An accurate two-phase approximate solution to the acute viral infection model
Perelson, Alan S
2009-01-01
During an acute viral infection, virus levels rise, reach a peak and then decline. Data and numerical solutions suggest the growth and decay phases are linear on a log scale. While viral dynamic models are typically nonlinear with analytical solutions difficult to obtain, the exponential nature of the solutions suggests approximations can be found. We derive a two-phase approximate solution to the target cell limited influenza model and illustrate the accuracy using data and previously established parameter values of six patients infected with influenza A. For one patient, the subsequent fall in virus concentration was not consistent with our predictions during the decay phase and an alternate approximation is derived. We find expressions for the rate and length of initial viral growth in terms of the parameters, the extent each parameter is involved in viral peaks, and the single parameter responsible for virus decay. We discuss applications of this analysis in antiviral treatments and investigating host and virus heterogeneities.
Lewis, E.R.; Schwartz, S.
2010-03-15
Light scattering by aerosols plays an important role in Earth’s radiative balance, and quantification of this phenomenon is important in understanding and accounting for anthropogenic influences on Earth’s climate. Light scattering by an aerosol particle is determined by its radius and index of refraction, and for aerosol particles that are hygroscopic, both of these quantities vary with relative humidity RH. Here exact expressions are derived for the dependences of the radius ratio (relative to the volume-equivalent dry radius) and index of refraction on RH for aqueous solutions of single solutes. Both of these quantities depend on the apparent molal volume of the solute in solution and on the practical osmotic coefficient of the solution, which in turn depend on concentration and thus implicitly on RH. Simple but accurate approximations are also presented for the RH dependences of both radius ratio and index of refraction for several atmospherically important inorganic solutes over the entire range of RH values for which these substances can exist as solution drops. For all substances considered, the radius ratio is accurate to within a few percent, and the index of refraction to within ~0.02, over this range of RH. Such parameterizations will be useful in radiation transfer models and climate models.
NASA Technical Reports Server (NTRS)
Kylling, Arve; Stamnes, Knut
1992-01-01
The present solutions to the linear transport equation pertain to monoenergetic particles' interaction with a multiple scattering/absorbing layered medium with a general anisotropic internal source term. Attention is given to a novel exponential-linear approximation to the internal source, as a function of scattering depth, which furnishes an at-once efficient and accurate solution to the linear transport equation through its reduction of the spatial mesh size. The great superiority of the proposed method is demonstrated by the numerical results obtained in the illustrative cases of (1) an embedded thermal source and (2) a rapidly varying beam pseudosource.
Differential equation based method for accurate approximations in optimization
NASA Technical Reports Server (NTRS)
Pritchard, Jocelyn I.; Adelman, Howard M.
1990-01-01
A method to efficiently and accurately approximate the effect of design changes on structural response is described. The key to this method is to interpret sensitivity equations as differential equations that may be solved explicitly for closed form approximations, hence, the method is denoted the Differential Equation Based (DEB) method. Approximations were developed for vibration frequencies, mode shapes and static displacements. The DEB approximation method was applied to a cantilever beam and results compared with the commonly-used linear Taylor series approximations and exact solutions. The test calculations involved perturbing the height, width, cross-sectional area, tip mass, and bending inertia of the beam. The DEB method proved to be very accurate, and in most cases, was more accurate than the linear Taylor series approximation. The method is applicable to simultaneous perturbation of several design variables. Also, the approximations may be used to calculate other system response quantities. For example, the approximations for displacements are used to approximate bending stresses.
Strong shock implosion, approximate solution
NASA Astrophysics Data System (ADS)
Fujimoto, Y.; Mishkin, E. A.; Alejaldre, C.
1983-01-01
The self-similar, center-bound motion of a strong spherical, or cylindrical, shock wave moving through an ideal gas with a constant, γ= cp/ cv, is considered and a linearized, approximate solution is derived. An X, Y phase plane of the self-similar solution is defined and the representative curved of the system behind the shock front is replaced by a straight line connecting the mappings of the shock front with that of its tail. The reduced pressure P(ξ), density R(ξ) and velocity U1(ξ) are found in closed, quite accurate, form. Comparison with numerically obtained results, for γ= {5}/{3} and γ= {7}/{5}, is shown.
Differential equation based method for accurate approximations in optimization
NASA Technical Reports Server (NTRS)
Pritchard, Jocelyn I.; Adelman, Howard M.
1990-01-01
This paper describes a method to efficiently and accurately approximate the effect of design changes on structural response. The key to this new method is to interpret sensitivity equations as differential equations that may be solved explicitly for closed form approximations, hence, the method is denoted the Differential Equation Based (DEB) method. Approximations were developed for vibration frequencies, mode shapes and static displacements. The DEB approximation method was applied to a cantilever beam and results compared with the commonly-used linear Taylor series approximations and exact solutions. The test calculations involved perturbing the height, width, cross-sectional area, tip mass, and bending inertia of the beam. The DEB method proved to be very accurate, and in msot cases, was more accurate than the linear Taylor series approximation. The method is applicable to simultaneous perturbation of several design variables. Also, the approximations may be used to calculate other system response quantities. For example, the approximations for displacement are used to approximate bending stresses.
