An Implementation Method of the Fractional-Order PID Control System Considering the Memory Constraint and its Application to the Temperature Control of Heat Plate

NASA Astrophysics Data System (ADS)

Sasano, Koji; Okajima, Hiroshi; Matsunaga, Nobutomo

Recently, the fractional order PID (FO-PID) control, which is the extension of the PID control, has been focused on. Even though the FO-PID requires the high-order filter, it is difficult to realize the high-order filter due to the memory limitation of digital computer. For implementation of FO-PID, approximation of the fractional integrator and differentiator are required. Short memory principle (SMP) is one of the effective approximation methods. However, there is a disadvantage that the approximated filter with SMP cannot eliminate the steady-state error. For this problem, we introduce the distributed implementation of the integrator and the dynamic quantizer to make the efficient use of permissible memory. The objective of this study is to clarify how to implement the accurate FO-PID with limited memories. In this paper, we propose the implementation method of FO-PID with memory constraint using dynamic quantizer. And the trade off between approximation of fractional elements and quantized data size are examined so as to close to the ideal FO-PID responses. The effectiveness of proposed method is evaluated by numerical example and experiment in the temperature control of heat plate.

Piecewise-homotopy analysis method (P-HAM) for first order nonlinear ODE

NASA Astrophysics Data System (ADS)

Chin, F. Y.; Lem, K. H.; Chong, F. S.

2013-09-01

In homotopy analysis method (HAM), the determination for the value of the auxiliary parameter h is based on the valid region of the h-curve in which the horizontal segment of the h-curve will decide the valid h-region. All h-value taken from the valid region, provided that the order of deformation is large enough, will in principle yield an approximation series that converges to the exact solution. However it is found out that the h-value chosen within this valid region does not always promise a good approximation under finite order. This paper suggests an improved method called Piecewise-HAM (P-HAM). In stead of a single h-value, this method suggests using many h-values. Each of the h-values comes from an individual h-curve while each h-curve is plotted by fixing the time t at a different value. Each h-value is claimed to produce a good approximation only about a neighborhood centered at the corresponding t which the h-curve is based on. Each segment of these good approximations is then joined to form the approximation curve. By this, the convergence region is enhanced further. The P-HAM is illustrated and supported by examples.

Legendre-tau approximation for functional differential equations. Part 2: The linear quadratic optimal control problem

NASA Technical Reports Server (NTRS)

Ito, K.; Teglas, R.

1984-01-01

The numerical scheme based on the Legendre-tau approximation is proposed to approximate the feedback solution to the linear quadratic optimal control problem for hereditary differential systems. The convergence property is established using Trotter ideas. The method yields very good approximations at low orders and provides an approximation technique for computing closed-loop eigenvalues of the feedback system. A comparison with existing methods (based on averaging and spline approximations) is made.

Legendre-tau approximation for functional differential equations. II - The linear quadratic optimal control problem

NASA Technical Reports Server (NTRS)

Ito, Kazufumi; Teglas, Russell

1987-01-01

The numerical scheme based on the Legendre-tau approximation is proposed to approximate the feedback solution to the linear quadratic optimal control problem for hereditary differential systems. The convergence property is established using Trotter ideas. The method yields very good approximations at low orders and provides an approximation technique for computing closed-loop eigenvalues of the feedback system. A comparison with existing methods (based on averaging and spline approximations) is made.

Scattering from very rough layers under the geometric optics approximation: further investigation.

PubMed

Pinel, Nicolas; Bourlier, Christophe

2008-06-01

Scattering from very rough homogeneous layers is studied in the high-frequency limit (under the geometric optics approximation) by taking the shadowing effect into account. To do so, the iterated Kirchhoff approximation, recently developed by Pinel et al. [Waves Random Complex Media17, 283 (2007)] and reduced to the geometric optics approximation, is used and investigated in more detail. The contributions from the higher orders of scattering inside the rough layer are calculated under the iterated Kirchhoff approximation. The method can be applied to rough layers of either very rough or perfectly flat lower interfaces, separating either lossless or lossy media. The results are compared with the PILE (propagation-inside-layer expansion) method, recently developed by Déchamps et al. [J. Opt. Soc. Am. A23, 359 (2006)], and accelerated by the forward-backward method with spectral acceleration. They highlight that there is very good agreement between the developed method and the reference numerical method for all scattering orders and that the method can be applied to root-mean-square (RMS) heights at least down to 0.25lambda.

An accurate method for evaluating the kernel of the integral equation relating lift to downwash in unsteady potential flow

NASA Technical Reports Server (NTRS)

Desmarais, R. N.

1982-01-01

The method is capable of generating approximations of arbitrary accuracy. It is based on approximating the algebraic part of the nonelementary integrals in the kernel by exponential functions and then integrating termwise. The exponent spacing in the approximation is a geometric sequence. The coefficients and exponent multiplier of the exponential approximation are computed by least squares so the method is completely automated. Exponential approximates generated in this manner are two orders of magnitude more accurate than the exponential approximation that is currently most often used for this purpose. The method can be used to generate approximations to attain any desired trade-off between accuracy and computing cost.

Cosmological collapse and the improved Zel'dovich approximation.

NASA Astrophysics Data System (ADS)

Salopek, D. S.; Stewart, J. M.; Croudace, K. M.; Parry, J.

Using a general relativistic formulation, the authors show how to compute the higher order terms in the Zel'dovich approximation which describes cosmological collapse. They evolve the 3-metric in a spatial gradient expansion. Their method is an advance over earlier work because it is local at each order. Using the improved Zel'dovich approximation, they compute the epoch of collapse.

Second order upwind Lagrangian particle method for Euler equations

DOE PAGES

Samulyak, Roman; Chen, Hsin -Chiang; Yu, Kwangmin

2016-06-01

A new second order upwind Lagrangian particle method for solving Euler equations for compressible inviscid fluid or gas flows is proposed. Similar to smoothed particle hydrodynamics (SPH), the method represents fluid cells with Lagrangian particles and is suitable for the simulation of complex free surface / multiphase flows. The main contributions of our method, which is different from SPH in all other aspects, are (a) significant improvement of approximation of differential operators based on a polynomial fit via weighted least squares approximation and the convergence of prescribed order, (b) an upwind second-order particle-based algorithm with limiter, providing accuracy and longmore » term stability, and (c) accurate resolution of states at free interfaces. In conclusion, numerical verification tests demonstrating the convergence order for fixed domain and free surface problems are presented.« less

Second order upwind Lagrangian particle method for Euler equations

DOE Office of Scientific and Technical Information (OSTI.GOV)

Samulyak, Roman; Chen, Hsin -Chiang; Yu, Kwangmin

A new second order upwind Lagrangian particle method for solving Euler equations for compressible inviscid fluid or gas flows is proposed. Similar to smoothed particle hydrodynamics (SPH), the method represents fluid cells with Lagrangian particles and is suitable for the simulation of complex free surface / multiphase flows. The main contributions of our method, which is different from SPH in all other aspects, are (a) significant improvement of approximation of differential operators based on a polynomial fit via weighted least squares approximation and the convergence of prescribed order, (b) an upwind second-order particle-based algorithm with limiter, providing accuracy and longmore » term stability, and (c) accurate resolution of states at free interfaces. In conclusion, numerical verification tests demonstrating the convergence order for fixed domain and free surface problems are presented.« less

A Second-Order Phase Transition as a Limit of the First-Order Phase Transitions —Coherent Anomalies and Critical Phenomena in the Potts Models—

NASA Astrophysics Data System (ADS)

Katori, Makoto

1988-12-01

A new scheme of the coherent-anomaly method (CAM) is proposed to study critical phenomena in the models for which a mean-field description gives spurious first-order phase transition. A canonical series of mean-field-type approximations are constructed so that the spurious discontinuity should vanish asymptotically as the approximate critical temperature approachs the true value. The true value of the critical exponents β and γ are related to the coherent-anomaly exponents defined among the classical approximations. The formulation is demonstrated in the two-dimensional q-state Potts models for q{=}3 and 4. The result shows that the present method enables us to estimate the critical exponents with high accuracy by using the date of the cluster-mean-field approximations.

Computational methods for estimation of parameters in hyperbolic systems

NASA Technical Reports Server (NTRS)

Banks, H. T.; Ito, K.; Murphy, K. A.

1983-01-01

Approximation techniques for estimating spatially varying coefficients and unknown boundary parameters in second order hyperbolic systems are discussed. Methods for state approximation (cubic splines, tau-Legendre) and approximation of function space parameters (interpolatory splines) are outlined and numerical findings for use of the resulting schemes in model "one dimensional seismic inversion' problems are summarized.

Some problems of nonlinear waves in solid propellant rocket motors

NASA Technical Reports Server (NTRS)

Culick, F. E. C.

1979-01-01

An approximate technique for analyzing nonlinear waves in solid propellant rocket motors is presented which inexpensively provides accurate results up to amplitudes of ten percent. The connection with linear stability analysis is shown. The method is extended to third order in the amplitude of wave motion in order to study nonlinear stability, or triggering. Application of the approximate method to the behavior of pulses is described.

Lagrangian particle method for compressible fluid dynamics

DOE Office of Scientific and Technical Information (OSTI.GOV)

Samulyak, Roman; Wang, Xingyu; Chen, Hsin -Chiang

A new Lagrangian particle method for solving Euler equations for compressible inviscid fluid or gas flows is proposed. Similar to smoothed particle hydrodynamics (SPH), the method represents fluid cells with Lagrangian particles and is suitable for the simulation of complex free surface / multi-phase flows. The main contributions of our method, which is different from SPH in all other aspects, are (a) significant improvement of approximation of differential operators based on a polynomial fit via weighted least squares approximation and the convergence of prescribed order, (b) a second-order particle-based algorithm that reduces to the first-order upwind method at local extremalmore » points, providing accuracy and long term stability, and (c) more accurate resolution of entropy discontinuities and states at free inter-faces. While the method is consistent and convergent to a prescribed order, the conservation of momentum and energy is not exact and depends on the convergence order . The method is generalizable to coupled hyperbolic-elliptic systems. As a result, numerical verification tests demonstrating the convergence order are presented as well as examples of complex multiphase flows.« less

Lagrangian particle method for compressible fluid dynamics

DOE PAGES

Samulyak, Roman; Wang, Xingyu; Chen, Hsin -Chiang

2018-02-09

A new Lagrangian particle method for solving Euler equations for compressible inviscid fluid or gas flows is proposed. Similar to smoothed particle hydrodynamics (SPH), the method represents fluid cells with Lagrangian particles and is suitable for the simulation of complex free surface / multi-phase flows. The main contributions of our method, which is different from SPH in all other aspects, are (a) significant improvement of approximation of differential operators based on a polynomial fit via weighted least squares approximation and the convergence of prescribed order, (b) a second-order particle-based algorithm that reduces to the first-order upwind method at local extremalmore » points, providing accuracy and long term stability, and (c) more accurate resolution of entropy discontinuities and states at free inter-faces. While the method is consistent and convergent to a prescribed order, the conservation of momentum and energy is not exact and depends on the convergence order . The method is generalizable to coupled hyperbolic-elliptic systems. As a result, numerical verification tests demonstrating the convergence order are presented as well as examples of complex multiphase flows.« less

Jacobi-Gauss-Lobatto collocation method for the numerical solution of 1+1 nonlinear Schrödinger equations

NASA Astrophysics Data System (ADS)

Doha, E. H.; Bhrawy, A. H.; Abdelkawy, M. A.; Van Gorder, Robert A.

2014-03-01

A Jacobi-Gauss-Lobatto collocation (J-GL-C) method, used in combination with the implicit Runge-Kutta method of fourth order, is proposed as a numerical algorithm for the approximation of solutions to nonlinear Schrödinger equations (NLSE) with initial-boundary data in 1+1 dimensions. Our procedure is implemented in two successive steps. In the first one, the J-GL-C is employed for approximating the functional dependence on the spatial variable, using (N-1) nodes of the Jacobi-Gauss-Lobatto interpolation which depends upon two general Jacobi parameters. The resulting equations together with the two-point boundary conditions induce a system of 2(N-1) first-order ordinary differential equations (ODEs) in time. In the second step, the implicit Runge-Kutta method of fourth order is applied to solve this temporal system. The proposed J-GL-C method, used in combination with the implicit Runge-Kutta method of fourth order, is employed to obtain highly accurate numerical approximations to four types of NLSE, including the attractive and repulsive NLSE and a Gross-Pitaevskii equation with space-periodic potential. The numerical results obtained by this algorithm have been compared with various exact solutions in order to demonstrate the accuracy and efficiency of the proposed method. Indeed, for relatively few nodes used, the absolute error in our numerical solutions is sufficiently small.

Symmetric Positive 4th Order Tensors & Their Estimation from Diffusion Weighted MRI⋆

PubMed Central

Barmpoutis, Angelos; Jian, Bing; Vemuri, Baba C.; Shepherd, Timothy M.

2009-01-01

In Diffusion Weighted Magnetic Resonance Image (DW-MRI) processing a 2nd order tensor has been commonly used to approximate the diffusivity function at each lattice point of the DW-MRI data. It is now well known that this 2nd-order approximation fails to approximate complex local tissue structures, such as fibers crossings. In this paper we employ a 4th order symmetric positive semi-definite (PSD) tensor approximation to represent the diffusivity function and present a novel technique to estimate these tensors from the DW-MRI data guaranteeing the PSD property. There have been several published articles in literature on higher order tensor approximations of the diffusivity function but none of them guarantee the positive semi-definite constraint, which is a fundamental constraint since negative values of the diffusivity coefficients are not meaningful. In our methods, we parameterize the 4th order tensors as a sum of squares of quadratic forms by using the so called Gram matrix method from linear algebra and its relation to the Hilbert’s theorem on ternary quartics. This parametric representation is then used in a nonlinear-least squares formulation to estimate the PSD tensors of order 4 from the data. We define a metric for the higher-order tensors and employ it for regularization across the lattice. Finally, performance of this model is depicted on synthetic data as well as real DW-MRI from an isolated rat hippocampus. PMID:17633709

Approximate analysis for repeated eigenvalue problems with applications to controls-structure integrated design

NASA Technical Reports Server (NTRS)

Kenny, Sean P.; Hou, Gene J. W.

1994-01-01

A method for eigenvalue and eigenvector approximate analysis for the case of repeated eigenvalues with distinct first derivatives is presented. The approximate analysis method developed involves a reparameterization of the multivariable structural eigenvalue problem in terms of a single positive-valued parameter. The resulting equations yield first-order approximations to changes in the eigenvalues and the eigenvectors associated with the repeated eigenvalue problem. This work also presents a numerical technique that facilitates the definition of an eigenvector derivative for the case of repeated eigenvalues with repeated eigenvalue derivatives (of all orders). Examples are given which demonstrate the application of such equations for sensitivity and approximate analysis. Emphasis is placed on the application of sensitivity analysis to large-scale structural and controls-structures optimization problems.

Parametric instability analysis of truncated conical shells using the Haar wavelet method

NASA Astrophysics Data System (ADS)

Dai, Qiyi; Cao, Qingjie

2018-05-01

In this paper, the Haar wavelet method is employed to analyze the parametric instability of truncated conical shells under static and time dependent periodic axial loads. The present work is based on the Love first-approximation theory for classical thin shells. The displacement field is expressed as the Haar wavelet series in the axial direction and trigonometric functions in the circumferential direction. Then the partial differential equations are reduced into a system of coupled Mathieu-type ordinary differential equations describing dynamic instability behavior of the shell. Using Bolotin's method, the first-order and second-order approximations of principal instability regions are determined. The correctness of present method is examined by comparing the results with those in the literature and very good agreement is observed. The difference between the first-order and second-order approximations of principal instability regions for tensile and compressive loads is also investigated. Finally, numerical results are presented to bring out the influences of various parameters like static load factors, boundary conditions and shell geometrical characteristics on the domains of parametric instability of conical shells.

Active magnetic refrigerants based on Gd-Si-Ge material and refrigeration apparatus and process

DOEpatents

Gschneidner, Jr., Karl A.; Pecharsky, Vitalij K.

1998-04-28

Active magnetic regenerator and method using Gd.sub.5 (Si.sub.x Ge.sub.1-x).sub.4, where x is equal to or less than 0.5, as a magnetic refrigerant that exhibits a reversible ferromagnetic/antiferromagnetic or ferromagnetic-II/ferromagnetic-I first order phase transition and extraordinary magneto-thermal properties, such as a giant magnetocaloric effect, that renders the refrigerant more efficient and useful than existing magnetic refrigerants for commercialization of magnetic regenerators. The reversible first order phase transition is tunable from approximately 30 K to approximately 290 K (near room temperature) and above by compositional adjustments. The active magnetic regenerator and method can function for refrigerating, air conditioning, and liquefying low temperature cryogens with significantly improved efficiency and operating temperature range from approximately 10 K to 300 K and above. Also an active magnetic regenerator and method using Gd.sub.5 (Si.sub.x Ge.sub.1-x).sub.4, where x is equal to or greater than 0.5, as a magnetic heater/refrigerant that exhibits a reversible ferromagnetic/paramagnetic second order phase transition with large magneto-thermal properties, such as a large magnetocaloric effect that permits the commercialization of a magnetic heat pump and/or refrigerant. This second order phase transition is tunable from approximately 280 K (near room temperature) to approximately 350 K by composition adjustments. The active magnetic regenerator and method can function for low level heating for climate control for buildings, homes and automobile, and chemical processing.

Active magnetic refrigerants based on Gd-Si-Ge material and refrigeration apparatus and process

DOEpatents

Gschneidner, K.A. Jr.; Pecharsky, V.K.

1998-04-28

Active magnetic regenerator and method using Gd{sub 5} (Si{sub x}Ge{sub 1{minus}x}){sub 4}, where x is equal to or less than 0.5, as a magnetic refrigerant that exhibits a reversible ferromagnetic/antiferromagnetic or ferromagnetic-II/ferromagnetic-I first order phase transition and extraordinary magneto-thermal properties, such as a giant magnetocaloric effect, that renders the refrigerant more efficient and useful than existing magnetic refrigerants for commercialization of magnetic regenerators. The reversible first order phase transition is tunable from approximately 30 K to approximately 290 K (near room temperature) and above by compositional adjustments. The active magnetic regenerator and method can function for refrigerating, air conditioning, and liquefying low temperature cryogens with significantly improved efficiency and operating temperature range from approximately 10 K to 300 K and above. Also an active magnetic regenerator and method using Gd{sub 5} (Si{sub x} Ge{sub 1{minus}x}){sub 4}, where x is equal to or greater than 0.5, as a magnetic heater/refrigerant that exhibits a reversible ferromagnetic/paramagnetic second order phase transition with large magneto-thermal properties, such as a large magnetocaloric effect that permits the commercialization of a magnetic heat pump and/or refrigerant. This second order phase transition is tunable from approximately 280 K (near room temperature) to approximately 350 K by composition adjustments. The active magnetic regenerator and method can function for low level heating for climate control for buildings, homes and automobile, and chemical processing. 27 figs.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Broda, Jill Terese

The neutron flux across the nuclear reactor core is of interest to reactor designers and others. The diffusion equation, an integro-differential equation in space and energy, is commonly used to determine the flux level. However, the solution of a simplified version of this equation when automated is very time consuming. Since the flux level changes with time, in general, this calculation must be made repeatedly. Therefore solution techniques that speed the calculation while maintaining accuracy are desirable. One factor that contributes to the solution time is the spatial flux shape approximation used. It is common practice to use the samemore » order flux shape approximation in each energy group even though this method may not be the most efficient. The one-dimensional, two-energy group diffusion equation was solved, for the node average flux and core k-effective, using two sets of spatial shape approximations for each of three reactor types. A fourth-order approximation in both energy groups forms the first set of approximations used. The second set used combines a second-order approximation with a fourth-order approximation in energy group two. Comparison of the results from the two approximation sets show that the use of a different order spatial flux shape approximation results in considerable loss in accuracy for the pressurized water reactor modeled. However, the loss in accuracy is small for the heavy water and graphite reactors modeled. The use of different order approximations in each energy group produces mixed results. Further investigation into the accuracy and computing time is required before any quantitative advantage of the use of the second-order approximation in energy group one and the fourth-order approximation in energy group two can be determined.« less

New Operational Matrices for Solving Fractional Differential Equations on the Half-Line

PubMed Central

2015-01-01

In this paper, the fractional-order generalized Laguerre operational matrices (FGLOM) of fractional derivatives and fractional integration are derived. These operational matrices are used together with spectral tau method for solving linear fractional differential equations (FDEs) of order ν (0 < ν < 1) on the half line. An upper bound of the absolute errors is obtained for the approximate and exact solutions. Fractional-order generalized Laguerre pseudo-spectral approximation is investigated for solving nonlinear initial value problem of fractional order ν. The extension of the fractional-order generalized Laguerre pseudo-spectral method is given to solve systems of FDEs. We present the advantages of using the spectral schemes based on fractional-order generalized Laguerre functions and compare them with other methods. Several numerical examples are implemented for FDEs and systems of FDEs including linear and nonlinear terms. We demonstrate the high accuracy and the efficiency of the proposed techniques. PMID:25996369

New operational matrices for solving fractional differential equations on the half-line.

PubMed

Bhrawy, Ali H; Taha, Taha M; Alzahrani, Ebraheem O; Alzahrani, Ebrahim O; Baleanu, Dumitru; Alzahrani, Abdulrahim A

2015-01-01

In this paper, the fractional-order generalized Laguerre operational matrices (FGLOM) of fractional derivatives and fractional integration are derived. These operational matrices are used together with spectral tau method for solving linear fractional differential equations (FDEs) of order ν (0 < ν < 1) on the half line. An upper bound of the absolute errors is obtained for the approximate and exact solutions. Fractional-order generalized Laguerre pseudo-spectral approximation is investigated for solving nonlinear initial value problem of fractional order ν. The extension of the fractional-order generalized Laguerre pseudo-spectral method is given to solve systems of FDEs. We present the advantages of using the spectral schemes based on fractional-order generalized Laguerre functions and compare them with other methods. Several numerical examples are implemented for FDEs and systems of FDEs including linear and nonlinear terms. We demonstrate the high accuracy and the efficiency of the proposed techniques.

Modified Taylor series method for solving nonlinear differential equations with mixed boundary conditions defined on finite intervals.

PubMed

Vazquez-Leal, Hector; Benhammouda, Brahim; Filobello-Nino, Uriel Antonio; Sarmiento-Reyes, Arturo; Jimenez-Fernandez, Victor Manuel; Marin-Hernandez, Antonio; Herrera-May, Agustin Leobardo; Diaz-Sanchez, Alejandro; Huerta-Chua, Jesus

2014-01-01

In this article, we propose the application of a modified Taylor series method (MTSM) for the approximation of nonlinear problems described on finite intervals. The issue of Taylor series method with mixed boundary conditions is circumvented using shooting constants and extra derivatives of the problem. In order to show the benefits of this proposal, three different kinds of problems are solved: three-point boundary valued problem (BVP) of third-order with a hyperbolic sine nonlinearity, two-point BVP for a second-order nonlinear differential equation with an exponential nonlinearity, and a two-point BVP for a third-order nonlinear differential equation with a radical nonlinearity. The result shows that the MTSM method is capable to generate easily computable and highly accurate approximations for nonlinear equations. 34L30.

Semismooth Newton method for gradient constrained minimization problem

NASA Astrophysics Data System (ADS)

Anyyeva, Serbiniyaz; Kunisch, Karl

2012-08-01

In this paper we treat a gradient constrained minimization problem, particular case of which is the elasto-plastic torsion problem. In order to get the numerical approximation to the solution we have developed an algorithm in an infinite dimensional space framework using the concept of the generalized (Newton) differentiation. Regularization was done in order to approximate the problem with the unconstrained minimization problem and to make the pointwise maximum function Newton differentiable. Using semismooth Newton method, continuation method was developed in function space. For the numerical implementation the variational equations at Newton steps are discretized using finite elements method.

A stable second order method for training back propagation networks

NASA Technical Reports Server (NTRS)

Nachtsheim, Philip R.

1993-01-01

A simple method for improving the learning rate of the back-propagation algorithm is described. The basis of the method is that approximate second order corrections can be incorporated in the output units. The extended method leads to significant improvements in the convergence rate.

An accurate and efficient method for evaluating the kernel of the integral equation relating pressure to normalwash in unsteady potential flow

NASA Technical Reports Server (NTRS)

Desmarais, R. N.

1982-01-01

This paper describes an accurate economical method for generating approximations to the kernel of the integral equation relating unsteady pressure to normalwash in nonplanar flow. The method is capable of generating approximations of arbitrary accuracy. It is based on approximating the algebraic part of the non elementary integrals in the kernel by exponential approximations and then integrating termwise. The exponent spacing in the approximation is a geometric sequence. The coefficients and exponent multiplier of the exponential approximation are computed by least squares so the method is completely automated. Exponential approximates generated in this manner are two orders of magnitude more accurate than the exponential approximation that is currently most often used for this purpose. Coefficients for 8, 12, 24, and 72 term approximations are tabulated in the report. Also, since the method is automated, it can be used to generate approximations to attain any desired trade-off between accuracy and computing cost.

An approximation method for configuration optimization of trusses

NASA Technical Reports Server (NTRS)

Hansen, Scott R.; Vanderplaats, Garret N.

1988-01-01

Two- and three-dimensional elastic trusses are designed for minimum weight by varying the areas of the members and the location of the joints. Constraints on member stresses and Euler buckling are imposed and multiple static loading conditions are considered. The method presented here utilizes an approximate structural analysis based on first order Taylor series expansions of the member forces. A numerical optimizer minimizes the weight of the truss using information from the approximate structural analysis. Comparisons with results from other methods are made. It is shown that the method of forming an approximate structural analysis based on linearized member forces leads to a highly efficient method of truss configuration optimization.

Stable multi-domain spectral penalty methods for fractional partial differential equations

NASA Astrophysics Data System (ADS)

Xu, Qinwu; Hesthaven, Jan S.

2014-01-01

We propose stable multi-domain spectral penalty methods suitable for solving fractional partial differential equations with fractional derivatives of any order. First, a high order discretization is proposed to approximate fractional derivatives of any order on any given grids based on orthogonal polynomials. The approximation order is analyzed and verified through numerical examples. Based on the discrete fractional derivative, we introduce stable multi-domain spectral penalty methods for solving fractional advection and diffusion equations. The equations are discretized in each sub-domain separately and the global schemes are obtained by weakly imposed boundary and interface conditions through a penalty term. Stability of the schemes are analyzed and numerical examples based on both uniform and nonuniform grids are considered to highlight the flexibility and high accuracy of the proposed schemes.

Nuclear magnetic resonance signal dynamics of liquids in the presence of distant dipolar fields, revisited

PubMed Central

Barros, Wilson; Gochberg, Daniel F.; Gore, John C.

2009-01-01

The description of the nuclear magnetic resonance magnetization dynamics in the presence of long-range dipolar interactions, which is based upon approximate solutions of Bloch–Torrey equations including the effect of a distant dipolar field, has been revisited. New experiments show that approximate analytic solutions have a broader regime of validity as well as dependencies on pulse-sequence parameters that seem to have been overlooked. In order to explain these experimental results, we developed a new method consisting of calculating the magnetization via an iterative formalism where both diffusion and distant dipolar field contributions are treated as integral operators incorporated into the Bloch–Torrey equations. The solution can be organized as a perturbative series, whereby access to higher order terms allows one to set better boundaries on validity regimes for analytic first-order approximations. Finally, the method legitimizes the use of simple analytic first-order approximations under less demanding experimental conditions, it predicts new pulse-sequence parameter dependencies for the range of validity, and clarifies weak points in previous calculations. PMID:19425789

Efficient High-Order Accurate Methods using Unstructured Grids for Hydrodynamics and Acoustics

DTIC Science & Technology

2007-08-31

Leer. On upstream differencing and godunov-type schemes for hyperbolic conservation laws. SIAM Review, 25(1):35-61, 1983. [46] F . Eleuterio Toro ...early stage [4-61. The basic idea can be surmised from simple approximation theory. If a continuous function f is to be approximated over a set of...a2f 4h4 a4ff(x+eh) = f (x)+-- + _ •-+• e +0 +... (1) where 0 < e < 1 for approximations inside the interval of width h. For a second-order approximation

On singlet s-wave electron-hydrogen scattering.

NASA Technical Reports Server (NTRS)

Madan, R. N.

1973-01-01

Discussion of various zeroth-order approximations to s-wave scattering of electrons by hydrogen atoms below the first excitation threshold. The formalism previously developed by the author (1967, 1968) is applied to Feshbach operators to derive integro-differential equations, with the optical-potential set equal to zero, for the singlet and triplet cases. Phase shifts of s-wave scattering are computed in the zeroth-order approximation of the Feshbach operator method and in the static-exchange approximation. It is found that the convergence of numerical computations is faster in the former approximation than in the latter.

Uniformly high-order accurate non-oscillatory schemes, 1

NASA Technical Reports Server (NTRS)

Harten, A.; Osher, S.

1985-01-01

The construction and the analysis of nonoscillatory shock capturing methods for the approximation of hyperbolic conservation laws was begun. These schemes share many desirable properties with total variation diminishing schemes (TVD), but TVD schemes have at most first order accuracy, in the sense of truncation error, at extreme of the solution. A uniformly second order approximation was constucted, which is nonoscillatory in the sense that the number of extrema of the discrete solution is not increasing in time. This is achieved via a nonoscillatory piecewise linear reconstruction of the solution from its cell averages, time evolution through an approximate solution of the resulting initial value problem, and averaging of this approximate solution over each cell.

A comparison of matrix methods for calculating eigenvalues in acoustically lined ducts

NASA Technical Reports Server (NTRS)

Watson, W.; Lansing, D. L.

1976-01-01

Three approximate methods - finite differences, weighted residuals, and finite elements - were used to solve the eigenvalue problem which arises in finding the acoustic modes and propagation constants in an absorptively lined two-dimensional duct without airflow. The matrix equations derived for each of these methods were solved for the eigenvalues corresponding to various values of wall impedance. Two matrix orders, 20 x 20 and 40 x 40, were used. The cases considered included values of wall admittance for which exact eigenvalues were known and for which several nearly equal roots were present. Ten of the lower order eigenvalues obtained from the three approximate methods were compared with solutions calculated from the exact characteristic equation in order to make an assessment of the relative accuracy and reliability of the three methods. The best results were given by the finite element method using a cubic polynomial. Excellent accuracy was consistently obtained, even for nearly equal eigenvalues, by using a 20 x 20 order matrix.

Eigenvalue and eigenvector sensitivity and approximate analysis for repeated eigenvalue problems

NASA Technical Reports Server (NTRS)

Hou, Gene J. W.; Kenny, Sean P.

1991-01-01

A set of computationally efficient equations for eigenvalue and eigenvector sensitivity analysis are derived, and a method for eigenvalue and eigenvector approximate analysis in the presence of repeated eigenvalues is presented. The method developed for approximate analysis involves a reparamaterization of the multivariable structural eigenvalue problem in terms of a single positive-valued parameter. The resulting equations yield first-order approximations of changes in both the eigenvalues and eigenvectors associated with the repeated eigenvalue problem. Examples are given to demonstrate the application of such equations for sensitivity and approximate analysis.

Monotonically improving approximate answers to relational algebra queries

NASA Technical Reports Server (NTRS)

Smith, Kenneth P.; Liu, J. W. S.

1989-01-01

We present here a query processing method that produces approximate answers to queries posed in standard relational algebra. This method is monotone in the sense that the accuracy of the approximate result improves with the amount of time spent producing the result. This strategy enables us to trade the time to produce the result for the accuracy of the result. An approximate relational model that characterizes appromimate relations and a partial order for comparing them is developed. Relational operators which operate on and return approximate relations are defined.

A priori error estimates for an hp-version of the discontinuous Galerkin method for hyperbolic conservation laws

NASA Technical Reports Server (NTRS)

Bey, Kim S.; Oden, J. Tinsley

1993-01-01

A priori error estimates are derived for hp-versions of the finite element method for discontinuous Galerkin approximations of a model class of linear, scalar, first-order hyperbolic conservation laws. These estimates are derived in a mesh dependent norm in which the coefficients depend upon both the local mesh size h(sub K) and a number p(sub k) which can be identified with the spectral order of the local approximations over each element.

a Numerical Comparison of Langrange and Kane's Methods of AN Arm Segment

NASA Astrophysics Data System (ADS)

Rambely, Azmin Sham; Halim, Norhafiza Ab.; Ahmad, Rokiah Rozita

A 2-D model of a two-link kinematic chain is developed using two dynamics equations of motion, namely Kane's and Lagrange Methods. The dynamics equations are reduced to first order differential equation and solved using modified Euler and fourth order Runge Kutta to approximate the shoulder and elbow joint angles during a smash performance in badminton. Results showed that Runge-Kutta produced a better and exact approximation than that of modified Euler and both dynamic equations produced better absolute errors.

On the superconvergence of Galerkin methods for hyperbolic IBVP

NASA Technical Reports Server (NTRS)

Gottlieb, David; Gustafsson, Bertil; Olsson, Pelle; Strand, BO

1993-01-01

Finite element Galerkin methods for periodic first order hyperbolic equations exhibit superconvergence on uniform grids at the nodes, i.e., there is an error estimate 0(h(sup 2r)) instead of the expected approximation order 0(h(sup r)). It will be shown that no matter how the approximating subspace S(sup h) is chosen, the superconvergence property is lost if there are characteristics leaving the domain. The implications of this result when constructing compact implicit difference schemes is also discussed.

Errors from approximation of ODE systems with reduced order models

DOE Office of Scientific and Technical Information (OSTI.GOV)

Vassilevska, Tanya

2016-12-30

This is a code to calculate the error from approximation of systems of ordinary differential equations (ODEs) by using Proper Orthogonal Decomposition (POD) Reduced Order Models (ROM) methods and to compare and analyze the errors for two POD ROM variants. The first variant is the standard POD ROM, the second variant is a modification of the method using the values of the time derivatives (a.k.a. time-derivative snapshots). The code compares the errors from the two variants under different conditions.

Density-functional expansion methods: evaluation of LDA, GGA, and meta-GGA functionals and different integral approximations.

PubMed

Giese, Timothy J; York, Darrin M

2010-12-28

We extend the Kohn-Sham potential energy expansion (VE) to include variations of the kinetic energy density and use the VE formulation with a 6-31G* basis to perform a "Jacob's ladder" comparison of small molecule properties using density functionals classified as being either LDA, GGA, or meta-GGA. We show that the VE reproduces standard Kohn-Sham DFT results well if all integrals are performed without further approximation, and there is no substantial improvement in using meta-GGA functionals relative to GGA functionals. The advantages of using GGA versus LDA functionals becomes apparent when modeling hydrogen bonds. We furthermore examine the effect of using integral approximations to compute the zeroth-order energy and first-order matrix elements, and the results suggest that the origin of the short-range repulsive potential within self-consistent charge density-functional tight-binding methods mainly arises from the approximations made to the first-order matrix elements.

