemented approach. This is the case for example, for contaminant transport modeling in streams. Nowadays, a traditional and one of the most common used water quality model such as QUAL2k, preserves its original algorithm, which omits nonlinear terms through linearization techniques, in spite of the continuous algorithmic development and computer power enhancement. For that reason, the main objective of this research was to generate a flexible tool for non-linear water quality modeling. The solution implemented here was based on two genetic algorithms, used in a nested way in order to find two different types of solutions sets: the first set is composed by the concentrations of the physical-chemical variables used in the modeling approach (16 variables), which satisfies the non-linear equation system. The second set, is the typical solution of the inverse problem, the parameters and constants values for the model when it is applied to a particular stream. From a total of sixteen (16) variables, thirteen (13) was modeled by using non-linear coupled equation systems and three (3) was modeled in an independent way. The model used here had a requirement of fifty (50) parameters. The nested genetic algorithm used for the numerical solution of a non-linear equation system proved to serve as a flexible tool to handle with the intrinsic non-linearity that emerges from the interactions occurring between multiple variables involved in water quality studies. However because there is a strong data limitation in

4. Numerical solutions of Einstein's equations for cosmological spacetimes with spatial topology S3 and symmetry group U(1)

Beyer, F.; Escobar, L.; Frauendiener, J.

2016-02-01

In this paper we consider the single patch pseudospectral scheme for tensorial and spinorial evolution problems on the 2-sphere presented by Beyer et al. [Classical Quantum Gravity 32, 175013 (2015); Classical Quantum Gravity31, 075019 (2014)], which is based on the spin-weighted spherical harmonics transform. We apply and extend this method to Einstein's equations and certain classes of spherical cosmological spacetimes. More specifically, we use the hyperbolic reductions of Einstein's equations obtained in the generalized wave map gauge formalism combined with Geroch's symmetry reduction, and focus on cosmological spacetimes with spatial S3 -topologies and symmetry groups U(1) or U (1 )×U (1 ) . We discuss analytical and numerical issues related to our implementation. We test our code by reproducing the exact inhomogeneous cosmological solutions of the vacuum Einstein field equations obtained by Beyer and Hennig [Classical Quantum Gravity 31, 095010 (2014)].

5. Numerical solution to the glancing sidewall oblique shock wave/turbulent boundary layer interaction in three dimension

NASA Technical Reports Server (NTRS)

Anderson, B. H.; Benson, T. J.

1983-01-01

A supersonic three-dimensional viscous forward-marching computer design code called PEPSIS is used to obtain a numerical solution of the three-dimensional problem of the interaction of a glancing sidewall oblique shock wave and a turbulent boundary layer. Very good results are obtained for a test case that was run to investigate the use of the wall-function boundary-condition approximation for a highly complex three-dimensional shock-boundary layer interaction. Two additional test cases (coarse mesh and medium mesh) are run to examine the question of near-wall resolution when no-slip boundary conditions are applied. A comparison with experimental data shows that the PEPSIS code gives excellent results in general and is practical for three-dimensional supersonic inlet calculations.

6. Numerical solution to the glancing sidewall oblique shock wave/turbulent boundary layer interaction in three-dimension

NASA Technical Reports Server (NTRS)

Anderson, B. H.; Benson, T. J.

1983-01-01

A supersonic three-dimensional viscous forward-marching computer design code called PEPSIS is used to obtain a numerical solution of the three-dimensional problem of the interaction of a glancing sidewall oblique shock wave and a turbulent boundary layer. Very good results are obtained for a test case that was run to investigate the use of the wall-function boundary-condition approximation for a highly complex three-dimensional shock-boundary layer interaction. Two additional test cases (coarse mesh and medium mesh) are run to examine the question of near-wall resolution when no-slip boundary conditions are applied. A comparison with experimental data shows that the PEPSIS code gives excellent results in general and is practical for three-dimensional supersonic inlet calculations.

7. A finite-difference, frequency-domain numerical scheme for the solution of the linearized unsteady Euler equations

NASA Technical Reports Server (NTRS)

Scott, James R.; Atassi, Hafiz M.

1991-01-01

A numerical method is developed for solving periodic, three-dimensional, vortical flows around lifting airfoils in subsonic flow. The first-order method, that is presented, fully accounts for the distortion effects of the nonuniform mean flow on the convected upstream vortical disturbances. The unsteady velocity is split into a vortical component which is a known function of the upstream flow conditions and the Lagrangian coordinates of the mean flow, and an irrotational field whose potential satisfies a nonconstant-coefficient, inhomogeneous, convective wave equation. Using an elliptic coordinate transformation, the unsteady boundary value problem is solved in the frequency domain on grids which are determined as a function of the Mach number and reduced frequency. Extensive comparisons are made with known solutions to unsteady vortical flow problems, and it is seen that the agreement is generally very good for reduced frequencies ranging from 0 up to 4.

8. The effects of the Asselin time filter on numerical solutions to the linearized shallow-water wave equations

NASA Technical Reports Server (NTRS)

Schlesinger, R. E.; Johnson, D. R.; Uccellini, L. W.

1983-01-01

In the present investigation, a one-dimensional linearized analysis is used to determine the effect of Asselin's (1972) time filter on both the computational stability and phase error of numerical solutions for the shallow water wave equations, in cases with diffusion but without rotation. An attempt has been made to establish the approximate optimal values of the filtering parameter nu for each of the 'lagged', Dufort-Frankel, and Crank-Nicholson diffusion schemes, suppressing the computational wave mode without materially altering the physical wave mode. It is determined that in the presence of diffusion, the optimum filter length depends on whether waves are undergoing significant propagation. When moderate propagation is present, with or without diffusion, the Asselin filter has little effect on the spatial phase lag of the physical mode for the leapfrog advection scheme of the three diffusion schemes considered.