Predict amine solution properties accurately
Cheng, S.; Meisen, A.; Chakma, A.
1996-02-01
Improved process design begins with using accurate physical property data. Especially in the preliminary design stage, physical property data such as density viscosity, thermal conductivity and specific heat can affect the overall performance of absorbers, heat exchangers, reboilers and pump. These properties can also influence temperature profiles in heat transfer equipment and thus control or affect the rate of amine breakdown. Aqueous-amine solution physical property data are available in graphical form. However, it is not convenient to use with computer-based calculations. Developed equations allow improved correlations of derived physical property estimates with published data. Expressions are given which can be used to estimate physical properties of methyldiethanolamine (MDEA), monoethanolamine (MEA) and diglycolamine (DGA) solutions.
Approximate analytic solutions to the NPDD: Short exposure approximations
NASA Astrophysics Data System (ADS)
Close, Ciara E.; Sheridan, John T.
2014-04-01
There have been many attempts to accurately describe the photochemical processes that take places in photopolymer materials. As the models have become more accurate, solving them has become more numerically intensive and more 'opaque'. Recent models incorporate the major photochemical reactions taking place as well as the diffusion effects resulting from the photo-polymerisation process, and have accurately described these processes in a number of different materials. It is our aim to develop accessible mathematical expressions which provide physical insights and simple quantitative predictions of practical value to material designers and users. In this paper, starting with the Non-Local Photo-Polymerisation Driven Diffusion (NPDD) model coupled integro-differential equations, we first simplify these equations and validate the accuracy of the resulting approximate model. This new set of governing equations are then used to produce accurate analytic solutions (polynomials) describing the evolution of the monomer and polymer concentrations, and the grating refractive index modulation, in the case of short low intensity sinusoidal exposures. The physical significance of the results and their consequences for holographic data storage (HDS) are then discussed.
Shock Emergence in Supernovae: Limiting Cases and Accurate Approximations
NASA Astrophysics Data System (ADS)
Ro, Stephen; Matzner, Christopher D.
2013-08-01
We examine the dynamics of accelerating normal shocks in stratified planar atmospheres, providing accurate fitting formulae for the scaling index relating shock velocity to the initial density and for the post-shock acceleration factor as functions of the polytropic and adiabatic indices which parameterize the problem. In the limit of a uniform initial atmosphere, there are analytical formulae for these quantities. In the opposite limit of a very steep density gradient, the solutions match the outcome of shock acceleration in exponential atmospheres.
SHOCK EMERGENCE IN SUPERNOVAE: LIMITING CASES AND ACCURATE APPROXIMATIONS
Ro, Stephen; Matzner, Christopher D.
2013-08-10
We examine the dynamics of accelerating normal shocks in stratified planar atmospheres, providing accurate fitting formulae for the scaling index relating shock velocity to the initial density and for the post-shock acceleration factor as functions of the polytropic and adiabatic indices which parameterize the problem. In the limit of a uniform initial atmosphere, there are analytical formulae for these quantities. In the opposite limit of a very steep density gradient, the solutions match the outcome of shock acceleration in exponential atmospheres.
On constructing accurate approximations of first integrals for difference equations
NASA Astrophysics Data System (ADS)
Rafei, M.; Van Horssen, W. T.
2013-04-01
In this paper, a perturbation method based on invariance factors and multiple scales will be presented for weakly nonlinear, regularly perturbed systems of ordinary difference equations. Asymptotic approximations of first integrals will be constructed on long iteration-scales, that is, on iteration-scales of order ɛ-1, where ɛ is a small parameter. It will be shown that all invariance factors have to satisfy a functional equation. To show how this perturbation method works, the method is applied to a Van der Pol equation, and a Rayleigh equation. It will be explicitly shown for the first time in the literature how these multiple scales should be introduced for systems of difference equations to obtain very accurate approximations of first integrals on long iteration-scales.
Approximate Solutions Of Equations Of Steady Diffusion
NASA Technical Reports Server (NTRS)
Edmonds, Larry D.
1992-01-01
Rigorous analysis yields reliable criteria for "best-fit" functions. Improved "curve-fitting" method yields approximate solutions to differential equations of steady-state diffusion. Method applies to problems in which rates of diffusion depend linearly or nonlinearly on concentrations of diffusants, approximate solutions analytic or numerical, and boundary conditions of Dirichlet type, of Neumann type, or mixture of both types. Applied to equations for diffusion of charge carriers in semiconductors in which mobilities and lifetimes of charge carriers depend on concentrations.
Approximate solutions for certain bidomain problems in electrocardiography
NASA Astrophysics Data System (ADS)
Johnston, Peter R.