Physical Applications of a Simple Approximation of Bessel Functions of Integer Order

ERIC Educational Resources Information Center

Barsan, V.; Cojocaru, S.

2007-01-01

Applications of a simple approximation of Bessel functions of integer order, in terms of trigonometric functions, are discussed for several examples from electromagnetism and optics. The method may be applied in the intermediate regime, bridging the "small values regime" and the "asymptotic" one, and covering, in this way, an area of great…

Modified multiple time scale method for solving strongly nonlinear damped forced vibration systems

NASA Astrophysics Data System (ADS)

Razzak, M. A.; Alam, M. Z.; Sharif, M. N.

2018-03-01

In this paper, modified multiple time scale (MTS) method is employed to solve strongly nonlinear forced vibration systems. The first-order approximation is only considered in order to avoid complexicity. The formulations and the determination of the solution procedure are very easy and straightforward. The classical multiple time scale (MS) and multiple scales Lindstedt-Poincare method (MSLP) do not give desire result for the strongly damped forced vibration systems with strong damping effects. The main aim of this paper is to remove these limitations. Two examples are considered to illustrate the effectiveness and convenience of the present procedure. The approximate external frequencies and the corresponding approximate solutions are determined by the present method. The results give good coincidence with corresponding numerical solution (considered to be exact) and also provide better result than other existing results. For weak nonlinearities with weak damping effect, the absolute relative error measures (first-order approximate external frequency) in this paper is only 0.07% when amplitude A = 1.5 , while the relative error gives MSLP method is surprisingly 28.81%. Furthermore, for strong nonlinearities with strong damping effect, the absolute relative error found in this article is only 0.02%, whereas the relative error obtained by MSLP method is 24.18%. Therefore, the present method is not only valid for weakly nonlinear damped forced systems, but also gives better result for strongly nonlinear systems with both small and strong damping effect.

Blending Velocities In Task Space In Computing Robot Motions

NASA Technical Reports Server (NTRS)

Volpe, Richard A.

1995-01-01

Blending of linear and angular velocities between sequential specified points in task space constitutes theoretical basis of improved method of computing trajectories followed by robotic manipulators. In method, generalized velocity-vector-blending technique provides relatively simple, common conceptual framework for blending linear, angular, and other parametric velocities. Velocity vectors originate from straight-line segments connecting specified task-space points, called "via frames" and represent specified robot poses. Linear-velocity-blending functions chosen from among first-order, third-order-polynomial, and cycloidal options. Angular velocities blended by use of first-order approximation of previous orientation-matrix-blending formulation. Angular-velocity approximation yields small residual error, quantified and corrected. Method offers both relative simplicity and speed needed for generation of robot-manipulator trajectories in real time.

A study on Marangoni convection by the variational iteration method

NASA Astrophysics Data System (ADS)

Karaoǧlu, Onur; Oturanç, Galip

2012-09-01

In this paper, we will consider the use of the variational iteration method and Padé approximant for finding approximate solutions for a Marangoni convection induced flow over a free surface due to an imposed temperature gradient. The solutions are compared with the numerical (fourth-order Runge Kutta) solutions.

High-Order Semi-Discrete Central-Upwind Schemes for Multi-Dimensional Hamilton-Jacobi Equations

NASA Technical Reports Server (NTRS)

Bryson, Steve; Levy, Doron; Biegel, Bran R. (Technical Monitor)

2002-01-01

We present high-order semi-discrete central-upwind numerical schemes for approximating solutions of multi-dimensional Hamilton-Jacobi (HJ) equations. This scheme is based on the use of fifth-order central interpolants like those developed in [1], in fluxes presented in [3]. These interpolants use the weighted essentially nonoscillatory (WENO) approach to avoid spurious oscillations near singularities, and become "central-upwind" in the semi-discrete limit. This scheme provides numerical approximations whose error is as much as an order of magnitude smaller than those in previous WENO-based fifth-order methods [2, 1]. Thee results are discussed via examples in one, two and three dimensions. We also pregnant explicit N-dimensional formulas for the fluxes, discuss their monotonicity and tl!e connection between this method and that in [2].

Periodic solutions of second-order nonlinear difference equations containing a small parameter. IV - Multi-discrete time method

NASA Technical Reports Server (NTRS)

Mickens, Ronald E.

1987-01-01

It is shown that a discrete multi-time method can be constructed to obtain approximations to the periodic solutions of a special class of second-order nonlinear difference equations containing a small parameter. Three examples illustrating the method are presented.

Optimizing some 3-stage W-methods for the time integration of PDEs

NASA Astrophysics Data System (ADS)

Gonzalez-Pinto, S.; Hernandez-Abreu, D.; Perez-Rodriguez, S.

2017-07-01

The optimization of some W-methods for the time integration of time-dependent PDEs in several spatial variables is considered. In [2, Theorem 1] several three-parametric families of three-stage W-methods for the integration of IVPs in ODEs were studied. Besides, the optimization of several specific methods for PDEs when the Approximate Matrix Factorization Splitting (AMF) is used to define the approximate Jacobian matrix (W ≈ fy(yn)) was carried out. Also, some convergence and stability properties were presented [2]. The derived methods were optimized on the base that the underlying explicit Runge-Kutta method is the one having the largest Monotonicity interval among the thee-stage order three Runge-Kutta methods [1]. Here, we propose an optimization of the methods by imposing some additional order condition [7] to keep order three for parabolic PDE problems [6] but at the price of reducing substantially the length of the nonlinear Monotonicity interval of the underlying explicit Runge-Kutta method.

Second-order singular pertubative theory for gravitational lenses

NASA Astrophysics Data System (ADS)

Alard, C.

2018-03-01

The extension of the singular perturbative approach to the second order is presented in this paper. The general expansion to the second order is derived. The second-order expansion is considered as a small correction to the first-order expansion. Using this approach, it is demonstrated that in practice the second-order expansion is reducible to a first order expansion via a re-definition of the first-order pertubative fields. Even if in usual applications the second-order correction is small the reducibility of the second-order expansion to the first-order expansion indicates a potential degeneracy issue. In general, this degeneracy is hard to break. A useful and simple second-order approximation is the thin source approximation, which offers a direct estimation of the correction. The practical application of the corrections derived in this paper is illustrated by using an elliptical NFW lens model. The second-order pertubative expansion provides a noticeable improvement, even for the simplest case of thin source approximation. To conclude, it is clear that for accurate modelization of gravitational lenses using the perturbative method the second-order perturbative expansion should be considered. In particular, an evaluation of the degeneracy due to the second-order term should be performed, for which the thin source approximation is particularly useful.

Second order accurate finite difference approximations for the transonic small disturbance equation and the full potential equation

NASA Technical Reports Server (NTRS)

Mostrel, M. M.

1988-01-01

New shock-capturing finite difference approximations for solving two scalar conservation law nonlinear partial differential equations describing inviscid, isentropic, compressible flows of aerodynamics at transonic speeds are presented. A global linear stability theorem is applied to these schemes in order to derive a necessary and sufficient condition for the finite element method. A technique is proposed to render the described approximations total variation-stable by applying the flux limiters to the nonlinear terms of the difference equation dimension by dimension. An entropy theorem applying to the approximations is proved, and an implicit, forward Euler-type time discretization of the approximation is presented. Results of some numerical experiments using the approximations are reported.

Hybrid perturbation methods based on statistical time series models

NASA Astrophysics Data System (ADS)

San-Juan, Juan Félix; San-Martín, Montserrat; Pérez, Iván; López, Rosario

2016-04-01

In this work we present a new methodology for orbit propagation, the hybrid perturbation theory, based on the combination of an integration method and a prediction technique. The former, which can be a numerical, analytical or semianalytical theory, generates an initial approximation that contains some inaccuracies derived from the fact that, in order to simplify the expressions and subsequent computations, not all the involved forces are taken into account and only low-order terms are considered, not to mention the fact that mathematical models of perturbations not always reproduce physical phenomena with absolute precision. The prediction technique, which can be based on either statistical time series models or computational intelligence methods, is aimed at modelling and reproducing missing dynamics in the previously integrated approximation. This combination results in the precision improvement of conventional numerical, analytical and semianalytical theories for determining the position and velocity of any artificial satellite or space debris object. In order to validate this methodology, we present a family of three hybrid orbit propagators formed by the combination of three different orders of approximation of an analytical theory and a statistical time series model, and analyse their capability to process the effect produced by the flattening of the Earth. The three considered analytical components are the integration of the Kepler problem, a first-order and a second-order analytical theories, whereas the prediction technique is the same in the three cases, namely an additive Holt-Winters method.

Approximate series solution of multi-dimensional, time fractional-order (heat-like) diffusion equations using FRDTM.

PubMed

Singh, Brajesh K; Srivastava, Vineet K

2015-04-01

The main goal of this paper is to present a new approximate series solution of the multi-dimensional (heat-like) diffusion equation with time-fractional derivative in Caputo form using a semi-analytical approach: fractional-order reduced differential transform method (FRDTM). The efficiency of FRDTM is confirmed by considering four test problems of the multi-dimensional time fractional-order diffusion equation. FRDTM is a very efficient, effective and powerful mathematical tool which provides exact or very close approximate solutions for a wide range of real-world problems arising in engineering and natural sciences, modelled in terms of differential equations.

Approximate series solution of multi-dimensional, time fractional-order (heat-like) diffusion equations using FRDTM

PubMed Central

Singh, Brajesh K.; Srivastava, Vineet K.

2015-01-01

The main goal of this paper is to present a new approximate series solution of the multi-dimensional (heat-like) diffusion equation with time-fractional derivative in Caputo form using a semi-analytical approach: fractional-order reduced differential transform method (FRDTM). The efficiency of FRDTM is confirmed by considering four test problems of the multi-dimensional time fractional-order diffusion equation. FRDTM is a very efficient, effective and powerful mathematical tool which provides exact or very close approximate solutions for a wide range of real-world problems arising in engineering and natural sciences, modelled in terms of differential equations. PMID:26064639

Fourth order difference methods for hyperbolic IBVP's

NASA Technical Reports Server (NTRS)

Gustafsson, Bertil; Olsson, Pelle

1994-01-01

Fourth order difference approximations of initial-boundary value problems for hyperbolic partial differential equations are considered. We use the method of lines approach with both explicit and compact implicit difference operators in space. The explicit operator satisfies an energy estimate leading to strict stability. For the implicit operator we develop boundary conditions and give a complete proof of strong stability using the Laplace transform technique. We also present numerical experiments for the linear advection equation and Burgers' equation with discontinuities in the solution or in its derivative. The first equation is used for modeling contact discontinuities in fluid dynamics, the second one for modeling shocks and rarefaction waves. The time discretization is done with a third order Runge-Kutta TVD method. For solutions with discontinuities in the solution itself we add a filter based on second order viscosity. In case of the non-linear Burger's equation we use a flux splitting technique that results in an energy estimate for certain different approximations, in which case also an entropy condition is fulfilled. In particular we shall demonstrate that the unsplit conservative form produces a non-physical shock instead of the physically correct rarefaction wave. In the numerical experiments we compare our fourth order methods with a standard second order one and with a third order TVD-method. The results show that the fourth order methods are the only ones that give good results for all the considered test problems.

Polynomial probability distribution estimation using the method of moments

PubMed Central

Mattsson, Lars; Rydén, Jesper

2017-01-01

We suggest a procedure for estimating Nth degree polynomial approximations to unknown (or known) probability density functions (PDFs) based on N statistical moments from each distribution. The procedure is based on the method of moments and is setup algorithmically to aid applicability and to ensure rigor in use. In order to show applicability, polynomial PDF approximations are obtained for the distribution families Normal, Log-Normal, Weibull as well as for a bimodal Weibull distribution and a data set of anonymized household electricity use. The results are compared with results for traditional PDF series expansion methods of Gram–Charlier type. It is concluded that this procedure is a comparatively simple procedure that could be used when traditional distribution families are not applicable or when polynomial expansions of probability distributions might be considered useful approximations. In particular this approach is practical for calculating convolutions of distributions, since such operations become integrals of polynomial expressions. Finally, in order to show an advanced applicability of the method, it is shown to be useful for approximating solutions to the Smoluchowski equation. PMID:28394949

Polynomial probability distribution estimation using the method of moments.

PubMed

Munkhammar, Joakim; Mattsson, Lars; Rydén, Jesper

2017-01-01

We suggest a procedure for estimating Nth degree polynomial approximations to unknown (or known) probability density functions (PDFs) based on N statistical moments from each distribution. The procedure is based on the method of moments and is setup algorithmically to aid applicability and to ensure rigor in use. In order to show applicability, polynomial PDF approximations are obtained for the distribution families Normal, Log-Normal, Weibull as well as for a bimodal Weibull distribution and a data set of anonymized household electricity use. The results are compared with results for traditional PDF series expansion methods of Gram-Charlier type. It is concluded that this procedure is a comparatively simple procedure that could be used when traditional distribution families are not applicable or when polynomial expansions of probability distributions might be considered useful approximations. In particular this approach is practical for calculating convolutions of distributions, since such operations become integrals of polynomial expressions. Finally, in order to show an advanced applicability of the method, it is shown to be useful for approximating solutions to the Smoluchowski equation.

Density-functional expansion methods: Grand challenges.

PubMed

Giese, Timothy J; York, Darrin M

2012-03-01

We discuss the source of errors in semiempirical density functional expansion (VE) methods. In particular, we show that VE methods are capable of well-reproducing their standard Kohn-Sham density functional method counterparts, but suffer from large errors upon using one or more of these approximations: the limited size of the atomic orbital basis, the Slater monopole auxiliary basis description of the response density, and the one- and two-body treatment of the core-Hamiltonian matrix elements. In the process of discussing these approximations and highlighting their symptoms, we introduce a new model that supplements the second-order density-functional tight-binding model with a self-consistent charge-dependent chemical potential equalization correction; we review our recently reported method for generalizing the auxiliary basis description of the atomic orbital response density; and we decompose the first-order potential into a summation of additive atomic components and many-body corrections, and from this examination, we provide new insights and preliminary results that motivate and inspire new approximate treatments of the core-Hamiltonian.

An Efficient Algorithm for Perturbed Orbit Integration Combining Analytical Continuation and Modified Chebyshev Picard Iteration

NASA Astrophysics Data System (ADS)

Elgohary, T.; Kim, D.; Turner, J.; Junkins, J.

2014-09-01

Several methods exist for integrating the motion in high order gravity fields. Some recent methods use an approximate starting orbit, and an efficient method is needed for generating warm starts that account for specific low order gravity approximations. By introducing two scalar Lagrange-like invariants and employing Leibniz product rule, the perturbed motion is integrated by a novel recursive formulation. The Lagrange-like invariants allow exact arbitrary order time derivatives. Restricting attention to the perturbations due to the zonal harmonics J2 through J6, we illustrate an idea. The recursively generated vector-valued time derivatives for the trajectory are used to develop a continuation series-based solution for propagating position and velocity. Numerical comparisons indicate performance improvements of ~ 70X over existing explicit Runge-Kutta methods while maintaining mm accuracy for the orbit predictions. The Modified Chebyshev Picard Iteration (MCPI) is an iterative path approximation method to solve nonlinear ordinary differential equations. The MCPI utilizes Picard iteration with orthogonal Chebyshev polynomial basis functions to recursively update the states. The key advantages of the MCPI are as follows: 1) Large segments of a trajectory can be approximated by evaluating the forcing function at multiple nodes along the current approximation during each iteration. 2) It can readily handle general gravity perturbations as well as non-conservative forces. 3) Parallel applications are possible. The Picard sequence converges to the solution over large time intervals when the forces are continuous and differentiable. According to the accuracy of the starting solutions, however, the MCPI may require significant number of iterations and function evaluations compared to other integrators. In this work, we provide an efficient methodology to establish good starting solutions from the continuation series method; this warm start improves the performance of the MCPI significantly and will likely be useful for other applications where efficiently computed approximate orbit solutions are needed.

Intermediate boundary conditions for LOD, ADI and approximate factorization methods

NASA Technical Reports Server (NTRS)

Leveque, R. J.

1985-01-01

A general approach to determining the correct intermediate boundary conditions for dimensional splitting methods is presented. The intermediate solution U is viewed as a second order accurate approximation to a modified equation. Deriving the modified equation and using the relationship between this equation and the original equation allows us to determine the correct boundary conditions for U*. This technique is illustrated by applying it to locally one dimensional (LOD) and alternating direction implicit (ADI) methods for the heat equation in two and three space dimensions. The approximate factorization method is considered in slightly more generality.

The Relation of Finite Element and Finite Difference Methods

NASA Technical Reports Server (NTRS)

Vinokur, M.

1976-01-01

Finite element and finite difference methods are examined in order to bring out their relationship. It is shown that both methods use two types of discrete representations of continuous functions. They differ in that finite difference methods emphasize the discretization of independent variable, while finite element methods emphasize the discretization of dependent variable (referred to as functional approximations). An important point is that finite element methods use global piecewise functional approximations, while finite difference methods normally use local functional approximations. A general conclusion is that finite element methods are best designed to handle complex boundaries, while finite difference methods are superior for complex equations. It is also shown that finite volume difference methods possess many of the advantages attributed to finite element methods.

Identification of stochastic interactions in nonlinear models of structural mechanics

NASA Astrophysics Data System (ADS)

Kala, Zdeněk

2017-07-01

In the paper, the polynomial approximation is presented by which the Sobol sensitivity analysis can be evaluated with all sensitivity indices. The nonlinear FEM model is approximated. The input area is mapped using simulations runs of Latin Hypercube Sampling method. The domain of the approximation polynomial is chosen so that it were possible to apply large number of simulation runs of Latin Hypercube Sampling method. The method presented also makes possible to evaluate higher-order sensitivity indices, which could not be identified in case of nonlinear FEM.

Validation of zero-order feedback strategies for medium range air-to-air interception in a horizontal plane

NASA Technical Reports Server (NTRS)

Shinar, J.

1982-01-01

A zero order feedback solution of a variable speed interception game between two aircraft in the horizontal plane, obtained by using the method of forced singular perturbation (FSP), is compared with the exact open loop solution. The comparison indicates that for initial distances of separation larger than eight turning radii of the evader, the accuracy of the feedback approximation is better than one percent. The result validates the zero order FSP approximation for medium range air combat analysis.

The nonconforming virtual element method for eigenvalue problems

DOE Office of Scientific and Technical Information (OSTI.GOV)

Gardini, Francesca; Manzini, Gianmarco; Vacca, Giuseppe

We analyse the nonconforming Virtual Element Method (VEM) for the approximation of elliptic eigenvalue problems. The nonconforming VEM allow to treat in the same formulation the two- and three-dimensional case.We present two possible formulations of the discrete problem, derived respectively by the nonstabilized and stabilized approximation of the L 2-inner product, and we study the convergence properties of the corresponding discrete eigenvalue problems. The proposed schemes provide a correct approximation of the spectrum and we prove optimal-order error estimates for the eigenfunctions and the usual double order of convergence of the eigenvalues. Finally we show a large set of numericalmore » tests supporting the theoretical results, including a comparison with the conforming Virtual Element choice.« less

Anharmonic phonon-phonon scattering modeling of three-dimensional atomistic transport: An efficient quantum treatment

NASA Astrophysics Data System (ADS)

Lee, Y.; Bescond, M.; Logoteta, D.; Cavassilas, N.; Lannoo, M.; Luisier, M.

2018-05-01

We propose an efficient method to quantum mechanically treat anharmonic interactions in the atomistic nonequilibrium Green's function simulation of phonon transport. We demonstrate that the so-called lowest-order approximation, implemented through a rescaling technique and analytically continued by means of the Padé approximants, can be used to accurately model third-order anharmonic effects. Although the paper focuses on a specific self-energy, the method is applicable to a very wide class of physical interactions. We apply this approach to the simulation of anharmonic phonon transport in realistic Si and Ge nanowires with uniform or discontinuous cross sections. The effect of increasing the temperature above 300 K is also investigated. In all the considered cases, we are able to obtain a good agreement with the routinely adopted self-consistent Born approximation, at a remarkably lower computational cost. In the more complicated case of high temperatures (≫300 K), we find that the first-order Richardson extrapolation applied to the sequence of the Padé approximants N -1 /N results in a significant acceleration of the convergence.

Electroencephalography in ellipsoidal geometry with fourth-order harmonics.

PubMed

Alcocer-Sosa, M; Gutierrez, D

2016-08-01

We present a solution to the electroencephalographs (EEG) forward problem of computing the scalp electric potentials for the case when the head's geometry is modeled using a four-shell ellipsoidal geometry and the brain sources with an equivalent current dipole (ECD). The proposed solution includes terms up to the fourth-order ellipsoidal harmonics and we compare this new approximation against those that only considered up to second- and third-order harmonics. Our comparisons use as reference a solution in which a tessellated volume approximates the head and the forward problem is solved through the boundary element method (BEM). We also assess the solution to the inverse problem of estimating the magnitude of an ECD through different harmonic approximations. Our results show that the fourth-order solution provides a better estimate of the ECD in comparison to lesser order ones.

The lowest-order weak Galerkin finite element method for the Darcy equation on quadrilateral and hybrid meshes

NASA Astrophysics Data System (ADS)

Liu, Jiangguo; Tavener, Simon; Wang, Zhuoran

2018-04-01

This paper investigates the lowest-order weak Galerkin finite element method for solving the Darcy equation on quadrilateral and hybrid meshes consisting of quadrilaterals and triangles. In this approach, the pressure is approximated by constants in element interiors and on edges. The discrete weak gradients of these constant basis functions are specified in local Raviart-Thomas spaces, specifically RT0 for triangles and unmapped RT[0] for quadrilaterals. These discrete weak gradients are used to approximate the classical gradient when solving the Darcy equation. The method produces continuous normal fluxes and is locally mass-conservative, regardless of mesh quality, and has optimal order convergence in pressure, velocity, and normal flux, when the quadrilaterals are asymptotically parallelograms. Implementation is straightforward and results in symmetric positive-definite discrete linear systems. We present numerical experiments and comparisons with other existing methods.

Orbitally invariant internally contracted multireference unitary coupled cluster theory and its perturbative approximation: theory and test calculations of second order approximation.

PubMed

Chen, Zhenhua; Hoffmann, Mark R

2012-07-07

A unitary wave operator, exp (G), G(+) = -G, is considered to transform a multiconfigurational reference wave function Φ to the potentially exact, within basis set limit, wave function Ψ = exp (G)Φ. To obtain a useful approximation, the Hausdorff expansion of the similarity transformed effective Hamiltonian, exp (-G)Hexp (G), is truncated at second order and the excitation manifold is limited; an additional separate perturbation approximation can also be made. In the perturbation approximation, which we refer to as multireference unitary second-order perturbation theory (MRUPT2), the Hamiltonian operator in the highest order commutator is approximated by a Mo̸ller-Plesset-type one-body zero-order Hamiltonian. If a complete active space self-consistent field wave function is used as reference, then the energy is invariant under orbital rotations within the inactive, active, and virtual orbital subspaces for both the second-order unitary coupled cluster method and its perturbative approximation. Furthermore, the redundancies of the excitation operators are addressed in a novel way, which is potentially more efficient compared to the usual full diagonalization of the metric of the excited configurations. Despite the loss of rigorous size-extensivity possibly due to the use of a variational approach rather than a projective one in the solution of the amplitudes, test calculations show that the size-extensivity errors are very small. Compared to other internally contracted multireference perturbation theories, MRUPT2 only needs reduced density matrices up to three-body even with a non-complete active space reference wave function when two-body excitations within the active orbital subspace are involved in the wave operator, exp (G). Both the coupled cluster and perturbation theory variants are amenable to large, incomplete model spaces. Applications to some widely studied model systems that can be problematic because of geometry dependent quasidegeneracy, H4, P4, and BeH(2), are performed in order to test the new methods on problems where full configuration interaction results are available.

Weak periodic solutions of xẍ + 1 = 0 and the Harmonic Balance Method

NASA Astrophysics Data System (ADS)

García-Saldaña, J. D.; Gasull, A.

2017-02-01

We prove that the differential equation xẍ + 1 = 0 has continuous weak periodic solutions and compute their periods. Then, we use the Harmonic Balance Method until order six to approximate these periods and to illustrate how the accuracy of the method increases with the order. Our computations rely on the Gröbner basis approach.

Speeding up spin-component-scaled third-order pertubation theory with the chain of spheres approximation: the COSX-SCS-MP3 method

NASA Astrophysics Data System (ADS)

Izsák, Róbert; Neese, Frank

2013-07-01

The 'chain of spheres' approximation, developed earlier for the efficient evaluation of the self-consistent field exchange term, is introduced here into the evaluation of the external exchange term of higher order correlation methods. Its performance is studied in the specific case of the spin-component-scaled third-order Møller--Plesset perturbation (SCS-MP3) theory. The results indicate that the approximation performs excellently in terms of both computer time and achievable accuracy. Significant speedups over a conventional method are obtained for larger systems and basis sets. Owing to this development, SCS-MP3 calculations on molecules of the size of penicillin (42 atoms) with a polarised triple-zeta basis set can be performed in ∼3 hours using 16 cores of an Intel Xeon E7-8837 processor with a 2.67 GHz clock speed, which represents a speedup by a factor of 8-9 compared to the previously most efficient algorithm. Thus, the increased accuracy offered by SCS-MP3 can now be explored for at least medium-sized molecules.

Seismic waves in heterogeneous material: subcell resolution of the discontinuous Galerkin method

NASA Astrophysics Data System (ADS)

Castro, Cristóbal E.; Käser, Martin; Brietzke, Gilbert B.

2010-07-01

We present an important extension of the arbitrary high-order discontinuous Galerkin (DG) finite-element method to model 2-D elastic wave propagation in highly heterogeneous material. In this new approach we include space-variable coefficients to describe smooth or discontinuous material variations inside each element using the same numerical approximation strategy as for the velocity-stress variables in the formulation of the elastic wave equation. The combination of the DG method with a time integration scheme based on the solution of arbitrary accuracy derivatives Riemann problems still provides an explicit, one-step scheme which achieves arbitrary high-order accuracy in space and time. Compared to previous formulations the new scheme contains two additional terms in the form of volume integrals. We show that the increasing computational cost per element can be overcompensated due to the improved material representation inside each element as coarser meshes can be used which reduces the total number of elements and therefore computational time to reach a desired error level. We confirm the accuracy of the proposed scheme performing convergence tests and several numerical experiments considering smooth and highly heterogeneous material. As the approximation of the velocity and stress variables in the wave equation and of the material properties in the model can be chosen independently, we investigate the influence of the polynomial material representation on the accuracy of the synthetic seismograms with respect to computational cost. Moreover, we study the behaviour of the new method on strong material discontinuities, in the case where the mesh is not aligned with such a material interface. In this case second-order linear material approximation seems to be the best choice, with higher-order intra-cell approximation leading to potential instable behaviour. For all test cases we validate our solution against the well-established standard fourth-order finite difference and spectral element method.

A diffusion approximation for ocean wave scatterings by randomly distributed ice floes

NASA Astrophysics Data System (ADS)

Zhao, Xin; Shen, Hayley

2016-11-01

This study presents a continuum approach using a diffusion approximation method to solve the scattering of ocean waves by randomly distributed ice floes. In order to model both strong and weak scattering, the proposed method decomposes the wave action density function into two parts: the transmitted part and the scattered part. For a given wave direction, the transmitted part of the wave action density is defined as the part of wave action density in the same direction before the scattering; and the scattered part is a first order Fourier series approximation for the directional spreading caused by scattering. An additional approximation is also adopted for simplification, in which the net directional redistribution of wave action by a single scatterer is assumed to be the reflected wave action of a normally incident wave into a semi-infinite ice cover. Other required input includes the mean shear modulus, diameter and thickness of ice floes, and the ice concentration. The directional spreading of wave energy from the diffusion approximation is found to be in reasonable agreement with the previous solution using the Boltzmann equation. The diffusion model provides an alternative method to implement wave scattering into an operational wave model.

Minimal-Approximation-Based Decentralized Backstepping Control of Interconnected Time-Delay Systems.

PubMed

Choi, Yun Ho; Yoo, Sung Jin

2016-12-01

A decentralized adaptive backstepping control design using minimal function approximators is proposed for nonlinear large-scale systems with unknown unmatched time-varying delayed interactions and unknown backlash-like hysteresis nonlinearities. Compared with existing decentralized backstepping methods, the contribution of this paper is to design a simple local control law for each subsystem, consisting of an actual control with one adaptive function approximator, without requiring the use of multiple function approximators and regardless of the order of each subsystem. The virtual controllers for each subsystem are used as intermediate signals for designing a local actual control at the last step. For each subsystem, a lumped unknown function including the unknown nonlinear terms and the hysteresis nonlinearities is derived at the last step and is estimated by one function approximator. Thus, the proposed approach only uses one function approximator to implement each local controller, while existing decentralized backstepping control methods require the number of function approximators equal to the order of each subsystem and a calculation of virtual controllers to implement each local actual controller. The stability of the total controlled closed-loop system is analyzed using the Lyapunov stability theorem.

A compatible high-order meshless method for the Stokes equations with applications to suspension flows

NASA Astrophysics Data System (ADS)

Trask, Nathaniel; Maxey, Martin; Hu, Xiaozhe

2018-02-01

A stable numerical solution of the steady Stokes problem requires compatibility between the choice of velocity and pressure approximation that has traditionally proven problematic for meshless methods. In this work, we present a discretization that couples a staggered scheme for pressure approximation with a divergence-free velocity reconstruction to obtain an adaptive, high-order, finite difference-like discretization that can be efficiently solved with conventional algebraic multigrid techniques. We use analytic benchmarks to demonstrate equal-order convergence for both velocity and pressure when solving problems with curvilinear geometries. In order to study problems in dense suspensions, we couple the solution for the flow to the equations of motion for freely suspended particles in an implicit monolithic scheme. The combination of high-order accuracy with fully-implicit schemes allows the accurate resolution of stiff lubrication forces directly from the solution of the Stokes problem without the need to introduce sub-grid lubrication models.

Approximating genomic reliabilities for national genomic evaluation

USDA-ARS?s Scientific Manuscript database

With the introduction of standard methods for approximating effective daughter/data contribution by Interbull in 2001, conventional EDC or reliabilities contributed by daughter phenotypes are directly comparable across countries and used in routine conventional evaluations. In order to make publishe...

A high-order time-parallel scheme for solving wave propagation problems via the direct construction of an approximate time-evolution operator

DOE Office of Scientific and Technical Information (OSTI.GOV)

Haut, T. S.; Babb, T.; Martinsson, P. G.

2015-06-16

Our manuscript demonstrates a technique for efficiently solving the classical wave equation, the shallow water equations, and, more generally, equations of the form ∂u/∂t=Lu∂u/∂t=Lu, where LL is a skew-Hermitian differential operator. The idea is to explicitly construct an approximation to the time-evolution operator exp(τL)exp(τL) for a relatively large time-step ττ. Recently developed techniques for approximating oscillatory scalar functions by rational functions, and accelerated algorithms for computing functions of discretized differential operators are exploited. Principal advantages of the proposed method include: stability even for large time-steps, the possibility to parallelize in time over many characteristic wavelengths and large speed-ups over existingmore » methods in situations where simulation over long times are required. Numerical examples involving the 2D rotating shallow water equations and the 2D wave equation in an inhomogenous medium are presented, and the method is compared to the 4th order Runge–Kutta (RK4) method and to the use of Chebyshev polynomials. The new method achieved high accuracy over long-time intervals, and with speeds that are orders of magnitude faster than both RK4 and the use of Chebyshev polynomials.« less

Solution of an eigenvalue problem for the Laplace operator on a spherical surface. M.S. Thesis - Maryland Univ.

NASA Technical Reports Server (NTRS)

Walden, H.

1974-01-01

Methods for obtaining approximate solutions for the fundamental eigenvalue of the Laplace-Beltrami operator (also referred to as the membrane eigenvalue problem for the vibration equation) on the unit spherical surface are developed. Two specific types of spherical surface domains are considered: (1) the interior of a spherical triangle, i.e., the region bounded by arcs of three great circles, and (2) the exterior of a great circle arc extending for less than pi radians on the sphere (a spherical surface with a slit). In both cases, zero boundary conditions are imposed. In order to solve the resulting second-order elliptic partial differential equations in two independent variables, a finite difference approximation is derived. The symmetric (generally five-point) finite difference equations that develop are written in matrix form and then solved by the iterative method of point successive overrelaxation. Upon convergence of this iterative method, the fundamental eigenvalue is approximated by iteration utilizing the power method as applied to the finite Rayleigh quotient.

Reduced size first-order subsonic and supersonic aeroelastic modeling

NASA Technical Reports Server (NTRS)

Karpel, Mordechay

1990-01-01

Various aeroelastic, aeroservoelastic, dynamic-response, and sensitivity analyses are based on a time-domain first-order (state-space) formulation of the equations of motion. The formulation of this paper is based on the minimum-state (MS) aerodynamic approximation method, which yields a low number of aerodynamic augmenting states. Modifications of the MS and the physical weighting procedures make the modeling method even more attractive. The flexibility of constraint selection is increased without increasing the approximation problem size; the accuracy of dynamic residualization of high-frequency modes is improved; and the resulting model is less sensitive to parametric changes in subsequent analyses. Applications to subsonic and supersonic cases demonstrate the generality, flexibility, accuracy, and efficiency of the method.

Variational path integral molecular dynamics and hybrid Monte Carlo algorithms using a fourth order propagator with applications to molecular systems

NASA Astrophysics Data System (ADS)

Kamibayashi, Yuki; Miura, Shinichi

2016-08-01

In the present study, variational path integral molecular dynamics and associated hybrid Monte Carlo (HMC) methods have been developed on the basis of a fourth order approximation of a density operator. To reveal various parameter dependence of physical quantities, we analytically solve one dimensional harmonic oscillators by the variational path integral; as a byproduct, we obtain the analytical expression of the discretized density matrix using the fourth order approximation for the oscillators. Then, we apply our methods to realistic systems like a water molecule and a para-hydrogen cluster. In the HMC, we adopt two level description to avoid the time consuming Hessian evaluation. For the systems examined in this paper, the HMC method is found to be about three times more efficient than the molecular dynamics method if appropriate HMC parameters are adopted; the advantage of the HMC method is suggested to be more evident for systems described by many body interaction.