9. EQUILIBRIUM CONFIGURATIONS OF SYNCHRONOUS BINARIES: NUMERICAL SOLUTIONS AND APPLICATION TO KUIPER BELT BINARY 2001 QG{sub 298}

SciTech Connect

Gnat, Orly; Sari, Re'em

2010-08-20

We present numerical computations of the equilibrium configurations of tidally locked homogeneous binaries rotating in circular orbits. Unlike the classical Roche approximations, we self-consistently account for the tidal and rotational deformations of both components, and relax the assumptions of ellipsoidal configurations and Keplerian rotation. We find numerical solutions for mass ratios q between 10{sup -3} and 1, starting at a small angular velocity for which tidal and rotational deformations are small, and following a sequence of increasing angular velocities. Each series terminates at an appropriate 'Roche limit', above which no equilibrium solution can be found. Even though the Roche limit is crossed before the 'Roche lobe' is filled, any further increase in the angular velocity will result in mass-loss. For close, comparable-mass binaries, we find that local deviations from ellipsoidal forms may be as large as 10%-20%, and departures from Keplerian rotation are significant. We compute the light curves that arise from our equilibrium configurations, assuming their distance is >>1 AU (e.g., in the Kuiper Belt). We consider both backscatter (proportional to the projected area) and diffuse (Lambert) reflections. Backscatter reflection always yields two minima of equal depths. Diffuse reflection, which is sensitive to the surface curvature, generally gives rise to unequal minima. We find detectable intensity differences of up to 10% between our light curves and those arising from the Roche approximations. Finally, we apply our models to Kuiper Belt binary 2001 QG{sub 298}, and find a nearly edge-on binary with a mass ratio q = 0.93{sup +0.07}{sub -0.03}, angular velocity {omega}{sup 2}/G{rho} = 0.333 {+-} 0.001 (statistical errors only), and pure diffuse reflection. For the observed period of 2001 QG{sub 298}, these parameters imply a bulk density {rho} = 0.72 {+-} 0.04 g cm{sup -3}.

10. Singular boundary value problem for the integrodifferential equation in an insurance model with stochastic premiums: Analysis and numerical solution

Belkina, T. A.; Konyukhova, N. B.; Kurochkin, S. V.

2012-10-01

A singular boundary value problem for a second-order linear integrodifferential equation with Volterra and non-Volterra integral operators is formulated and analyzed. The equation is defined on ℝ+, has a weak singularity at zero and a strong singularity at infinity, and depends on several positive parameters. Under natural constraints on the coefficients of the equation, existence and uniqueness theorems for this problem with given limit boundary conditions at singular points are proved, asymptotic representations of the solution are given, and an algorithm for its numerical determination is described. Numerical computations are performed and their interpretation is given. The problem arises in the study of the survival probability of an insurance company over infinite time (as a function of its initial surplus) in a dynamic insurance model that is a modification of the classical Cramer-Lundberg model with a stochastic process rate of premium under a certain investment strategy in the financial market. A comparative analysis of the results with those produced by the model with deterministic premiums is given.

11. A two-zone method with an enhanced accuracy for a numerical solution of the diffusion equation

Cheon, Jin-Sik; Koo, Yang-Hyun; Lee, Byung-Ho; Oh, Je-Yong; Sohn, Dong-Seong

2006-12-01

A variational principle is applied to the diffusion equation to numerically obtain the fission gas release from a spherical grain. The two-zone method, originally proposed by Matthews and Wood, is modified to overcome its insufficient accuracy for a low release. The results of the variational approaches are examined by observing the gas concentration along the grain radius. At the early stage, the concentration near the grain boundary is higher than that at the inner points of the grain in the cases of the two-zone method as well as the finite element analysis with the number of the elements at as many as 10. The accuracy of the two-zone method is considerably enhanced by relocating the nodal points of the two zones. The trial functions are derived as a function of the released fraction. During the calculations, the number of degrees of freedom needs to be reduced to guarantee physically admissible concentration profiles. Numerical verifications are performed extensively. By taking a computational time comparable to the algorithm by Forsberg and Massih, the present method provides a solution with reasonable accuracy in the whole range of the released fraction.

12. Numerical Uncertainty Analysis for Computational Fluid Dynamics using Student T Distribution -- Application of CFD Uncertainty Analysis Compared to Exact Analytical Solution

NASA Technical Reports Server (NTRS)

Groves, Curtis E.; Ilie, marcel; Shallhorn, Paul A.

2014-01-01

Computational Fluid Dynamics (CFD) is the standard numerical tool used by Fluid Dynamists to estimate solutions to many problems in academia, government, and industry. CFD is known to have errors and uncertainties and there is no universally adopted method to estimate such quantities. This paper describes an approach to estimate CFD uncertainties strictly numerically using inputs and the Student-T distribution. The approach is compared to an exact analytical solution of fully developed, laminar flow between infinite, stationary plates. It is shown that treating all CFD input parameters as oscillatory uncertainty terms coupled with the Student-T distribution can encompass the exact solution.

13. A high order accurate difference scheme for complex flow fields

SciTech Connect

Dexun Fu; Yanwen Ma

1997-06-01

A high order accurate finite difference method for direct numerical simulation of coherent structure in the mixing layers is presented. The reason for oscillation production in numerical solutions is analyzed. It is caused by a nonuniform group velocity of wavepackets. A method of group velocity control for the improvement of the shock resolution is presented. In numerical simulation the fifth-order accurate upwind compact difference relation is used to approximate the derivatives in the convection terms of the compressible N-S equations, a sixth-order accurate symmetric compact difference relation is used to approximate the viscous terms, and a three-stage R-K method is used to advance in time. In order to improve the shock resolution the scheme is reconstructed with the method of diffusion analogy which is used to control the group velocity of wavepackets. 18 refs., 12 figs., 1 tab.