2008-10-01
The simulation of problems in electrocardiography using the bidomain model for cardiac tissue often creates issues with satisfaction of the boundary conditions required to obtain a solution. Recent studies have proposed approximate methods for solving such problems by satisfying the boundary conditions only approximately. This paper presents an analysis of their approximations using a similar method, but one which ensures that the boundary conditions are satisfied during the whole solution process. Also considered are additional functional forms, used in the approximate solutions, which are more appropriate to specific boundary conditions. The analysis shows that the approximations introduced by Patel and Roth [Phys. Rev. E 72, 051931 (2005)] generally give accurate results. However, there are certain situations where functional forms based on the geometry of the problem under consideration can give improved approximations. It is also demonstrated that the recent methods are equivalent to different approaches to solving the same problems introduced 20years earlier.
Approximate Solutions in Planted 3-SAT
NASA Astrophysics Data System (ADS)
Hsu, Benjamin; Laumann, Christopher; Moessner, Roderich; Sondhi, Shivaji
2013-03-01
In many computational settings, there exists many instances where finding a solution requires a computing time that grows exponentially in the number of variables. Concrete examples occur in combinatorial optimization problems and cryptography in computer science or glassy systems in physics. However, while exact solutions are often known to require exponential time, a related and important question is the running time required to find approximate solutions. Treating this problem as a problem in statistical physics at finite temperature, we examine the computational running time in finding approximate solutions in 3-satisfiability for randomly generated 3-SAT instances which are guaranteed to have a solution. Analytic predictions are corroborated by numerical evidence using stochastic local search algorithms. A first order transition is found in the running time of these algorithms.
Analytic Approximate Solution for Falkner-Skan Equation
Marinca, Bogdan
2014-01-01
This paper deals with the Falkner-Skan nonlinear differential equation. An analytic approximate technique, namely, optimal homotopy asymptotic method (OHAM), is employed to propose a procedure to solve a boundary-layer problem. Our method does not depend upon small parameters and provides us with a convenient way to optimally control the convergence of the approximate solutions. The obtained results reveal that this procedure is very effective, simple, and accurate. A very good agreement was found between our approximate results and numerical solutions, which prove that OHAM is very efficient in practice, ensuring a very rapid convergence after only one iteration. PMID:24883417
Approximated solutions to Born-Infeld dynamics
NASA Astrophysics Data System (ADS)
Ferraro, Rafael; Nigro, Mauro
2016-02-01
The Born-Infeld equation in the plane is usefully captured in complex language. The general exact solution can be written as a combination of holomorphic and anti-holomorphic functions. However, this solution only expresses the potential in an implicit way. We rework the formulation to obtain the complex potential in an explicit way, by means of a perturbative procedure. We take care of the secular behavior common to this kind of approach, by resorting to a symmetry the equation has at the considered order of approximation. We apply the method to build approximated solutions to Born-Infeld electrodynamics. We solve for BI electromagnetic waves traveling in opposite directions. We study the propagation at interfaces, with the aim of searching for effects susceptible to experimental detection. In particular, we show that a reflected wave is produced when a wave is incident on a semi-space containing a magnetostatic field.
Approximate solutions of the hyperbolic Kepler equation
NASA Astrophysics Data System (ADS)
Avendano, Martín; Martín-Molina, Verónica; Ortigas-Galindo, Jorge
2015-12-01
We provide an approximate zero widetilde{S}(g,L) for the hyperbolic Kepler's equation S-g {{arcsinh}}(S)-L=0 for gin (0,1) and Lin [0,∞ ). We prove, by using Smale's α -theory, that Newton's method starting at our approximate zero produces a sequence that converges to the actual solution S( g, L) at quadratic speed, i.e. if S_n is the value obtained after n iterations, then |S_n-S|≤ 0.5^{2^n-1}|widetilde{S}-S|. The approximate zero widetilde{S}(g,L) is a piecewise-defined function involving several linear expressions and one with cubic and square roots. In bounded regions of (0,1) × [0,∞ ) that exclude a small neighborhood of g=1, L=0, we also provide a method to construct simpler starters involving only constants.
Exponentially accurate approximations to piece-wise smooth periodic functions
NASA Technical Reports Server (NTRS)
Greer, James; Banerjee, Saheb
1995-01-01
A family of simple, periodic basis functions with 'built-in' discontinuities are introduced, and their properties are analyzed and discussed. Some of their potential usefulness is illustrated in conjunction with the Fourier series representations of functions with discontinuities. In particular, it is demonstrated how they can be used to construct a sequence of approximations which converges exponentially in the maximum norm to a piece-wise smooth function. The theory is illustrated with several examples and the results are discussed in the context of other sequences of functions which can be used to approximate discontinuous functions.
Simple accurate approximations for the optical properties of metallic nanospheres and nanoshells.