Approximated analytical solution to an Ebola optimal control problem

NASA Astrophysics Data System (ADS)

Hincapié-Palacio, Doracelly; Ospina, Juan; Torres, Delfim F. M.

2016-11-01

An analytical expression for the optimal control of an Ebola problem is obtained. The analytical solution is found as a first-order approximation to the Pontryagin Maximum Principle via the Euler-Lagrange equation. An implementation of the method is given using the computer algebra system Maple. Our analytical solutions confirm the results recently reported in the literature using numerical methods.

UNAERO: A package of FORTRAN subroutines for approximating unsteady aerodynamics in the time domain

NASA Technical Reports Server (NTRS)

Dunn, H. J.

1985-01-01

This report serves as an instruction and maintenance manual for a collection of CDC CYBER FORTRAN IV subroutines for approximating the unsteady aerodynamic forces in the time domain. The result is a set of constant-coefficient first-order differential equations that approximate the dynamics of the vehicle. Provisions are included for adjusting the number of modes used for calculating the approximations so that an accurate approximation is generated. The number of data points at different values of reduced frequency can also be varied to adjust the accuracy of the approximation over the reduced-frequency range. The denominator coefficients of the approximation may be calculated by means of a gradient method or a least-squares approximation technique. Both the approximation methods use weights on the residual error. A new set of system equations, at a different dynamic pressure, can be generated without the approximations being recalculated.

Path integrals with higher order actions: Application to realistic chemical systems

NASA Astrophysics Data System (ADS)

Lindoy, Lachlan P.; Huang, Gavin S.; Jordan, Meredith J. T.

2018-02-01

Quantum thermodynamic parameters can be determined using path integral Monte Carlo (PIMC) simulations. These simulations, however, become computationally demanding as the quantum nature of the system increases, although their efficiency can be improved by using higher order approximations to the thermal density matrix, specifically the action. Here we compare the standard, primitive approximation to the action (PA) and three higher order approximations, the Takahashi-Imada action (TIA), the Suzuki-Chin action (SCA) and the Chin action (CA). The resulting PIMC methods are applied to two realistic potential energy surfaces, for H2O and HCN-HNC, both of which are spectroscopically accurate and contain three-body interactions. We further numerically optimise, for each potential, the SCA parameter and the two free parameters in the CA, obtaining more significant improvements in efficiency than seen previously in the literature. For both H2O and HCN-HNC, accounting for all required potential and force evaluations, the optimised CA formalism is approximately twice as efficient as the TIA formalism and approximately an order of magnitude more efficient than the PA. The optimised SCA formalism shows similar efficiency gains to the CA for HCN-HNC but has similar efficiency to the TIA for H2O at low temperature. In H2O and HCN-HNC systems, the optimal value of the a1 CA parameter is approximately 1/3 , corresponding to an equal weighting of all force terms in the thermal density matrix, and similar to previous studies, the optimal α parameter in the SCA was ˜0.31. Importantly, poor choice of parameter significantly degrades the performance of the SCA and CA methods. In particular, for the CA, setting a1 = 0 is not efficient: the reduction in convergence efficiency is not offset by the lower number of force evaluations. We also find that the harmonic approximation to the CA parameters, whilst providing a fourth order approximation to the action, is not optimal for these realistic potentials: numerical optimisation leads to better approximate cancellation of the fifth order terms, with deviation between the harmonic and numerically optimised parameters more marked in the more quantum H2O system. This suggests that numerically optimising the CA or SCA parameters, which can be done at high temperature, will be important in fully realising the efficiency gains of these formalisms for realistic potentials.

A 2D Gaussian-Beam-Based Method for Modeling the Dichroic Surfaces of Quasi-Optical Systems

NASA Astrophysics Data System (ADS)

Elis, Kevin; Chabory, Alexandre; Sokoloff, Jérôme; Bolioli, Sylvain

2016-08-01

In this article, we propose an approach in the spectral domain to treat the interaction of a field with a dichroic surface in two dimensions. For a Gaussian beam illumination of the surface, the reflected and transmitted fields are approximated by one reflected and one transmitted Gaussian beams. Their characteristics are determined by means of a matching in the spectral domain, which requires a second-order approximation of the dichroic surface response when excited by plane waves. This approximation is of the same order as the one used in Gaussian beam shooting algorithm to model curved interfaces associated with lenses, reflector, etc. The method uses general analytical formulations for the GBs that depend either on a paraxial or far-field approximation. Numerical experiments are led to test the efficiency of the method in terms of accuracy and computation time. They include a parametric study and a case for which the illumination is provided by a horn antenna. For the latter, the incident field is firstly expressed as a sum of Gaussian beams by means of Gabor frames.

Numerical solution of the unsteady Navier-Stokes equation

NASA Technical Reports Server (NTRS)

Osher, Stanley J.; Engquist, Bjoern

1985-01-01

The construction and the analysis of nonoscillatory shock capturing methods for the approximation of hyperbolic conservation laws are discussed. These schemes share many desirable properties with total variation diminishing schemes, but TVD schemes have at most first-order accuracy, in the sense of truncation error, at extrema of the solution. In this paper a uniformly second-order approximation is constructed, which is nonoscillatory in the sense that the number of extrema of the discrete solution is not increasing in time. This is achieved via a nonoscillatory piecewise linear reconstruction of the solution from its cell averages, time evolution through an approximate solution of the resulting initial value problem, and averaging of this approximate solution over each cell.

On High-Order Upwind Methods for Advection

NASA Technical Reports Server (NTRS)

Huynh, H. T.

2017-01-01

In the fourth installment of the celebrated series of five papers entitled "Towards the ultimate conservative difference scheme", Van Leer (1977) introduced five schemes for advection, the first three are piecewise linear, and the last two, piecewise parabolic. Among the five, scheme I, which is the least accurate, extends with relative ease to systems of equations in multiple dimensions. As a result, it became the most popular and is widely known as the MUSCL scheme (monotone upstream-centered schemes for conservation laws). Schemes III and V have the same accuracy, are the most accurate, and are closely related to current high-order methods. Scheme III uses a piecewise linear approximation that is discontinuous across cells, and can be considered as a precursor of the discontinuous Galerkin methods. Scheme V employs a piecewise quadratic approximation that is, as opposed to the case of scheme III, continuous across cells. This method is the basis for the on-going "active flux scheme" developed by Roe and collaborators. Here, schemes III and V are shown to be equivalent in the sense that they yield identical (reconstructed) solutions, provided the initial condition for scheme III is defined from that of scheme V in a manner dependent on the CFL number. This equivalence is counter intuitive since it is generally believed that piecewise linear and piecewise parabolic methods cannot produce the same solutions due to their different degrees of approximation. The finding also shows a key connection between the approaches of discontinuous and continuous polynomial approximations. In addition to the discussed equivalence, a framework using both projection and interpolation that extends schemes III and V into a single family of high-order schemes is introduced. For these high-order extensions, it is demonstrated via Fourier analysis that schemes with the same number of degrees of freedom ?? per cell, in spite of the different piecewise polynomial degrees, share the same sets of eigenvalues and thus, have the same stability and accuracy. Moreover, these schemes are accurate to order 2??-1, which is higher than the expected order of ??.

Diagrammatic expansion for positive spectral functions beyond GW: Application to vertex corrections in the electron gas

NASA Astrophysics Data System (ADS)

Stefanucci, G.; Pavlyukh, Y.; Uimonen, A.-M.; van Leeuwen, R.

2014-09-01

We present a diagrammatic approach to construct self-energy approximations within many-body perturbation theory with positive spectral properties. The method cures the problem of negative spectral functions which arises from a straightforward inclusion of vertex diagrams beyond the GW approximation. Our approach consists of a two-step procedure: We first express the approximate many-body self-energy as a product of half-diagrams and then identify the minimal number of half-diagrams to add in order to form a perfect square. The resulting self-energy is an unconventional sum of self-energy diagrams in which the internal lines of half a diagram are time-ordered Green's functions, whereas those of the other half are anti-time-ordered Green's functions, and the lines joining the two halves are either lesser or greater Green's functions. The theory is developed using noninteracting Green's functions and subsequently extended to self-consistent Green's functions. Issues related to the conserving properties of diagrammatic approximations with positive spectral functions are also addressed. As a major application of the formalism we derive the minimal set of additional diagrams to make positive the spectral function of the GW approximation with lowest-order vertex corrections and screened interactions. The method is then applied to vertex corrections in the three-dimensional homogeneous electron gas by using a combination of analytical frequency integrations and numerical Monte Carlo momentum integrations to evaluate the diagrams.

An updated Lagrangian discontinuous Galerkin hydrodynamic method for gas dynamics

DOE Office of Scientific and Technical Information (OSTI.GOV)

Wu, Tong; Shashkov, Mikhail Jurievich; Morgan, Nathaniel Ray

Here, we present a new Lagrangian discontinuous Galerkin (DG) hydrodynamic method for gas dynamics. The new method evolves conserved unknowns in the current configuration, which obviates the Jacobi matrix that maps the element in a reference coordinate system or the initial coordinate system to the current configuration. The density, momentum, and total energy (ρ, ρu, E) are approximated with conservative higher-order Taylor expansions over the element and are limited toward a piecewise constant field near discontinuities using a limiter. Two new limiting methods are presented for enforcing the bounds on the primitive variables of density, velocity, and specific internal energymore » (ρ, u, e). The nodal velocity, and the corresponding forces, are calculated by solving an approximate Riemann problem at the element nodes. An explicit second-order method is used to temporally advance the solution. This new Lagrangian DG hydrodynamic method conserves mass, momentum, and total energy. 1D Cartesian coordinates test problem results are presented to demonstrate the accuracy and convergence order of the new DG method with the new limiters.« less

An updated Lagrangian discontinuous Galerkin hydrodynamic method for gas dynamics

DOE PAGES

Wu, Tong; Shashkov, Mikhail Jurievich; Morgan, Nathaniel Ray; ...

2018-04-09

Here, we present a new Lagrangian discontinuous Galerkin (DG) hydrodynamic method for gas dynamics. The new method evolves conserved unknowns in the current configuration, which obviates the Jacobi matrix that maps the element in a reference coordinate system or the initial coordinate system to the current configuration. The density, momentum, and total energy (ρ, ρu, E) are approximated with conservative higher-order Taylor expansions over the element and are limited toward a piecewise constant field near discontinuities using a limiter. Two new limiting methods are presented for enforcing the bounds on the primitive variables of density, velocity, and specific internal energymore » (ρ, u, e). The nodal velocity, and the corresponding forces, are calculated by solving an approximate Riemann problem at the element nodes. An explicit second-order method is used to temporally advance the solution. This new Lagrangian DG hydrodynamic method conserves mass, momentum, and total energy. 1D Cartesian coordinates test problem results are presented to demonstrate the accuracy and convergence order of the new DG method with the new limiters.« less

Exponential-fitted methods for integrating stiff systems of ordinary differential equations: Applications to homogeneous gas-phase chemical kinetics

NASA Technical Reports Server (NTRS)

Pratt, D. T.

1984-01-01

Conventional algorithms for the numerical integration of ordinary differential equations (ODEs) are based on the use of polynomial functions as interpolants. However, the exact solutions of stiff ODEs behave like decaying exponential functions, which are poorly approximated by polynomials. An obvious choice of interpolant are the exponential functions themselves, or their low-order diagonal Pade (rational function) approximants. A number of explicit, A-stable, integration algorithms were derived from the use of a three-parameter exponential function as interpolant, and their relationship to low-order, polynomial-based and rational-function-based implicit and explicit methods were shown by examining their low-order diagonal Pade approximants. A robust implicit formula was derived by exponential fitting the trapezoidal rule. Application of these algorithms to integration of the ODEs governing homogenous, gas-phase chemical kinetics was demonstrated in a developmental code CREK1D, which compares favorably with the Gear-Hindmarsh code LSODE in spite of the use of a primitive stepsize control strategy.

a Bounded Finite-Difference Discretization of a Two-Dimensional Diffusion Equation with Logistic Nonlinear Reaction

NASA Astrophysics Data System (ADS)

Macías-Díaz, J. E.

In the present manuscript, we introduce a finite-difference scheme to approximate solutions of the two-dimensional version of Fisher's equation from population dynamics, which is a model for which the existence of traveling-wave fronts bounded within (0,1) is a well-known fact. The method presented here is a nonstandard technique which, in the linear regime, approximates the solutions of the original model with a consistency of second order in space and first order in time. The theory of M-matrices is employed here in order to elucidate conditions under which the method is able to preserve the positivity and the boundedness of solutions. In fact, our main result establishes relatively flexible conditions under which the preservation of the positivity and the boundedness of new approximations is guaranteed. Some simulations of the propagation of a traveling-wave solution confirm the analytical results derived in this work; moreover, the experiments evince a good agreement between the numerical result and the analytical solutions.

Decentralized Quasi-Newton Methods

NASA Astrophysics Data System (ADS)

Eisen, Mark; Mokhtari, Aryan; Ribeiro, Alejandro

2017-05-01

We introduce the decentralized Broyden-Fletcher-Goldfarb-Shanno (D-BFGS) method as a variation of the BFGS quasi-Newton method for solving decentralized optimization problems. The D-BFGS method is of interest in problems that are not well conditioned, making first order decentralized methods ineffective, and in which second order information is not readily available, making second order decentralized methods impossible. D-BFGS is a fully distributed algorithm in which nodes approximate curvature information of themselves and their neighbors through the satisfaction of a secant condition. We additionally provide a formulation of the algorithm in asynchronous settings. Convergence of D-BFGS is established formally in both the synchronous and asynchronous settings and strong performance advantages relative to first order methods are shown numerically.

The uniform asymptotic swallowtail approximation - Practical methods for oscillating integrals with four coalescing saddle points

NASA Technical Reports Server (NTRS)

Connor, J. N. L.; Curtis, P. R.; Farrelly, D.

1984-01-01

Methods that can be used in the numerical implementation of the uniform swallowtail approximation are described. An explicit expression for that approximation is presented to the lowest order, showing that there are three problems which must be overcome in practice before the approximation can be applied to any given problem. It is shown that a recently developed quadrature method can be used for the accurate numerical evaluation of the swallowtail canonical integral and its partial derivatives. Isometric plots of these are presented to illustrate some of their properties. The problem of obtaining the arguments of the swallowtail integral from an analytical function of its argument is considered, describing two methods of solving this problem. The asymptotic evaluation of the butterfly canonical integral is addressed.

Higher-order force moments of active particles

NASA Astrophysics Data System (ADS)

Nasouri, Babak; Elfring, Gwynn J.

2018-04-01

Active particles moving through fluids generate disturbance flows due to their activity. For simplicity, the induced flow field is often modeled by the leading terms in a far-field approximation of the Stokes equations, whose coefficients are the force, torque, and stresslet (zeroth- and first-order force moments) of the active particle. This level of approximation is quite useful, but may also fail to predict more complex behaviors that are observed experimentally. In this study, to provide a better approximation, we evaluate the contribution of the second-order force moments to the flow field and, by reciprocal theorem, present explicit formulas for the stresslet dipole, rotlet dipole, and potential dipole for an arbitrarily shaped active particle. As examples of this method, we derive modified Faxén laws for active spherical particles and resolve higher-order moments for active rod-like particles.

Improved phase shift approach to the energy correction of the infinite order sudden approximation

DOE Office of Scientific and Technical Information (OSTI.GOV)

Chang, B.; Eno, L.; Rabitz, H.

1980-07-15

A new method is presented for obtaining energy corrections to the infinite order sudden (IOS) approximation by incorporating the effect of the internal molecular Hamiltonian into the IOS wave function. This is done by utilizing the JWKB approximation to transform the Schroedinger equation into a differential equation for the phase. It is found that the internal Hamiltonian generates an effective potential from which a new improved phase shift is obtained. This phase shift is then used in place of the IOS phase shift to generate new transition probabilities. As an illustration the resulting improved phase shift (IPS) method is appliedmore » to the Secrest--Johnson model for the collinear collision of an atom and diatom. In the vicinity of the sudden limit, the IPS method gives results for transition probabilities, P/sub n/..-->..n+..delta..n, in significantly better agreement with the 'exact' close coupling calculations than the IOS method, particularly for large ..delta..n. However, when the IOS results are not even qualitatively correct, the IPS method is unable to satisfactorily provide improvements.« less

Newton's method applied to finite-difference approximations for the steady-state compressible Navier-Stokes equations

NASA Technical Reports Server (NTRS)

Bailey, Harry E.; Beam, Richard M.

1991-01-01

Finite-difference approximations for steady-state compressible Navier-Stokes equations, whose two spatial dimensions are written in generalized curvilinear coordinates and strong conservation-law form, are presently solved by means of Newton's method in order to obtain a lifting-airfoil flow field under subsonic and transonnic conditions. In addition to ascertaining the computational requirements of an initial guess ensuring convergence and the degree of computational efficiency obtainable via the approximate Newton method's freezing of the Jacobian matrices, attention is given to the need for auxiliary methods assessing the temporal stability of steady-state solutions. It is demonstrated that nonunique solutions of the finite-difference equations are obtainable by Newton's method in conjunction with a continuation method.

Transient analysis of an adaptive system for optimization of design parameters

NASA Technical Reports Server (NTRS)

Bayard, D. S.

1992-01-01

Averaging methods are applied to analyzing and optimizing the transient response associated with the direct adaptive control of an oscillatory second-order minimum-phase system. The analytical design methods developed for a second-order plant can be applied with some approximation to a MIMO flexible structure having a single dominant mode.

A numerical scheme based on radial basis function finite difference (RBF-FD) technique for solving the high-dimensional nonlinear Schrödinger equations using an explicit time discretization: Runge-Kutta method

NASA Astrophysics Data System (ADS)

Dehghan, Mehdi; Mohammadi, Vahid

2017-08-01

In this research, we investigate the numerical solution of nonlinear Schrödinger equations in two and three dimensions. The numerical meshless method which will be used here is RBF-FD technique. The main advantage of this method is the approximation of the required derivatives based on finite difference technique at each local-support domain as Ωi. At each Ωi, we require to solve a small linear system of algebraic equations with a conditionally positive definite matrix of order 1 (interpolation matrix). This scheme is efficient and its computational cost is same as the moving least squares (MLS) approximation. A challengeable issue is choosing suitable shape parameter for interpolation matrix in this way. In order to overcome this matter, an algorithm which was established by Sarra (2012), will be applied. This algorithm computes the condition number of the local interpolation matrix using the singular value decomposition (SVD) for obtaining the smallest and largest singular values of that matrix. Moreover, an explicit method based on Runge-Kutta formula of fourth-order accuracy will be applied for approximating the time variable. It also decreases the computational costs at each time step since we will not solve a nonlinear system. On the other hand, to compare RBF-FD method with another meshless technique, the moving kriging least squares (MKLS) approximation is considered for the studied model. Our results demonstrate the ability of the present approach for solving the applicable model which is investigated in the current research work.

Similarity-transformed equation-of-motion vibrational coupled-cluster theory.

PubMed

Faucheaux, Jacob A; Nooijen, Marcel; Hirata, So

2018-02-07

A similarity-transformed equation-of-motion vibrational coupled-cluster (STEOM-XVCC) method is introduced as a one-mode theory with an effective vibrational Hamiltonian, which is similarity transformed twice so that its lower-order operators are dressed with higher-order anharmonic effects. The first transformation uses an exponential excitation operator, defining the equation-of-motion vibrational coupled-cluster (EOM-XVCC) method, and the second uses an exponential excitation-deexcitation operator. From diagonalization of this doubly similarity-transformed Hamiltonian in the small one-mode excitation space, the method simultaneously computes accurate anharmonic vibrational frequencies of all fundamentals, which have unique significance in vibrational analyses. We establish a diagrammatic method of deriving the working equations of STEOM-XVCC and prove their connectedness and thus size-consistency as well as the exact equality of its frequencies with the corresponding roots of EOM-XVCC. We furthermore elucidate the similarities and differences between electronic and vibrational STEOM methods and between STEOM-XVCC and vibrational many-body Green's function theory based on the Dyson equation, which is also an anharmonic one-mode theory. The latter comparison inspires three approximate STEOM-XVCC methods utilizing the common approximations made in the Dyson equation: the diagonal approximation, a perturbative expansion of the Dyson self-energy, and the frequency-independent approximation. The STEOM-XVCC method including up to the simultaneous four-mode excitation operator in a quartic force field and its three approximate variants are formulated and implemented in computer codes with the aid of computer algebra, and they are applied to small test cases with varied degrees of anharmonicity.

Similarity-transformed equation-of-motion vibrational coupled-cluster theory

NASA Astrophysics Data System (ADS)

Faucheaux, Jacob A.; Nooijen, Marcel; Hirata, So

2018-02-01

A similarity-transformed equation-of-motion vibrational coupled-cluster (STEOM-XVCC) method is introduced as a one-mode theory with an effective vibrational Hamiltonian, which is similarity transformed twice so that its lower-order operators are dressed with higher-order anharmonic effects. The first transformation uses an exponential excitation operator, defining the equation-of-motion vibrational coupled-cluster (EOM-XVCC) method, and the second uses an exponential excitation-deexcitation operator. From diagonalization of this doubly similarity-transformed Hamiltonian in the small one-mode excitation space, the method simultaneously computes accurate anharmonic vibrational frequencies of all fundamentals, which have unique significance in vibrational analyses. We establish a diagrammatic method of deriving the working equations of STEOM-XVCC and prove their connectedness and thus size-consistency as well as the exact equality of its frequencies with the corresponding roots of EOM-XVCC. We furthermore elucidate the similarities and differences between electronic and vibrational STEOM methods and between STEOM-XVCC and vibrational many-body Green's function theory based on the Dyson equation, which is also an anharmonic one-mode theory. The latter comparison inspires three approximate STEOM-XVCC methods utilizing the common approximations made in the Dyson equation: the diagonal approximation, a perturbative expansion of the Dyson self-energy, and the frequency-independent approximation. The STEOM-XVCC method including up to the simultaneous four-mode excitation operator in a quartic force field and its three approximate variants are formulated and implemented in computer codes with the aid of computer algebra, and they are applied to small test cases with varied degrees of anharmonicity.

Aeroservoelastic modeling and applications using minimum-state approximations of the unsteady aerodynamics

NASA Technical Reports Server (NTRS)

Tiffany, Sherwood H.; Karpel, Mordechay

1989-01-01

Various control analysis, design, and simulation techniques for aeroelastic applications require the equations of motion to be cast in a linear time-invariant state-space form. Unsteady aerodynamics forces have to be approximated as rational functions of the Laplace variable in order to put them in this framework. For the minimum-state method, the number of denominator roots in the rational approximation. Results are shown of applying various approximation enhancements (including optimization, frequency dependent weighting of the tabular data, and constraint selection) with the minimum-state formulation to the active flexible wing wind-tunnel model. The results demonstrate that good models can be developed which have an order of magnitude fewer augmenting aerodynamic equations more than traditional approaches. This reduction facilitates the design of lower order control systems, analysis of control system performance, and near real-time simulation of aeroservoelastic phenomena.

Physical foundation of the fluid particle dynamics method for colloid dynamics simulation.

PubMed

Furukawa, Akira; Tateno, Michio; Tanaka, Hajime

2018-05-16

Colloid dynamics is significantly influenced by many-body hydrodynamic interactions mediated by a suspending fluid. However, theoretical and numerical treatments of such interactions are extremely difficult. To overcome this situation, we developed a fluid particle dynamics (FPD) method [H. Tanaka and T. Araki, Phys. Rev. Lett., 2000, 35, 3523], which is based on two key approximations: (i) a colloidal particle is treated as a highly viscous particle and (ii) the viscosity profile is described by a smooth interfacial profile function. Approximation (i) makes our method free from the solid-fluid boundary condition, significantly simplifying the treatment of many-body hydrodynamic interactions while satisfying the incompressible condition without the Stokes approximation. Approximation (ii) allows us to incorporate an extra degree of freedom in a fluid, e.g., orientational order and concentration, as an additional field variable. Here, we consider two fundamental problems associated with these approximations. One is the introduction of thermal noise and the other is the incorporation of coupling of the colloid surface with an order parameter introduced into a fluid component, which is crucial when considering colloidal particles suspended in a complex fluid. Here, we show that our FPD method makes it possible to simulate colloid dynamics properly while including full hydrodynamic interactions, inertia effects, incompressibility, thermal noise, and additional degrees of freedom of a fluid, which may be relevant for wide applications in colloidal and soft matter science.

On the Total Variation of High-Order Semi-Discrete Central Schemes for Conservation Laws

NASA Technical Reports Server (NTRS)

Bryson, Steve; Levy, Doron

2004-01-01

We discuss a new fifth-order, semi-discrete, central-upwind scheme for solving one-dimensional systems of conservation laws. This scheme combines a fifth-order WENO reconstruction, a semi-discrete central-upwind numerical flux, and a strong stability preserving Runge-Kutta method. We test our method with various examples, and give particular attention to the evolution of the total variation of the approximations.

Semi-Analytic Reconstruction of Flux in Finite Volume Formulations

NASA Technical Reports Server (NTRS)

Gnoffo, Peter A.

2006-01-01

Semi-analytic reconstruction uses the analytic solution to a second-order, steady, ordinary differential equation (ODE) to simultaneously evaluate the convective and diffusive flux at all interfaces of a finite volume formulation. The second-order ODE is itself a linearized approximation to the governing first- and second- order partial differential equation conservation laws. Thus, semi-analytic reconstruction defines a family of formulations for finite volume interface fluxes using analytic solutions to approximating equations. Limiters are not applied in a conventional sense; rather, diffusivity is adjusted in the vicinity of changes in sign of eigenvalues in order to achieve a sufficiently small cell Reynolds number in the analytic formulation across critical points. Several approaches for application of semi-analytic reconstruction for the solution of one-dimensional scalar equations are introduced. Results are compared with exact analytic solutions to Burger s Equation as well as a conventional, upwind discretization using Roe s method. One approach, the end-point wave speed (EPWS) approximation, is further developed for more complex applications. One-dimensional vector equations are tested on a quasi one-dimensional nozzle application. The EPWS algorithm has a more compact difference stencil than Roe s algorithm but reconstruction time is approximately a factor of four larger than for Roe. Though both are second-order accurate schemes, Roe s method approaches a grid converged solution with fewer grid points. Reconstruction of flux in the context of multi-dimensional, vector conservation laws including effects of thermochemical nonequilibrium in the Navier-Stokes equations is developed.

A Least-Squares-Based Weak Galerkin Finite Element Method for Second Order Elliptic Equations

DOE Office of Scientific and Technical Information (OSTI.GOV)

Mu, Lin; Wang, Junping; Ye, Xiu

Here, in this article, we introduce a least-squares-based weak Galerkin finite element method for the second order elliptic equation. This new method is shown to provide very accurate numerical approximations for both the primal and the flux variables. In contrast to other existing least-squares finite element methods, this new method allows us to use discontinuous approximating functions on finite element partitions consisting of arbitrary polygon/polyhedron shapes. We also develop a Schur complement algorithm for the resulting discretization problem by eliminating all the unknowns that represent the solution information in the interior of each element. Optimal order error estimates for bothmore » the primal and the flux variables are established. An extensive set of numerical experiments are conducted to demonstrate the robustness, reliability, flexibility, and accuracy of the least-squares-based weak Galerkin finite element method. Finally, the numerical examples cover a wide range of applied problems, including singularly perturbed reaction-diffusion equations and the flow of fluid in porous media with strong anisotropy and heterogeneity.« less

A Least-Squares-Based Weak Galerkin Finite Element Method for Second Order Elliptic Equations

DOE PAGES

Mu, Lin; Wang, Junping; Ye, Xiu

2017-08-17

Here, in this article, we introduce a least-squares-based weak Galerkin finite element method for the second order elliptic equation. This new method is shown to provide very accurate numerical approximations for both the primal and the flux variables. In contrast to other existing least-squares finite element methods, this new method allows us to use discontinuous approximating functions on finite element partitions consisting of arbitrary polygon/polyhedron shapes. We also develop a Schur complement algorithm for the resulting discretization problem by eliminating all the unknowns that represent the solution information in the interior of each element. Optimal order error estimates for bothmore » the primal and the flux variables are established. An extensive set of numerical experiments are conducted to demonstrate the robustness, reliability, flexibility, and accuracy of the least-squares-based weak Galerkin finite element method. Finally, the numerical examples cover a wide range of applied problems, including singularly perturbed reaction-diffusion equations and the flow of fluid in porous media with strong anisotropy and heterogeneity.« less

An Exact Model-Based Method for Near-Field Sources Localization with Bistatic MIMO System.

PubMed

Singh, Parth Raj; Wang, Yide; Chargé, Pascal

2017-03-30

In this paper, we propose an exact model-based method for near-field sources localization with a bistatic multiple input, multiple output (MIMO) radar system, and compare it with an approximated model-based method. The aim of this paper is to propose an efficient way to use the exact model of the received signals of near-field sources in order to eliminate the systematic error introduced by the use of approximated model in most existing near-field sources localization techniques. The proposed method uses parallel factor (PARAFAC) decomposition to deal with the exact model. Thanks to the exact model, the proposed method has better precision and resolution than the compared approximated model-based method. The simulation results show the performance of the proposed method.

Nonlinear programming extensions to rational function approximation methods for unsteady aerodynamic forces

NASA Technical Reports Server (NTRS)

Tiffany, Sherwood H.; Adams, William M., Jr.

1988-01-01

The approximation of unsteady generalized aerodynamic forces in the equations of motion of a flexible aircraft are discussed. Two methods of formulating these approximations are extended to include the same flexibility in constraining the approximations and the same methodology in optimizing nonlinear parameters as another currently used extended least-squares method. Optimal selection of nonlinear parameters is made in each of the three methods by use of the same nonlinear, nongradient optimizer. The objective of the nonlinear optimization is to obtain rational approximations to the unsteady aerodynamics whose state-space realization is lower order than that required when no optimization of the nonlinear terms is performed. The free linear parameters are determined using the least-squares matrix techniques of a Lagrange multiplier formulation of an objective function which incorporates selected linear equality constraints. State-space mathematical models resulting from different approaches are described and results are presented that show comparative evaluations from application of each of the extended methods to a numerical example.

MLFMA-accelerated Nyström method for ultrasonic scattering - Numerical results and experimental validation

NASA Astrophysics Data System (ADS)

Gurrala, Praveen; Downs, Andrew; Chen, Kun; Song, Jiming; Roberts, Ron

2018-04-01

Full wave scattering models for ultrasonic waves are necessary for the accurate prediction of voltage signals received from complex defects/flaws in practical nondestructive evaluation (NDE) measurements. We propose the high-order Nyström method accelerated by the multilevel fast multipole algorithm (MLFMA) as an improvement to the state-of-the-art full-wave scattering models that are based on boundary integral equations. We present numerical results demonstrating improvements in simulation time and memory requirement. Particularly, we demonstrate the need for higher order geom-etry and field approximation in modeling NDE measurements. Also, we illustrate the importance of full-wave scattering models using experimental pulse-echo data from a spherical inclusion in a solid, which cannot be modeled accurately by approximation-based scattering models such as the Kirchhoff approximation.

Some spectral approximation of one-dimensional fourth-order problems

NASA Technical Reports Server (NTRS)

Bernardi, Christine; Maday, Yvon

1989-01-01

Some spectral type collocation method well suited for the approximation of fourth-order systems are proposed. The model problem is the biharmonic equation, in one and two dimensions when the boundary conditions are periodic in one direction. It is proved that the standard Gauss-Lobatto nodes are not the best choice for the collocation points. Then, a new set of nodes related to some generalized Gauss type quadrature formulas is proposed. Also provided is a complete analysis of these formulas including some new issues about the asymptotic behavior of the weights and we apply these results to the analysis of the collocation method.

Turbofan Duct Propagation Model

NASA Technical Reports Server (NTRS)

Lan, Justin H.; Posey, Joe W. (Technical Monitor)

2001-01-01

The CDUCT code utilizes a parabolic approximation to the convected Helmholtz equation in order to efficiently model acoustic propagation in acoustically treated, complex shaped ducts. The parabolic approximation solves one-way wave propagation with a marching method which neglects backwards reflected waves. The derivation of the parabolic approximation is presented. Several code validation cases are given. An acoustic lining design process for an example aft fan duct is discussed. It is noted that the method can efficiently model realistic three-dimension effects, acoustic lining, and flow within the computational capabilities of a typical computer workstation.

Distorted-wave methods for electron capture in ion-atom collisions

DOE Office of Scientific and Technical Information (OSTI.GOV)

Burgdoerfer, J.; Taulbjerg, K.

1986-05-01

Distorted-wave methods for electron capture are discussed with emphasis on the surface term in the T matrix and on the properties of the associated integral equations. The surface term is generally nonvanishing if the distorted waves are sufficiently accurate to include parts of the considered physical process. Two examples are considered in detail. If distorted waves of the strong-potential Born-approximation (SPB) type are employed the surface term supplies the first-Born-approximation part of the T matrix. The surface term is shown to vanish in the continuum-distorted-wave (CDW) method. The integral kernel is in either case free of the dangerous disconnected termsmore » discussed by Greider and Dodd but the CDW theory is peculiar in the sense that its first-order approximation (CDW1) excludes a specific on-shell portion of the double-scattering term that is closely connected with the classical Thomas process. The latter is described by the second-order term in the CDW series. The distorted-wave Born approximation with SPB waves is shown to be free of divergences. In the limit of asymmetric collisions the DWB suggests a modification of the SPB approximation to avoid the divergence problem recently identified by Dewangan and Eichler.« less

An Iterative Solver in the Presence and Absence of Multiplicity for Nonlinear Equations

PubMed Central

Özkum, Gülcan

2013-01-01

We develop a high-order fixed point type method to approximate a multiple root. By using three functional evaluations per full cycle, a new class of fourth-order methods for this purpose is suggested and established. The methods from the class require the knowledge of the multiplicity. We also present a method in the absence of multiplicity for nonlinear equations. In order to attest the efficiency of the obtained methods, we employ numerical comparisons alongside obtaining basins of attraction to compare them in the complex plane according to their convergence speed and chaotic behavior. PMID:24453914

On cell entropy inequality for discontinuous Galerkin methods

NASA Technical Reports Server (NTRS)

Jiang, Guangshan; Shu, Chi-Wang

1993-01-01

We prove a cell entropy inequality for a class of high order discontinuous Galerkin finite element methods approximating conservation laws, which implies convergence for the one dimensional scalar convex case.