14. Two-domain description of solute transport in heterogeneous porous media: Comparison between theoretical predictions and numerical experiments

Cherblanc, F.; Ahmadi, A.; Quintard, M.

2007-05-01

This paper deals with two-equation models describing solute transport in highly heterogeneous porous systems and more particularly dual permeability structures composed of high- and low-permeability regions. A macroscopic two-equation model has been previously proposed in the literature based on the volume averaging technique [Ahmadi A, Quintard M, Whitaker S. Transport in chemically and mechanically heterogeneous porous media V: two-equation model for solute transport with adsorption, Adv Water Resour 1998;22:59-86; Cherblanc F, Ahmadi A, Quintard M. Two-medium description of dispersion in heterogeneous porous media: calculation of macroscopic properties. Water Resour Res 2003;39(6):1154-73]. Through this theoretical upscaling method, both convection and dispersion mechanisms are taken into account in both regions, allowing one to deal with a large range of heterogeneous systems. In this paper, the numerical tools associated with this model are developed in order to test the theory by comparing macroscopic concentration fields to those obtained by Darcy-scale numerical experiments. The heterogeneous structures considered are made up of low-permeability nodules embedded in a continuous high-permeability region. Several permeability ratios are used, leading to very different macroscopic behaviours. Taking advantage of the Darcy-scale simulations, the role of convection and dispersion in the mass exchange between the two regions is investigated. Large-scale averaged concentration fields and elution curves are extracted from the Darcy-scale numerical experiments and compared to the theoretical predictions given by the two-equation model. Very good agreement is found between experimental and theoretical results. A permeability ratio around 100 presents a behaviour characteristic of "mobile-mobile" systems emphasizing the relevance of this two-equation description. Eventually, the theory is used to set-up a criterion for the existence of local equilibrium conditions

15. Combined Numerical/Analytical Perturbation Solutions of the Navier-Stokes Equations for Aerodynamic Ejector/Mixer Nozzle Flows

NASA Technical Reports Server (NTRS)

DeChant, Lawrence Justin

1998-01-01

In spite of rapid advances in both scalar and parallel computational tools, the large number of variables involved in both design and inverse problems make the use of sophisticated fluid flow models impractical, With this restriction, it is concluded that an important family of methods for mathematical/computational development are reduced or approximate fluid flow models. In this study a combined perturbation/numerical modeling methodology is developed which provides a rigorously derived family of solutions. The mathematical model is computationally more efficient than classical boundary layer but provides important two-dimensional information not available using quasi-1-d approaches. An additional strength of the current methodology is its ability to locally predict static pressure fields in a manner analogous to more sophisticated parabolized Navier Stokes (PNS) formulations. To resolve singular behavior, the model utilizes classical analytical solution techniques. Hence, analytical methods have been combined with efficient numerical methods to yield an efficient hybrid fluid flow model. In particular, the main objective of this research has been to develop a system of analytical and numerical ejector/mixer nozzle models, which require minimal empirical input. A computer code, DREA Differential Reduced Ejector/mixer Analysis has been developed with the ability to run sufficiently fast so that it may be used either as a subroutine or called by an design optimization routine. Models are of direct use to the High Speed Civil Transport Program (a joint government/industry project seeking to develop an economically.viable U.S. commercial supersonic transport vehicle) and are currently being adopted by both NASA and industry. Experimental validation of these models is provided by comparison to results obtained from open literature and Limited Exclusive Right Distribution (LERD) sources, as well as dedicated experiments performed at Texas A&M. These experiments have

16. A Methodology for Confirmatory Testing of Numerical Models of Groundwater Flow and Solute Transport in Fractured Crystalline Rock

Hartley, L.; Follin, S.; Rhen, I.; Selroos, J.

2008-12-01

Three-dimensional, regional, numerical models of groundwater flow and solute transport in fractured crystalline rock are used for two sites in Sweden that are considered for geological disposal of spent nuclear fuel. The models are used to underpin the conceptual modeling that is based on multi-disciplinary data and include descriptions of the geometry of geological features (deformation zones and fracture networks), transient hydrological and chemical boundary conditions, strong spatial heterogeneity in the hydraulic properties, density driven flow, solute transport including rock matrix diffusion, and mixing of different water types in a palaeo-hydrogeological perspective (last 10,000 years). From a credibility point of view, comparisons between measured and simulated data are important and provide a means to address our ability to understand complex hydrogeological systems, and hence what particular applications of a hydrogeological model of a physical system that are justified, e.g. in subsequent repository performance assessment studies. For instance, it has been suggested that an understanding of the hydrochemical evolution throughout geological time is a powerful tool to predict the future evolution of groundwater flow and its chemical composition. The general approach applied in the numerical modeling was to first parameterize the deformation zones and fracture networks hydraulically using fracture and inflow data from single-hole tests. Second, the confirmatory step relies on using essentially the same groundwater flow and solute transport model in terms of grid discretization and parameter settings for matching three types of independent field data: 1) large-scale cross-hole (interference) tests, 2) long-term monitoring of groundwater levels, and 3) hydrochemical composition of fracture water and matrix pore water in deep boreholes. We demonstrate here the modelling approach of the second step - confirmatory testing - using data from the site

17. More accurate determination of the quantity of ice crystallized at low cooling rates in the glycerol and 1,2-propanediol aqueous solutions: comparison with equilibrium.