Schebarchov, Dmitri; Auguié, Baptiste; Le Ru, Eric C
2013-03-28
This work aims to provide simple and accurate closed-form approximations to predict the scattering and absorption spectra of metallic nanospheres and nanoshells supporting localised surface plasmon resonances. Particular attention is given to the validity and accuracy of these expressions in the range of nanoparticle sizes relevant to plasmonics, typically limited to around 100 nm in diameter. Using recent results on the rigorous radiative correction of electrostatic solutions, we propose a new set of long-wavelength polarizability approximations for both nanospheres and nanoshells. The improvement offered by these expressions is demonstrated with direct comparisons to other approximations previously obtained in the literature, and their absolute accuracy is tested against the exact Mie theory. PMID:23358525
Method of successive approximations for the solution of certain problems in aerodynamics
NASA Technical Reports Server (NTRS)
Shvets, M E
1951-01-01
A method of successive approximations for the solution of problems in the fields of diffusion, boundary-layer flow, and heat-transfer is illustrated by solving problems in each of these fields. In most of the examples, the approximate solutions are compared with known accurate solutions and the agreement is shown to be good.
Accurate numerical solutions of conservative nonlinear oscillators
NASA Astrophysics Data System (ADS)
Khan, Najeeb Alam; Nasir Uddin, Khan; Nadeem Alam, Khan
2014-12-01
The objective of this paper is to present an investigation to analyze the vibration of a conservative nonlinear oscillator in the form u" + lambda u + u^(2n-1) + (1 + epsilon^2 u^(4m))^(1/2) = 0 for any arbitrary power of n and m. This method converts the differential equation to sets of algebraic equations and solve numerically. We have presented for three different cases: a higher order Duffing equation, an equation with irrational restoring force and a plasma physics equation. It is also found that the method is valid for any arbitrary order of n and m. Comparisons have been made with the results found in the literature the method gives accurate results.
High order accurate solutions of viscous problems
NASA Technical Reports Server (NTRS)
Hayder, M. E.; Turkel, Eli
1993-01-01
We consider a fourth order extension to MacCormack's scheme. The original extension was fourth order only for the inviscid terms but was second order for the viscous terms. We show how to modify the viscous terms so that the scheme is uniformly fourth order in the spatial derivatives. Applications are given to some boundary layer flows. In addition, for applications to shear flows the effect of the outflow boundary conditions are very important. We compare the accuracy of several of these different boundary conditions for both boundary layer and shear flows. Stretching at the outflow usually increases the oscillations in the numerical solution but the addition of a filtered sponge layer (with or without stretching) reduces such oscillations. The oscillations are generated by insufficient resolution of the shear layer. When the shear layer is sufficiently resolved then oscillations are not generated and there is less of a need for a nonreflecting boundary condition.
NASA Astrophysics Data System (ADS)
Shu, Yu-Chen; Chern, I.-Liang; Chang, Chien C.
2014-10-01
Most elliptic interface solvers become complicated for complex interface problems at those “exceptional points” where there are not enough neighboring interior points for high order interpolation. Such complication increases especially in three dimensions. Usually, the solvers are thus reduced to low order accuracy. In this paper, we classify these exceptional points and propose two recipes to maintain order of accuracy there, aiming at improving the previous coupling interface method [26]. Yet the idea is also applicable to other interface solvers. The main idea is to have at least first order approximations for second order derivatives at those exceptional points. Recipe 1 is to use the finite difference approximation for the second order derivatives at a nearby interior grid point, whenever this is possible. Recipe 2 is to flip domain signatures and introduce a ghost state so that a second-order method can be applied. This ghost state is a smooth extension of the solution at the exceptional point from the other side of the interface. The original state is recovered by a post-processing using nearby states and jump conditions. The choice of recipes is determined by a classification scheme of the exceptional points. The method renders the solution and its gradient uniformly second-order accurate in the entire computed domain. Numerical examples are provided to illustrate the second order accuracy of the presently proposed method in approximating the gradients of the original states for some complex interfaces which we had tested previous in two and three dimensions, and a real molecule (1D63) which is double-helix shape and composed of hundreds of atoms.
Shu, Yu-Chen; Chern, I-Liang; Chang, Chien C.
2014-10-15
Most elliptic interface solvers become complicated for complex interface problems at those “exceptional points” where there are not enough neighboring interior points for high order interpolation. Such complication increases especially in three dimensions. Usually, the solvers are thus reduced to low order accuracy. In this paper, we classify these exceptional points and propose two recipes to maintain order of accuracy there, aiming at improving the previous coupling interface method [26]. Yet the idea is also applicable to other interface solvers. The main idea is to have at least first order approximations for second order derivatives at those exceptional points. Recipe 1 is to use the finite difference approximation for the second order derivatives at a nearby interior grid point, whenever this is possible. Recipe 2 is to flip domain signatures and introduce a ghost state so that a second-order method can be applied. This ghost state is a smooth extension of the solution at the exceptional point from the other side of the interface. The original state is recovered by a post-processing using nearby states and jump conditions. The choice of recipes is determined by a classification scheme of the exceptional points. The method renders the solution and its gradient uniformly second-order accurate in the entire computed domain. Numerical examples are provided to illustrate the second order accuracy of the presently proposed method in approximating the gradients of the original states for some complex interfaces which we had tested previous in two and three dimensions, and a real molecule ( (1D63)) which is double-helix shape and composed of hundreds of atoms.