WEAK GALERKIN METHODS FOR SECOND ORDER ELLIPTIC INTERFACE PROBLEMS

PubMed Central

MU, LIN; WANG, JUNPING; WEI, GUOWEI; YE, XIU; ZHAO, SHAN

2013-01-01

Weak Galerkin methods refer to general finite element methods for partial differential equations (PDEs) in which differential operators are approximated by their weak forms as distributions. Such weak forms give rise to desirable flexibilities in enforcing boundary and interface conditions. A weak Galerkin finite element method (WG-FEM) is developed in this paper for solving elliptic PDEs with discontinuous coefficients and interfaces. Theoretically, it is proved that high order numerical schemes can be designed by using the WG-FEM with polynomials of high order on each element. Extensive numerical experiments have been carried to validate the WG-FEM for solving second order elliptic interface problems. High order of convergence is numerically confirmed in both L2 and L∞ norms for the piecewise linear WG-FEM. Special attention is paid to solve many interface problems, in which the solution possesses a certain singularity due to the nonsmoothness of the interface. A challenge in research is to design nearly second order numerical methods that work well for problems with low regularity in the solution. The best known numerical scheme in the literature is of order O(h) to O(h1.5) for the solution itself in L∞ norm. It is demonstrated that the WG-FEM of the lowest order, i.e., the piecewise constant WG-FEM, is capable of delivering numerical approximations that are of order O(h1.75) to O(h2) in the L∞ norm for C1 or Lipschitz continuous interfaces associated with a C1 or H2 continuous solution. PMID:24072935

Kernel Wiener filter and its application to pattern recognition.

PubMed

Yoshino, Hirokazu; Dong, Chen; Washizawa, Yoshikazu; Yamashita, Yukihiko

2010-11-01

The Wiener filter (WF) is widely used for inverse problems. From an observed signal, it provides the best estimated signal with respect to the squared error averaged over the original and the observed signals among linear operators. The kernel WF (KWF), extended directly from WF, has a problem that an additive noise has to be handled by samples. Since the computational complexity of kernel methods depends on the number of samples, a huge computational cost is necessary for the case. By using the first-order approximation of kernel functions, we realize KWF that can handle such a noise not by samples but as a random variable. We also propose the error estimation method for kernel filters by using the approximations. In order to show the advantages of the proposed methods, we conducted the experiments to denoise images and estimate errors. We also apply KWF to classification since KWF can provide an approximated result of the maximum a posteriori classifier that provides the best recognition accuracy. The noise term in the criterion can be used for the classification in the presence of noise or a new regularization to suppress changes in the input space, whereas the ordinary regularization for the kernel method suppresses changes in the feature space. In order to show the advantages of the proposed methods, we conducted experiments of binary and multiclass classifications and classification in the presence of noise.

Optimal sixteenth order convergent method based on quasi-Hermite interpolation for computing roots.

PubMed

Zafar, Fiza; Hussain, Nawab; Fatimah, Zirwah; Kharal, Athar

2014-01-01

We have given a four-step, multipoint iterative method without memory for solving nonlinear equations. The method is constructed by using quasi-Hermite interpolation and has order of convergence sixteen. As this method requires four function evaluations and one derivative evaluation at each step, it is optimal in the sense of the Kung and Traub conjecture. The comparisons are given with some other newly developed sixteenth-order methods. Interval Newton's method is also used for finding the enough accurate initial approximations. Some figures show the enclosure of finitely many zeroes of nonlinear equations in an interval. Basins of attractions show the effectiveness of the method.

Laplace transform homotopy perturbation method for the approximation of variational problems.

PubMed

Filobello-Nino, U; Vazquez-Leal, H; Rashidi, M M; Sedighi, H M; Perez-Sesma, A; Sandoval-Hernandez, M; Sarmiento-Reyes, A; Contreras-Hernandez, A D; Pereyra-Diaz, D; Hoyos-Reyes, C; Jimenez-Fernandez, V M; Huerta-Chua, J; Castro-Gonzalez, F; Laguna-Camacho, J R

2016-01-01

This article proposes the application of Laplace Transform-Homotopy Perturbation Method and some of its modifications in order to find analytical approximate solutions for the linear and nonlinear differential equations which arise from some variational problems. As case study we will solve four ordinary differential equations, and we will show that the proposed solutions have good accuracy, even we will obtain an exact solution. In the sequel, we will see that the square residual error for the approximate solutions, belongs to the interval [0.001918936920, 0.06334882582], which confirms the accuracy of the proposed methods, taking into account the complexity and difficulty of variational problems.

Fast and Analytical EAP Approximation from a 4th-Order Tensor.

PubMed

Ghosh, Aurobrata; Deriche, Rachid

2012-01-01

Generalized diffusion tensor imaging (GDTI) was developed to model complex apparent diffusivity coefficient (ADC) using higher-order tensors (HOTs) and to overcome the inherent single-peak shortcoming of DTI. However, the geometry of a complex ADC profile does not correspond to the underlying structure of fibers. This tissue geometry can be inferred from the shape of the ensemble average propagator (EAP). Though interesting methods for estimating a positive ADC using 4th-order diffusion tensors were developed, GDTI in general was overtaken by other approaches, for example, the orientation distribution function (ODF), since it is considerably difficult to recuperate the EAP from a HOT model of the ADC in GDTI. In this paper, we present a novel closed-form approximation of the EAP using Hermite polynomials from a modified HOT model of the original GDTI-ADC. Since the solution is analytical, it is fast, differentiable, and the approximation converges well to the true EAP. This method also makes the effort of computing a positive ADC worthwhile, since now both the ADC and the EAP can be used and have closed forms. We demonstrate our approach with 4th-order tensors on synthetic data and in vivo human data.

Fast and Analytical EAP Approximation from a 4th-Order Tensor

PubMed Central

Ghosh, Aurobrata; Deriche, Rachid

2012-01-01

Generalized diffusion tensor imaging (GDTI) was developed to model complex apparent diffusivity coefficient (ADC) using higher-order tensors (HOTs) and to overcome the inherent single-peak shortcoming of DTI. However, the geometry of a complex ADC profile does not correspond to the underlying structure of fibers. This tissue geometry can be inferred from the shape of the ensemble average propagator (EAP). Though interesting methods for estimating a positive ADC using 4th-order diffusion tensors were developed, GDTI in general was overtaken by other approaches, for example, the orientation distribution function (ODF), since it is considerably difficult to recuperate the EAP from a HOT model of the ADC in GDTI. In this paper, we present a novel closed-form approximation of the EAP using Hermite polynomials from a modified HOT model of the original GDTI-ADC. Since the solution is analytical, it is fast, differentiable, and the approximation converges well to the true EAP. This method also makes the effort of computing a positive ADC worthwhile, since now both the ADC and the EAP can be used and have closed forms. We demonstrate our approach with 4th-order tensors on synthetic data and in vivo human data. PMID:23365552

Time-stable boundary conditions for finite-difference schemes solving hyperbolic systems: Methodology and application to high-order compact schemes

NASA Technical Reports Server (NTRS)

Carpenter, Mark H.; Gottlieb, David; Abarbanel, Saul

1993-01-01

We present a systematic method for constructing boundary conditions (numerical and physical) of the required accuracy, for compact (Pade-like) high-order finite-difference schemes for hyperbolic systems. First, a roper summation-by-parts formula is found for the approximate derivative. A 'simultaneous approximation term' (SAT) is then introduced to treat the boundary conditions. This procedure leads to time-stable schemes even in the system case. An explicit construction of the fourth-order compact case is given. Numerical studies are presented to verify the efficacy of the approach.

Double power series method for approximating cosmological perturbations

NASA Astrophysics Data System (ADS)

Wren, Andrew J.; Malik, Karim A.

2017-04-01

We introduce a double power series method for finding approximate analytical solutions for systems of differential equations commonly found in cosmological perturbation theory. The method was set out, in a noncosmological context, by Feshchenko, Shkil' and Nikolenko (FSN) in 1966, and is applicable to cases where perturbations are on subhorizon scales. The FSN method is essentially an extension of the well known Wentzel-Kramers-Brillouin (WKB) method for finding approximate analytical solutions for ordinary differential equations. The FSN method we use is applicable well beyond perturbation theory to solve systems of ordinary differential equations, linear in the derivatives, that also depend on a small parameter, which here we take to be related to the inverse wave-number. We use the FSN method to find new approximate oscillating solutions in linear order cosmological perturbation theory for a flat radiation-matter universe. Together with this model's well-known growing and decaying Mészáros solutions, these oscillating modes provide a complete set of subhorizon approximations for the metric potential, radiation and matter perturbations. Comparison with numerical solutions of the perturbation equations shows that our approximations can be made accurate to within a typical error of 1%, or better. We also set out a heuristic method for error estimation. A Mathematica notebook which implements the double power series method is made available online.

A NURBS-enhanced finite volume solver for steady Euler equations

NASA Astrophysics Data System (ADS)

Meng, Xucheng; Hu, Guanghui

2018-04-01

In Hu and Yi (2016) [20], a non-oscillatory k-exact reconstruction method was proposed towards the high-order finite volume methods for steady Euler equations, which successfully demonstrated the high-order behavior in the simulations. However, the degeneracy of the numerical accuracy of the approximate solutions to problems with curved boundary can be observed obviously. In this paper, the issue is resolved by introducing the Non-Uniform Rational B-splines (NURBS) method, i.e., with given discrete description of the computational domain, an approximate NURBS curve is reconstructed to provide quality quadrature information along the curved boundary. The advantages of using NURBS include i). both the numerical accuracy of the approximate solutions and convergence rate of the numerical methods are improved simultaneously, and ii). the NURBS curve generation is independent of other modules of the numerical framework, which makes its application very flexible. It is also shown in the paper that by introducing more elements along the normal direction for the reconstruction patch of the boundary element, significant improvement in the convergence to steady state can be achieved. The numerical examples confirm the above features very well.

Theoretical studies of floating-reference method for NIR blood glucose sensing

NASA Astrophysics Data System (ADS)

Shi, Zhenzhi; Yang, Yue; Zhao, Huijuan; Chen, Wenliang; Liu, Rong; Xu, Kexin

2011-03-01

Non-invasive blood glucose monitoring using NIR light has been suffered from the variety of optical background that is mainly caused by the change of human body, such as the change of temperature, water concentration, and so on. In order to eliminate these internal influence and external interference a so called floating-reference method has been proposed to provide an internal reference. From the analysis of the diffuse reflectance spectrum, a position has been found where diffuse reflection of light is not sensitive to the glucose concentrations. Our previous work has proved the existence of reference position using diffusion equation. However, since glucose monitoring generally use the NIR light in region of 1000-2000nm, diffusion equation is not valid because of the high absorption coefficient and small source-detector separations. In this paper, steady-state high-order approximate model is used to further investigate the existence of the floating reference position in semi-infinite medium. Based on the analysis of different optical parameters on the impact of spatially resolved reflectance of light, we find that the existence of the floating-reference position is the result of the interaction of optical parameters. Comparing to the results of Monte Carlo simulation, the applicable region of diffusion approximation and higher-order approximation for the calculation of floating-reference position is discussed at the wavelength of 1000nm-1800nm, using the intralipid solution of different concentrations. The results indicate that when the reduced albedo is greater than 0.93, diffusion approximation results are more close to simulation results, otherwise the high order approximation is more applicable.

High-Order Implicit-Explicit Multi-Block Time-stepping Method for Hyperbolic PDEs

NASA Technical Reports Server (NTRS)

Nielsen, Tanner B.; Carpenter, Mark H.; Fisher, Travis C.; Frankel, Steven H.

2014-01-01

This work seeks to explore and improve the current time-stepping schemes used in computational fluid dynamics (CFD) in order to reduce overall computational time. A high-order scheme has been developed using a combination of implicit and explicit (IMEX) time-stepping Runge-Kutta (RK) schemes which increases numerical stability with respect to the time step size, resulting in decreased computational time. The IMEX scheme alone does not yield the desired increase in numerical stability, but when used in conjunction with an overlapping partitioned (multi-block) domain significant increase in stability is observed. To show this, the Overlapping-Partition IMEX (OP IMEX) scheme is applied to both one-dimensional (1D) and two-dimensional (2D) problems, the nonlinear viscous Burger's equation and 2D advection equation, respectively. The method uses two different summation by parts (SBP) derivative approximations, second-order and fourth-order accurate. The Dirichlet boundary conditions are imposed using the Simultaneous Approximation Term (SAT) penalty method. The 6-stage additive Runge-Kutta IMEX time integration schemes are fourth-order accurate in time. An increase in numerical stability 65 times greater than the fully explicit scheme is demonstrated to be achievable with the OP IMEX method applied to 1D Burger's equation. Results from the 2D, purely convective, advection equation show stability increases on the order of 10 times the explicit scheme using the OP IMEX method. Also, the domain partitioning method in this work shows potential for breaking the computational domain into manageable sizes such that implicit solutions for full three-dimensional CFD simulations can be computed using direct solving methods rather than the standard iterative methods currently used.

Adaptive finite-volume WENO schemes on dynamically redistributed grids for compressible Euler equations

NASA Astrophysics Data System (ADS)

Pathak, Harshavardhana S.; Shukla, Ratnesh K.

2016-08-01

A high-order adaptive finite-volume method is presented for simulating inviscid compressible flows on time-dependent redistributed grids. The method achieves dynamic adaptation through a combination of time-dependent mesh node clustering in regions characterized by strong solution gradients and an optimal selection of the order of accuracy and the associated reconstruction stencil in a conservative finite-volume framework. This combined approach maximizes spatial resolution in discontinuous regions that require low-order approximations for oscillation-free shock capturing. Over smooth regions, high-order discretization through finite-volume WENO schemes minimizes numerical dissipation and provides excellent resolution of intricate flow features. The method including the moving mesh equations and the compressible flow solver is formulated entirely on a transformed time-independent computational domain discretized using a simple uniform Cartesian mesh. Approximations for the metric terms that enforce discrete geometric conservation law while preserving the fourth-order accuracy of the two-point Gaussian quadrature rule are developed. Spurious Cartesian grid induced shock instabilities such as carbuncles that feature in a local one-dimensional contact capturing treatment along the cell face normals are effectively eliminated through upwind flux calculation using a rotated Hartex-Lax-van Leer contact resolving (HLLC) approximate Riemann solver for the Euler equations in generalized coordinates. Numerical experiments with the fifth and ninth-order WENO reconstructions at the two-point Gaussian quadrature nodes, over a range of challenging test cases, indicate that the redistributed mesh effectively adapts to the dynamic flow gradients thereby improving the solution accuracy substantially even when the initial starting mesh is non-adaptive. The high adaptivity combined with the fifth and especially the ninth-order WENO reconstruction allows remarkably sharp capture of discontinuous propagating shocks with simultaneous resolution of smooth yet complex small scale unsteady flow features to an exceptional detail.

Discrete conservation laws and the convergence of long time simulations of the mkdv equation

NASA Astrophysics Data System (ADS)

Gorria, C.; Alejo, M. A.; Vega, L.

2013-02-01

Pseudospectral collocation methods and finite difference methods have been used for approximating an important family of soliton like solutions of the mKdV equation. These solutions present a structural instability which make difficult to approximate their evolution in long time intervals with enough accuracy. The standard numerical methods do not guarantee the convergence to the proper solution of the initial value problem and often fail by approaching solutions associated to different initial conditions. In this frame the numerical schemes that preserve the discrete invariants related to some conservation laws of this equation produce better results than the methods which only take care of a high consistency order. Pseudospectral spatial discretization appear as the most robust of the numerical methods, but finite difference schemes are useful in order to analyze the rule played by the conservation of the invariants in the convergence.

A Runge-Kutta discontinuous finite element method for high speed flows

NASA Technical Reports Server (NTRS)

Bey, Kim S.; Oden, J. T.

1991-01-01

A Runge-Kutta discontinuous finite element method is developed for hyperbolic systems of conservation laws in two space variables. The discontinuous Galerkin spatial approximation to the conservation laws results in a system of ordinary differential equations which are marched in time using Runge-Kutta methods. Numerical results for the two-dimensional Burger's equation show that the method is (p+1)-order accurate in time and space, where p is the degree of the polynomial approximation of the solution within an element and is capable of capturing shocks over a single element without oscillations. Results for this problem also show that the accuracy of the solution in smooth regions is unaffected by the local projection and that the accuracy in smooth regions increases as p increases. Numerical results for the Euler equations show that the method captures shocks without oscillations and with higher resolution than a first-order scheme.

Approximation of the exponential integral (well function) using sampling methods

NASA Astrophysics Data System (ADS)

Baalousha, Husam Musa

2015-04-01

Exponential integral (also known as well function) is often used in hydrogeology to solve Theis and Hantush equations. Many methods have been developed to approximate the exponential integral. Most of these methods are based on numerical approximations and are valid for a certain range of the argument value. This paper presents a new approach to approximate the exponential integral. The new approach is based on sampling methods. Three different sampling methods; Latin Hypercube Sampling (LHS), Orthogonal Array (OA), and Orthogonal Array-based Latin Hypercube (OA-LH) have been used to approximate the function. Different argument values, covering a wide range, have been used. The results of sampling methods were compared with results obtained by Mathematica software, which was used as a benchmark. All three sampling methods converge to the result obtained by Mathematica, at different rates. It was found that the orthogonal array (OA) method has the fastest convergence rate compared with LHS and OA-LH. The root mean square error RMSE of OA was in the order of 1E-08. This method can be used with any argument value, and can be used to solve other integrals in hydrogeology such as the leaky aquifer integral.

Parallel Implementation of a High Order Implicit Collocation Method for the Heat Equation

NASA Technical Reports Server (NTRS)

Kouatchou, Jules; Halem, Milton (Technical Monitor)

2000-01-01

We combine a high order compact finite difference approximation and collocation techniques to numerically solve the two dimensional heat equation. The resulting method is implicit arid can be parallelized with a strategy that allows parallelization across both time and space. We compare the parallel implementation of the new method with a classical implicit method, namely the Crank-Nicolson method, where the parallelization is done across space only. Numerical experiments are carried out on the SGI Origin 2000.

An accurate method for solving a class of fractional Sturm-Liouville eigenvalue problems

NASA Astrophysics Data System (ADS)

Kashkari, Bothayna S. H.; Syam, Muhammed I.

2018-06-01

This article is devoted to both theoretical and numerical study of the eigenvalues of nonsingular fractional second-order Sturm-Liouville problem. In this paper, we implement a fractional-order Legendre Tau method to approximate the eigenvalues. This method transforms the Sturm-Liouville problem to a sparse nonsingular linear system which is solved using the continuation method. Theoretical results for the considered problem are provided and proved. Numerical results are presented to show the efficiency of the proposed method.

Accelerated solution of discrete ordinates approximation to the Boltzmann transport equation via model reduction

DOE PAGES

Tencer, John; Carlberg, Kevin; Larsen, Marvin; ...

2017-06-17

Radiation heat transfer is an important phenomenon in many physical systems of practical interest. When participating media is important, the radiative transfer equation (RTE) must be solved for the radiative intensity as a function of location, time, direction, and wavelength. In many heat-transfer applications, a quasi-steady assumption is valid, thereby removing time dependence. The dependence on wavelength is often treated through a weighted sum of gray gases (WSGG) approach. The discrete ordinates method (DOM) is one of the most common methods for approximating the angular (i.e., directional) dependence. The DOM exactly solves for the radiative intensity for a finite numbermore » of discrete ordinate directions and computes approximations to integrals over the angular space using a quadrature rule; the chosen ordinate directions correspond to the nodes of this quadrature rule. This paper applies a projection-based model-reduction approach to make high-order quadrature computationally feasible for the DOM for purely absorbing applications. First, the proposed approach constructs a reduced basis from (high-fidelity) solutions of the radiative intensity computed at a relatively small number of ordinate directions. Then, the method computes inexpensive approximations of the radiative intensity at the (remaining) quadrature points of a high-order quadrature using a reduced-order model constructed from the reduced basis. Finally, this results in a much more accurate solution than might have been achieved using only the ordinate directions used to compute the reduced basis. One- and three-dimensional test problems highlight the efficiency of the proposed method.« less

Accelerated solution of discrete ordinates approximation to the Boltzmann transport equation via model reduction

DOE Office of Scientific and Technical Information (OSTI.GOV)

Tencer, John; Carlberg, Kevin; Larsen, Marvin

Radiation heat transfer is an important phenomenon in many physical systems of practical interest. When participating media is important, the radiative transfer equation (RTE) must be solved for the radiative intensity as a function of location, time, direction, and wavelength. In many heat-transfer applications, a quasi-steady assumption is valid, thereby removing time dependence. The dependence on wavelength is often treated through a weighted sum of gray gases (WSGG) approach. The discrete ordinates method (DOM) is one of the most common methods for approximating the angular (i.e., directional) dependence. The DOM exactly solves for the radiative intensity for a finite numbermore » of discrete ordinate directions and computes approximations to integrals over the angular space using a quadrature rule; the chosen ordinate directions correspond to the nodes of this quadrature rule. This paper applies a projection-based model-reduction approach to make high-order quadrature computationally feasible for the DOM for purely absorbing applications. First, the proposed approach constructs a reduced basis from (high-fidelity) solutions of the radiative intensity computed at a relatively small number of ordinate directions. Then, the method computes inexpensive approximations of the radiative intensity at the (remaining) quadrature points of a high-order quadrature using a reduced-order model constructed from the reduced basis. Finally, this results in a much more accurate solution than might have been achieved using only the ordinate directions used to compute the reduced basis. One- and three-dimensional test problems highlight the efficiency of the proposed method.« less

Fully decoupled monolithic projection method for natural convection problems

NASA Astrophysics Data System (ADS)

Pan, Xiaomin; Kim, Kyoungyoun; Lee, Changhoon; Choi, Jung-Il

2017-04-01

To solve time-dependent natural convection problems, we propose a fully decoupled monolithic projection method. The proposed method applies the Crank-Nicolson scheme in time and the second-order central finite difference in space. To obtain a non-iterative monolithic method from the fully discretized nonlinear system, we first adopt linearizations of the nonlinear convection terms and the general buoyancy term with incurring second-order errors in time. Approximate block lower-upper decompositions, along with an approximate factorization technique, are additionally employed to a global linearly coupled system, which leads to several decoupled subsystems, i.e., a fully decoupled monolithic procedure. We establish global error estimates to verify the second-order temporal accuracy of the proposed method for velocity, pressure, and temperature in terms of a discrete l2-norm. Moreover, according to the energy evolution, the proposed method is proved to be stable if the time step is less than or equal to a constant. In addition, we provide numerical simulations of two-dimensional Rayleigh-Bénard convection and periodic forced flow. The results demonstrate that the proposed method significantly mitigates the time step limitation, reduces the computational cost because only one Poisson equation is required to be solved, and preserves the second-order temporal accuracy for velocity, pressure, and temperature. Finally, the proposed method reasonably predicts a three-dimensional Rayleigh-Bénard convection for different Rayleigh numbers.

Comparison of Response Surface and Kriging Models in the Multidisciplinary Design of an Aerospike Nozzle

NASA Technical Reports Server (NTRS)

Simpson, Timothy W.

1998-01-01

The use of response surface models and kriging models are compared for approximating non-random, deterministic computer analyses. After discussing the traditional response surface approach for constructing polynomial models for approximation, kriging is presented as an alternative statistical-based approximation method for the design and analysis of computer experiments. Both approximation methods are applied to the multidisciplinary design and analysis of an aerospike nozzle which consists of a computational fluid dynamics model and a finite element analysis model. Error analysis of the response surface and kriging models is performed along with a graphical comparison of the approximations. Four optimization problems are formulated and solved using both approximation models. While neither approximation technique consistently outperforms the other in this example, the kriging models using only a constant for the underlying global model and a Gaussian correlation function perform as well as the second order polynomial response surface models.

Numerical solution of distributed order fractional differential equations

NASA Astrophysics Data System (ADS)

Katsikadelis, John T.

2014-02-01

In this paper a method for the numerical solution of distributed order FDEs (fractional differential equations) of a general form is presented. The method applies to both linear and nonlinear equations. The Caputo type fractional derivative is employed. The distributed order FDE is approximated with a multi-term FDE, which is then solved by adjusting appropriately the numerical method developed for multi-term FDEs by Katsikadelis. Several example equations are solved and the response of mechanical systems described by such equations is studied. The convergence and the accuracy of the method for linear and nonlinear equations are demonstrated through well corroborated numerical results.

Estimation of Phase in Fringe Projection Technique Using High-order Instantaneous Moments Based Method

NASA Astrophysics Data System (ADS)

Gorthi, Sai Siva; Rajshekhar, G.; Rastogi, Pramod

2010-04-01

For three-dimensional (3D) shape measurement using fringe projection techniques, the information about the 3D shape of an object is encoded in the phase of a recorded fringe pattern. The paper proposes a high-order instantaneous moments based method to estimate phase from a single fringe pattern in fringe projection. The proposed method works by approximating the phase as a piece-wise polynomial and subsequently determining the polynomial coefficients using high-order instantaneous moments to construct the polynomial phase. Simulation results are presented to show the method's potential.

Exponential Methods for the Time Integration of Schroedinger Equation

DOE Office of Scientific and Technical Information (OSTI.GOV)

Cano, B.; Gonzalez-Pachon, A.

2010-09-30

We consider exponential methods of second order in time in order to integrate the cubic nonlinear Schroedinger equation. We are interested in taking profit of the special structure of this equation. Therefore, we look at symmetry, symplecticity and approximation of invariants of the proposed methods. That will allow to integrate till long times with reasonable accuracy. Computational efficiency is also our aim. Therefore, we make numerical computations in order to compare the methods considered and so as to conclude that explicit Lawson schemes projected on the norm of the solution are an efficient tool to integrate this equation.

On the acceleration of charged particles at relativistic shock fronts

NASA Technical Reports Server (NTRS)

Kirk, J. G.; Schneider, P.

1987-01-01

The diffusive acceleration of highly relativistic particles at a shock is reconsidered. Using the same physical assumptions as Blandford and Ostriker (1978), but dropping the restriction to nonrelativistic shock velocities, the authors find approximate solutions of the particle kinetic equation by generalizing the diffusion approximation to higher order terms in the anisotropy of the particle distribution. The general solution of the transport equation on either side of the shock is constructed, which involves the solution of an eigenvalue problem. By matching the two solutions at the shock, the spectral index of the resulting power law is found by taking into account a sufficiently large number of eigenfunctions. Low-order truncation corresponds to the standard diffusion approximation and to a somewhat more general method described by Peacock (1981). In addition to the energy spectrum, the method yields the angular distribution of the particles and its spatial dependence.

A point-value enhanced finite volume method based on approximate delta functions

NASA Astrophysics Data System (ADS)

Xuan, Li-Jun; Majdalani, Joseph

2018-02-01

We revisit the concept of an approximate delta function (ADF), introduced by Huynh (2011) [1], in the form of a finite-order polynomial that holds identical integral properties to the Dirac delta function when used in conjunction with a finite-order polynomial integrand over a finite domain. We show that the use of generic ADF polynomials can be effective at recovering and generalizing several high-order methods, including Taylor-based and nodal-based Discontinuous Galerkin methods, as well as the Correction Procedure via Reconstruction. Based on the ADF concept, we then proceed to formulate a Point-value enhanced Finite Volume (PFV) method, which stores and updates the cell-averaged values inside each element as well as the unknown quantities and, if needed, their derivatives on nodal points. The sharing of nodal information with surrounding elements saves the number of degrees of freedom compared to other compact methods at the same order. To ensure conservation, cell-averaged values are updated using an identical approach to that adopted in the finite volume method. Here, the updating of nodal values and their derivatives is achieved through an ADF concept that leverages all of the elements within the domain of integration that share the same nodal point. The resulting scheme is shown to be very stable at successively increasing orders. Both accuracy and stability of the PFV method are verified using a Fourier analysis and through applications to the linear wave and nonlinear Burgers' equations in one-dimensional space.

High-Order Entropy Stable Finite Difference Schemes for Nonlinear Conservation Laws: Finite Domains

NASA Technical Reports Server (NTRS)

Fisher, Travis C.; Carpenter, Mark H.

2013-01-01

Developing stable and robust high-order finite difference schemes requires mathematical formalism and appropriate methods of analysis. In this work, nonlinear entropy stability is used to derive provably stable high-order finite difference methods with formal boundary closures for conservation laws. Particular emphasis is placed on the entropy stability of the compressible Navier-Stokes equations. A newly derived entropy stable weighted essentially non-oscillatory finite difference method is used to simulate problems with shocks and a conservative, entropy stable, narrow-stencil finite difference approach is used to approximate viscous terms.

High order filtering methods for approximating hyperbolic systems of conservation laws

NASA Technical Reports Server (NTRS)

Lafon, F.; Osher, S.

1991-01-01

The essentially nonoscillatory (ENO) schemes, while potentially useful in the computation of discontinuous solutions of hyperbolic conservation-law systems, are computationally costly relative to simple central-difference methods. A filtering technique is presented which employs central differencing of arbitrarily high-order accuracy except where a local test detects the presence of spurious oscillations and calls upon the full ENO apparatus to remove them. A factor-of-three speedup is thus obtained over the full-ENO method for a wide range of problems, with high-order accuracy in regions of smooth flow.

Finite Differences and Collocation Methods for the Solution of the Two Dimensional Heat Equation

NASA Technical Reports Server (NTRS)

Kouatchou, Jules

1999-01-01

In this paper we combine finite difference approximations (for spatial derivatives) and collocation techniques (for the time component) to numerically solve the two dimensional heat equation. We employ respectively a second-order and a fourth-order schemes for the spatial derivatives and the discretization method gives rise to a linear system of equations. We show that the matrix of the system is non-singular. Numerical experiments carried out on serial computers, show the unconditional stability of the proposed method and the high accuracy achieved by the fourth-order scheme.

The Adams formulas for numerical integration of differential equations from 1st to 20th order

NASA Technical Reports Server (NTRS)

Kirkpatrick, J. C.

1976-01-01

The Adams Bashforth predictor coefficients and the Adams Moulton corrector coefficients for the integration of differential equations are presented for methods of 1st to 20th order. The order of the method as presented refers to the highest order difference formula used in Newton's backward difference interpolation formula, on which the Adams method is based. The Adams method is a polynomial approximation method derived from Newton's backward difference interpolation formula. The Newton formula is derived and expanded to 20th order. The Adams predictor and corrector formulas are derived and expressed in terms of differences of the derivatives, as well as in terms of the derivatives themselves. All coefficients are given to 18 significant digits. For the difference formula only, the ratio coefficients are given to 10th order.

Reduced linear noise approximation for biochemical reaction networks with time-scale separation: The stochastic tQSSA+

NASA Astrophysics Data System (ADS)

Herath, Narmada; Del Vecchio, Domitilla

2018-03-01

Biochemical reaction networks often involve reactions that take place on different time scales, giving rise to "slow" and "fast" system variables. This property is widely used in the analysis of systems to obtain dynamical models with reduced dimensions. In this paper, we consider stochastic dynamics of biochemical reaction networks modeled using the Linear Noise Approximation (LNA). Under time-scale separation conditions, we obtain a reduced-order LNA that approximates both the slow and fast variables in the system. We mathematically prove that the first and second moments of this reduced-order model converge to those of the full system as the time-scale separation becomes large. These mathematical results, in particular, provide a rigorous justification to the accuracy of LNA models derived using the stochastic total quasi-steady state approximation (tQSSA). Since, in contrast to the stochastic tQSSA, our reduced-order model also provides approximations for the fast variable stochastic properties, we term our method the "stochastic tQSSA+". Finally, we demonstrate the application of our approach on two biochemical network motifs found in gene-regulatory and signal transduction networks.

Weak Galerkin method for the Biot’s consolidation model

DOE PAGES

Hu, Xiaozhe; Mu, Lin; Ye, Xiu

2017-08-23

In this study, we develop a weak Galerkin (WG) finite element method for the Biot’s consolidation model in the classical displacement–pressure two-field formulation. Weak Galerkin linear finite elements are used for both displacement and pressure approximations in spatial discretizations. Backward Euler scheme is used for temporal discretization in order to obtain an implicit fully discretized scheme. We study the well-posedness of the linear system at each time step and also derive the overall optimal-order convergence of the WG formulation. Such WG scheme is designed on general shape regular polytopal meshes and provides stable and oscillation-free approximation for the pressure withoutmore » special treatment. Lastlyl, numerical experiments are presented to demonstrate the efficiency and accuracy of the proposed weak Galerkin finite element method.« less

Weak Galerkin method for the Biot’s consolidation model

DOE Office of Scientific and Technical Information (OSTI.GOV)

Hu, Xiaozhe; Mu, Lin; Ye, Xiu

In this study, we develop a weak Galerkin (WG) finite element method for the Biot’s consolidation model in the classical displacement–pressure two-field formulation. Weak Galerkin linear finite elements are used for both displacement and pressure approximations in spatial discretizations. Backward Euler scheme is used for temporal discretization in order to obtain an implicit fully discretized scheme. We study the well-posedness of the linear system at each time step and also derive the overall optimal-order convergence of the WG formulation. Such WG scheme is designed on general shape regular polytopal meshes and provides stable and oscillation-free approximation for the pressure withoutmore » special treatment. Lastlyl, numerical experiments are presented to demonstrate the efficiency and accuracy of the proposed weak Galerkin finite element method.« less

Efficient algorithms for analyzing the singularly perturbed boundary value problems of fractional order

NASA Astrophysics Data System (ADS)

Sayevand, K.; Pichaghchi, K.

2018-04-01

In this paper, we were concerned with the description of the singularly perturbed boundary value problems in the scope of fractional calculus. We should mention that, one of the main methods used to solve these problems in classical calculus is the so-called matched asymptotic expansion method. However we shall note that, this was not achievable via the existing classical definitions of fractional derivative, because they do not obey the chain rule which one of the key elements of the matched asymptotic expansion method. In order to accommodate this method to fractional derivative, we employ a relatively new derivative so-called the local fractional derivative. Using the properties of local fractional derivative, we extend the matched asymptotic expansion method to the scope of fractional calculus and introduce a reliable new algorithm to develop approximate solutions of the singularly perturbed boundary value problems of fractional order. In the new method, the original problem is partitioned into inner and outer solution equations. The reduced equation is solved with suitable boundary conditions which provide the terminal boundary conditions for the boundary layer correction. The inner solution problem is next solved as a solvable boundary value problem. The width of the boundary layer is approximated using appropriate resemblance function. Some theoretical results are established and proved. Some illustrating examples are solved and the results are compared with those of matched asymptotic expansion method and homotopy analysis method to demonstrate the accuracy and efficiency of the method. It can be observed that, the proposed method approximates the exact solution very well not only in the boundary layer, but also away from the layer.

Numerical analysis for trajectory controllability of a coupled multi-order fractional delay differential system via the shifted Jacobi method

NASA Astrophysics Data System (ADS)

Priya, B. Ganesh; Muthukumar, P.