PubMed

Boutron, P

1984-04-01

It is generally assumed that when cells are cooled at rates close to those corresponding to the maximum of survival, once supercooling has ceased, above the eutectic melting temperature the extracellular ice is in equilibrium with the residual solution. This did not seem evident to us due to the difficulty of ice crystallization in cryoprotective solutions. The maximum quantities of ice crystallized in glycerol and 1,2-propanediol solutions have been calculated from the area of the solidification and fusion peaks obtained with a Perkin-Elmer DSC-2 differential scanning calorimeter. The accuracy has been improved by several corrections: better defined baseline, thermal variation of the heat of fusion of the ice, heat of solution of the water from its melting with the residual solution. More ice crystallizes in the glycerol than in the 1,2-propanediol solutions, of which the amorphous residue contains about 40 to 55% 1,2-propanediol. The equilibrium values are unknown in the presence of 1,2-propanediol. With glycerol, in our experiments, the maximum is first lower than the equilibrium but approaches it as the concentration increases. It is not completely determined by the colligative properties of the solutes.

18. Numerical model of water flow and solute accumulation in vertisols using HYDRUS 2D/3D code

Weiss, Tomáš; Dahan, Ofer; Turkeltub, Tuvia

2015-04-01

Keywords: dessication-crack-induced-salinization, preferential flow, conceptual model, numerical model, vadose zone, vertisols, soil water retention function, HYDRUS 2D/3D Vertisols cover a hydrologically very significant area of semi-arid regions often through which water infiltrates to groundwater aquifers. Understanding of water flow and solute accumulation is thus very relevant to agricultural activity and water resources management. Previous works suggest a conceptual model of dessication-crack-induced-salinization where salinization of sediment in the deep section of the vadose zone (up to 4 m) is induced by subsurface evaporation due to convective air flow in the dessication cracks. It suggests that the salinization is induced by the hydraulic gradient between the dry sediment in the vicinity of cracks (low potential) and the relatively wet sediment further from the main cracks (high potential). This paper presents a modified previously suggested conceptual model and a numerical model. The model uses a simple uniform flow approach but unconventionally prescribes the boundary conditions and the hydraulic parameters of soil. The numerical model is bound to one location close to a dairy farm waste lagoon, but the application of the suggested conceptual model could be possibly extended to all semi-arid regions with vertisols. Simulations were conducted using several modeling approaches with an ultimate goal of fitting the simulation results to the controlling variables measured in the field: temporal variation in water content across thick layer of unsaturated clay sediment (>10 m), sediment salinity and salinity the water draining down the vadose zone to the water table. The development of the model was engineered in several steps; all computed as forward solutions by try-and-error approach. The model suggests very deep instant infiltration of fresh water up to 12 m, which is also supported by the field data. The paper suggests prescribing a special atmospheric

19. Solutions of the two-dimensional Hubbard model: Benchmarks and results from a wide range of numerical algorithms

SciTech Connect

LeBlanc, J. P. F.; Antipov, Andrey E.; Becca, Federico; Bulik, Ireneusz W.; Chan, Garnet Kin-Lic; Chung, Chia -Min; Deng, Youjin; Ferrero, Michel; Henderson, Thomas M.; Jiménez-Hoyos, Carlos A.; Kozik, E.; Liu, Xuan -Wen; Millis, Andrew J.; Prokof’ev, N. V.; Qin, Mingpu; Scuseria, Gustavo E.; Shi, Hao; Svistunov, B. V.; Tocchio, Luca F.; Tupitsyn, I. S.; White, Steven R.; Zhang, Shiwei; Zheng, Bo -Xiao; Zhu, Zhenyue; Gull, Emanuel

2015-12-14

Numerical results for ground-state and excited-state properties (energies, double occupancies, and Matsubara-axis self-energies) of the single-orbital Hubbard model on a two-dimensional square lattice are presented, in order to provide an assessment of our ability to compute accurate results in the thermodynamic limit. Many methods are employed, including auxiliary-field quantum Monte Carlo, bare and bold-line diagrammatic Monte Carlo, method of dual fermions, density matrix embedding theory, density matrix renormalization group, dynamical cluster approximation, diffusion Monte Carlo within a fixed-node approximation, unrestricted coupled cluster theory, and multireference projected Hartree-Fock methods. Comparison of results obtained by different methods allows for the identification of uncertainties and systematic errors. The importance of extrapolation to converged thermodynamic-limit values is emphasized. Furthermore, cases where agreement between different methods is obtained establish benchmark results that may be useful in the validation of new approaches and the improvement of existing methods.

20. Solutions of the two-dimensional Hubbard model: Benchmarks and results from a wide range of numerical algorithms

DOE PAGES

LeBlanc, J. P. F.; Antipov, Andrey E.; Becca, Federico; Bulik, Ireneusz W.; Chan, Garnet Kin-Lic; Chung, Chia -Min; Deng, Youjin; Ferrero, Michel; Henderson, Thomas M.; Jiménez-Hoyos, Carlos A.; et al

2015-12-14

Numerical results for ground-state and excited-state properties (energies, double occupancies, and Matsubara-axis self-energies) of the single-orbital Hubbard model on a two-dimensional square lattice are presented, in order to provide an assessment of our ability to compute accurate results in the thermodynamic limit. Many methods are employed, including auxiliary-field quantum Monte Carlo, bare and bold-line diagrammatic Monte Carlo, method of dual fermions, density matrix embedding theory, density matrix renormalization group, dynamical cluster approximation, diffusion Monte Carlo within a fixed-node approximation, unrestricted coupled cluster theory, and multireference projected Hartree-Fock methods. Comparison of results obtained by different methods allows for the identification ofmore » uncertainties and systematic errors. The importance of extrapolation to converged thermodynamic-limit values is emphasized. Furthermore, cases where agreement between different methods is obtained establish benchmark results that may be useful in the validation of new approaches and the improvement of existing methods.« less

1. A THREE-DIMENSIONAL NUMERICAL SOLUTION FOR THE SHAPE OF A ROTATIONALLY DISTORTED POLYTROPE OF INDEX UNITY

SciTech Connect

Kong, Dali; Zhang, Keke; Schubert, Gerald; Anderson, John E-mail: K.Zhang@exeter.ac.uk