Approximate Solutions for Flow with a Stretching Boundary due to Partial Slip
Filobello-Nino, U.; Vazquez-Leal, H.; Sarmiento-Reyes, A.; Benhammouda, B.; Jimenez-Fernandez, V. M.; Pereyra-Diaz, D.; Perez-Sesma, A.; Cervantes-Perez, J.; Huerta-Chua, J.; Sanchez-Orea, J.; Contreras-Hernandez, A. D.
2014-01-01
The homotopy perturbation method (HPM) is coupled with versions of Laplace-Padé and Padé methods to provide an approximate solution to the nonlinear differential equation that describes the behaviour of a flow with a stretching flat boundary due to partial slip. Comparing results between approximate and numerical solutions, we concluded that our results are capable of providing an accurate solution and are extremely efficient. PMID:27433526
Approximate Solution for Choked Flow in Gas Seal Pads
NASA Technical Reports Server (NTRS)
Fleming, David P.
2004-01-01
Previous analyses of high pressure seals have considered adiabatic flow with friction but neglected effects of seal rotation. Most of this work analyzed a one-dimensional flow field. This works well to calculate stiffness and leakage of full circular seals, either face seals or annular ring seals. However, it cannot provide accurate results for a rectangular seal pad with its strongly two-dimensional flow field and its reliance on hydrodynamic forces to maintain a full fluid film. On the other hand, solutions of Reynolds lubrication equation have been obtained for the two-dimensional flow in a seal pad. But these solutions do not account for choking which occurs at high seal pressure ratios, nor do they consider the pressure loss that occurs in the entrance region of the flow field. The aim of the present work is to build on the Reynolds equation solution by use of an approximate choked flow analysis. This will account for the pressure losses in the flow entrance region, ensure that fluid velocities remain subsonic, and enable fluid inertial effects within the pad film to be accounted for. Results show that, in general, fluid inertia acts to decrease pad film load capacity and leakage, and increase film stiffness.
NASA Astrophysics Data System (ADS)
Colalongo, Luigi; Ghittorelli, Matteo; Torricelli, Fabrizio; Kovács-Vajna, Zsolt Miklos
2015-12-01
Surface-potential-based mathematical models are among the most accurate and physically based compact models of Thin-Film Transistors (TFTs) and, in turn, of Organic Thin-Film Transistors (OTFTs), available today. However, the need for iterative computations of the surface potential limits their computational efficiency and diffusion in CAD applications. The existing closed-form approximations of the surface potential are based on regional approximations and empirical smoothing functions that could result not enough accurate to model OTFTs and, in particular, transconductances and transcapacitances. In this paper we present an accurate and computationally efficient closed-form approximation of the surface potential, based on the Lagrange Reversion Theorem, that can be exploited in advanced surface-potential-based OTFTs and TFTs device models.
On the use and error of approximation in the Domenico (1987) solution.
West, Michael R; Kueper, Bernard H; Ungs, Michael J
2007-01-01
A mathematical solution for solute transport in a three-dimensional porous medium with a patch source under steady-state, uniform ground water flow conditions was developed by Domenico (1987). The solution derivation strategy used an approximate approach to solve the boundary value problem, resulting in a nonexact solution. Variations of the Domenico (1987) solution are incorporated into the software programs BIOSCREEN and BIOCHLOR, which are frequently used to evaluate subsurface contaminant transport problems. This article mathematically elucidates the error in the approximation and presents simulations that compare different versions of the Domenico (1987) solution to an exact analytical solution to demonstrate the potential error inherent in the approximate expressions. Results suggest that the accuracy of the approximate solutions is highly variable and dependent on the selection of input parameters. For solute transport in a medium-grained sand aquifer, the Domenico (1987) solution underpredicts solute concentrations along the centerline of the plume by as much as 80% depending on the case of interest. Increasing the dispersivity, time, or dimensionality of the system leads to increased error. Because more accurate exact analytical solutions exist, we suggest that the Domenico (1987) solution, and its predecessor and successor approximate solutions, need not be employed as the basis for screening tools at contaminated sites. PMID:17335477
NASA Astrophysics Data System (ADS)
Lima, F. M. S.
2009-11-01
In a previous work, O'Connell (Phys. Teach. 40, 24 (2002)) investigated the time dependence of the tension in the string of a simple pendulum oscillating within the small-angle regime. In spite of the approximation sin θ ≈ θ being accurate only for amplitudes below 7°, his experimental results are for a pendulum oscillating with an amplitude of about 18°, therefore beyond the small-angle regime. This lapse may also be found in some textbooks, laboratory manuals and internet. By noting that the exact analytical solution for this problem involves the so-called Jacobi elliptic functions, which are unknown to most students (even instructors), I take into account a sinusoidal approximate solution for the pendulum equation I introduced in a recent work (Eur. J. Phys. 29 1091 (2008)) for deriving a simple trigonometric approximation for the tension valid for all possible amplitudes. This approximation is compared to both the O'Connell and the exact results, revealing that it is accurate enough for analysing large-angle pendulum experiments.