2018-02-01

This paper deals with the trajectory controllability for a class of multi-order fractional linear systems subject to a constant delay in state vector. The solution for the coupled fractional delay differential equation is established by the Mittag-Leffler function. The necessary and sufficient condition for the trajectory controllability is formulated and proved by the generalized Gronwall's inequality. The approximate trajectory for the proposed system is obtained through the shifted Jacobi operational matrix method. The numerical simulation of the approximate solution shows the theoretical results. Finally, some remarks and comments on the existing results of constrained controllability for the fractional dynamical system are also presented.

Testing higher-order Lagrangian perturbation theory against numerical simulation. 1: Pancake models

NASA Technical Reports Server (NTRS)

Buchert, T.; Melott, A. L.; Weiss, A. G.

1993-01-01

We present results showing an improvement of the accuracy of perturbation theory as applied to cosmological structure formation for a useful range of quasi-linear scales. The Lagrangian theory of gravitational instability of an Einstein-de Sitter dust cosmogony investigated and solved up to the third order is compared with numerical simulations. In this paper we study the dynamics of pancake models as a first step. In previous work the accuracy of several analytical approximations for the modeling of large-scale structure in the mildly non-linear regime was analyzed in the same way, allowing for direct comparison of the accuracy of various approximations. In particular, the Zel'dovich approximation (hereafter ZA) as a subclass of the first-order Lagrangian perturbation solutions was found to provide an excellent approximation to the density field in the mildly non-linear regime (i.e. up to a linear r.m.s. density contrast of sigma is approximately 2). The performance of ZA in hierarchical clustering models can be greatly improved by truncating the initial power spectrum (smoothing the initial data). We here explore whether this approximation can be further improved with higher-order corrections in the displacement mapping from homogeneity. We study a single pancake model (truncated power-spectrum with power-spectrum with power-index n = -1) using cross-correlation statistics employed in previous work. We found that for all statistical methods used the higher-order corrections improve the results obtained for the first-order solution up to the stage when sigma (linear theory) is approximately 1. While this improvement can be seen for all spatial scales, later stages retain this feature only above a certain scale which is increasing with time. However, third-order is not much improvement over second-order at any stage. The total breakdown of the perturbation approach is observed at the stage, where sigma (linear theory) is approximately 2, which corresponds to the onset of hierarchical clustering. This success is found at a considerable higher non-linearity than is usual for perturbation theory. Whether a truncation of the initial power-spectrum in hierarchical models retains this improvement will be analyzed in a forthcoming work.

Fourth order exponential time differencing method with local discontinuous Galerkin approximation for coupled nonlinear Schrodinger equations

DOE PAGES

Liang, Xiao; Khaliq, Abdul Q. M.; Xing, Yulong

2015-01-23

In this paper, we study a local discontinuous Galerkin method combined with fourth order exponential time differencing Runge-Kutta time discretization and a fourth order conservative method for solving the nonlinear Schrödinger equations. Based on different choices of numerical fluxes, we propose both energy-conserving and energy-dissipative local discontinuous Galerkin methods, and have proven the error estimates for the semi-discrete methods applied to linear Schrödinger equation. The numerical methods are proven to be highly efficient and stable for long-range soliton computations. Finally, extensive numerical examples are provided to illustrate the accuracy, efficiency and reliability of the proposed methods.

Projection-free approximate balanced truncation of large unstable systems

NASA Astrophysics Data System (ADS)

Flinois, Thibault L. B.; Morgans, Aimee S.; Schmid, Peter J.

2015-08-01

In this article, we show that the projection-free, snapshot-based, balanced truncation method can be applied directly to unstable systems. We prove that even for unstable systems, the unmodified balanced proper orthogonal decomposition algorithm theoretically yields a converged transformation that balances the Gramians (including the unstable subspace). We then apply the method to a spatially developing unstable system and show that it results in reduced-order models of similar quality to the ones obtained with existing methods. Due to the unbounded growth of unstable modes, a practical restriction on the final impulse response simulation time appears, which can be adjusted depending on the desired order of the reduced-order model. Recommendations are given to further reduce the cost of the method if the system is large and to improve the performance of the method if it does not yield acceptable results in its unmodified form. Finally, the method is applied to the linearized flow around a cylinder at Re = 100 to show that it actually is able to accurately reproduce impulse responses for more realistic unstable large-scale systems in practice. The well-established approximate balanced truncation numerical framework therefore can be safely applied to unstable systems without any modifications. Additionally, balanced reduced-order models can readily be obtained even for large systems, where the computational cost of existing methods is prohibitive.

On shifted Jacobi spectral method for high-order multi-point boundary value problems

NASA Astrophysics Data System (ADS)

Doha, E. H.; Bhrawy, A. H.; Hafez, R. M.

2012-10-01

This paper reports a spectral tau method for numerically solving multi-point boundary value problems (BVPs) of linear high-order ordinary differential equations. The construction of the shifted Jacobi tau approximation is based on conventional differentiation. This use of differentiation allows the imposition of the governing equation at the whole set of grid points and the straight forward implementation of multiple boundary conditions. Extension of the tau method for high-order multi-point BVPs with variable coefficients is treated using the shifted Jacobi Gauss-Lobatto quadrature. Shifted Jacobi collocation method is developed for solving nonlinear high-order multi-point BVPs. The performance of the proposed methods is investigated by considering several examples. Accurate results and high convergence rates are achieved.

Applied Routh approximation

NASA Technical Reports Server (NTRS)

Merrill, W. C.

1978-01-01

The Routh approximation technique for reducing the complexity of system models was applied in the frequency domain to a 16th order, state variable model of the F100 engine and to a 43d order, transfer function model of a launch vehicle boost pump pressure regulator. The results motivate extending the frequency domain formulation of the Routh method to the time domain in order to handle the state variable formulation directly. The time domain formulation was derived and a characterization that specifies all possible Routh similarity transformations was given. The characterization was computed by solving two eigenvalue-eigenvector problems. The application of the time domain Routh technique to the state variable engine model is described, and some results are given. Additional computational problems are discussed, including an optimization procedure that can improve the approximation accuracy by taking advantage of the transformation characterization.

Numerical solution of second order ODE directly by two point block backward differentiation formula

NASA Astrophysics Data System (ADS)

Zainuddin, Nooraini; Ibrahim, Zarina Bibi; Othman, Khairil Iskandar; Suleiman, Mohamed; Jamaludin, Noraini

2015-12-01

Direct Two Point Block Backward Differentiation Formula, (BBDF2) for solving second order ordinary differential equations (ODEs) will be presented throughout this paper. The method is derived by differentiating the interpolating polynomial using three back values. In BBDF2, two approximate solutions are produced simultaneously at each step of integration. The method derived is implemented by using fixed step size and the numerical results that follow demonstrate the advantage of the direct method as compared to the reduction method.

Spectral Element Method for the Simulation of Unsteady Compressible Flows

NASA Technical Reports Server (NTRS)

Diosady, Laslo Tibor; Murman, Scott M.

2013-01-01

This work uses a discontinuous-Galerkin spectral-element method (DGSEM) to solve the compressible Navier-Stokes equations [1{3]. The inviscid ux is computed using the approximate Riemann solver of Roe [4]. The viscous fluxes are computed using the second form of Bassi and Rebay (BR2) [5] in a manner consistent with the spectral-element approximation. The method of lines with the classical 4th-order explicit Runge-Kutta scheme is used for time integration. Results for polynomial orders up to p = 15 (16th order) are presented. The code is parallelized using the Message Passing Interface (MPI). The computations presented in this work are performed using the Sandy Bridge nodes of the NASA Pleiades supercomputer at NASA Ames Research Center. Each Sandy Bridge node consists of 2 eight-core Intel Xeon E5-2670 processors with a clock speed of 2.6Ghz and 2GB per core memory. On a Sandy Bridge node the Tau Benchmark [6] runs in a time of 7.6s.

A method for reducing the order of nonlinear dynamic systems

NASA Astrophysics Data System (ADS)

Masri, S. F.; Miller, R. K.; Sassi, H.; Caughey, T. K.

1984-06-01

An approximate method that uses conventional condensation techniques for linear systems together with the nonparametric identification of the reduced-order model generalized nonlinear restoring forces is presented for reducing the order of discrete multidegree-of-freedom dynamic systems that possess arbitrary nonlinear characteristics. The utility of the proposed method is demonstrated by considering a redundant three-dimensional finite-element model half of whose elements incorporate hysteretic properties. A nonlinear reduced-order model, of one-third the order of the original model, is developed on the basis of wideband stationary random excitation and the validity of the reduced-order model is subsequently demonstrated by its ability to predict with adequate accuracy the transient response of the original nonlinear model under a different nonstationary random excitation.

An infinite-order two-component relativistic Hamiltonian by a simple one-step transformation.

PubMed

Ilias, Miroslav; Saue, Trond

2007-02-14

The authors report the implementation of a simple one-step method for obtaining an infinite-order two-component (IOTC) relativistic Hamiltonian using matrix algebra. They apply the IOTC Hamiltonian to calculations of excitation and ionization energies as well as electric and magnetic properties of the radon atom. The results are compared to corresponding calculations using identical basis sets and based on the four-component Dirac-Coulomb Hamiltonian as well as Douglas-Kroll-Hess and zeroth-order regular approximation Hamiltonians, all implemented in the DIRAC program package, thus allowing a comprehensive comparison of relativistic Hamiltonians within the finite basis approximation.

Effective quadrature formula in solving linear integro-differential equations of order two

NASA Astrophysics Data System (ADS)

Eshkuvatov, Z. K.; Kammuji, M.; Long, N. M. A. Nik; Yunus, Arif A. M.

2017-08-01

In this note, we solve general form of Fredholm-Volterra integro-differential equations (IDEs) of order 2 with boundary condition approximately and show that proposed method is effective and reliable. Initially, IDEs is reduced into integral equation of the third kind by using standard integration techniques and identity between multiple and single integrals then truncated Legendre series are used to estimate the unknown function. For the kernel integrals, we have applied Gauss-Legendre quadrature formula and collocation points are chosen as the roots of the Legendre polynomials. Finally, reduce the integral equations of the third kind into the system of algebraic equations and Gaussian elimination method is applied to get approximate solutions. Numerical examples and comparisons with other methods reveal that the proposed method is very effective and dominated others in many cases. General theory of existence of the solution is also discussed.

A New Method for Interpreting Nonstationary Running Correlations and Its Application to the ENSO-EAWM Relationship

NASA Astrophysics Data System (ADS)

Geng, Xin; Zhang, Wenjun; Jin, Fei-Fei; Stuecker, Malte F.

2018-01-01

We here propose a new statistical method to interpret nonstationary running correlations by decomposing them into a stationary part and a first-order Taylor expansion approximation for the nonstationary part. Then, this method is applied to investigate the nonstationary behavior of the El Niño-Southern Oscillation (ENSO)-East Asian winter monsoon (EAWM) relationship, which exhibits prominent multidecadal variations. It is demonstrated that the first-order approximation of the nonstationary part can be expressed to a large extent by the impact of the nonlinear interaction between the Atlantic Multidecadal Oscillation (AMO) and ENSO (AMO*Niño3.4) on the EAWM. Therefore, the nonstationarity in the ENSO-EAWM relationship comes predominantly from the impact of an AMO modulation on the ENSO-EAWM teleconnection via this key nonlinear interaction. This general method can be applied to investigate nonstationary relationships that are often observed between various different climate phenomena.

Multigrid methods for isogeometric discretization

PubMed Central

Gahalaut, K.P.S.; Kraus, J.K.; Tomar, S.K.

2013-01-01

We present (geometric) multigrid methods for isogeometric discretization of scalar second order elliptic problems. The smoothing property of the relaxation method, and the approximation property of the intergrid transfer operators are analyzed. These properties, when used in the framework of classical multigrid theory, imply uniform convergence of two-grid and multigrid methods. Supporting numerical results are provided for the smoothing property, the approximation property, convergence factor and iterations count for V-, W- and F-cycles, and the linear dependence of V-cycle convergence on the smoothing steps. For two dimensions, numerical results include the problems with variable coefficients, simple multi-patch geometry, a quarter annulus, and the dependence of convergence behavior on refinement levels ℓ, whereas for three dimensions, only the constant coefficient problem in a unit cube is considered. The numerical results are complete up to polynomial order p=4, and for C0 and Cp-1 smoothness. PMID:24511168

High Order Approximations for Compressible Fluid Dynamics on Unstructured and Cartesian Meshes

NASA Technical Reports Server (NTRS)

Barth, Timothy (Editor); Deconinck, Herman (Editor)

1999-01-01

The development of high-order accurate numerical discretization techniques for irregular domains and meshes is often cited as one of the remaining challenges facing the field of computational fluid dynamics. In structural mechanics, the advantages of high-order finite element approximation are widely recognized. This is especially true when high-order element approximation is combined with element refinement (h-p refinement). In computational fluid dynamics, high-order discretization methods are infrequently used in the computation of compressible fluid flow. The hyperbolic nature of the governing equations and the presence of solution discontinuities makes high-order accuracy difficult to achieve. Consequently, second-order accurate methods are still predominately used in industrial applications even though evidence suggests that high-order methods may offer a way to significantly improve the resolution and accuracy for these calculations. To address this important topic, a special course was jointly organized by the Applied Vehicle Technology Panel of NATO's Research and Technology Organization (RTO), the von Karman Institute for Fluid Dynamics, and the Numerical Aerospace Simulation Division at the NASA Ames Research Center. The NATO RTO sponsored course entitled "Higher Order Discretization Methods in Computational Fluid Dynamics" was held September 14-18, 1998 at the von Karman Institute for Fluid Dynamics in Belgium and September 21-25, 1998 at the NASA Ames Research Center in the United States. During this special course, lecturers from Europe and the United States gave a series of comprehensive lectures on advanced topics related to the high-order numerical discretization of partial differential equations with primary emphasis given to computational fluid dynamics (CFD). Additional consideration was given to topics in computational physics such as the high-order discretization of the Hamilton-Jacobi, Helmholtz, and elasticity equations. This volume consists of five articles prepared by the special course lecturers. These articles should be of particular relevance to those readers with an interest in numerical discretization techniques which generalize to very high-order accuracy. The articles of Professors Abgrall and Shu consider the mathematical formulation of high-order accurate finite volume schemes utilizing essentially non-oscillatory (ENO) and weighted essentially non-oscillatory (WENO) reconstruction together with upwind flux evaluation. These formulations are particularly effective in computing numerical solutions of conservation laws containing solution discontinuities. Careful attention is given by the authors to implementational issues and techniques for improving the overall efficiency of these methods. The article of Professor Cockburn discusses the discontinuous Galerkin finite element method. This method naturally extends to high-order accuracy and has an interpretation as a finite volume method. Cockburn addresses two important issues associated with the discontinuous Galerkin method: controlling spurious extrema near solution discontinuities via "limiting" and the extension to second order advective-diffusive equations (joint work with Shu). The articles of Dr. Henderson and Professor Schwab consider the mathematical formulation and implementation of the h-p finite element methods using hierarchical basis functions and adaptive mesh refinement. These methods are particularly useful in computing high-order accurate solutions containing perturbative layers and corner singularities. Additional flexibility is obtained using a mortar FEM technique whereby nonconforming elements are interfaced together. Numerous examples are given by Henderson applying the h-p FEM method to the simulation of turbulence and turbulence transition.

On simulating flow with multiple time scales using a method of averages

DOE Office of Scientific and Technical Information (OSTI.GOV)

Margolin, L.G.

1997-12-31

The author presents a new computational method based on averaging to efficiently simulate certain systems with multiple time scales. He first develops the method in a simple one-dimensional setting and employs linear stability analysis to demonstrate numerical stability. He then extends the method to multidimensional fluid flow. His method of averages does not depend on explicit splitting of the equations nor on modal decomposition. Rather he combines low order and high order algorithms in a generalized predictor-corrector framework. He illustrates the methodology in the context of a shallow fluid approximation to an ocean basin circulation. He finds that his newmore » method reproduces the accuracy of a fully explicit second-order accurate scheme, while costing less than a first-order accurate scheme.« less

Numerical investigation of sixth order Boussinesq equation

NASA Astrophysics Data System (ADS)

Kolkovska, N.; Vucheva, V.

2017-10-01

We propose a family of conservative finite difference schemes for the Boussinesq equation with sixth order dispersion terms. The schemes are of second order of approximation. The method is conditionally stable with a mild restriction τ = O(h) on the step sizes. Numerical tests are performed for quadratic and cubic nonlinearities. The numerical experiments show second order of convergence of the discrete solution to the exact one.

Numerical solution of sixth-order boundary-value problems using Legendre wavelet collocation method

NASA Astrophysics Data System (ADS)

Sohaib, Muhammad; Haq, Sirajul; Mukhtar, Safyan; Khan, Imad

2018-03-01

An efficient method is proposed to approximate sixth order boundary value problems. The proposed method is based on Legendre wavelet in which Legendre polynomial is used. The mechanism of the method is to use collocation points that converts the differential equation into a system of algebraic equations. For validation two test problems are discussed. The results obtained from proposed method are quite accurate, also close to exact solution, and other different methods. The proposed method is computationally more effective and leads to more accurate results as compared to other methods from literature.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Ochiai, Yoshihiro

Heat-conduction analysis under steady state without heat generation can easily be treated by the boundary element method. However, in the case with heat conduction with heat generation can approximately be solved without a domain integral by an improved multiple-reciprocity boundary element method. The convention multiple-reciprocity boundary element method is not suitable for complicated heat generation. In the improved multiple-reciprocity boundary element method, on the other hand, the domain integral in each step is divided into point, line, and area integrals. In order to solve the problem, the contour lines of heat generation, which approximate the actual heat generation, are used.

An efficient computer based wavelets approximation method to solve Fuzzy boundary value differential equations

NASA Astrophysics Data System (ADS)

Alam Khan, Najeeb; Razzaq, Oyoon Abdul

2016-03-01

In the present work a wavelets approximation method is employed to solve fuzzy boundary value differential equations (FBVDEs). Essentially, a truncated Legendre wavelets series together with the Legendre wavelets operational matrix of derivative are utilized to convert FB- VDE into a simple computational problem by reducing it into a system of fuzzy algebraic linear equations. The capability of scheme is investigated on second order FB- VDE considered under generalized H-differentiability. Solutions are represented graphically showing competency and accuracy of this method.

Three-Dimensional Navier-Stokes Method with Two-Equation Turbulence Models for Efficient Numerical Simulation of Hypersonic Flows

NASA Technical Reports Server (NTRS)

Bardina, J. E.

1994-01-01

A new computational efficient 3-D compressible Reynolds-averaged implicit Navier-Stokes method with advanced two equation turbulence models for high speed flows is presented. All convective terms are modeled using an entropy satisfying higher-order Total Variation Diminishing (TVD) scheme based on implicit upwind flux-difference split approximations and arithmetic averaging procedure of primitive variables. This method combines the best features of data management and computational efficiency of space marching procedures with the generality and stability of time dependent Navier-Stokes procedures to solve flows with mixed supersonic and subsonic zones, including streamwise separated flows. Its robust stability derives from a combination of conservative implicit upwind flux-difference splitting with Roe's property U to provide accurate shock capturing capability that non-conservative schemes do not guarantee, alternating symmetric Gauss-Seidel 'method of planes' relaxation procedure coupled with a three-dimensional two-factor diagonal-dominant approximate factorization scheme, TVD flux limiters of higher-order flux differences satisfying realizability, and well-posed characteristic-based implicit boundary-point a'pproximations consistent with the local characteristics domain of dependence. The efficiency of the method is highly increased with Newton Raphson acceleration which allows convergence in essentially one forward sweep for supersonic flows. The method is verified by comparing with experiment and other Navier-Stokes methods. Here, results of adiabatic and cooled flat plate flows, compression corner flow, and 3-D hypersonic shock-wave/turbulent boundary layer interaction flows are presented. The robust 3-D method achieves a better computational efficiency of at least one order of magnitude over the CNS Navier-Stokes code. It provides cost-effective aerodynamic predictions in agreement with experiment, and the capability of predicting complex flow structures in complex geometries with good accuracy.

An h-p Taylor-Galerkin finite element method for compressible Euler equations

NASA Technical Reports Server (NTRS)

Demkowicz, L.; Oden, J. T.; Rachowicz, W.; Hardy, O.

1991-01-01

An extension of the familiar Taylor-Galerkin method to arbitrary h-p spatial approximations is proposed. Boundary conditions are analyzed, and a linear stability result for arbitrary meshes is given, showing the unconditional stability for the parameter of implicitness alpha not less than 0.5. The wedge and blunt body problems are solved with both linear, quadratic, and cubic elements and h-adaptivity, showing the feasibility of higher orders of approximation for problems with shocks.

Transport of phase space densities through tetrahedral meshes using discrete flow mapping

NASA Astrophysics Data System (ADS)

Bajars, Janis; Chappell, David J.; Søndergaard, Niels; Tanner, Gregor

2017-01-01

Discrete flow mapping was recently introduced as an efficient ray based method determining wave energy distributions in complex built up structures. Wave energy densities are transported along ray trajectories through polygonal mesh elements using a finite dimensional approximation of a ray transfer operator. In this way the method can be viewed as a smoothed ray tracing method defined over meshed surfaces. Many applications require the resolution of wave energy distributions in three-dimensional domains, such as in room acoustics, underwater acoustics and for electromagnetic cavity problems. In this work we extend discrete flow mapping to three-dimensional domains by propagating wave energy densities through tetrahedral meshes. The geometric simplicity of the tetrahedral mesh elements is utilised to efficiently compute the ray transfer operator using a mixture of analytic and spectrally accurate numerical integration. The important issue of how to choose a suitable basis approximation in phase space whilst maintaining a reasonable computational cost is addressed via low order local approximations on tetrahedral faces in the position coordinate and high order orthogonal polynomial expansions in momentum space.

Implementation of Kalman filter algorithm on models reduced using singular pertubation approximation method and its application to measurement of water level

NASA Astrophysics Data System (ADS)

Rachmawati, Vimala; Khusnul Arif, Didik; Adzkiya, Dieky

2018-03-01

The systems contained in the universe often have a large order. Thus, the mathematical model has many state variables that affect the computation time. In addition, generally not all variables are known, so estimations are needed to measure the magnitude of the system that cannot be measured directly. In this paper, we discuss the model reduction and estimation of state variables in the river system to measure the water level. The model reduction of a system is an approximation method of a system with a lower order without significant errors but has a dynamic behaviour that is similar to the original system. The Singular Perturbation Approximation method is one of the model reduction methods where all state variables of the equilibrium system are partitioned into fast and slow modes. Then, The Kalman filter algorithm is used to estimate state variables of stochastic dynamic systems where estimations are computed by predicting state variables based on system dynamics and measurement data. Kalman filters are used to estimate state variables in the original system and reduced system. Then, we compare the estimation results of the state and computational time between the original and reduced system.

Wavelet-based adaptation methodology combined with finite difference WENO to solve ideal magnetohydrodynamics

NASA Astrophysics Data System (ADS)

Do, Seongju; Li, Haojun; Kang, Myungjoo

2017-06-01

In this paper, we present an accurate and efficient wavelet-based adaptive weighted essentially non-oscillatory (WENO) scheme for hydrodynamics and ideal magnetohydrodynamics (MHD) equations arising from the hyperbolic conservation systems. The proposed method works with the finite difference weighted essentially non-oscillatory (FD-WENO) method in space and the third order total variation diminishing (TVD) Runge-Kutta (RK) method in time. The philosophy of this work is to use the lifted interpolating wavelets as not only detector for singularities but also interpolator. Especially, flexible interpolations can be performed by an inverse wavelet transformation. When the divergence cleaning method introducing auxiliary scalar field ψ is applied to the base numerical schemes for imposing divergence-free condition to the magnetic field in a MHD equation, the approximations to derivatives of ψ require the neighboring points. Moreover, the fifth order WENO interpolation requires large stencil to reconstruct high order polynomial. In such cases, an efficient interpolation method is necessary. The adaptive spatial differentiation method is considered as well as the adaptation of grid resolutions. In order to avoid the heavy computation of FD-WENO, in the smooth regions fixed stencil approximation without computing the non-linear WENO weights is used, and the characteristic decomposition method is replaced by a component-wise approach. Numerical results demonstrate that with the adaptive method we are able to resolve the solutions that agree well with the solution of the corresponding fine grid.

Numerical scheme approximating solution and parameters in a beam equation

NASA Astrophysics Data System (ADS)

Ferdinand, Robert R.

2003-12-01

We present a mathematical model which describes vibration in a metallic beam about its equilibrium position. This model takes the form of a nonlinear second-order (in time) and fourth-order (in space) partial differential equation with boundary and initial conditions. A finite-element Galerkin approximation scheme is used to estimate model solution. Infinite-dimensional model parameters are then estimated numerically using an inverse method procedure which involves the minimization of a least-squares cost functional. Numerical results are presented and future work to be done is discussed.

The constraint method: A new finite element technique. [applied to static and dynamic loads on plates

NASA Technical Reports Server (NTRS)

Tsai, C.; Szabo, B. A.

1973-01-01

An approch to the finite element method which utilizes families of conforming finite elements based on complete polynomials is presented. Finite element approximations based on this method converge with respect to progressively reduced element sizes as well as with respect to progressively increasing orders of approximation. Numerical results of static and dynamic applications of plates are presented to demonstrate the efficiency of the method. Comparisons are made with plate elements in NASTRAN and the high-precision plate element developed by Cowper and his co-workers. Some considerations are given to implementation of the constraint method into general purpose computer programs such as NASTRAN.

MIST - MINIMUM-STATE METHOD FOR RATIONAL APPROXIMATION OF UNSTEADY AERODYNAMIC FORCE COEFFICIENT MATRICES

NASA Technical Reports Server (NTRS)

Karpel, M.

1994-01-01

Various control analysis, design, and simulation techniques of aeroservoelastic systems require the equations of motion to be cast in a linear, time-invariant state-space form. In order to account for unsteady aerodynamics, rational function approximations must be obtained to represent them in the first order equations of the state-space formulation. A computer program, MIST, has been developed which determines minimum-state approximations of the coefficient matrices of the unsteady aerodynamic forces. The Minimum-State Method facilitates the design of lower-order control systems, analysis of control system performance, and near real-time simulation of aeroservoelastic phenomena such as the outboard-wing acceleration response to gust velocity. Engineers using this program will be able to calculate minimum-state rational approximations of the generalized unsteady aerodynamic forces. Using the Minimum-State formulation of the state-space equations, they will be able to obtain state-space models with good open-loop characteristics while reducing the number of aerodynamic equations by an order of magnitude more than traditional approaches. These low-order state-space mathematical models are good for design and simulation of aeroservoelastic systems. The computer program, MIST, accepts tabular values of the generalized aerodynamic forces over a set of reduced frequencies. It then determines approximations to these tabular data in the LaPlace domain using rational functions. MIST provides the capability to select the denominator coefficients in the rational approximations, to selectably constrain the approximations without increasing the problem size, and to determine and emphasize critical frequency ranges in determining the approximations. MIST has been written to allow two types data weighting options. The first weighting is a traditional normalization of the aerodynamic data to the maximum unit value of each aerodynamic coefficient. The second allows weighting the importance of different tabular values in determining the approximations based upon physical characteristics of the system. Specifically, the physical weighting capability is such that each tabulated aerodynamic coefficient, at each reduced frequency value, is weighted according to the effect of an incremental error of this coefficient on aeroelastic characteristics of the system. In both cases, the resulting approximations yield a relatively low number of aerodynamic lag states in the subsequent state-space model. MIST is written in ANSI FORTRAN 77 for DEC VAX series computers running VMS. It requires approximately 1Mb of RAM for execution. The standard distribution medium for this package is a 9-track 1600 BPI magnetic tape in DEC VAX FILES-11 format. It is also available on a TK50 tape cartridge in DEC VAX BACKUP format. MIST was developed in 1991. DEC VAX and VMS are trademarks of Digital Equipment Corporation. FORTRAN 77 is a registered trademark of Lahey Computer Systems, Inc.

Method of implementing digital phase-locked loops

NASA Technical Reports Server (NTRS)

Stephens, Scott A. (Inventor); Thomas, Jess Brooks, Jr. (Inventor)

1993-01-01

In a new formulation for digital phase-locked loops, loop-filter constants are determined from loop roots that can each be selectively placed in the s-plane on the basis of a new set of parameters, each with simple and direct physical meaning in terms of loop noise bandwidth, root-specific decay rate, or root-specific damping. Loops of first to fourth order are treated in the continuous-update approximation (BLT yields 0) and in a discrete-update formulation with arbitrary BLT. Deficiencies of the continuous-update approximation in large-BLT applications are avoided in the new discrete-update formulation. A new method for direct, transient-free acquisition with third- and fourth-order loops can improve the versatility and reliability of acquisition with such loops.

Computational methods of robust controller design for aerodynamic flutter suppression

NASA Technical Reports Server (NTRS)

Anderson, L. R.

1981-01-01

The development of Riccati iteration, a tool for the design and analysis of linear control systems is examined. First, Riccati iteration is applied to the problem of pole placement and order reduction in two-time scale control systems. Order reduction, yielding a good approximation to the original system, is demonstrated using a 16th order linear model of a turbofan engine. Next, a numerical method for solving the Riccati equation is presented and demonstrated for a set of eighth order random examples. A literature review of robust controller design methods follows which includes a number of methods for reducing the trajectory and performance index sensitivity in linear regulators. Lastly, robust controller design for large parameter variations is discussed.

A higher order numerical method for time fractional partial differential equations with nonsmooth data

NASA Astrophysics Data System (ADS)

Xing, Yanyuan; Yan, Yubin

2018-03-01

Gao et al. [11] (2014) introduced a numerical scheme to approximate the Caputo fractional derivative with the convergence rate O (k 3 - α), 0 < α < 1 by directly approximating the integer-order derivative with some finite difference quotients in the definition of the Caputo fractional derivative, see also Lv and Xu [20] (2016), where k is the time step size. Under the assumption that the solution of the time fractional partial differential equation is sufficiently smooth, Lv and Xu [20] (2016) proved by using energy method that the corresponding numerical method for solving time fractional partial differential equation has the convergence rate O (k 3 - α), 0 < α < 1 uniformly with respect to the time variable t. However, in general the solution of the time fractional partial differential equation has low regularity and in this case the numerical method fails to have the convergence rate O (k 3 - α), 0 < α < 1 uniformly with respect to the time variable t. In this paper, we first obtain a similar approximation scheme to the Riemann-Liouville fractional derivative with the convergence rate O (k 3 - α), 0 < α < 1 as in Gao et al. [11] (2014) by approximating the Hadamard finite-part integral with the piecewise quadratic interpolation polynomials. Based on this scheme, we introduce a time discretization scheme to approximate the time fractional partial differential equation and show by using Laplace transform methods that the time discretization scheme has the convergence rate O (k 3 - α), 0 < α < 1 for any fixed tn > 0 for smooth and nonsmooth data in both homogeneous and inhomogeneous cases. Numerical examples are given to show that the theoretical results are consistent with the numerical results.

Edge-augmented Fourier partial sums with applications to Magnetic Resonance Imaging (MRI)

NASA Astrophysics Data System (ADS)

Larriva-Latt, Jade; Morrison, Angela; Radgowski, Alison; Tobin, Joseph; Iwen, Mark; Viswanathan, Aditya

2017-08-01

Certain applications such as Magnetic Resonance Imaging (MRI) require the reconstruction of functions from Fourier spectral data. When the underlying functions are piecewise-smooth, standard Fourier approximation methods suffer from the Gibbs phenomenon - with associated oscillatory artifacts in the vicinity of edges and an overall reduced order of convergence in the approximation. This paper proposes an edge-augmented Fourier reconstruction procedure which uses only the first few Fourier coefficients of an underlying piecewise-smooth function to accurately estimate jump information and then incorporate it into a Fourier partial sum approximation. We provide both theoretical and empirical results showing the improved accuracy of the proposed method, as well as comparisons demonstrating superior performance over existing state-of-the-art sparse optimization-based methods.

High order filtering methods for approximating hyberbolic systems of conservation laws

NASA Technical Reports Server (NTRS)

Lafon, F.; Osher, S.

1990-01-01

In the computation of discontinuous solutions of hyperbolic systems of conservation laws, the recently developed essentially non-oscillatory (ENO) schemes appear to be very useful. However, they are computationally costly compared to simple central difference methods. A filtering method which is developed uses simple central differencing of arbitrarily high order accuracy, except when a novel local test indicates the development of spurious oscillations. At these points, the full ENO apparatus is used, maintaining the high order of accuracy, but removing spurious oscillations. Numerical results indicate the success of the method. High order of accuracy was obtained in regions of smooth flow without spurious oscillations for a wide range of problems and a significant speed up of generally a factor of almost three over the full ENO method.

A model of head-related transfer functions based on a state-space analysis

NASA Astrophysics Data System (ADS)

Adams, Norman Herkamp

This dissertation develops and validates a novel state-space method for binaural auditory display. Binaural displays seek to immerse a listener in a 3D virtual auditory scene with a pair of headphones. The challenge for any binaural display is to compute the two signals to supply to the headphones. The present work considers a general framework capable of synthesizing a wide variety of auditory scenes. The framework models collections of head-related transfer functions (HRTFs) simultaneously. This framework improves the flexibility of contemporary displays, but it also compounds the steep computational cost of the display. The cost is reduced dramatically by formulating the collection of HRTFs in the state-space and employing order-reduction techniques to design efficient approximants. Order-reduction techniques based on the Hankel-operator are found to yield accurate low-cost approximants. However, the inter-aural time difference (ITD) of the HRTFs degrades the time-domain response of the approximants. Fortunately, this problem can be circumvented by employing a state-space architecture that allows the ITD to be modeled outside of the state-space. Accordingly, three state-space architectures are considered. Overall, a multiple-input, single-output (MISO) architecture yields the best compromise between performance and flexibility. The state-space approximants are evaluated both empirically and psychoacoustically. An array of truncated FIR filters is used as a pragmatic reference system for comparison. For a fixed cost bound, the state-space systems yield lower approximation error than FIR arrays for D>10, where D is the number of directions in the HRTF collection. A series of headphone listening tests are also performed to validate the state-space approach, and to estimate the minimum order N of indiscriminable approximants. For D = 50, the state-space systems yield order thresholds less than half those of the FIR arrays. Depending upon the stimulus uncertainty, a minimum state-space order of 7≤N≤23 appears to be adequate. In conclusion, the proposed state-space method enables a more flexible and immersive binaural display with low computational cost.