2013-02-15

We present a new three-dimensional numerical method for calculating the non-spherical shape and internal structure of a model of a rapidly rotating gaseous body with a polytropic index of unity. The calculation is based on a finite-element method and accounts for the full effects of rotation. After validating the numerical approach against the asymptotic solution of Chandrasekhar that is valid only for a slowly rotating gaseous body, we apply it to models of Jupiter and a rapidly rotating, highly flattened star ({alpha} Eridani). In the case of Jupiter, the two-dimensional distributions of density and pressure are determined via a hybrid inverse approach by adjusting an a priori unknown coefficient in the equation of state until the model shape matches the observed shape of Jupiter. After obtaining the two-dimensional distribution of density, we then compute the zonal gravity coefficients and the total mass from the non-spherical model that takes full account of rotation-induced shape change. Our non-spherical model with a polytropic index of unity is able to produce the known mass of Jupiter with about 4% accuracy and the zonal gravitational coefficient J {sub 2} of Jupiter with better than 2% accuracy, a reasonable result considering that there is only one parameter in the model. For {alpha} Eridani, we calculate its rotationally distorted shape and internal structure based on the observationally deduced rotation rate and size of the star by using a similar hybrid inverse approach. Our model of the star closely approximates the observed flattening.

2. FANTOM: Two- and three-dimensional numerical modelling of creeping flows for the solution of geological problems

Thieulot, Cedric

2011-09-01

A new finite element code for the solution of the Stokes and heat transport equations is presented. It has purposely been designed to address geological flow problems in two and three dimensions at crustal and lithospheric scales. A variety of rheologies has been implemented including nonlinear thermally activated creep and brittle (or plastic) frictional models. A cloud of particles is used to track materials in the simulation domain which allows to record the integrated history of deformation; its density is variable and dynamically adapted. The code is built on the Arbitrary Lagrangian-Eulerian kinematical description: the computational grid deforms vertically and allows for a true free surface while the computational domain remains of constant width in the horizontal direction. The code can be run in sequential or parallel mode. The parallelisation is based on the MPI paradigm and the domain decomposition algorithm is presented. The solution to the large system of algebraic equations resulting from the finite element discretisation and linearisation of the set of coupled partial differential equations to be solved is obtained by means of an efficient (sequential or massively parallel) direct solver. Details of implementation concerning plasticity, cloud handing, nonlinear convergence, strain accumulation and pressure smoothing are given. The sequential 2D version of the code is used to run the numerical sandbox benchmark at normal and high resolutions. The 2D parallel version is used in the case of a thermo-mechanically coupled extension experiment in which the mantle is present. The 3D parallel version is used to run a crustal scale orogeny experiment. The overall performance of the code, as well as the respect of the incompressibility constraint, of the yield stress criterion and of the volume conservation are discussed, along with parallel scalability. Benchmark results of scalar field advection, the Rayleigh-Taylor experiment and the falling block

3. High altitude chemically reacting gas particle mixtures. Volume 1: A theoretical analysis and development of the numerical solution. [rocket nozzle and orbital plume flow fields

NASA Technical Reports Server (NTRS)

Smith, S. D.

1984-01-01

The overall contractual effort and the theory and numerical solution for the Reacting and Multi-Phase (RAMP2) computer code are described. The code can be used to model the dominant phenomena which affect the prediction of liquid and solid rocket nozzle and orbital plume flow fields. Fundamental equations for steady flow of reacting gas-particle mixtures, method of characteristics, mesh point construction, and numerical integration of the conservation equations are considered herein.

4. Heat Generation/Absorption Effects in a Boundary Layer Stretched Flow of Maxwell Nanofluid: Analytic and Numeric Solutions

PubMed Central

Awais, Muhammad; Hayat, Tasawar; Irum, Sania; Alsaedi, Ahmed

2015-01-01

Analysis has been done to investigate the heat generation/absorption effects in a steady flow of non-Newtonian nanofluid over a surface which is stretching linearly in its own plane. An upper convected Maxwell model (UCM) has been utilized as the non-Newtonian fluid model in view of the fact that it can predict relaxation time phenomenon which the Newtonian model cannot. Behavior of the relaxations phenomenon has been presented in terms of Deborah number. Transport phenomenon with convective cooling process has been analyzed. Brownian motion “Db” and thermophoresis effects “Dt” occur in the transport equations. The momentum, energy and nanoparticle concentration profiles are examined with respect to the involved rheological parameters namely the Deborah number, source/sink parameter, the Brownian motion parameters, thermophoresis parameter and Biot number. Both numerical and analytic solutions are presented and found in nice agreement. Comparison with the published data is also made to ensure the validity. Stream lines for Maxwell and Newtonian fluid models are presented in the analysis. PMID:26115101

5. Numerical solution of the moving boundary-value problem based on the model of piston-like oil displacement

2016-06-01

Prediction of the motion of the oil-water contact boundary has great importance in the problems of design of oilfield development by flooding. In this paper we consider a piston-like model of oil-water displacement, which takes into account differences in viscosity and density of the two fluids. Oil reservoir assumed to be homogeneous and infinite, fixed thickness, with constant values of porosity and permeability coefficients. Filtration of liquids is described by Darcy's law. It is assumed, that both fluids are weakly compressible and the pressure in the reservoir satisfies the quasi-stationary diffusion equation. Piston-like displacement model leads to the discontinuity of the tangential component of the velocity vector at the boundary of oil-water contact. Use the Cauchy integral reduces the problem of finding the current boundaries of oil-water contact to the system of singular integral equations for the tangential and normal components of the velocity vector and the Cauchy problem for the integration of the differential equations of motion of the boundary of oil-water contact. An algorithm for the numerical solution of this problem is developed. The monitoring of oil-water boundary motion for different regular and irregular schemes of flooding is carried out.