Rational approximations to solutions of linear differential equations
Chudnovsky, D. V.; Chudnovsky, G. V.
1983-01-01
Rational approximations of Padé and Padé type to solutions of differential equations are considered. One of the main results is a theorem stating that a simultaneous approximation to arbitrary solutions of linear differential equations over C(x) cannot be “better” than trivial ones implied by the Dirichlet box principle. This constitutes, in particular, the solution in the linear case of Kolchin's problem that the “Roth's theorem” holds for arbitrary solutions of algebraic differential equations. Complete effective proofs for several valuations are presented based on the Wronskian methods and graded subrings of Picard-Vessiot extensions. PMID:16593357
Hamilton's Principle and Approximate Solutions to Problems in Classical Mechanics
ERIC Educational Resources Information Center
Schlitt, D. W.
1977-01-01
Shows how to use the Ritz method for obtaining approximate solutions to problems expressed in variational form directly from the variational equation. Application of this method to classical mechanics is given. (MLH)
Techniques for correcting approximate finite difference solutions. [considering transonic flow
NASA Technical Reports Server (NTRS)
Nixon, D.
1978-01-01
A method of correcting finite-difference solutions for the effect of truncation error or the use of an approximate basic equation is presented. Applications to transonic flow problems are described and examples are given.
Approximate Analytical Solutions for Hypersonic Flow Over Slender Power Law Bodies
NASA Technical Reports Server (NTRS)
Mirels, Harold
1959-01-01
Approximate analytical solutions are presented for two-dimensional and axisymmetric hypersonic flow over slender power law bodies. Both zero order (M approaches infinity) and first order (small but nonvanishing values of 1/(M(Delta)(sup 2) solutions are presented, where M is free-stream Mach number and Delta is a characteristic slope. These solutions are compared with exact numerical integration of the equations of motion and appear to be accurate particularly when the shock is relatively close to the body.
Vibration of clamped right triangular thin plates: Accurate simplified solutions
NASA Astrophysics Data System (ADS)
Saliba, H. T.
1994-12-01
Use of the superposition techniques in the free-vibration analyses of thin plates, as they were first introduced by Gorman, has provided simple and effective solutions to a vast number of rectangular plate problems. A modified superposition method is presented that is a noticeable improvement over existing techniques. It deals only with simple support conditions, leading to a simple, highly accurate, and very economical solution to the free-vibration problem of simply-supported right angle triangular plates. The modified method is also applicable to clamped-edge conditions.
Approximate solutions for non-linear iterative fractional differential equations
NASA Astrophysics Data System (ADS)
Damag, Faten H.; Kiliçman, Adem; Ibrahim, Rabha W.
2016-06-01
This paper establishes approximate solution for non-linear iterative fractional differential equations: d/γv (s ) d sγ =ℵ (s ,v ,v (v )), where γ ∈ (0, 1], s ∈ I := [0, 1]. Our method is based on some convergence tools for analytic solution in a connected region. We show that the suggested solution is unique and convergent by some well known geometric functions.
An approximate solution for interlaminar stresses in composite laminates
NASA Technical Reports Server (NTRS)
Rose, Cheryl A.; Herakovich, Carl T.
1993-01-01
An efficient approximate solution for interlaminar stresses in finite width, symmetric and unsymmetric laminated composites subjected to axial and/or bending loads is presented. The solution is based upon statically admissible stress fields which take into consideration local property mismatch effects and global equilibrium requirements. Unknown constants in the assumed stress states are determined through minimization of the laminate complementary energy. Typical results are presented for through-thickness and interlaminar stress distributions for angle-ply and cross-ply laminates subjected to axial loading. It is shown that the present formulation represents an improved, efficient approximate solution for interlaminar stresses.
Vibration of clamped right triangular thin plates: Accurate simplified solutions
NASA Astrophysics Data System (ADS)
Saliba, H. T.
1994-12-01
Use of the superposition techniques in the free-vibration analyses of thin plates, as they were first introduced by Gorman, has provided simple and effective solutions to a vast number of rectangular plate problems. The method has also been extended to nonrectangular plates such as triangular and trapezoidal plates. However, serious difficulties were encountered in some of these analyses. These difficulties were discussed and obviated in Salibra, 1990. This reference, however, dealt only with simple support conditions, leading to a simple, highly accurate, and very economical solution to the free-vibration problem of simply supported right angle triangular plates. The purpose of this Note is to show that the modified superposition method of Salibra, 1990 is also applicable to clamped-edge conditions. This is accomplished through the application of this method to the title problem.