A-posteriori error estimation for the finite point method with applications to compressible flow

NASA Astrophysics Data System (ADS)

Ortega, Enrique; Flores, Roberto; Oñate, Eugenio; Idelsohn, Sergio

2017-08-01

An a-posteriori error estimate with application to inviscid compressible flow problems is presented. The estimate is a surrogate measure of the discretization error, obtained from an approximation to the truncation terms of the governing equations. This approximation is calculated from the discrete nodal differential residuals using a reconstructed solution field on a modified stencil of points. Both the error estimation methodology and the flow solution scheme are implemented using the Finite Point Method, a meshless technique enabling higher-order approximations and reconstruction procedures on general unstructured discretizations. The performance of the proposed error indicator is studied and applications to adaptive grid refinement are presented.

Restoring the Pauli principle in the random phase approximation ground state

NASA Astrophysics Data System (ADS)

Kosov, D. S.

2017-12-01

Random phase approximation ground state contains electronic configurations where two (and more) identical electrons can occupy the same molecular spin-orbital violating the Pauli exclusion principle. This overcounting of electronic configurations happens due to quasiboson approximation in the treatment of electron-hole pair operators. We describe the method to restore the Pauli principle in the RPA wavefunction. The proposed theory is illustrated by the calculations of molecular dipole moments and electronic kinetic energies. The Hartree-Fock based RPA, which is corrected for the Pauli principle, gives the results of comparable accuracy with Møller-Plesset second order perturbation theory and coupled-cluster singles and doubles method.

Stabilizing potentials in bound state analytic continuation methods for electronic resonances in polyatomic molecules

DOE PAGES

White, Alec F.; Head-Gordon, Martin; McCurdy, C. William

2017-01-30

The computation of Siegert energies by analytic continuation of bound state energies has recently been applied to shape resonances in polyatomic molecules by several authors. Here, we critically evaluate a recently proposed analytic continuation method based on low order (type III) Padé approximants as well as an analytic continuation method based on high order (type II) Padé approximants. We compare three classes of stabilizing potentials: Coulomb potentials, Gaussian potentials, and attenuated Coulomb potentials. These methods are applied to a model potential where the correct answer is known exactly and to the 2Π g shape resonance of N 2 - whichmore » has been studied extensively by other methods. Both the choice of stabilizing potential and method of analytic continuation prove to be important to the accuracy of the results. We then conclude that an attenuated Coulomb potential is the most effective of the three for bound state analytic continuation methods. With the proper potential, such methods show promise for algorithmic determination of the positions and widths of molecular shape resonances.« less

Stabilizing potentials in bound state analytic continuation methods for electronic resonances in polyatomic molecules

DOE Office of Scientific and Technical Information (OSTI.GOV)

White, Alec F.; Head-Gordon, Martin; McCurdy, C. William

The computation of Siegert energies by analytic continuation of bound state energies has recently been applied to shape resonances in polyatomic molecules by several authors. Here, we critically evaluate a recently proposed analytic continuation method based on low order (type III) Padé approximants as well as an analytic continuation method based on high order (type II) Padé approximants. We compare three classes of stabilizing potentials: Coulomb potentials, Gaussian potentials, and attenuated Coulomb potentials. These methods are applied to a model potential where the correct answer is known exactly and to the 2Π g shape resonance of N 2 - whichmore » has been studied extensively by other methods. Both the choice of stabilizing potential and method of analytic continuation prove to be important to the accuracy of the results. We then conclude that an attenuated Coulomb potential is the most effective of the three for bound state analytic continuation methods. With the proper potential, such methods show promise for algorithmic determination of the positions and widths of molecular shape resonances.« less

Momentum-space cluster dual-fermion method

NASA Astrophysics Data System (ADS)

Iskakov, Sergei; Terletska, Hanna; Gull, Emanuel

2018-03-01

Recent years have seen the development of two types of nonlocal extensions to the single-site dynamical mean field theory. On one hand, cluster approximations, such as the dynamical cluster approximation, recover short-range momentum-dependent correlations nonperturbatively. On the other hand, diagrammatic extensions, such as the dual-fermion theory, recover long-ranged corrections perturbatively. The correct treatment of both strong short-ranged and weak long-ranged correlations within the same framework is therefore expected to lead to a quick convergence of results, and offers the potential of obtaining smooth self-energies in nonperturbative regimes of phase space. In this paper, we present an exact cluster dual-fermion method based on an expansion around the dynamical cluster approximation. Unlike previous formulations, our method does not employ a coarse-graining approximation to the interaction, which we show to be the leading source of error at high temperature, and converges to the exact result independently of the size of the underlying cluster. We illustrate the power of the method with results for the second-order cluster dual-fermion approximation to the single-particle self-energies and double occupancies.

Atomic approximation to the projection on electronic states in the Douglas-Kroll-Hess approach to the relativistic Kohn-Sham method.

PubMed

Matveev, Alexei V; Rösch, Notker

2008-06-28

We suggest an approximate relativistic model for economical all-electron calculations on molecular systems that exploits an atomic ansatz for the relativistic projection transformation. With such a choice, the projection transformation matrix is by definition both transferable and independent of the geometry. The formulation is flexible with regard to the level at which the projection transformation is approximated; we employ the free-particle Foldy-Wouthuysen and the second-order Douglas-Kroll-Hess variants. The (atomic) infinite-order decoupling scheme shows little effect on structural parameters in scalar-relativistic calculations; also, the use of a screened nuclear potential in the definition of the projection transformation shows hardly any effect in the context of the present work. Applications to structural and energetic parameters of various systems (diatomics AuH, AuCl, and Au(2), two structural isomers of Ir(4), and uranyl dication UO(2) (2+) solvated by 3-6 water ligands) show that the atomic approximation to the conventional second-order Douglas-Kroll-Hess projection (ADKH) transformation yields highly accurate results at substantial computational savings, in particular, when calculating energy derivatives of larger systems. The size-dependence of the intrinsic error of the ADKH method in extended systems of heavy elements is analyzed for the atomization energies of Pd(n) clusters (n

Tensor hypercontraction density fitting. I. Quartic scaling second- and third-order Møller-Plesset perturbation theory

NASA Astrophysics Data System (ADS)

Hohenstein, Edward G.; Parrish, Robert M.; Martínez, Todd J.

2012-07-01

Many approximations have been developed to help deal with the O(N4) growth of the electron repulsion integral (ERI) tensor, where N is the number of one-electron basis functions used to represent the electronic wavefunction. Of these, the density fitting (DF) approximation is currently the most widely used despite the fact that it is often incapable of altering the underlying scaling of computational effort with respect to molecular size. We present a method for exploiting sparsity in three-center overlap integrals through tensor decomposition to obtain a low-rank approximation to density fitting (tensor hypercontraction density fitting or THC-DF). This new approximation reduces the 4th-order ERI tensor to a product of five matrices, simultaneously reducing the storage requirement as well as increasing the flexibility to regroup terms and reduce scaling behavior. As an example, we demonstrate such a scaling reduction for second- and third-order perturbation theory (MP2 and MP3), showing that both can be carried out in O(N4) operations. This should be compared to the usual scaling behavior of O(N5) and O(N6) for MP2 and MP3, respectively. The THC-DF technique can also be applied to other methods in electronic structure theory, such as coupled-cluster and configuration interaction, promising significant gains in computational efficiency and storage reduction.

Approximate optimal guidance for the advanced launch system

NASA Technical Reports Server (NTRS)

Feeley, T. S.; Speyer, J. L.

1993-01-01

A real-time guidance scheme for the problem of maximizing the payload into orbit subject to the equations of motion for a rocket over a spherical, non-rotating earth is presented. An approximate optimal launch guidance law is developed based upon an asymptotic expansion of the Hamilton - Jacobi - Bellman or dynamic programming equation. The expansion is performed in terms of a small parameter, which is used to separate the dynamics of the problem into primary and perturbation dynamics. For the zeroth-order problem the small parameter is set to zero and a closed-form solution to the zeroth-order expansion term of Hamilton - Jacobi - Bellman equation is obtained. Higher-order terms of the expansion include the effects of the neglected perturbation dynamics. These higher-order terms are determined from the solution of first-order linear partial differential equations requiring only the evaluation of quadratures. This technique is preferred as a real-time, on-line guidance scheme to alternative numerical iterative optimization schemes because of the unreliable convergence properties of these iterative guidance schemes and because the quadratures needed for the approximate optimal guidance law can be performed rapidly and by parallel processing. Even if the approximate solution is not nearly optimal, when using this technique the zeroth-order solution always provides a path which satisfies the terminal constraints. Results for two-degree-of-freedom simulations are presented for the simplified problem of flight in the equatorial plane and compared to the guidance scheme generated by the shooting method which is an iterative second-order technique.

Arbitrary-level hanging nodes for adaptive hphp-FEM approximations in 3D

DOE Office of Scientific and Technical Information (OSTI.GOV)

Pavel Kus; Pavel Solin; David Andrs

2014-11-01

In this paper we discuss constrained approximation with arbitrary-level hanging nodes in adaptive higher-order finite element methods (hphp-FEM) for three-dimensional problems. This technique enables using highly irregular meshes, and it greatly simplifies the design of adaptive algorithms as it prevents refinements from propagating recursively through the finite element mesh. The technique makes it possible to design efficient adaptive algorithms for purely hexahedral meshes. We present a detailed mathematical description of the method and illustrate it with numerical examples.

General relaxation schemes in multigrid algorithms for higher order singularity methods

NASA Technical Reports Server (NTRS)

Oskam, B.; Fray, J. M. J.

1981-01-01

Relaxation schemes based on approximate and incomplete factorization technique (AF) are described. The AF schemes allow construction of a fast multigrid method for solving integral equations of the second and first kind. The smoothing factors for integral equations of the first kind, and comparison with similar results from the second kind of equations are a novel item. Application of the MD algorithm shows convergence to the level of truncation error of a second order accurate panel method.

Three-dimensional ordered particulate structures: Method to retrieve characteristics from photonic band gap data

NASA Astrophysics Data System (ADS)

Miskevich, Alexander A.; Loiko, Valery A.

2015-01-01

A method to retrieve characteristics of ordered particulate structures, such as photonic crystals, is proposed. It is based on the solution of the inverse problem using data on the photonic band gap (PBG). The quasicrystalline approximation (QCA) of the theory of multiple scattering of waves and the transfer matrix method (TMM) are used. Retrieval of the refractive index of particles is demonstrated. Refractive indices of the artificial opal particles are estimated using the published experimental data.

Travelling-wave solutions of a weakly nonlinear two-dimensional higher-order Kadomtsev-Petviashvili dynamical equation for dispersive shallow-water waves

NASA Astrophysics Data System (ADS)

Seadawy, Aly R.

2017-01-01

The propagation of three-dimensional nonlinear irrotational flow of an inviscid and incompressible fluid of the long waves in dispersive shallow-water approximation is analyzed. The problem formulation of the long waves in dispersive shallow-water approximation lead to fifth-order Kadomtsev-Petviashvili (KP) dynamical equation by applying the reductive perturbation theory. By using an extended auxiliary equation method, the solitary travelling-wave solutions of the two-dimensional nonlinear fifth-order KP dynamical equation are derived. An analytical as well as a numerical solution of the two-dimensional nonlinear KP equation are obtained and analyzed with the effects of external pressure flow.

The effect of compressive viscosity and thermal conduction on the longitudinal MHD waves

NASA Astrophysics Data System (ADS)

Bahari, K.; Shahhosaini, N.

2018-05-01

longitudinal Magnetohydrodynamic (MHD) oscillations have been studied in a slowly cooling coronal loop, in the presence of thermal conduction and compressive viscosity, in the linear MHD approximation. WKB method has been used to solve the governing equations. In the leading order approximation the dispersion relation has been obtained, and using the first order approximation the time dependent amplitude has been determined. Cooling causes the oscillations to amplify and damping mechanisms are more efficient in hot loops. In cool loops the oscillation amplitude increases with time but in hot loops the oscillation amplitude decreases with time. Our conclusion is that in hot loops the efficiency of the compressive viscosity in damping longitudinal waves is comparable to that of the thermal conduction.

The effect of compressive viscosity and thermal conduction on the longitudinal MHD waves

NASA Astrophysics Data System (ADS)

Bahari, K.; Shahhosaini, N.

2018-07-01

Longitudinal magnetohydrodynamic (MHD) oscillations have been studied in a slowly cooling coronal loop, in the presence of thermal conduction and compressive viscosity, in the linear MHD approximation. The WKB method has been used to solve the governing equations. In the leading order approximation the dispersion relation has been obtained, and using the first-order approximation the time-dependent amplitude has been determined. Cooling causes the oscillations to amplify and damping mechanisms are more efficient in hot loops. In cool loops the oscillation amplitude increases with time but in hot loops the oscillation amplitude decreases with time. Our conclusion is that in hot loops the efficiency of the compressive viscosity in damping longitudinal waves is comparable to that of the thermal conduction.

A model and variance reduction method for computing statistical outputs of stochastic elliptic partial differential equations

DOE Office of Scientific and Technical Information (OSTI.GOV)

Vidal-Codina, F., E-mail: fvidal@mit.edu; Nguyen, N.C., E-mail: cuongng@mit.edu; Giles, M.B., E-mail: mike.giles@maths.ox.ac.uk

We present a model and variance reduction method for the fast and reliable computation of statistical outputs of stochastic elliptic partial differential equations. Our method consists of three main ingredients: (1) the hybridizable discontinuous Galerkin (HDG) discretization of elliptic partial differential equations (PDEs), which allows us to obtain high-order accurate solutions of the governing PDE; (2) the reduced basis method for a new HDG discretization of the underlying PDE to enable real-time solution of the parameterized PDE in the presence of stochastic parameters; and (3) a multilevel variance reduction method that exploits the statistical correlation among the different reduced basismore » approximations and the high-fidelity HDG discretization to accelerate the convergence of the Monte Carlo simulations. The multilevel variance reduction method provides efficient computation of the statistical outputs by shifting most of the computational burden from the high-fidelity HDG approximation to the reduced basis approximations. Furthermore, we develop a posteriori error estimates for our approximations of the statistical outputs. Based on these error estimates, we propose an algorithm for optimally choosing both the dimensions of the reduced basis approximations and the sizes of Monte Carlo samples to achieve a given error tolerance. We provide numerical examples to demonstrate the performance of the proposed method.« less

Modeling and control of flexible structures

NASA Technical Reports Server (NTRS)

Gibson, J. S.; Mingori, D. L.

1988-01-01

This monograph presents integrated modeling and controller design methods for flexible structures. The controllers, or compensators, developed are optimal in the linear-quadratic-Gaussian sense. The performance objectives, sensor and actuator locations and external disturbances influence both the construction of the model and the design of the finite dimensional compensator. The modeling and controller design procedures are carried out in parallel to ensure compatibility of these two aspects of the design problem. Model reduction techniques are introduced to keep both the model order and the controller order as small as possible. A linear distributed, or infinite dimensional, model is the theoretical basis for most of the text, but finite dimensional models arising from both lumped-mass and finite element approximations also play an important role. A central purpose of the approach here is to approximate an optimal infinite dimensional controller with an implementable finite dimensional compensator. Both convergence theory and numerical approximation methods are given. Simple examples are used to illustrate the theory.

Computational Aspects of Sensitivity Calculations in Linear Transient Structural Analysis. Ph.D. Thesis

NASA Technical Reports Server (NTRS)

Greene, William H.

1989-01-01

A study has been performed focusing on the calculation of sensitivities of displacements, velocities, accelerations, and stresses in linear, structural, transient response problems. One significant goal was to develop and evaluate sensitivity calculation techniques suitable for large-order finite element analyses. Accordingly, approximation vectors such as vibration mode shapes are used to reduce the dimensionality of the finite element model. Much of the research focused on the accuracy of both response quantities and sensitivities as a function of number of vectors used. Two types of sensitivity calculation techniques were developed and evaluated. The first type of technique is an overall finite difference method where the analysis is repeated for perturbed designs. The second type of technique is termed semianalytical because it involves direct, analytical differentiation of the equations of motion with finite difference approximation of the coefficient matrices. To be computationally practical in large-order problems, the overall finite difference methods must use the approximation vectors from the original design in the analyses of the perturbed models.

Calibration Method to Eliminate Zeroth Order Effect in Lateral Shearing Interferometry

NASA Astrophysics Data System (ADS)

Fang, Chao; Xiang, Yang; Qi, Keqi; Chen, Dawei

2018-04-01

In this paper, a calibration method is proposed which eliminates the zeroth order effect in lateral shearing interferometry. An analytical expression of the calibration error function is deduced, and the relationship between the phase-restoration error and calibration error is established. The analytical results show that the phase-restoration error introduced by the calibration error is proportional to the phase shifting error and zeroth order effect. The calibration method is verified using simulations and experiments. The simulation results show that the phase-restoration error is approximately proportional to the phase shift error and zeroth order effect, when the phase shifting error is less than 2° and the zeroth order effect is less than 0.2. The experimental result shows that compared with the conventional method with 9-frame interferograms, the calibration method with 5-frame interferograms achieves nearly the same restoration accuracy.

Spectral/ hp element methods: Recent developments, applications, and perspectives

NASA Astrophysics Data System (ADS)

Xu, Hui; Cantwell, Chris D.; Monteserin, Carlos; Eskilsson, Claes; Engsig-Karup, Allan P.; Sherwin, Spencer J.

2018-02-01

The spectral/ hp element method combines the geometric flexibility of the classical h-type finite element technique with the desirable numerical properties of spectral methods, employing high-degree piecewise polynomial basis functions on coarse finite element-type meshes. The spatial approximation is based upon orthogonal polynomials, such as Legendre or Chebychev polynomials, modified to accommodate a C 0 - continuous expansion. Computationally and theoretically, by increasing the polynomial order p, high-precision solutions and fast convergence can be obtained and, in particular, under certain regularity assumptions an exponential reduction in approximation error between numerical and exact solutions can be achieved. This method has now been applied in many simulation studies of both fundamental and practical engineering flows. This paper briefly describes the formulation of the spectral/ hp element method and provides an overview of its application to computational fluid dynamics. In particular, it focuses on the use of the spectral/ hp element method in transitional flows and ocean engineering. Finally, some of the major challenges to be overcome in order to use the spectral/ hp element method in more complex science and engineering applications are discussed.

A high-order staggered meshless method for elliptic problems

DOE PAGES

Trask, Nathaniel; Perego, Mauro; Bochev, Pavel Blagoveston

2017-03-21

Here, we present a new meshless method for scalar diffusion equations, which is motivated by their compatible discretizations on primal-dual grids. Unlike the latter though, our approach is truly meshless because it only requires the graph of nearby neighbor connectivity of the discretization points. This graph defines a local primal-dual grid complex with a virtual dual grid, in the sense that specification of the dual metric attributes is implicit in the method's construction. Our method combines a topological gradient operator on the local primal grid with a generalized moving least squares approximation of the divergence on the local dual grid. We show that the resulting approximation of the div-grad operator maintains polynomial reproduction to arbitrary orders and yields a meshless method, which attainsmore » $$O(h^{m})$$ convergence in both $L^2$- and $H^1$-norms, similar to mixed finite element methods. We demonstrate this convergence on curvilinear domains using manufactured solutions in two and three dimensions. Application of the new method to problems with discontinuous coefficients reveals solutions that are qualitatively similar to those of compatible mesh-based discretizations.« less

A novel condition for stable nonlinear sampled-data models using higher-order discretized approximations with zero dynamics.

PubMed

Zeng, Cheng; Liang, Shan; Xiang, Shuwen

2017-05-01

Continuous-time systems are usually modelled by the form of ordinary differential equations arising from physical laws. However, the use of these models in practice and utilizing, analyzing or transmitting these data from such systems must first invariably be discretized. More importantly, for digital control of a continuous-time nonlinear system, a good sampled-data model is required. This paper investigates the new consistency condition which is weaker than the previous similar results presented. Moreover, given the stability of the high-order approximate model with stable zero dynamics, the novel condition presented stabilizes the exact sampled-data model of the nonlinear system for sufficiently small sampling periods. An insightful interpretation of the obtained results can be made in terms of the stable sampling zero dynamics, and the new consistency condition is surprisingly associated with the relative degree of the nonlinear continuous-time system. Our controller design, based on the higher-order approximate discretized model, extends the existing methods which mainly deal with the Euler approximation. Copyright © 2017 ISA. Published by Elsevier Ltd. All rights reserved.

Low temperature simulation of subliming boundary layer flow in Jupiter atmosphere

NASA Technical Reports Server (NTRS)

Chen, C. J.

1976-01-01

A low-temperature approximate simulation for the sublimation of a graphite heat shield under Jovian entry conditions is studied. A set of algebraic equations is derived to approximate the governing equation and boundary conditions, based on order-of-magnitude analysis. Characteristic quantities such as the wall temperature and the subliming velocity are predicted. Similarity parameters that are needed to simulate the most dominant phenomena of the Jovian entry flow are also given. An approximate simulation of the sublimation of the graphite heat shield is performed with an air-dry-ice model. The simulation with the air-dry-ice model may be carried out experimentally at a lower temperature of 3000 to 6000 K instead of the entry temperature of 14,000 K. The rate of graphite sublimation predicted by the present algebraic approximation agrees to the order of magnitude with extrapolated data. The limitations of the simulation method and its utility are discussed.

Iterative CT reconstruction using coordinate descent with ordered subsets of data

NASA Astrophysics Data System (ADS)

Noo, F.; Hahn, K.; Schöndube, H.; Stierstorfer, K.

2016-04-01

Image reconstruction based on iterative minimization of a penalized weighted least-square criteria has become an important topic of research in X-ray computed tomography. This topic is motivated by increasing evidence that such a formalism may enable a significant reduction in dose imparted to the patient while maintaining or improving image quality. One important issue associated with this iterative image reconstruction concept is slow convergence and the associated computational effort. For this reason, there is interest in finding methods that produce approximate versions of the targeted image with a small number of iterations and an acceptable level of discrepancy. We introduce here a novel method to produce such approximations: ordered subsets in combination with iterative coordinate descent. Preliminary results demonstrate that this method can produce, within 10 iterations and using only a constant image as initial condition, satisfactory reconstructions that retain the noise properties of the targeted image.

A third-order approximation method for three-dimensional wheel-rail contact

NASA Astrophysics Data System (ADS)

Negretti, Daniele

2012-03-01

Multibody train analysis is used increasingly by railway operators whenever a reliable and time-efficient method to evaluate the contact between wheel and rail is needed; particularly, the wheel-rail contact is one of the most important aspects that affects a reliable and time-efficient vehicle dynamics computation. The focus of the approach proposed here is to carry out such tasks by means of online wheel-rail elastic contact detection. In order to improve efficiency and save time, a main analytical approach is used for the definition of wheel and rail surfaces as well as for contact detection, then a final numerical evaluation is used to locate contact. The final numerical procedure consists in finding the zeros of a nonlinear function in a single variable. The overall method is based on the approximation of the wheel surface, which does not influence the contact location significantly, as shown in the paper.

Computation of turbulent pipe and duct flow using third order upwind scheme

NASA Technical Reports Server (NTRS)

Kawamura, T.

1986-01-01

The fully developed turbulence in a circular pipe and in a square duct is simulated directly without using turbulence models in the Navier-Stokes equations. The utilized method employs a third-order upwind scheme for the approximation to the nonlinear term and the second-order Adams-Bashforth method for the time derivative in the Navier-Stokes equation. The computational results appear to capture the large-scale turbulent structures at least qualitatively. The significance of the artificial viscosity inherent in the present scheme is discussed.

Biochemical simulations: stochastic, approximate stochastic and hybrid approaches.

PubMed

Pahle, Jürgen

2009-01-01

Computer simulations have become an invaluable tool to study the sometimes counterintuitive temporal dynamics of (bio-)chemical systems. In particular, stochastic simulation methods have attracted increasing interest recently. In contrast to the well-known deterministic approach based on ordinary differential equations, they can capture effects that occur due to the underlying discreteness of the systems and random fluctuations in molecular numbers. Numerous stochastic, approximate stochastic and hybrid simulation methods have been proposed in the literature. In this article, they are systematically reviewed in order to guide the researcher and help her find the appropriate method for a specific problem.

Biochemical simulations: stochastic, approximate stochastic and hybrid approaches

PubMed Central

2009-01-01

Computer simulations have become an invaluable tool to study the sometimes counterintuitive temporal dynamics of (bio-)chemical systems. In particular, stochastic simulation methods have attracted increasing interest recently. In contrast to the well-known deterministic approach based on ordinary differential equations, they can capture effects that occur due to the underlying discreteness of the systems and random fluctuations in molecular numbers. Numerous stochastic, approximate stochastic and hybrid simulation methods have been proposed in the literature. In this article, they are systematically reviewed in order to guide the researcher and help her find the appropriate method for a specific problem. PMID:19151097

Couple of the Variational Iteration Method and Fractional-Order Legendre Functions Method for Fractional Differential Equations

PubMed Central

Song, Junqiang; Leng, Hongze; Lu, Fengshun

2014-01-01

We present a new numerical method to get the approximate solutions of fractional differential equations. A new operational matrix of integration for fractional-order Legendre functions (FLFs) is first derived. Then a modified variational iteration formula which can avoid “noise terms” is constructed. Finally a numerical method based on variational iteration method (VIM) and FLFs is developed for fractional differential equations (FDEs). Block-pulse functions (BPFs) are used to calculate the FLFs coefficient matrices of the nonlinear terms. Five examples are discussed to demonstrate the validity and applicability of the technique. PMID:24511303

Accurate upwind methods for the Euler equations

NASA Technical Reports Server (NTRS)

Huynh, Hung T.

1993-01-01

A new class of piecewise linear methods for the numerical solution of the one-dimensional Euler equations of gas dynamics is presented. These methods are uniformly second-order accurate, and can be considered as extensions of Godunov's scheme. With an appropriate definition of monotonicity preservation for the case of linear convection, it can be shown that they preserve monotonicity. Similar to Van Leer's MUSCL scheme, they consist of two key steps: a reconstruction step followed by an upwind step. For the reconstruction step, a monotonicity constraint that preserves uniform second-order accuracy is introduced. Computational efficiency is enhanced by devising a criterion that detects the 'smooth' part of the data where the constraint is redundant. The concept and coding of the constraint are simplified by the use of the median function. A slope steepening technique, which has no effect at smooth regions and can resolve a contact discontinuity in four cells, is described. As for the upwind step, existing and new methods are applied in a manner slightly different from those in the literature. These methods are derived by approximating the Euler equations via linearization and diagonalization. At a 'smooth' interface, Harten, Lax, and Van Leer's one intermediate state model is employed. A modification for this model that can resolve contact discontinuities is presented. Near a discontinuity, either this modified model or a more accurate one, namely, Roe's flux-difference splitting. is used. The current presentation of Roe's method, via the conceptually simple flux-vector splitting, not only establishes a connection between the two splittings, but also leads to an admissibility correction with no conditional statement, and an efficient approximation to Osher's approximate Riemann solver. These reconstruction and upwind steps result in schemes that are uniformly second-order accurate and economical at smooth regions, and yield high resolution at discontinuities.

Construction of exponentially fitted symplectic Runge-Kutta-Nyström methods from partitioned Runge-Kutta methods

NASA Astrophysics Data System (ADS)

Monovasilis, Theodore; Kalogiratou, Zacharoula; Simos, T. E.

2014-10-01

In this work we derive exponentially fitted symplectic Runge-Kutta-Nyström (RKN) methods from symplectic exponentially fitted partitioned Runge-Kutta (PRK) methods methods (for the approximate solution of general problems of this category see [18] - [40] and references therein). We construct RKN methods from PRK methods with up to five stages and fourth algebraic order.

Examining the accuracy of the infinite order sudden approximation using sensitivity analysis

NASA Astrophysics Data System (ADS)

Eno, Larry; Rabitz, Herschel

1981-08-01

A method is developed for assessing the accuracy of scattering observables calculated within the framework of the infinite order sudden (IOS) approximation. In particular, we focus on the energy sudden assumption of the IOS method and our approach involves the determination of the sensitivity of the IOS scattering matrix SIOS with respect to a parameter which reintroduces the internal energy operator ?0 into the IOS Hamiltonian. This procedure is an example of sensitivity analysis of missing model components (?0 in this case) in the reference Hamiltonian. In contrast to simple first-order perturbation theory a finite result is obtained for the effect of ?0 on SIOS. As an illustration, our method of analysis is applied to integral state-to-state cross sections for the scattering of an atom and rigid rotor. Results are generated within the He+H2 system and a comparison is made between IOS and coupled states cross sections and the corresponding IOS sensitivities. It is found that the sensitivity coefficients are very useful indicators of the accuracy of the IOS results. Finally, further developments and applications are discussed.

B and F Projection Methods for Nearly Incompressible Linear and Nonlinear Elasticity and Plasticity using Higher-order NURBS Elements

DTIC Science & Technology

2007-08-01

Infinite plate with a hole: sequence of meshes produced by h-refinement. The geometry of the coarsest mesh...recalled with an emphasis on k -refinement. In Section 3, the use of high-order NURBS within a projection technique is studied in the geometri - cally linear...case with a B̄ method to investigate the choice of approximation and projection spaces with NURBS.

Limitations of the method of complex basis functions

DOE Office of Scientific and Technical Information (OSTI.GOV)

Baumel, R.T.; Crocker, M.C.; Nuttall, J.

1975-08-01

The method of complex basis functions proposed by Rescigno and Reinhardt is applied to the calculation of the amplitude in a model problem which can be treated analytically. It is found for an important class of potentials, including some of infinite range and also the square well, that the method does not provide a converging sequence of approximations. However, in some cases, approximations of relatively low order might be close to the correct result. The method is also applied to S-wave e-H elastic scattering above the ionization threshold, and spurious ''convergence'' to the wrong result is found. A procedure whichmore » might overcome the difficulties of the method is proposed.« less

Local CC2 response method based on the Laplace transform: analytic energy gradients for ground and excited states.

PubMed

Ledermüller, Katrin; Schütz, Martin

2014-04-28

A multistate local CC2 response method for the calculation of analytic energy gradients with respect to nuclear displacements is presented for ground and electronically excited states. The gradient enables the search for equilibrium geometries of extended molecular systems. Laplace transform is used to partition the eigenvalue problem in order to obtain an effective singles eigenvalue problem and adaptive, state-specific local approximations. This leads to an approximation in the energy Lagrangian, which however is shown (by comparison with the corresponding gradient method without Laplace transform) to be of no concern for geometry optimizations. The accuracy of the local approximation is tested and the efficiency of the new code is demonstrated by application calculations devoted to a photocatalytic decarboxylation process of present interest.

Second derivatives for approximate spin projection methods

DOE Office of Scientific and Technical Information (OSTI.GOV)

Thompson, Lee M.; Hratchian, Hrant P., E-mail: hhratchian@ucmerced.edu

2015-02-07

The use of broken-symmetry electronic structure methods is required in order to obtain correct behavior of electronically strained open-shell systems, such as transition states, biradicals, and transition metals. This approach often has issues with spin contamination, which can lead to significant errors in predicted energies, geometries, and properties. Approximate projection schemes are able to correct for spin contamination and can often yield improved results. To fully make use of these methods and to carry out exploration of the potential energy surface, it is desirable to develop an efficient second energy derivative theory. In this paper, we formulate the analytical secondmore » derivatives for the Yamaguchi approximate projection scheme, building on recent work that has yielded an efficient implementation of the analytical first derivatives.« less

ADM For Solving Linear Second-Order Fredholm Integro-Differential Equations

NASA Astrophysics Data System (ADS)

Karim, Mohd F.; Mohamad, Mahathir; Saifullah Rusiman, Mohd; Che-Him, Norziha; Roslan, Rozaini; Khalid, Kamil

2018-04-01

In this paper, we apply Adomian Decomposition Method (ADM) as numerically analyse linear second-order Fredholm Integro-differential Equations. The approximate solutions of the problems are calculated by Maple package. Some numerical examples have been considered to illustrate the ADM for solving this equation. The results are compared with the existing exact solution. Thus, the Adomian decomposition method can be the best alternative method for solving linear second-order Fredholm Integro-Differential equation. It converges to the exact solution quickly and in the same time reduces computational work for solving the equation. The result obtained by ADM shows the ability and efficiency for solving these equations.

Uniform high order spectral methods for one and two dimensional Euler equations

NASA Technical Reports Server (NTRS)

Cai, Wei; Shu, Chi-Wang

1991-01-01

Uniform high order spectral methods to solve multi-dimensional Euler equations for gas dynamics are discussed. Uniform high order spectral approximations with spectral accuracy in smooth regions of solutions are constructed by introducing the idea of the Essentially Non-Oscillatory (ENO) polynomial interpolations into the spectral methods. The authors present numerical results for the inviscid Burgers' equation, and for the one dimensional Euler equations including the interactions between a shock wave and density disturbance, Sod's and Lax's shock tube problems, and the blast wave problem. The interaction between a Mach 3 two dimensional shock wave and a rotating vortex is simulated.

Upwind methods for the Baer–Nunziato equations and higher-order reconstruction using artificial viscosity

DOE Office of Scientific and Technical Information (OSTI.GOV)

Fraysse, F., E-mail: francois.fraysse@rs2n.eu; E. T. S. de Ingeniería Aeronáutica y del Espacio, Universidad Politécnica de Madrid, Madrid; Redondo, C.

This article is devoted to the numerical discretisation of the hyperbolic two-phase flow model of Baer and Nunziato. A special attention is paid on the discretisation of intercell flux functions in the framework of Finite Volume and Discontinuous Galerkin approaches, where care has to be taken to efficiently approximate the non-conservative products inherent to the model equations. Various upwind approximate Riemann solvers have been tested on a bench of discontinuous test cases. New discretisation schemes are proposed in a Discontinuous Galerkin framework following the criterion of Abgrall and the path-conservative formalism. A stabilisation technique based on artificial viscosity is appliedmore » to the high-order Discontinuous Galerkin method and compared against classical TVD-MUSCL Finite Volume flux reconstruction.« less

Stochastic Optimal Prediction with Application to Averaged Euler Equations

DOE Office of Scientific and Technical Information (OSTI.GOV)

Bell, John; Chorin, Alexandre J.; Crutchfield, William

Optimal prediction (OP) methods compensate for a lack of resolution in the numerical solution of complex problems through the use of an invariant measure as a prior measure in the Bayesian sense. In first-order OP, unresolved information is approximated by its conditional expectation with respect to the invariant measure. In higher-order OP, unresolved information is approximated by a stochastic estimator, leading to a system of random or stochastic differential equations. We explain the ideas through a simple example, and then apply them to the solution of Averaged Euler equations in two space dimensions.