6. Heat Generation/Absorption Effects in a Boundary Layer Stretched Flow of Maxwell Nanofluid: Analytic and Numeric Solutions.

PubMed

Awais, Muhammad; Hayat, Tasawar; Irum, Sania; Alsaedi, Ahmed

2015-01-01

Analysis has been done to investigate the heat generation/absorption effects in a steady flow of non-Newtonian nanofluid over a surface which is stretching linearly in its own plane. An upper convected Maxwell model (UCM) has been utilized as the non-Newtonian fluid model in view of the fact that it can predict relaxation time phenomenon which the Newtonian model cannot. Behavior of the relaxations phenomenon has been presented in terms of Deborah number. Transport phenomenon with convective cooling process has been analyzed. Brownian motion "Db" and thermophoresis effects "Dt" occur in the transport equations. The momentum, energy and nanoparticle concentration profiles are examined with respect to the involved rheological parameters namely the Deborah number, source/sink parameter, the Brownian motion parameters, thermophoresis parameter and Biot number. Both numerical and analytic solutions are presented and found in nice agreement. Comparison with the published data is also made to ensure the validity. Stream lines for Maxwell and Newtonian fluid models are presented in the analysis. PMID:26115101

7. Oblique shock breakout in supernovae and gamma-ray bursts. II. Numerical solutions for non-relativistic pattern speeds

SciTech Connect

Salbi, Pegah; Matzner, Christopher D.; Ro, Stephen; Levin, Yuri

2014-07-20

Non-spherical explosions develop non-radial flows as the pattern of shock emergence progresses across the stellar surface. In supernovae, these flows can limit ejecta speeds, stifle shock breakout emission, and cause collisions outside the star. Similar phenomena occur in stellar and planetary collisions, tidal disruption events, accretion-induced collapses, and propagating detonations. We present two-dimensional, nested-grid Athena simulations of non-radial shock emergence in a frame comoving with the breakout pattern, focusing on the adiabatic, non-relativistic limit in a plane stratified envelope. We set boundary conditions using a known self-similar solution and explore the role of box size and resolution on the result. The shock front curves toward the stellar surface, and exhibits a kink from which weak discontinuities originate. Flow around the point of shock emergence is neither perfectly steady nor self-similar. Waves and vortices, which are not predominantly due to grid effects, emanate from this region. The post-shock flow is deflected along the stellar surface and its pressure disturbs the stellar atmosphere upstream of the emerging shock. We use the numerical results and their analytical limits to predict the effects of radiation transfer and gravity, which are not included in our simulations.

8. Dynamic design, numerical solution and effective verification of acceleration-level obstacle-avoidance scheme for robot manipulators

Xiao, Lin; Zhang, Yunong

2016-03-01

For avoiding obstacles and joint physical constraints of robot manipulators, this paper proposes and investigates a novel obstacle avoidance scheme (termed the acceleration-level obstacle-avoidance scheme). The scheme is based on a new obstacle-avoidance criterion that is designed by using the gradient neural network approach for the first time. In addition, joint physical constraints such as joint-angle limits, joint-velocity limits and joint-acceleration limits are incorporated into such a scheme, which is further reformulated as a quadratic programming (QP). Two important 'bridge' theorems are established so that such a QP can be converted equivalently to a linear variational inequality and then equivalently to a piecewise-linear projection equation (PLPE). A numerical algorithm based on a PLPE is thus developed and applied for an online solution of the resultant QP. Four path-tracking tasks based on the PA10 robot in the presence of point and window-shaped obstacles demonstrate and verify the effectiveness and accuracy of the acceleration-level obstacle-avoidance scheme. Besides, the comparisons between the non-obstacle-avoidance and obstacle-avoidance results further validate the superiority of the proposed scheme.

9. A new numerical solution for the MHD peristaltic flow of a bio-fluid with variable viscosity in a circular cylindrical tube via Adomian decomposition method

Ebaid, A.

2008-08-01

In this Letter, we considered a numerical treatment for the solution of the hydromagnetic peristaltic flow of a bio-fluid with variable viscosity in a circular cylindrical tube using Adomian decomposition method and a modified form of this method. The axial velocity is obtained in a closed form. Comparison is made between the results obtained by only three terms of Adomian series with those obtained previously by perturbation technique. It is observed that only few terms of the series expansion are required to obtain the numerical solution with good accuracy.

10. Extension of the AMBER force field for nitroxide radicals and combined QM/MM/PCM approach to the accurate determination of EPR parameters of DMPOH in solution

PubMed Central

Hermosilla, Laura; Prampolini, Giacomo; Calle, Paloma; García de la Vega, José Manuel; Brancato, Giuseppe; Barone, Vincenzo

2015-01-01

A computational strategy that combines both time-dependent and time-independent approaches is exploited to accurately model molecular dynamics and solvent effects on the isotropic hyperfine coupling constants of the DMPO-H nitroxide. Our recent general force field for nitroxides derived from AMBER ff99SB is further extended to systems involving hydrogen atoms in β-positions with respect to NO. The resulting force-field has been employed in a series of classical molecular dynamics simulations, comparing the computed EPR parameters from selected molecular configurations to the corresponding experimental data in different solvents. The effect of vibrational averaging on the spectroscopic parameters is also taken into account, by second order vibrational perturbation theory involving semi-diagonal third energy derivatives together first and second property derivatives. PMID:26584116

11. Dynamical Approach Study of Spurious Steady-State Numerical Solutions of Nonlinear Differential Equations. 2; Global Asymptotic Behavior of Time Discretizations; 2. Global Asymptotic Behavior of time Discretizations

NASA Technical Reports Server (NTRS)

Yee, H. C.; Sweby, P. K.