Accurate response surface approximations for weight equations based on structural optimization
NASA Astrophysics Data System (ADS)
Papila, Melih
Accurate weight prediction methods are vitally important for aircraft design optimization. Therefore, designers seek weight prediction techniques with low computational cost and high accuracy, and usually require a compromise between the two. The compromise can be achieved by combining stress analysis and response surface (RS) methodology. While stress analysis provides accurate weight information, RS techniques help to transmit effectively this information to the optimization procedure. The focus of this dissertation is structural weight equations in the form of RS approximations and their accuracy when fitted to results of structural optimizations that are based on finite element analyses. Use of RS methodology filters out the numerical noise in structural optimization results and provides a smooth weight function that can easily be used in gradient-based configuration optimization. In engineering applications RS approximations of low order polynomials are widely used, but the weight may not be modeled well by low-order polynomials, leading to bias errors. In addition, some structural optimization results may have high-amplitude errors (outliers) that may severely affect the accuracy of the weight equation. Statistical techniques associated with RS methodology are sought in order to deal with these two difficulties: (1) high-amplitude numerical noise (outliers) and (2) approximation model inadequacy. The investigation starts with reducing approximation error by identifying and repairing outliers. A potential reason for outliers in optimization results is premature convergence, and outliers of such nature may be corrected by employing different convergence settings. It is demonstrated that outlier repair can lead to accuracy improvements over the more standard approach of removing outliers. The adequacy of approximation is then studied by a modified lack-of-fit approach, and RS errors due to the approximation model are reduced by using higher order polynomials. In
Fullerton, G D; Keener, C R; Cameron, I L
1994-12-01
The authors describe empirical corrections to ideally dilute expressions for freezing point depression of aqueous solutions to arrive at new expressions accurate up to three molal concentration. The method assumes non-ideality is due primarily to solute/solvent interactions such that the correct free water mass Mwc is the mass of water in solution Mw minus I.M(s) where M(s) is the mass of solute and I an empirical solute/solvent interaction coefficient. The interaction coefficient is easily derived from the constant in the linear regression fit to the experimental plot of Mw/M(s) as a function of 1/delta T (inverse freezing point depression). The I-value, when substituted into the new thermodynamic expressions derived from the assumption of equivalent activity of water in solution and ice, provides accurate predictions of freezing point depression (+/- 0.05 degrees C) up to 2.5 molal concentration for all the test molecules evaluated; glucose, sucrose, glycerol and ethylene glycol. The concentration limit is the approximate monolayer water coverage limit for the solutes which suggests that direct solute/solute interactions are negligible below this limit. This is contrary to the view of many authors due to the common practice of including hydration forces (a soft potential added to the hard core atomic potential) in the interaction potential between solute particles. When this is recognized the two viewpoints are in fundamental agreement. PMID:7699200
Accurate numerical solution of compressible, linear stability equations
NASA Technical Reports Server (NTRS)
Malik, M. R.; Chuang, S.; Hussaini, M. Y.
1982-01-01
The present investigation is concerned with a fourth order accurate finite difference method and its application to the study of the temporal and spatial stability of the three-dimensional compressible boundary layer flow on a swept wing. This method belongs to the class of compact two-point difference schemes discussed by White (1974) and Keller (1974). The method was apparently first used for solving the two-dimensional boundary layer equations. Attention is given to the governing equations, the solution technique, and the search for eigenvalues. A general purpose subroutine is employed for solving a block tridiagonal system of equations. The computer time can be reduced significantly by exploiting the special structure of two matrices.
Efficient solution of parabolic equations by Krylov approximation methods
NASA Technical Reports Server (NTRS)
Gallopoulos, E.; Saad, Y.
1990-01-01
Numerical techniques for solving parabolic equations by the method of lines is addressed. The main motivation for the proposed approach is the possibility of exploiting a high degree of parallelism in a simple manner. The basic idea of the method is to approximate the action of the evolution operator on a given state vector by means of a projection process onto a Krylov subspace. Thus, the resulting approximation consists of applying an evolution operator of a very small dimension to a known vector which is, in turn, computed accurately by exploiting well-known rational approximations to the exponential. Because the rational approximation is only applied to a small matrix, the only operations required with the original large matrix are matrix-by-vector multiplications, and as a result the algorithm can easily be parallelized and vectorized. Some relevant approximation and stability issues are discussed. We present some numerical experiments with the method and compare its performance with a few explicit and implicit algorithms.