Tangent Adjoint Methods In a Higher-Order Space-Time Discontinuous-Galerkin Solver For Turbulent Flows

NASA Technical Reports Server (NTRS)

Diosady, Laslo; Murman, Scott; Blonigan, Patrick; Garai, Anirban

2017-01-01

Presented space-time adjoint solver for turbulent compressible flows. Confirmed failure of traditional sensitivity methods for chaotic flows. Assessed rate of exponential growth of adjoint for practical 3D turbulent simulation. Demonstrated failure of short-window sensitivity approximations.

Error analysis of finite difference schemes applied to hyperbolic initial boundary value problems

NASA Technical Reports Server (NTRS)

Skollermo, G.

1979-01-01

Finite difference methods for the numerical solution of mixed initial boundary value problems for hyperbolic equations are studied. The reported investigation has the objective to develop a technique for the total error analysis of a finite difference scheme, taking into account initial approximations, boundary conditions, and interior approximation. Attention is given to the Cauchy problem and the initial approximation, the homogeneous problem in an infinite strip with inhomogeneous boundary data, the reflection of errors in the boundaries, and two different boundary approximations for the leapfrog scheme with a fourth order accurate difference operator in space.

Versions of the collocation and least squares method for solving biharmonic equations in non-canonical domains

NASA Astrophysics Data System (ADS)

Belyaev, V. A.; Shapeev, V. P.

2017-10-01

New versions of the collocations and least squares method of high-order accuracy are proposed and implemented for the numerical solution of the boundary value problems for the biharmonic equation in non-canonical domains. The solution of the biharmonic equation is used for simulating the stress-strain state of an isotropic plate under the action of transverse load. The differential problem is projected into a space of fourth-degree polynomials by the CLS method. The boundary conditions for the approximate solution are put down exactly on the boundary of the computational domain. The versions of the CLS method are implemented on the grids which are constructed in two different ways. It is shown that the approximate solution of problems converges with high order. Thus it matches with high accuracy with the analytical solution of the test problems in the case of known solution in the numerical experiments on the convergence of the solution of various problems on a sequence of grids.

Two-dimensional electronic spectra from the hierarchical equations of motion method: Application to model dimers

NASA Astrophysics Data System (ADS)

Chen, Liping; Zheng, Renhui; Shi, Qiang; Yan, YiJing

2010-01-01

We extend our previous study of absorption line shapes of molecular aggregates using the Liouville space hierarchical equations of motion (HEOM) method [L. P. Chen, R. H. Zheng, Q. Shi, and Y. J. Yan, J. Chem. Phys. 131, 094502 (2009)] to calculate third order optical response functions and two-dimensional electronic spectra of model dimers. As in our previous work, we have focused on the applicability of several approximate methods related to the HEOM method. We show that while the second order perturbative quantum master equations are generally inaccurate in describing the peak shapes and solvation dynamics, they can give reasonable peak amplitude evolution even in the intermediate coupling regime. The stochastic Liouville equation results in good peak shapes, but does not properly describe the excited state dynamics due to the lack of detailed balance. A modified version of the high temperature approximation to the HEOM gives the best agreement with the exact result.

First-Order System Least-Squares for Second-Order Elliptic Problems with Discontinuous Coefficients

NASA Technical Reports Server (NTRS)

Manteuffel, Thomas A.; McCormick, Stephen F.; Starke, Gerhard

1996-01-01

The first-order system least-squares methodology represents an alternative to standard mixed finite element methods. Among its advantages is the fact that the finite element spaces approximating the pressure and flux variables are not restricted by the inf-sup condition and that the least-squares functional itself serves as an appropriate error measure. This paper studies the first-order system least-squares approach for scalar second-order elliptic boundary value problems with discontinuous coefficients. Ellipticity of an appropriately scaled least-squares bilinear form of the size of the jumps in the coefficients leading to adequate finite element approximation results. The occurrence of singularities at interface corners and cross-points is discussed. and a weighted least-squares functional is introduced to handle such cases. Numerical experiments are presented for two test problems to illustrate the performance of this approach.

Determination of intermediate perturbed orbits of Near-Earth asteroids from range and range rate measurements at three times

NASA Astrophysics Data System (ADS)

Shefer, V. A.

2014-12-01

Two methods that the author developed earlier for finding the intermediate perturbed orbit of a small celestial body from three pairs of range and range rate observations [1, 2] are applied to the determination of orbits of Near-Earth asteroids. The methods are based on using the superosculating orbits with three- and fourth-order tangency. The degrees of approximation of the real motion by the constructed intermediate orbits near the middle measurement time are two and three orders of magnitude higher than by the Keplerian orbit determined with the help of traditional methods. We calculated the orbits of the asteroids 99942 Apophis, 1566 Icarus, 4179 Toutatis, 2007 DN41 and 2012 DA14. For the sake of brevity, we call the method based on the orbit with third-order tangency as Algorithm A1 and the method based on the orbit with fourth-order tangency -- as Algorithm A2. The results of the calculations are compared with the results of the calculations by the version of the methods mentioned that allows us to construct the unperturbed Keplerian orbit. We call this version of the methods as Algorithm A. The observational data were simulated using the nominal trajectories of the selected asteroids. These trajectories were obtained by the numerical integration of the differential equations of motion subject to the perturbations from the eight major planets, Pluto, and the Moon. The integration was carried out with the help of the 15-order Everhart procedure [3]. The main results of the calculations are the following. When the reference time interval is shortened by half (for small sizes of this interval), the errors in the compared algorithms A, A1, A2 decrease approximately by the factors 4, 16, 64 in coordinates and by the factors 2, 8, 16 in velocities, respectively. Such behavior of the errors is most clearly seen with the asteroids 2007 DN41 and 2012 DA14. This leads to a significant increase in the accuracy of the real motion approximation by the intermediate orbits constructed using the A1 and A2 algorithms (2-4 orders of magnitude in coordinates and 4-7 orders of magnitude in velocities higher) compared to the accuracy of the approximation by Keplerian orbits with decreasing the reference arc of the trajectory. Here, the higher is the efficiency of the algorithms A1 and A2, the smaller are the values of the topocentric distances, i.e., the greater are the perturbations caused by the Earth's gravitation. The advantage of Algorithm A2 over Algorithm A1 in accuracy extends approximately one order of magnitude. The minimal methodic errors of the position vector by using the A1 and A2 algorithms range from several meters in the case of the asteroid Apophis to several millimeters in the case of the asteroid 2012 DA14. Hence, the numerical examples analyzed in this work lead us to conclude that the proposed in [1, 2] methods for determination of an intermediate perturbed orbit from range and range rate measurements at three time points allow for significantly raising the accuracy of the calculation of the initial asteroid orbits in comparison with the algorithm based on the finding the unperturbed Keplerian orbit. The shorter is the orbital arc specified by the extreme time points, the greater is the advantage of the algorithms suggested over the algorithms of the traditional approach in the accuracy. The advantage of the algorithms suggested in the accuracy increases with raising the perturbations too, which is especially important for calculation of the initial trajectories of the space objects detected in the Earth's neighbourhood. The work was supported by the Ministry of Education and Science of the Russian Federation, project no. 2014/223(1567).

Computational aspects of sensitivity calculations in linear transient structural analysis. Ph.D. Thesis - Virginia Polytechnic Inst. and State Univ.

NASA Technical Reports Server (NTRS)

Greene, William H.

1990-01-01

A study was performed focusing on the calculation of sensitivities of displacements, velocities, accelerations, and stresses in linear, structural, transient response problems. One significant goal of the study was to develop and evaluate sensitivity calculation techniques suitable for large-order finite element analyses. Accordingly, approximation vectors such as vibration mode shapes are used to reduce the dimensionality of the finite element model. Much of the research focused on the accuracy of both response quantities and sensitivities as a function of number of vectors used. Two types of sensitivity calculation techniques were developed and evaluated. The first type of technique is an overall finite difference method where the analysis is repeated for perturbed designs. The second type of technique is termed semi-analytical because it involves direct, analytical differentiation of the equations of motion with finite difference approximation of the coefficient matrices. To be computationally practical in large-order problems, the overall finite difference methods must use the approximation vectors from the original design in the analyses of the perturbed models. In several cases this fixed mode approach resulted in very poor approximations of the stress sensitivities. Almost all of the original modes were required for an accurate sensitivity and for small numbers of modes, the accuracy was extremely poor. To overcome this poor accuracy, two semi-analytical techniques were developed. The first technique accounts for the change in eigenvectors through approximate eigenvector derivatives. The second technique applies the mode acceleration method of transient analysis to the sensitivity calculations. Both result in accurate values of the stress sensitivities with a small number of modes and much lower computational costs than if the vibration modes were recalculated and then used in an overall finite difference method.

Methods for High-Order Multi-Scale and Stochastic Problems Analysis, Algorithms, and Applications

DTIC Science & Technology

2016-10-17

finite volume schemes, discontinuous Galerkin finite element method, and related methods, for solving computational fluid dynamics (CFD) problems and...approximation for finite element methods. (3) The development of methods of simulation and analysis for the study of large scale stochastic systems of...laws, finite element method, Bernstein-Bezier finite elements , weakly interacting particle systems, accelerated Monte Carlo, stochastic networks 16

Radiative corrections to quantum sticking on graphene

NASA Astrophysics Data System (ADS)

Sengupta, Sanghita; Clougherty, Dennis P.

2017-07-01

We study the sticking rate of atomic hydrogen to suspended graphene using four different methods that include contributions from processes with multiphonon emission. We compare the numerical results of the sticking rate obtained by: (i) the loop expansion of the atom self-energy; (ii) the noncrossing approximation (NCA); (iii) the independent boson model approximation (IBMA); and (iv) a leading-order soft-phonon resummation method (SPR). The loop expansion reveals an infrared problem, analogous to the infamous infrared problem in QED. The two-loop contribution to the sticking rate gives a result that tends to diverge for large membranes. The latter three methods remedy this infrared problem and give results that are finite in the limit of an infinite membrane. We find that for micromembranes (sizes ranging 100 nm to 10 μ m ), the latter three methods give results that are in good agreement with each other and yield sticking rates that are mildly suppressed relative to the lowest-order golden rule rate. Lastly, we find that the SPR sticking rate decreases slowly to zero with increasing membrane size, while both the NCA and IBMA rates tend to a nonzero constant in this limit. Thus, approximations to the sticking rate can be sensitive to the effects of soft-phonon emission for large membranes.

A Variational Nodal Approach to 2D/1D Pin Resolved Neutron Transport for Pressurized Water Reactors

DOE PAGES

Zhang, Tengfei; Lewis, E. E.; Smith, M. A.; ...

2017-04-18

A two-dimensional/one-dimensional (2D/1D) variational nodal approach is presented for pressurized water reactor core calculations without fuel-moderator homogenization. A 2D/1D approximation to the within-group neutron transport equation is derived and converted to an even-parity form. The corresponding nodal functional is presented and discretized to obtain response matrix equations. Within the nodes, finite elements in the x-y plane and orthogonal functions in z are used to approximate the spatial flux distribution. On the radial interfaces, orthogonal polynomials are employed; on the axial interfaces, piecewise constants corresponding to the finite elements eliminate the interface homogenization that has been a challenge for method ofmore » characteristics (MOC)-based 2D/1D approximations. The angular discretization utilizes an even-parity integral method within the nodes, and low-order spherical harmonics (P N) on the axial interfaces. The x-y surfaces are treated with high-order P N combined with quasi-reflected interface conditions. Furthermore, the method is applied to the C5G7 benchmark problems and compared to Monte Carlo reference calculations.« less

A Variational Nodal Approach to 2D/1D Pin Resolved Neutron Transport for Pressurized Water Reactors

DOE Office of Scientific and Technical Information (OSTI.GOV)

Zhang, Tengfei; Lewis, E. E.; Smith, M. A.

A two-dimensional/one-dimensional (2D/1D) variational nodal approach is presented for pressurized water reactor core calculations without fuel-moderator homogenization. A 2D/1D approximation to the within-group neutron transport equation is derived and converted to an even-parity form. The corresponding nodal functional is presented and discretized to obtain response matrix equations. Within the nodes, finite elements in the x-y plane and orthogonal functions in z are used to approximate the spatial flux distribution. On the radial interfaces, orthogonal polynomials are employed; on the axial interfaces, piecewise constants corresponding to the finite elements eliminate the interface homogenization that has been a challenge for method ofmore » characteristics (MOC)-based 2D/1D approximations. The angular discretization utilizes an even-parity integral method within the nodes, and low-order spherical harmonics (P N) on the axial interfaces. The x-y surfaces are treated with high-order P N combined with quasi-reflected interface conditions. Furthermore, the method is applied to the C5G7 benchmark problems and compared to Monte Carlo reference calculations.« less

A High-Order Immersed Boundary Method for Acoustic Wave Scattering and Low-Mach Number Flow-Induced Sound in Complex Geometries

PubMed Central

Seo, Jung Hee; Mittal, Rajat

2010-01-01

A new sharp-interface immersed boundary method based approach for the computation of low-Mach number flow-induced sound around complex geometries is described. The underlying approach is based on a hydrodynamic/acoustic splitting technique where the incompressible flow is first computed using a second-order accurate immersed boundary solver. This is followed by the computation of sound using the linearized perturbed compressible equations (LPCE). The primary contribution of the current work is the development of a versatile, high-order accurate immersed boundary method for solving the LPCE in complex domains. This new method applies the boundary condition on the immersed boundary to a high-order by combining the ghost-cell approach with a weighted least-squares error method based on a high-order approximating polynomial. The method is validated for canonical acoustic wave scattering and flow-induced noise problems. Applications of this technique to relatively complex cases of practical interest are also presented. PMID:21318129

Positivity-preserving cell-centered Lagrangian schemes for multi-material compressible flows: From first-order to high-orders. Part I: The one-dimensional case

NASA Astrophysics Data System (ADS)

Vilar, François; Shu, Chi-Wang; Maire, Pierre-Henri

2016-05-01

One of the main issues in the field of numerical schemes is to ally robustness with accuracy. Considering gas dynamics, numerical approximations may generate negative density or pressure, which may lead to nonlinear instability and crash of the code. This phenomenon is even more critical using a Lagrangian formalism, the grid moving and being deformed during the calculation. Furthermore, most of the problems studied in this framework contain very intense rarefaction and shock waves. In this paper, the admissibility of numerical solutions obtained by high-order finite-volume-scheme-based methods, such as the discontinuous Galerkin (DG) method, the essentially non-oscillatory (ENO) and the weighted ENO (WENO) finite volume schemes, is addressed in the one-dimensional Lagrangian gas dynamics framework. After briefly recalling how to derive Lagrangian forms of the 1D gas dynamics system of equations, a discussion on positivity-preserving approximate Riemann solvers, ensuring first-order finite volume schemes to be positive, is then given. This study is conducted for both ideal gas and non-ideal gas equations of state (EOS), such as the Jones-Wilkins-Lee (JWL) EOS or the Mie-Grüneisen (MG) EOS, and relies on two different techniques: either a particular definition of the local approximation of the acoustic impedances arising from the approximate Riemann solver, or an additional time step constraint relative to the cell volume variation. Then, making use of the work presented in [89,90,22], this positivity study is extended to high-orders of accuracy, where new time step constraints are obtained, and proper limitation is required. Through this new procedure, scheme robustness is highly improved and hence new problems can be tackled. Numerical results are provided to demonstrate the effectiveness of these methods. This paper is the first part of a series of two. The whole analysis presented here is extended to the two-dimensional case in [85], and proves to fit a wide range of numerical schemes in the literature, such as those presented in [19,64,15,82,84].

A three-dimensional parabolic equation model of sound propagation using higher-order operator splitting and Padé approximants.

PubMed

Lin, Ying-Tsong; Collis, Jon M; Duda, Timothy F

2012-11-01

An alternating direction implicit (ADI) three-dimensional fluid parabolic equation solution method with enhanced accuracy is presented. The method uses a square-root Helmholtz operator splitting algorithm that retains cross-multiplied operator terms that have been previously neglected. With these higher-order cross terms, the valid angular range of the parabolic equation solution is improved. The method is tested for accuracy against an image solution in an idealized wedge problem. Computational efficiency improvements resulting from the ADI discretization are also discussed.

Quantum Enhanced Inference in Markov Logic Networks

NASA Astrophysics Data System (ADS)

Wittek, Peter; Gogolin, Christian

2017-04-01

Markov logic networks (MLNs) reconcile two opposing schools in machine learning and artificial intelligence: causal networks, which account for uncertainty extremely well, and first-order logic, which allows for formal deduction. An MLN is essentially a first-order logic template to generate Markov networks. Inference in MLNs is probabilistic and it is often performed by approximate methods such as Markov chain Monte Carlo (MCMC) Gibbs sampling. An MLN has many regular, symmetric structures that can be exploited at both first-order level and in the generated Markov network. We analyze the graph structures that are produced by various lifting methods and investigate the extent to which quantum protocols can be used to speed up Gibbs sampling with state preparation and measurement schemes. We review different such approaches, discuss their advantages, theoretical limitations, and their appeal to implementations. We find that a straightforward application of a recent result yields exponential speedup compared to classical heuristics in approximate probabilistic inference, thereby demonstrating another example where advanced quantum resources can potentially prove useful in machine learning.

Application of an Extended Parabolic Equation to the Calculation of the Mean Field and the Transverse and Longitudinal Mutual Coherence Functions Within Atmospheric Turbulence

NASA Technical Reports Server (NTRS)

Manning, Robert M.

2005-01-01

Solutions are derived for the generalized mutual coherence function (MCF), i.e., the second order moment, of a random wave field propagating through a random medium within the context of the extended parabolic equation. Here, "generalized" connotes the consideration of both the transverse as well as the longitudinal second order moments (with respect to the direction of propagation). Such solutions will afford a comparison between the results of the parabolic equation within the pararaxial approximation and those of the wide-angle extended theory. To this end, a statistical operator method is developed which gives a general equation for an arbitrary spatial statistical moment of the wave field. The generality of the operator method allows one to obtain an expression for the second order field moment in the direction longitudinal to the direction of propagation. Analytical solutions to these equations are derived for the Kolmogorov and Tatarskii spectra of atmospheric permittivity fluctuations within the Markov approximation.

From the Boltzmann to the Lattice-Boltzmann Equation:. Beyond BGK Collision Models

NASA Astrophysics Data System (ADS)

Philippi, Paulo Cesar; Hegele, Luiz Adolfo; Surmas, Rodrigo; Siebert, Diogo Nardelli; Dos Santos, Luís Orlando Emerich

In this work, we present a derivation for the lattice-Boltzmann equation directly from the linearized Boltzmann equation, combining the following main features: multiple relaxation times and thermodynamic consistency in the description of non isothermal compressible flows. The method presented here is based on the discretization of increasingly order kinetic models of the Boltzmann equation. Following a Gross-Jackson procedure, the linearized collision term is developed in Hermite polynomial tensors and the resulting infinite series is diagonalized after a chosen integer N, establishing the order of approximation of the collision term. The velocity space is discretized, in accordance with a quadrature method based on prescribed abscissas (Philippi et al., Phys. Rev E 73, 056702, 2006). The problem of describing the energy transfer is discussed, in relation with the order of approximation of a two relaxation-times lattice Boltzmann model. The velocity-step, temperature-step and the shock tube problems are investigated, adopting lattices with 37, 53 and 81 velocities.

Quantum Enhanced Inference in Markov Logic Networks.

PubMed

Wittek, Peter; Gogolin, Christian

2017-04-19

Markov logic networks (MLNs) reconcile two opposing schools in machine learning and artificial intelligence: causal networks, which account for uncertainty extremely well, and first-order logic, which allows for formal deduction. An MLN is essentially a first-order logic template to generate Markov networks. Inference in MLNs is probabilistic and it is often performed by approximate methods such as Markov chain Monte Carlo (MCMC) Gibbs sampling. An MLN has many regular, symmetric structures that can be exploited at both first-order level and in the generated Markov network. We analyze the graph structures that are produced by various lifting methods and investigate the extent to which quantum protocols can be used to speed up Gibbs sampling with state preparation and measurement schemes. We review different such approaches, discuss their advantages, theoretical limitations, and their appeal to implementations. We find that a straightforward application of a recent result yields exponential speedup compared to classical heuristics in approximate probabilistic inference, thereby demonstrating another example where advanced quantum resources can potentially prove useful in machine learning.

Quantum Enhanced Inference in Markov Logic Networks

PubMed Central

Wittek, Peter; Gogolin, Christian

2017-01-01

Markov logic networks (MLNs) reconcile two opposing schools in machine learning and artificial intelligence: causal networks, which account for uncertainty extremely well, and first-order logic, which allows for formal deduction. An MLN is essentially a first-order logic template to generate Markov networks. Inference in MLNs is probabilistic and it is often performed by approximate methods such as Markov chain Monte Carlo (MCMC) Gibbs sampling. An MLN has many regular, symmetric structures that can be exploited at both first-order level and in the generated Markov network. We analyze the graph structures that are produced by various lifting methods and investigate the extent to which quantum protocols can be used to speed up Gibbs sampling with state preparation and measurement schemes. We review different such approaches, discuss their advantages, theoretical limitations, and their appeal to implementations. We find that a straightforward application of a recent result yields exponential speedup compared to classical heuristics in approximate probabilistic inference, thereby demonstrating another example where advanced quantum resources can potentially prove useful in machine learning. PMID:28422093

Dense grid sibling frames with linear phase filters

NASA Astrophysics Data System (ADS)

Abdelnour, Farras

2013-09-01

We introduce new 5-band dyadic sibling frames with dense time-frequency grid. Given a lowpass filter satisfying certain conditions, the remaining filters are obtained using spectral factorization. The analysis and synthesis filterbanks share the same lowpass and bandpass filters but have different and oversampled highpass filters. This leads to wavelets approximating shift-invariance. The filters are FIR, have linear phase, and the resulting wavelets have vanishing moments. The filters are designed using spectral factorization method. The proposed method leads to smooth limit functions with higher approximation order, and computationally stable filterbanks.

Comparative Analysis of Various Single-tone Frequency Estimation Techniques in High-order Instantaneous Moments Based Phase Estimation Method

NASA Astrophysics Data System (ADS)

Rajshekhar, G.; Gorthi, Sai Siva; Rastogi, Pramod

2010-04-01

For phase estimation in digital holographic interferometry, a high-order instantaneous moments (HIM) based method was recently developed which relies on piecewise polynomial approximation of phase and subsequent evaluation of the polynomial coefficients using the HIM operator. A crucial step in the method is mapping the polynomial coefficient estimation to single-tone frequency determination for which various techniques exist. The paper presents a comparative analysis of the performance of the HIM operator based method in using different single-tone frequency estimation techniques for phase estimation. The analysis is supplemented by simulation results.

Reduced-cost second-order algebraic-diagrammatic construction method for excitation energies and transition moments

NASA Astrophysics Data System (ADS)

Mester, Dávid; Nagy, Péter R.; Kállay, Mihály

2018-03-01

A reduced-cost implementation of the second-order algebraic-diagrammatic construction [ADC(2)] method is presented. We introduce approximations by restricting virtual natural orbitals and natural auxiliary functions, which results, on average, in more than an order of magnitude speedup compared to conventional, density-fitting ADC(2) algorithms. The present scheme is the successor of our previous approach [D. Mester, P. R. Nagy, and M. Kállay, J. Chem. Phys. 146, 194102 (2017)], which has been successfully applied to obtain singlet excitation energies with the linear-response second-order coupled-cluster singles and doubles model. Here we report further methodological improvements and the extension of the method to compute singlet and triplet ADC(2) excitation energies and transition moments. The various approximations are carefully benchmarked, and conservative truncation thresholds are selected which guarantee errors much smaller than the intrinsic error of the ADC(2) method. Using the canonical values as reference, we find that the mean absolute error for both singlet and triplet ADC(2) excitation energies is 0.02 eV, while that for oscillator strengths is 0.001 a.u. The rigorous cutoff parameters together with the significantly reduced operation count and storage requirements allow us to obtain accurate ADC(2) excitation energies and transition properties using triple-ζ basis sets for systems of up to one hundred atoms.

Fuzzy interval Finite Element/Statistical Energy Analysis for mid-frequency analysis of built-up systems with mixed fuzzy and interval parameters

NASA Astrophysics Data System (ADS)

Yin, Hui; Yu, Dejie; Yin, Shengwen; Xia, Baizhan

2016-10-01

This paper introduces mixed fuzzy and interval parametric uncertainties into the FE components of the hybrid Finite Element/Statistical Energy Analysis (FE/SEA) model for mid-frequency analysis of built-up systems, thus an uncertain ensemble combining non-parametric with mixed fuzzy and interval parametric uncertainties comes into being. A fuzzy interval Finite Element/Statistical Energy Analysis (FIFE/SEA) framework is proposed to obtain the uncertain responses of built-up systems, which are described as intervals with fuzzy bounds, termed as fuzzy-bounded intervals (FBIs) in this paper. Based on the level-cut technique, a first-order fuzzy interval perturbation FE/SEA (FFIPFE/SEA) and a second-order fuzzy interval perturbation FE/SEA method (SFIPFE/SEA) are developed to handle the mixed parametric uncertainties efficiently. FFIPFE/SEA approximates the response functions by the first-order Taylor series, while SFIPFE/SEA improves the accuracy by considering the second-order items of Taylor series, in which all the mixed second-order items are neglected. To further improve the accuracy, a Chebyshev fuzzy interval method (CFIM) is proposed, in which the Chebyshev polynomials is used to approximate the response functions. The FBIs are eventually reconstructed by assembling the extrema solutions at all cut levels. Numerical results on two built-up systems verify the effectiveness of the proposed methods.

The NonConforming Virtual Element Method for the Stokes Equations

DOE PAGES

Cangiani, Andrea; Gyrya, Vitaliy; Manzini, Gianmarco

2016-01-01

In this paper, we present the nonconforming virtual element method (VEM) for the numerical approximation of velocity and pressure in the steady Stokes problem. The pressure is approximated using discontinuous piecewise polynomials, while each component of the velocity is approximated using the nonconforming virtual element space. On each mesh element the local virtual space contains the space of polynomials of up to a given degree, plus suitable nonpolynomial functions. The virtual element functions are implicitly defined as the solution of local Poisson problems with polynomial Neumann boundary conditions. As typical in VEM approaches, the explicit evaluation of the non-polynomial functionsmore » is not required. This approach makes it possible to construct nonconforming (virtual) spaces for any polynomial degree regardless of the parity, for two- and three-dimensional problems, and for meshes with very general polygonal and polyhedral elements. We show that the nonconforming VEM is inf-sup stable and establish optimal a priori error estimates for the velocity and pressure approximations. Finally, numerical examples confirm the convergence analysis and the effectiveness of the method in providing high-order accurate approximations.« less

A class of reduced-order models in the theory of waves and stability.

PubMed

Chapman, C J; Sorokin, S V

2016-02-01

This paper presents a class of approximations to a type of wave field for which the dispersion relation is transcendental. The approximations have two defining characteristics: (i) they give the field shape exactly when the frequency and wavenumber lie on a grid of points in the (frequency, wavenumber) plane and (ii) the approximate dispersion relations are polynomials that pass exactly through points on this grid. Thus, the method is interpolatory in nature, but the interpolation takes place in (frequency, wavenumber) space, rather than in physical space. Full details are presented for a non-trivial example, that of antisymmetric elastic waves in a layer. The method is related to partial fraction expansions and barycentric representations of functions. An asymptotic analysis is presented, involving Stirling's approximation to the psi function, and a logarithmic correction to the polynomial dispersion relation.

A consistent hierarchy of generalized kinetic equation approximations to the master equation applied to surface catalysis.

PubMed

Herschlag, Gregory J; Mitran, Sorin; Lin, Guang

2015-06-21

We develop a hierarchy of approximations to the master equation for systems that exhibit translational invariance and finite-range spatial correlation. Each approximation within the hierarchy is a set of ordinary differential equations that considers spatial correlations of varying lattice distance; the assumption is that the full system will have finite spatial correlations and thus the behavior of the models within the hierarchy will approach that of the full system. We provide evidence of this convergence in the context of one- and two-dimensional numerical examples. Lower levels within the hierarchy that consider shorter spatial correlations are shown to be up to three orders of magnitude faster than traditional kinetic Monte Carlo methods (KMC) for one-dimensional systems, while predicting similar system dynamics and steady states as KMC methods. We then test the hierarchy on a two-dimensional model for the oxidation of CO on RuO2(110), showing that low-order truncations of the hierarchy efficiently capture the essential system dynamics. By considering sequences of models in the hierarchy that account for longer spatial correlations, successive model predictions may be used to establish empirical approximation of error estimates. The hierarchy may be thought of as a class of generalized phenomenological kinetic models since each element of the hierarchy approximates the master equation and the lowest level in the hierarchy is identical to a simple existing phenomenological kinetic models.

Lag and anticipating synchronization without time-delay coupling.

PubMed

Corron, Ned J; Blakely, Jonathan N; Pethel, Shawn D

2005-06-01

We describe a new method for achieving approximate lag and anticipating synchronization in unidirectionally coupled chaotic oscillators. The method uses a specific parameter mismatch between the drive and response that is a first-order approximation to true time-delay coupling. As a result, an adjustable lag or anticipation effect can be achieved without the need for a variable delay line, making the method simpler and more economical to implement in many physical systems. We present a stability analysis, demonstrate the method numerically, and report experimental observation of the effect in radio-frequency electronic oscillators. In the circuit experiments, both lag and anticipation are controlled by tuning a single capacitor in the response oscillator.

A Weak Galerkin Method for the Reissner–Mindlin Plate in Primary Form

DOE Office of Scientific and Technical Information (OSTI.GOV)

Mu, Lin; Wang, Junping; Ye, Xiu

We developed a new finite element method for the Reissner–Mindlin equations in its primary form by using the weak Galerkin approach. Like other weak Galerkin finite element methods, this one is highly flexible and robust by allowing the use of discontinuous approximating functions on arbitrary shape of polygons and, at the same time, is parameter independent on its stability and convergence. Furthermore, error estimates of optimal order in mesh size h are established for the corresponding weak Galerkin approximations. Numerical experiments are conducted for verifying the convergence theory, as well as suggesting some superconvergence and a uniform convergence of themore » method with respect to the plate thickness.« less

A Weak Galerkin Method for the Reissner–Mindlin Plate in Primary Form

DOE PAGES

Mu, Lin; Wang, Junping; Ye, Xiu

2017-10-04

We developed a new finite element method for the Reissner–Mindlin equations in its primary form by using the weak Galerkin approach. Like other weak Galerkin finite element methods, this one is highly flexible and robust by allowing the use of discontinuous approximating functions on arbitrary shape of polygons and, at the same time, is parameter independent on its stability and convergence. Furthermore, error estimates of optimal order in mesh size h are established for the corresponding weak Galerkin approximations. Numerical experiments are conducted for verifying the convergence theory, as well as suggesting some superconvergence and a uniform convergence of themore » method with respect to the plate thickness.« less

Extension of the self-consistent-charge density-functional tight-binding method: third-order expansion of the density functional theory total energy and introduction of a modified effective coulomb interaction.

PubMed

Yang, Yang; Yu, Haibo; York, Darrin; Cui, Qiang; Elstner, Marcus

2007-10-25

The standard self-consistent-charge density-functional-tight-binding (SCC-DFTB) method (Phys. Rev. B 1998, 58, 7260) is derived by a second-order expansion of the density functional theory total energy expression, followed by an approximation of the charge density fluctuations by charge monopoles and an effective damped Coulomb interaction between the atomic net charges. The central assumptions behind this effective charge-charge interaction are the inverse relation of atomic size and chemical hardness and the use of a fixed chemical hardness parameter independent of the atomic charge state. While these approximations seem to be unproblematic for many covalently bound systems, they are quantitatively insufficient for hydrogen-bonding interactions and (anionic) molecules with localized net charges. Here, we present an extension of the SCC-DFTB method to incorporate third-order terms in the charge density fluctuations, leading to chemical hardness parameters that are dependent on the atomic charge state and a modification of the Coulomb scaling to improve the electrostatic treatment within the second-order terms. These modifications lead to a significant improvement in the description of hydrogen-bonding interactions and proton affinities of biologically relevant molecules.

Improved first-order uncertainty method for water-quality modeling

USGS Publications Warehouse

Melching, C.S.; Anmangandla, S.

1992-01-01

Uncertainties are unavoidable in water-quality modeling and subsequent management decisions. Monte Carlo simulation and first-order uncertainty analysis (involving linearization at central values of the uncertain variables) have been frequently used to estimate probability distributions for water-quality model output due to their simplicity. Each method has its drawbacks: Monte Carlo simulation's is mainly computational time; and first-order analysis are mainly questions of accuracy and representativeness, especially for nonlinear systems and extreme conditions. An improved (advanced) first-order method is presented, where the linearization point varies to match the output level whose exceedance probability is sought. The advanced first-order method is tested on the Streeter-Phelps equation to estimate the probability distribution of critical dissolved-oxygen deficit and critical dissolved oxygen using two hypothetical examples from the literature. The advanced first-order method provides a close approximation of the exceedance probability for the Streeter-Phelps model output estimated by Monte Carlo simulation using less computer time - by two orders of magnitude - regardless of the probability distributions assumed for the uncertain model parameters.

An almost symmetric Strang splitting scheme for nonlinear evolution equations.

PubMed

Einkemmer, Lukas; Ostermann, Alexander

2014-07-01

In this paper we consider splitting methods for the time integration of parabolic and certain classes of hyperbolic partial differential equations, where one partial flow cannot be computed exactly. Instead, we use a numerical approximation based on the linearization of the vector field. This is of interest in applications as it allows us to apply splitting methods to a wider class of problems from the sciences. However, in the situation described, the classic Strang splitting scheme, while still being a method of second order, is not longer symmetric. This, in turn, implies that the construction of higher order methods by composition is limited to order three only. To remedy this situation, based on previous work in the context of ordinary differential equations, we construct a class of Strang splitting schemes that are symmetric up to a desired order. We show rigorously that, under suitable assumptions on the nonlinearity, these methods are of second order and can then be used to construct higher order methods by composition. In addition, we illustrate the theoretical results by conducting numerical experiments for the Brusselator system and the KdV equation.

An almost symmetric Strang splitting scheme for nonlinear evolution equations☆

PubMed Central

Einkemmer, Lukas; Ostermann, Alexander

2014-01-01

In this paper we consider splitting methods for the time integration of parabolic and certain classes of hyperbolic partial differential equations, where one partial flow cannot be computed exactly. Instead, we use a numerical approximation based on the linearization of the vector field. This is of interest in applications as it allows us to apply splitting methods to a wider class of problems from the sciences. However, in the situation described, the classic Strang splitting scheme, while still being a method of second order, is not longer symmetric. This, in turn, implies that the construction of higher order methods by composition is limited to order three only. To remedy this situation, based on previous work in the context of ordinary differential equations, we construct a class of Strang splitting schemes that are symmetric up to a desired order. We show rigorously that, under suitable assumptions on the nonlinearity, these methods are of second order and can then be used to construct higher order methods by composition. In addition, we illustrate the theoretical results by conducting numerical experiments for the Brusselator system and the KdV equation. PMID:25844017

On processed splitting methods and high-order actions in path-integral Monte Carlo simulations.