1995-01-01

The global asymptotic nonlinear behavior of 1 1 explicit and implicit time discretizations for four 2 x 2 systems of first-order autonomous nonlinear ordinary differential equations (ODES) is analyzed. The objectives are to gain a basic understanding of the difference in the dynamics of numerics between the scalars and systems of nonlinear autonomous ODEs and to set a baseline global asymptotic solution behavior of these schemes for practical computations in computational fluid dynamics. We show how 'numerical' basins of attraction can complement the bifurcation diagrams in gaining more detailed global asymptotic behavior of time discretizations for nonlinear differential equations (DEs). We show how in the presence of spurious asymptotes the basins of the true stable steady states can be segmented by the basins of the spurious stable and unstable asymptotes. One major consequence of this phenomenon which is not commonly known is that this spurious behavior can result in a dramatic distortion and, in most cases, a dramatic shrinkage and segmentation of the basin of attraction of the true solution for finite time steps. Such distortion, shrinkage and segmentation of the numerical basins of attraction will occur regardless of the stability of the spurious asymptotes, and will occur for unconditionally stable implicit linear multistep methods. In other words, for the same (common) steady-state solution the associated basin of attraction of the DE might be very different from the discretized counterparts and the numerical basin of attraction can be very different from numerical method to numerical method. The results can be used as an explanation for possible causes of error, and slow convergence and nonconvergence of steady-state numerical solutions when using the time-dependent approach for nonlinear hyperbolic or parabolic PDES.

12. Dynamical Approach Study of Spurious Steady-State Numerical Solutions of Nonlinear Differential Equations. Part 2; Global Asymptotic Behavior of Time Discretizations

NASA Technical Reports Server (NTRS)

Yee, H. C.; Sweby, P. K.

1995-01-01

The global asymptotic nonlinear behavior of 11 explicit and implicit time discretizations for four 2 x 2 systems of first-order autonomous nonlinear ordinary differential equations (ODEs) is analyzed. The objectives are to gain a basic understanding of the difference in the dynamics of numerics between the scalars and systems of nonlinear autonomous ODEs and to set a baseline global asymptotic solution behavior of these schemes for practical computations in computational fluid dynamics. We show how 'numerical' basins of attraction can complement the bifurcation diagrams in gaining more detailed global asymptotic behavior of time discretizations for nonlinear differential equations (DEs). We show how in the presence of spurious asymptotes the basins of the true stable steady states can be segmented by the basins of the spurious stable and unstable asymptotes. One major consequence of this phenomenon which is not commonly known is that this spurious behavior can result in a dramatic distortion and, in most cases, a dramatic shrinkage and segmentation of the basin of attraction of the true solution for finite time steps. Such distortion, shrinkage and segmentation of the numerical basins of attraction will occur regardless of the stability of the spurious asymptotes, and will occur for unconditionally stable implicit linear multistep methods. In other words, for the same (common) steady-state solution the associated basin of attraction of the DE might be very different from the discretized counterparts and the numerical basin of attraction can be very different from numerical method to numerical method. The results can be used as an explanation for possible causes of error, and slow convergence and nonconvergence of steady-state numerical solutions when using the time-dependent approach for nonlinear hyperbolic or parabolic PDEs.

13. Super-thermal particles in hot plasmas—Kinetic models, numerical solution strategies, and comparison to tokamak experiments

Lauber, Philipp

2013-12-01

The excitation of collective instabilities by super-thermal particles in hot plasmas and the related transport processes attract increasing interest due to their fundamental challenges for theoretical models and their practical importance for burning fusion plasmas. In fact, the physics of a self-heated thermonuclear plasma due to fusion-born 3.5 MeV α-particles is one of the most important outstanding fundamental research topics on the way to a fusion power plant with magnetic confinement. Within the last 10 years significant advances on both the theoretical and the experimental sides have been made leading to a more detailed and quantitative understanding of fast-particle-driven instabilities. On the theoretical side, the crucial step was to move from fluid models for the plasma background with a hybrid kinetic expression for the energetic particles to a fully kinetic model for all the plasma species, i.e. background ions, background electrons, and fast ions. This improvement allows one to describe consistently the resonant interaction between global plasma waves such as shear Alfvén and Alfvén-acoustic waves, and the particles via Landau damping, i.e. the dynamics parallel to the magnetic background field. Also, mode conversion mechanisms require the inclusion of background ion scales in a kinetic, non-perturbative way. This accurate treatment of the plasma background leads not only to changes in the linear mode properties such as frequency, growth/damping rate, and mode structure but also influences the non-linear dynamics. Due to major advances, innovations and installation of diagnostics in present day experiments, this comparison can be carried out in a more detailed and comprehensive way than a few years ago. For example, the measurement of damping rates via active external antennas, the imaging of 2D mode structures via electron-cyclotron-emission spectroscopy, and the direct detection of escaping fast ions allow to diagnose various kinetic features of

14. Accurate mass determination, quantification and determination of detection limits in liquid chromatography-high-resolution time-of-flight mass spectrometry: challenges and practical solutions.

PubMed

Vergeynst, Leendert; Van Langenhove, Herman; Joos, Pieter; Demeestere, Kristof

2013-07-30

Uniform guidelines for the data processing and validation of qualitative and quantitative multi-residue analysis using full-spectrum high-resolution mass spectrometry are scarce. Through systematic research, optimal mass accuracy and sensitivity are obtained after refining the post-processing of the HRMS data. For qualitative analysis, transforming the raw profile spectra to centroid spectra is recommended resulting in a 2.3 fold improved precision on the accurate mass determination of spectrum peaks. However, processing centroid data for quantitative purposes could lead to signal interruption when too narrow mass windows are applied for the construction of extracted ion chromatograms. Therefore, peak integration on the raw profile data is recommended. An optimal width of the mass window of 50 ppm, which is a trade-off between sensitivity and selectivity, was obtained for a TOF instrument providing a resolving power of 20,000 at full width at half maximum (FWHM). For the validation of HRMS analytical methods, widespread concepts such as the signal-to-noise ratios for the determination of decision limits and detection capabilities have shown to be not always applicable because in some cases almost no noise can be detected anymore. A statistical methodology providing a reliable alternative is extended and applied. PMID:23856232

15. CANM, a program for numerical solution of a system of nonlinear equations using the continuous analog of Newton's method

Abrashkevich, Alexander; Puzynin, I. V.

2004-01-01

A FORTRAN program is presented which solves a system of nonlinear simultaneous equations using the continuous analog of Newton's method (CANM). The user has the option of either to provide a subroutine which calculates the Jacobian matrix or allow the program to calculate it by a forward-difference approximation. Five iterative schemes using different algorithms of determining adaptive step size of the CANM process are implemented in the program. Program summaryTitle of program: CANM Catalogue number: ADSN Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADSN Program available from: CPC Program Library, Queen's University of Belfast, Northern Ireland Licensing provisions: none Computer for which the program is designed and others on which it has been tested: Computers: IBM RS/6000 Model 320H, SGI Origin2000, SGI Octane, HP 9000/755, Intel Pentium IV PC Installation: Department of Chemistry, University of Toronto, Toronto, Canada Operating systems under which the program has been tested: IRIX64 6.1, 6.4 and 6.5, AIX 3.4, HP-UX 9.01, Linux 2.4.7 Programming language used: FORTRAN 90 Memory required to execute with typical data: depends on the number of nonlinear equations in a system. Test run requires 80 KB No. of bits in distributed program including test data, etc.: 15283 Distribution format: tar gz format No. of lines in distributed program, including test data, etc.: 1794 Peripherals used: line printer, scratch disc store External subprograms used: DGECO and DGESL [1] Keywords: nonlinear equations, Newton's method, continuous analog of Newton's method, continuous parameter, evolutionary differential equation, Euler's method Nature of physical problem: System of nonlinear simultaneous equations F i(x 1,x 2,…,x n)=0,1⩽i⩽n, is numerically solved. It can be written in vector form as F( X)= 0, X∈ Rn, where F : Rn→ Rn is a twice continuously differentiable function with domain and range in n-dimensional Euclidean space. The solutions of such systems of

16. Accurate theoretical prediction of vibrational frequencies in an inhomogeneous dynamic environment: A case study of a glutamate molecule in water solution and in a protein-bound form

Speranskiy, Kirill; Kurnikova, Maria

2004-07-01

We propose a hierarchical approach to model vibrational frequencies of a ligand in a strongly fluctuating inhomogeneous environment such as a liquid solution or when bound to a macromolecule, e.g., a protein. Vibrational frequencies typically measured experimentally are ensemble averaged quantities which result (in part) from the influence of the strongly fluctuating solvent. Solvent fluctuations can be sampled effectively by a classical molecular simulation, which in our model serves as the first, low level of the hierarchy. At the second high level of the hierarchy a small subset of system coordinates is used to construct a patch of the potential surface (ab initio) relevant to the vibration in question. This subset of coordinates is under the influence of an instantaneous external force exerted by the environment. The force is calculated at the lower level of the hierarchy. The proposed methodology is applied to model vibrational frequencies of a glutamate in water and when bound to the Glutamate receptor protein and its mutant. Our results are in close agreement with the experimental values and frequency shifts measured by the Jayaraman group by the Fourier transform infrared spectroscopy [Q. Cheng et al., Biochem. 41, 1602 (2002)]. Our methodology proved useful in successfully reproducing vibrational frequencies of a ligand in such a soft, flexible, and strongly inhomogeneous protein as the Glutamate receptor.

17. A new numerical framework for solving conservation laws: The method of space-time conservation element and solution element

NASA Technical Reports Server (NTRS)

Chang, Sin-Chung; To, Wai-Ming

1991-01-01

A new numerical framework for solving conservation laws is being developed. It employs: (1) a nontraditional formulation of the conservation laws in which space and time are treated on the same footing, and (2) a nontraditional use of discrete variables such as numerical marching can be carried out by using a set of relations that represents both local and global flux conservation.

18. Analytical and numerical solutions of the potential and electric field generated by different electrode arrays in a tumor tissue under electrotherapy

PubMed Central

2011-01-01

Background Electrotherapy is a relatively well established and efficient method of tumor treatment. In this paper we focus on analytical and numerical calculations of the potential and electric field distributions inside a tumor tissue in a two-dimensional model (2D-model) generated by means of electrode arrays with shapes of different conic sections (ellipse, parabola and hyperbola). Methods Analytical calculations of the potential and electric field distributions based on 2D-models for different electrode arrays are performed by solving the Laplace equation, meanwhile the numerical solution is solved by means of finite element method in two dimensions. Results Both analytical and numerical solutions reveal significant differences between the electric field distributions generated by electrode arrays with shapes of circle and different conic sections (elliptic, parabolic and hyperbolic). Electrode arrays with circular, elliptical and hyperbolic shapes have the advantage of concentrating the electric field lines in the tumor. Conclusion The mathematical approach presented in this study provides a useful tool for the design of electrode arrays with different shapes of conic sections by means of the use of the unifying principle. At the same time, we verify the good correspondence between the analytical and numerical solutions for the potential and electric field distributions generated by the electrode array with different conic sections. PMID:21943385

19. Construction of Three Dimensional Solutions for the Maxwell Equations

NASA Technical Reports Server (NTRS)

Yefet, A.; Turkel, E.

1998-01-01

We consider numerical solutions for the three dimensional time dependent Maxwell equations. We construct a fourth order accurate compact implicit scheme and compare it to the Yee scheme for free space in a box.

20. Investigation of Nozzle Stability for the First Ovalization Mode by Numerical Solution of the Fluid Structure Interaction Problem