PubMed

Casas, Fernando

2010-10-21

Processed splitting methods are particularly well adapted to carry out path-integral Monte Carlo (PIMC) simulations: since one is mainly interested in estimating traces of operators, only the kernel of the method is necessary to approximate the thermal density matrix. Unfortunately, they suffer the same drawback as standard, nonprocessed integrators: kernels of effective order greater than two necessarily involve some negative coefficients. This problem can be circumvented, however, by incorporating modified potentials into the composition, thus rendering schemes of higher effective order. In this work we analyze a family of fourth-order schemes recently proposed in the PIMC setting, paying special attention to their linear stability properties, and justify their observed behavior in practice. We also propose a new fourth-order scheme requiring the same computational cost but with an enlarged stability interval.

Crystal structure and phase stability in Fe{sub 1{minus}x}Co{sub x} from AB initio theory

DOE Office of Scientific and Technical Information (OSTI.GOV)

Soederlind, P.; Abrikosov, I.A.; James, P.

1996-06-01

For alloys between Fe and Co, their magnetic properties determine their structure. From the occupation of d states, a phase diagram is expected which depend largely on the spin polarization. A method more elaborate than canonical band models is used to calculate the spin moment and crystal structure energies. This method was the multisublattice generalization of the coherent potential approximation in conjunction with the Linear-Muffin-Tin-Orbital method in the atomic sphere approximation. To treat itinerant magnetism, the Vosko-Wilk-Nusair parameterization was used for the local spin density approximation. The fcc, bcc, and hcp phases were studied as completely random alloys, while themore » {alpha}{prime} phase for off-stoichiometries were considered as partially ordered. Results are compared with experiment and canonical band model.« less

A meshless method for solving two-dimensional variable-order time fractional advection-diffusion equation

NASA Astrophysics Data System (ADS)

Tayebi, A.; Shekari, Y.; Heydari, M. H.

2017-07-01

Several physical phenomena such as transformation of pollutants, energy, particles and many others can be described by the well-known convection-diffusion equation which is a combination of the diffusion and advection equations. In this paper, this equation is generalized with the concept of variable-order fractional derivatives. The generalized equation is called variable-order time fractional advection-diffusion equation (V-OTFA-DE). An accurate and robust meshless method based on the moving least squares (MLS) approximation and the finite difference scheme is proposed for its numerical solution on two-dimensional (2-D) arbitrary domains. In the time domain, the finite difference technique with a θ-weighted scheme and in the space domain, the MLS approximation are employed to obtain appropriate semi-discrete solutions. Since the newly developed method is a meshless approach, it does not require any background mesh structure to obtain semi-discrete solutions of the problem under consideration, and the numerical solutions are constructed entirely based on a set of scattered nodes. The proposed method is validated in solving three different examples including two benchmark problems and an applied problem of pollutant distribution in the atmosphere. In all such cases, the obtained results show that the proposed method is very accurate and robust. Moreover, a remarkable property so-called positive scheme for the proposed method is observed in solving concentration transport phenomena.

Random function theory revisited - Exact solutions versus the first order smoothing conjecture

NASA Technical Reports Server (NTRS)

Lerche, I.; Parker, E. N.

1975-01-01

We remark again that the mathematical conjecture known as first order smoothing or the quasi-linear approximation does not give the correct dependence on correlation length (time) in many cases, although it gives the correct limit as the correlation length (time) goes to zero. In this sense, then, the method is unreliable.

ASP: Automated symbolic computation of approximate symmetries of differential equations

NASA Astrophysics Data System (ADS)

Jefferson, G. F.; Carminati, J.

2013-03-01

A recent paper (Pakdemirli et al. (2004) [12]) compared three methods of determining approximate symmetries of differential equations. Two of these methods are well known and involve either a perturbation of the classical Lie symmetry generator of the differential system (Baikov, Gazizov and Ibragimov (1988) [7], Ibragimov (1996) [6]) or a perturbation of the dependent variable/s and subsequent determination of the classical Lie point symmetries of the resulting coupled system (Fushchych and Shtelen (1989) [11]), both up to a specified order in the perturbation parameter. The third method, proposed by Pakdemirli, Yürüsoy and Dolapçi (2004) [12], simplifies the calculations required by Fushchych and Shtelen's method through the assignment of arbitrary functions to the non-linear components prior to computing symmetries. All three methods have been implemented in the new MAPLE package ASP (Automated Symmetry Package) which is an add-on to the MAPLE symmetry package DESOLVII (Vu, Jefferson and Carminati (2012) [25]). To our knowledge, this is the first computer package to automate all three methods of determining approximate symmetries for differential systems. Extensions to the theory have also been suggested for the third method and which generalise the first method to systems of differential equations. Finally, a number of approximate symmetries and corresponding solutions are compared with results in the literature.

Locating the Discontinuities of a Bounded Function by the Partial Sums of its Fourier Series I: Periodical Case

NASA Technical Reports Server (NTRS)

Kvernadze, George; Hagstrom,Thomas; Shapiro, Henry

1997-01-01

A key step for some methods dealing with the reconstruction of a function with jump discontinuities is the accurate approximation of the jumps and their locations. Various methods have been suggested in the literature to obtain this valuable information. In the present paper, we develop an algorithm based on identities which determine the jumps of a 2(pi)-periodic bounded not-too-highly oscillating function by the partial sums of its differentiated Fourier series. The algorithm enables one to approximate the locations of discontinuities and the magnitudes of jumps of a bounded function. We study the accuracy of approximation and establish asymptotic expansions for the approximations of a 27(pi)-periodic piecewise smooth function with one discontinuity. By an appropriate linear combination, obtained via derivatives of different order, we significantly improve the accuracy. Next, we use Richardson's extrapolation method to enhance the accuracy even more. For a function with multiple discontinuities we establish simple formulae which "eliminate" all discontinuities of the function but one. Then we treat the function as if it had one singularity following the method described above.

Approximation of a radial diffusion model with a multiple-rate model for hetero-disperse particle mixtures

PubMed Central

Ju, Daeyoung; Young, Thomas M.; Ginn, Timothy R.

2012-01-01

An innovative method is proposed for approximation of the set of radial diffusion equations governing mass exchange between aqueous bulk phase and intra-particle phase for a hetero-disperse mixture of particles such as occur in suspension in surface water, in riverine/estuarine sediment beds, in soils and in aquifer materials. For this purpose the temporal variation of concentration at several uniformly distributed points within a normalized representative particle with spherical, cylindrical or planar shape is fitted with a 2-domain linear reversible mass exchange model. The approximation method is then superposed in order to generalize the model to a hetero-disperse mixture of particles. The method can reduce the computational effort needed in solving the intra-particle mass exchange of a hetero-disperse mixture of particles significantly and also the error due to the approximation is shown to be relatively small. The method is applied to describe desorption batch experiment of 1,2-Dichlorobenzene from four different soils with known particle size distributions and it could produce good agreement with experimental data. PMID:18304692

A semi-analytical method for the computation of the Lyapunov exponents of fractional-order systems

NASA Astrophysics Data System (ADS)

Caponetto, Riccardo; Fazzino, Stefano

2013-01-01

Fractional-order differential equations are interesting for their applications in the construction of mathematical models in finance, materials science or diffusion. In this paper, an application of a well known transformation technique, Differential Transform Method (DTM), to the area of fractional differential equation is employed for calculating Lyapunov exponents of fractional order systems. It is known that the Lyapunov exponents, first introduced by Oseledec, play a crucial role in characterizing the behaviour of dynamical systems. They can be used to analyze the sensitive dependence on initial conditions and the presence of chaotic attractors. The results reveal that the proposed method is very effective and simple and leads to accurate, approximately convergent solutions.

Preparation and magnetic properties of multiferroic CuMnO2 nanoparticles.

PubMed

Kurokawa, Akinobu; Yanoh, Tkuya; Yano, Shinya; Ichiyanagi, Yuko

2014-03-01

CuMnO2 nanoparticles with diameters of 64 nm were synthesized by a novel wet chemical method. An optimized two-step annealing method was developed through the analysis of thermogravimetric differential thermal analysis (TG-DTA) measurements in order to obtain single-phase CuMnO2. A sharp exothermic peak was observed in the DTA curve at approximately 500 K where structural changes of the copper oxides and manganese oxides in the precursor are expected to occur. It is believed that Cu+ ions were oxidized to Cu2+ ions and that Mn2+ ions were oxidized to Mn3+ ions in the Cu-Mn-O system. Deoxidization reactions were also found at approximately 1200 K. The optimized annealing temperature for the first step was determined to be 623 K in air. The optimized annealing temperature for the second step was 1173 K in an Ar atmosphere. Magnetization measurements suggested an antiferromagnetic spin ordering at approximately 50 K. It was expected that Mn3+ spin interactions induced magnetic phase transition affected by definite temperature.

Accurate Estimate of Some Propagation Characteristics for the First Higher Order Mode in Graded Index Fiber with Simple Analytic Chebyshev Method

NASA Astrophysics Data System (ADS)

Dutta, Ivy; Chowdhury, Anirban Roy; Kumbhakar, Dharmadas

2013-03-01

Using Chebyshev power series approach, accurate description for the first higher order (LP11) mode of graded index fibers having three different profile shape functions are presented in this paper and applied to predict their propagation characteristics. These characteristics include fractional power guided through the core, excitation efficiency and Petermann I and II spot sizes with their approximate analytic formulations. We have shown that where two and three Chebyshev points in LP11 mode approximation present fairly accurate results, the values based on our calculations involving four Chebyshev points match excellently with available exact numerical results.

Kurtosis Approach Nonlinear Blind Source Separation

NASA Technical Reports Server (NTRS)

Duong, Vu A.; Stubbemd, Allen R.

2005-01-01

In this paper, we introduce a new algorithm for blind source signal separation for post-nonlinear mixtures. The mixtures are assumed to be linearly mixed from unknown sources first and then distorted by memoryless nonlinear functions. The nonlinear functions are assumed to be smooth and can be approximated by polynomials. Both the coefficients of the unknown mixing matrix and the coefficients of the approximated polynomials are estimated by the gradient descent method conditional on the higher order statistical requirements. The results of simulation experiments presented in this paper demonstrate the validity and usefulness of our approach for nonlinear blind source signal separation Keywords: Independent Component Analysis, Kurtosis, Higher order statistics.

Numerical computations on one-dimensional inverse scattering problems

NASA Technical Reports Server (NTRS)

Dunn, M. H.; Hariharan, S. I.

1983-01-01

An approximate method to determine the index of refraction of a dielectric obstacle is presented. For simplicity one dimensional models of electromagnetic scattering are treated. The governing equations yield a second order boundary value problem, in which the index of refraction appears as a functional parameter. The availability of reflection coefficients yield two additional boundary conditions. The index of refraction by a k-th order spline which can be written as a linear combination of B-splines is approximated. For N distinct reflection coefficients, the resulting N boundary value problems yield a system of N nonlinear equations in N unknowns which are the coefficients of the B-splines.

Solar neutrino masses and mixing from bilinear R-parity broken supersymmetry: Analytical versus numerical results

NASA Astrophysics Data System (ADS)

Díaz, M.; Hirsch, M.; Porod, W.; Romão, J.; Valle, J.

2003-07-01

We give an analytical calculation of solar neutrino masses and mixing at one-loop order within bilinear R-parity breaking supersymmetry, and compare our results to the exact numerical calculation. Our method is based on a systematic perturbative expansion of R-parity violating vertices to leading order. We find in general quite good agreement between the approximate and full numerical calculations, but the approximate expressions are much simpler to implement. Our formalism works especially well for the case of the large mixing angle Mikheyev-Smirnov-Wolfenstein solution, now strongly favored by the recent KamLAND reactor neutrino data.

Occurrence probability of structured motifs in random sequences.

PubMed

Robin, S; Daudin, J-J; Richard, H; Sagot, M-F; Schbath, S

2002-01-01

The problem of extracting from a set of nucleic acid sequences motifs which may have biological function is more and more important. In this paper, we are interested in particular motifs that may be implicated in the transcription process. These motifs, called structured motifs, are composed of two ordered parts separated by a variable distance and allowing for substitutions. In order to assess their statistical significance, we propose approximations of the probability of occurrences of such a structured motif in a given sequence. An application of our method to evaluate candidate promoters in E. coli and B. subtilis is presented. Simulations show the goodness of the approximations.

Spike solutions in Gierer#x2013;Meinhardt model with a time dependent anomaly exponent

NASA Astrophysics Data System (ADS)

Nec, Yana

2018-01-01

Experimental evidence of complex dispersion regimes in natural systems, where the growth of the mean square displacement in time cannot be characterised by a single power, has been accruing for the past two decades. In such processes the exponent γ(t) in ⟨r2⟩ ∼ tγ(t) at times might be approximated by a piecewise constant function, or it can be a continuous function. Variable order differential equations are an emerging mathematical tool with a strong potential to model these systems. However, variable order differential equations are not tractable by the classic differential equations theory. This contribution illustrates how a classic method can be adapted to gain insight into a system of this type. Herein a variable order Gierer-Meinhardt model is posed, a generic reaction- diffusion system of a chemical origin. With a fixed order this system possesses a solution in the form of a constellation of arbitrarily situated localised pulses, when the components' diffusivity ratio is asymptotically small. The pattern was shown to exist subject to multiple step-like transitions between normal diffusion and sub-diffusion, as well as between distinct sub-diffusive regimes. The analytical approximation obtained permits qualitative analysis of the impact thereof. Numerical solution for typical cross-over scenarios revealed such features as earlier equilibration and non-monotonic excursions before attainment of equilibrium. The method is general and allows for an approximate numerical solution with any reasonably behaved γ(t).

A Family of Ellipse Methods for Solving Non-Linear Equations

ERIC Educational Resources Information Center

Gupta, K. C.; Kanwar, V.; Kumar, Sanjeev

2009-01-01

This note presents a method for the numerical approximation of simple zeros of a non-linear equation in one variable. In order to do so, the method uses an ellipse rather than a tangent approach. The main advantage of our method is that it does not fail even if the derivative of the function is either zero or very small in the vicinity of the…

A simple finite element method for linear hyperbolic problems

DOE PAGES

Mu, Lin; Ye, Xiu

2017-09-14

Here, we introduce a simple finite element method for solving first order hyperbolic equations with easy implementation and analysis. Our new method, with a symmetric, positive definite system, is designed to use discontinuous approximations on finite element partitions consisting of arbitrary shape of polygons/polyhedra. Error estimate is established. Extensive numerical examples are tested that demonstrate the robustness and flexibility of the method.

A simple finite element method for linear hyperbolic problems

DOE Office of Scientific and Technical Information (OSTI.GOV)

Mu, Lin; Ye, Xiu

Here, we introduce a simple finite element method for solving first order hyperbolic equations with easy implementation and analysis. Our new method, with a symmetric, positive definite system, is designed to use discontinuous approximations on finite element partitions consisting of arbitrary shape of polygons/polyhedra. Error estimate is established. Extensive numerical examples are tested that demonstrate the robustness and flexibility of the method.

Lognormal Approximations of Fault Tree Uncertainty Distributions.

PubMed

El-Shanawany, Ashraf Ben; Ardron, Keith H; Walker, Simon P

2018-01-26

Fault trees are used in reliability modeling to create logical models of fault combinations that can lead to undesirable events. The output of a fault tree analysis (the top event probability) is expressed in terms of the failure probabilities of basic events that are input to the model. Typically, the basic event probabilities are not known exactly, but are modeled as probability distributions: therefore, the top event probability is also represented as an uncertainty distribution. Monte Carlo methods are generally used for evaluating the uncertainty distribution, but such calculations are computationally intensive and do not readily reveal the dominant contributors to the uncertainty. In this article, a closed-form approximation for the fault tree top event uncertainty distribution is developed, which is applicable when the uncertainties in the basic events of the model are lognormally distributed. The results of the approximate method are compared with results from two sampling-based methods: namely, the Monte Carlo method and the Wilks method based on order statistics. It is shown that the closed-form expression can provide a reasonable approximation to results obtained by Monte Carlo sampling, without incurring the computational expense. The Wilks method is found to be a useful means of providing an upper bound for the percentiles of the uncertainty distribution while being computationally inexpensive compared with full Monte Carlo sampling. The lognormal approximation method and Wilks's method appear attractive, practical alternatives for the evaluation of uncertainty in the output of fault trees and similar multilinear models. © 2018 Society for Risk Analysis.

Approximate method for calculating transonic flow about lifting wing-body configurations: Computer program and user's manual

NASA Technical Reports Server (NTRS)

Barnwell, R. W.; Davis, R. M.

1975-01-01

A user's manual is presented for a computer program which calculates inviscid flow about lifting configurations in the free-stream Mach-number range from zero to low supersonic. Angles of attack of the order of the configuration thickness-length ratio and less can be calculated. An approximate formulation was used which accounts for shock waves, leading-edge separation and wind-tunnel wall effects.

Comments on localized and integral localized approximations in spherical coordinates

NASA Astrophysics Data System (ADS)

Gouesbet, Gérard; Lock, James A.

2016-08-01

Localized approximation procedures are efficient ways to evaluate beam shape coefficients of laser beams, and are particularly useful when other methods are ineffective or inefficient. Comments on these procedures are, however, required in order to help researchers make correct decisions concerning their use. This paper has the flavor of a short review and takes the opportunity to attract the attention of the readers to a required refinement of terminology.

Fourth order Douglas implicit scheme for solving three dimension reaction diffusion equation with non-linear source term

NASA Astrophysics Data System (ADS)

Hasnain, Shahid; Saqib, Muhammad; Mashat, Daoud Suleiman

2017-07-01

This research paper represents a numerical approximation to non-linear three dimension reaction diffusion equation with non-linear source term from population genetics. Since various initial and boundary value problems exist in three dimension reaction diffusion phenomena, which are studied numerically by different numerical methods, here we use finite difference schemes (Alternating Direction Implicit and Fourth Order Douglas Implicit) to approximate the solution. Accuracy is studied in term of L2, L∞ and relative error norms by random selected grids along time levels for comparison with analytical results. The test example demonstrates the accuracy, efficiency and versatility of the proposed schemes. Numerical results showed that Fourth Order Douglas Implicit scheme is very efficient and reliable for solving 3-D non-linear reaction diffusion equation.

Approximate solution of space and time fractional higher order phase field equation

NASA Astrophysics Data System (ADS)

Shamseldeen, S.

2018-03-01

This paper is concerned with a class of space and time fractional partial differential equation (STFDE) with Riesz derivative in space and Caputo in time. The proposed STFDE is considered as a generalization of a sixth-order partial phase field equation. We describe the application of the optimal homotopy analysis method (OHAM) to obtain an approximate solution for the suggested fractional initial value problem. An averaged-squared residual error function is defined and used to determine the optimal convergence control parameter. Two numerical examples are studied, considering periodic and non-periodic initial conditions, to justify the efficiency and the accuracy of the adopted iterative approach. The dependence of the solution on the order of the fractional derivative in space and time and model parameters is investigated.

Overall Performance Evaluation of Tubular Scraper Conveyors Using a TOPSIS-Based Multiattribute Decision-Making Method

PubMed Central

Yao, Yanping; Kou, Ziming; Meng, Wenjun; Han, Gang

2014-01-01

Properly evaluating the overall performance of tubular scraper conveyors (TSCs) can increase their overall efficiency and reduce economic investments, but such methods have rarely been studied. This study evaluated the overall performance of TSCs based on the technique for order of preference by similarity to ideal solution (TOPSIS). Three conveyors of the same type produced in the same factory were investigated. Their scraper space, material filling coefficient, and vibration coefficient of the traction components were evaluated. A mathematical model of the multiattribute decision matrix was constructed; a weighted judgment matrix was obtained using the DELPHI method. The linguistic positive-ideal solution (LPIS), the linguistic negative-ideal solution (LNIS), and the distance from each solution to the LPIS and the LNIS, that is, the approximation degrees, were calculated. The optimal solution was determined by ordering the approximation degrees for each solution. The TOPSIS-based results were compared with the measurement results provided by the manufacturer. The ordering result based on the three evaluated parameters was highly consistent with the result provided by the manufacturer. The TOPSIS-based method serves as a suitable evaluation tool for the overall performance of TSCs. It facilitates the optimal deployment of TSCs for industrial purposes. PMID:24991646

High-order cyclo-difference techniques: An alternative to finite differences

NASA Technical Reports Server (NTRS)

Carpenter, Mark H.; Otto, John C.

1993-01-01

The summation-by-parts energy norm is used to establish a new class of high-order finite-difference techniques referred to here as 'cyclo-difference' techniques. These techniques are constructed cyclically from stable subelements, and require no numerical boundary conditions; when coupled with the simultaneous approximation term (SAT) boundary treatment, they are time asymptotically stable for an arbitrary hyperbolic system. These techniques are similar to spectral element techniques and are ideally suited for parallel implementation, but do not require special collocation points or orthogonal basis functions. The principal focus is on methods of sixth-order formal accuracy or less; however, these methods could be extended in principle to any arbitrary order of accuracy.

Examining the accuracy of the infinite order sudden approximation using sensitivity analysis

DOE Office of Scientific and Technical Information (OSTI.GOV)

Eno, L.; Rabitz, H.

1981-08-15

A method is developed for assessing the accuracy of scattering observables calculated within the framework of the infinite order sudden (IOS) approximation. In particular, we focus on the energy sudden assumption of the IOS method and our approach involves the determination of the sensitivity of the IOS scattering matrix S/sup IOS/ with respect to a parameter which reintroduces the internal energy operator h/sub 0/ into the IOS Hamiltonian. This procedure is an example of sensitivity analysis of missing model components (h/sub 0/ in this case) in the reference Hamiltonian. In contrast to simple first-order perturbation theory a finite result ismore » obtained for the effect of h/sub 0/ on S/sup IOS/. As an illustration, our method of analysis is applied to integral state-to-state cross sections for the scattering of an atom and rigid rotor. Results are generated within the He+H/sub 2/ system and a comparison is made between IOS and coupled states cross sections and the corresponding IOS sensitivities. It is found that the sensitivity coefficients are very useful indicators of the accuracy of the IOS results. Finally, further developments and applications are discussed.« less

Design and simulation of origami structures with smooth folds

PubMed Central

Peraza Hernandez, E. A.; Lagoudas, D. C.

2017-01-01

Origami has enabled new approaches to the fabrication and functionality of multiple structures. Current methods for origami design are restricted to the idealization of folds as creases of zeroth-order geometric continuity. Such an idealization is not proper for origami structures of non-negligible fold thickness or maximum curvature at the folds restricted by material limitations. For such structures, folds are not properly represented as creases but rather as bent regions of higher-order geometric continuity. Such fold regions of arbitrary order of continuity are termed as smooth folds. This paper presents a method for solving the following origami design problem: given a goal shape represented as a polygonal mesh (termed as the goal mesh), find the geometry of a single planar sheet, its pattern of smooth folds, and the history of folding motion allowing the sheet to approximate the goal mesh. The parametrization of the planar sheet and the constraints that allow for a valid pattern of smooth folds are presented. The method is tested against various goal meshes having diverse geometries. The results show that every determined sheet approximates its corresponding goal mesh in a known folded configuration having fold angles obtained from the geometry of the goal mesh. PMID:28484322

Design and simulation of origami structures with smooth folds.

PubMed

Peraza Hernandez, E A; Hartl, D J; Lagoudas, D C

2017-04-01

Origami has enabled new approaches to the fabrication and functionality of multiple structures. Current methods for origami design are restricted to the idealization of folds as creases of zeroth-order geometric continuity. Such an idealization is not proper for origami structures of non-negligible fold thickness or maximum curvature at the folds restricted by material limitations. For such structures, folds are not properly represented as creases but rather as bent regions of higher-order geometric continuity. Such fold regions of arbitrary order of continuity are termed as smooth folds . This paper presents a method for solving the following origami design problem: given a goal shape represented as a polygonal mesh (termed as the goal mesh ), find the geometry of a single planar sheet, its pattern of smooth folds, and the history of folding motion allowing the sheet to approximate the goal mesh. The parametrization of the planar sheet and the constraints that allow for a valid pattern of smooth folds are presented. The method is tested against various goal meshes having diverse geometries. The results show that every determined sheet approximates its corresponding goal mesh in a known folded configuration having fold angles obtained from the geometry of the goal mesh.

Propagation of Computational Uncertainty Using the Modern Design of Experiments

NASA Technical Reports Server (NTRS)

DeLoach, Richard

2007-01-01

This paper describes the use of formally designed experiments to aid in the error analysis of a computational experiment. A method is described by which the underlying code is approximated with relatively low-order polynomial graduating functions represented by truncated Taylor series approximations to the true underlying response function. A resource-minimal approach is outlined by which such graduating functions can be estimated from a minimum number of case runs of the underlying computational code. Certain practical considerations are discussed, including ways and means of coping with high-order response functions. The distributional properties of prediction residuals are presented and discussed. A practical method is presented for quantifying that component of the prediction uncertainty of a computational code that can be attributed to imperfect knowledge of independent variable levels. This method is illustrated with a recent assessment of uncertainty in computational estimates of Space Shuttle thermal and structural reentry loads attributable to ice and foam debris impact on ascent.

Parallel Computing of Upwelling in a Rotating Stratified Flow

NASA Astrophysics Data System (ADS)

Cui, A.; Street, R. L.

1997-11-01

A code for the three-dimensional, unsteady, incompressible, and turbulent flow has been implemented on the IBM SP2, using message passing. The effects of rotation and variable density are included. A finite volume method is used to discretize the Navier-Stokes equations in general curvilinear coordinates on a non-staggered grid. All the spatial derivatives are approximated using second-order central differences with the exception of the convection terms, which are handled with special upwind-difference schemes. The semi-implicit, second-order accurate, time-advancement scheme employs the Adams-Bashforth method for the explicit terms and Crank-Nicolson for the implicit terms. A multigrid method, with the four-color ZEBRA as smoother, is used to solve the Poisson equation for pressure, while the momentum equations are solved with an approximate factorization technique. The code was successfully validated for a variety test cases. Simulations of a laboratory model of coastal upwelling in a rotating annulus are in progress and will be presented.

Harmonic-phase path-integral approximation of thermal quantum correlation functions

NASA Astrophysics Data System (ADS)

Robertson, Christopher; Habershon, Scott

2018-03-01

We present an approximation to the thermal symmetric form of the quantum time-correlation function in the standard position path-integral representation. By transforming to a sum-and-difference position representation and then Taylor-expanding the potential energy surface of the system to second order, the resulting expression provides a harmonic weighting function that approximately recovers the contribution of the phase to the time-correlation function. This method is readily implemented in a Monte Carlo sampling scheme and provides exact results for harmonic potentials (for both linear and non-linear operators) and near-quantitative results for anharmonic systems for low temperatures and times that are likely to be relevant to condensed phase experiments. This article focuses on one-dimensional examples to provide insights into convergence and sampling properties, and we also discuss how this approximation method may be extended to many-dimensional systems.

Multipolar Ewald methods, 1: theory, accuracy, and performance.

PubMed

Giese, Timothy J; Panteva, Maria T; Chen, Haoyuan; York, Darrin M

2015-02-10

The Ewald, Particle Mesh Ewald (PME), and Fast Fourier–Poisson (FFP) methods are developed for systems composed of spherical multipole moment expansions. A unified set of equations is derived that takes advantage of a spherical tensor gradient operator formalism in both real space and reciprocal space to allow extension to arbitrary multipole order. The implementation of these methods into a novel linear-scaling modified “divide-and-conquer” (mDC) quantum mechanical force field is discussed. The evaluation times and relative force errors are compared between the three methods, as a function of multipole expansion order. Timings and errors are also compared within the context of the quantum mechanical force field, which encounters primary errors related to the quality of reproducing electrostatic forces for a given density matrix and secondary errors resulting from the propagation of the approximate electrostatics into the self-consistent field procedure, which yields a converged, variational, but nonetheless approximate density matrix. Condensed-phase simulations of an mDC water model are performed with the multipolar PME method and compared to an electrostatic cutoff method, which is shown to artificially increase the density of water and heat of vaporization relative to full electrostatic treatment.

Matrix form of Legendre polynomials for solving linear integro-differential equations of high order

NASA Astrophysics Data System (ADS)

Kammuji, M.; Eshkuvatov, Z. K.; Yunus, Arif A. M.

2017-04-01

This paper presents an effective approximate solution of high order of Fredholm-Volterra integro-differential equations (FVIDEs) with boundary condition. Legendre truncated series is used as a basis functions to estimate the unknown function. Matrix operation of Legendre polynomials is used to transform FVIDEs with boundary conditions into matrix equation of Fredholm-Volterra type. Gauss Legendre quadrature formula and collocation method are applied to transfer the matrix equation into system of linear algebraic equations. The latter equation is solved by Gauss elimination method. The accuracy and validity of this method are discussed by solving two numerical examples and comparisons with wavelet and methods.

Multicriteria approaches for a private equity fund

NASA Astrophysics Data System (ADS)

Tammer, Christiane; Tannert, Johannes

2012-09-01

We develop a new model for a Private Equity Fund based on stochastic differential equations. In order to find efficient strategies for the fund manager we formulate a multicriteria optimization problem for a Private Equity Fund. Using the e-constraint method we solve this multicriteria optimization problem. Furthermore, a genetic algorithm is applied in order to get an approximation of the efficient frontier.

Oscillator strengths, first-order properties, and nuclear gradients for local ADC(2).

PubMed

Schütz, Martin

2015-06-07

We describe theory and implementation of oscillator strengths, orbital-relaxed first-order properties, and nuclear gradients for the local algebraic diagrammatic construction scheme through second order. The formalism is derived via time-dependent linear response theory based on a second-order unitary coupled cluster model. The implementation presented here is a modification of our previously developed algorithms for Laplace transform based local time-dependent coupled cluster linear response (CC2LR); the local approximations thus are state specific and adaptive. The symmetry of the Jacobian leads to considerable simplifications relative to the local CC2LR method; as a result, a gradient evaluation is about four times less expensive. Test calculations show that in geometry optimizations, usually very similar geometries are obtained as with the local CC2LR method (provided that a second-order method is applicable). As an exemplary application, we performed geometry optimizations on the low-lying singlet states of chlorophyllide a.

The application of the Routh approximation method to turbofan engine models

NASA Technical Reports Server (NTRS)

Merrill, W. C.

1977-01-01

The Routh approximation technique is applied in the frequency domain to a 16th order state variable turbofan engine model. The results obtained motivate the extension of the frequency domain formulation of the Routh method to the time domain to handle the state variable formulation directly. The time domain formulation is derived, and a characterization, which specifies all possible Routh similarity transformations, is given. The characterization is computed by the solution of two eigenvalue eigenvector problems. The application of the time domain Routh technique to the state variable engine model is described, and some results are given.

An approximate viscous shock layer technique for calculating chemically reacting hypersonic flows about blunt-nosed bodies

NASA Technical Reports Server (NTRS)

Cheatwood, F. Mcneil; Dejarnette, Fred R.

1991-01-01

An approximate axisymmetric method was developed which can reliably calculate fully viscous hypersonic flows over blunt nosed bodies. By substituting Maslen's second order pressure expression for the normal momentum equation, a simplified form of the viscous shock layer (VSL) equations is obtained. This approach can solve both the subsonic and supersonic regions of the shock layer without a starting solution for the shock shape. The approach is applicable to perfect gas, equilibrium, and nonequilibrium flowfields. Since the method is fully viscous, the problems associated with a boundary layer solution with an inviscid layer solution are avoided. This procedure is significantly faster than the parabolized Navier-Stokes (PNS) or VSL solvers and would be useful in a preliminary design environment. Problems associated with a previously developed approximate VSL technique are addressed before extending the method to nonequilibrium calculations. Perfect gas (laminar and turbulent), equilibrium, and nonequilibrium solutions were generated for airflows over several analytic body shapes. Surface heat transfer, skin friction, and pressure predictions are comparable to VSL results. In addition, computed heating rates are in good agreement with experimental data. The present technique generates its own shock shape as part of its solution, and therefore could be used to provide more accurate initial shock shapes for higher order procedures which require starting solutions.

POSTPROCESSING MIXED FINITE ELEMENT METHODS FOR SOLVING CAHN-HILLIARD EQUATION: METHODS AND ERROR ANALYSIS

PubMed Central

Wang, Wansheng; Chen, Long; Zhou, Jie

2015-01-01

A postprocessing technique for mixed finite element methods for the Cahn-Hilliard equation is developed and analyzed. Once the mixed finite element approximations have been computed at a fixed time on the coarser mesh, the approximations are postprocessed by solving two decoupled Poisson equations in an enriched finite element space (either on a finer grid or a higher-order space) for which many fast Poisson solvers can be applied. The nonlinear iteration is only applied to a much smaller size problem and the computational cost using Newton and direct solvers is negligible compared with the cost of the linear problem. The analysis presented here shows that this technique remains the optimal rate of convergence for both the concentration and the chemical potential approximations. The corresponding error estimate obtained in our paper, especially the negative norm error estimates, are non-trivial and different with the existing results in the literatures. PMID:27110063

Approximate Dispersion Relations for Waves on Arbitrary Shear Flows

NASA Astrophysics Data System (ADS)

Ellingsen, S. À.; Li, Y.

2017-12-01

An approximate dispersion relation is derived and presented for linear surface waves atop a shear current whose magnitude and direction can vary arbitrarily with depth. The approximation, derived to first order of deviation from potential flow, is shown to produce good approximations at all wavelengths for a wide range of naturally occuring shear flows as well as widely used model flows. The relation reduces in many cases to a 3-D generalization of the much used approximation by Skop (1987), developed further by Kirby and Chen (1989), but is shown to be more robust, succeeding in situations where the Kirby and Chen model fails. The two approximations incur the same numerical cost and difficulty. While the Kirby and Chen approximation is excellent for a wide range of currents, the exact criteria for its applicability have not been known. We explain the apparently serendipitous success of the latter and derive proper conditions of applicability for both approximate dispersion relations. Our new model has a greater range of applicability. A second order approximation is also derived. It greatly improves accuracy, which is shown to be important in difficult cases. It has an advantage over the corresponding second-order expression proposed by Kirby and Chen that its criterion of accuracy is explicitly known, which is not currently the case for the latter to our knowledge. Our second-order term is also arguably significantly simpler to implement, and more physically transparent, than its sibling due to Kirby and Chen.