Linaro, Daniele; Storace, Marco; Giugliano, Michele
2011-01-01
Stochastic channel gating is the major source of intrinsic neuronal noise whose functional consequences at the microcircuit- and network-levels have been only partly explored. A systematic study of this channel noise in large ensembles of biophysically detailed model neurons calls for the availability of fast numerical methods. In fact, exact techniques employ the microscopic simulation of the random opening and closing of individual ion channels, usually based on Markov models, whose computational loads are prohibitive for next generation massive computer models of the brain. In this work, we operatively define a procedure for translating any Markov model describing voltage- or ligand-gated membrane ion-conductances into an effective stochastic version, whose computer simulation is efficient, without compromising accuracy. Our approximation is based on an improved Langevin-like approach, which employs stochastic differential equations and no Montecarlo methods. As opposed to an earlier proposal recently debated in the literature, our approximation reproduces accurately the statistical properties of the exact microscopic simulations, under a variety of conditions, from spontaneous to evoked response features. In addition, our method is not restricted to the Hodgkin-Huxley sodium and potassium currents and is general for a variety of voltage- and ligand-gated ion currents. As a by-product, the analysis of the properties emerging in exact Markov schemes by standard probability calculus enables us for the first time to analytically identify the sources of inaccuracy of the previous proposal, while providing solid ground for its modification and improvement we present here. PMID:21423712
Novel determination of differential-equation solutions: universal approximation method
NASA Astrophysics Data System (ADS)
Leephakpreeda, Thananchai
2002-09-01
In a conventional approach to numerical computation, finite difference and finite element methods are usually implemented to determine the solution of a set of differential equations (DEs). This paper presents a novel approach to solve DEs by applying the universal approximation method through an artificial intelligence utility in a simple way. In this proposed method, neural network model (NNM) and fuzzy linguistic model (FLM) are applied as universal approximators for any nonlinear continuous functions. With this outstanding capability, the solutions of DEs can be approximated by the appropriate NNM or FLM within an arbitrary accuracy. The adjustable parameters of such NNM and FLM are determined by implementing the optimization algorithm. This systematic search yields sub-optimal adjustable parameters of NNM and FLM with the satisfactory conditions and with the minimum residual errors of the governing equations subject to the constraints of boundary conditions of DEs. The simulation results are investigated for the viability of efficiently determining the solutions of the ordinary and partial nonlinear DEs.
Fall with linear drag and Wien's displacement law: approximate solution and Lambert function
NASA Astrophysics Data System (ADS)
Vial, Alexandre
2012-07-01
We present an approximate solution for the downward time of travel in the case of a mass falling with a linear drag force. We show how a quasi-analytical solution implying the Lambert function can be found. We also show that solving the previous problem is equivalent to the search for Wien's displacement law. These results can be of interest for undergraduate students, as they show that some transcendental equations found in physics may be solved without purely numerical methods. Moreover, as will be seen in the case of Wien's displacement law, solutions based on series expansion can be very accurate even with few terms.
A method for the accurate and smooth approximation of standard thermodynamic functions
NASA Astrophysics Data System (ADS)
Coufal, O.
2013-01-01
A method is proposed for the calculation of approximations of standard thermodynamic functions. The method is consistent with the physical properties of standard thermodynamic functions. This means that the approximation functions are, in contrast to the hitherto used approximations, continuous and smooth in every temperature interval in which no phase transformations take place. The calculation algorithm was implemented by the SmoothSTF program in the C++ language which is part of this paper. Program summaryProgram title:SmoothSTF Catalogue identifier: AENH_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AENH_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 3807 No. of bytes in distributed program, including test data, etc.: 131965 Distribution format: tar.gz Programming language: C++. Computer: Any computer with gcc version 4.3.2 compiler. Operating system: Debian GNU Linux 6.0. The program can be run in operating systems in which the gcc compiler can be installed, see http://gcc.gnu.org/install/specific.html. RAM: 256 MB are sufficient for the table of standard thermodynamic functions with 500 lines Classification: 4.9. Nature of problem: Standard thermodynamic functions (STF) of individual substances are given by thermal capacity at constant pressure, entropy and enthalpy. STF are continuous and smooth in every temperature interval in which no phase transformations take place. The temperature dependence of STF as expressed by the table of its values is for further application approximated by temperature functions. In the paper, a method is proposed for calculating approximation functions which, in contrast to the hitherto used approximations, are continuous and smooth in every temperature interval. Solution method: The approximation functions are
Fast and accurate approximate inference of transcript expression from RNA-seq data
Hensman, James; Papastamoulis, Panagiotis; Glaus, Peter; Honkela, Antti; Rattray, Magnus
2015-01-01
Motivation: Assigning RNA-seq reads to their transcript of origin is a fundamental task in transcript expression estimation. Where ambiguities in assignments exist due to transcripts sharing sequence, e.g. alternative isoforms or alleles, the problem can be solved through probabilistic inference. Bayesian methods have been shown to provide accurate transcript abundance estimates compared with competing methods. However, exact Bayesian inference is intractable and approximate methods such as Markov chain Monte Carlo and Variational Bayes (VB) are typically used. While providing a high degree of accuracy and modelling flexibility, standard implementations can be prohibitively slow for large datasets and complex transcriptome annotations. Results: We propose a novel approximate inference scheme based on VB and apply it to an existing model of transcript expression inference from RNA-seq data. Recent advances in VB algorithmics are used to improve the convergence of the algorithm beyond the standard Variational Bayes Expectation Maximization algorithm. We apply our algorithm to simulated and biological datasets, demonstrating a significant increase in speed with only very small loss in accuracy of expression level estimation. We carry out a comparative study against seven popular alternative methods and demonstrate that our new algorithm provides excellent accuracy and inter-replicate consistency while remaining competitive in computation time. Availability and implementation: The methods were implemented in R and C++, and are available as part of the BitSeq project at github.com/BitSeq. The method is also available through the BitSeq Bioconductor package. The source code to reproduce all simulation results can be accessed via github.com/BitSeq/BitSeqVB_benchmarking. Contact: james.hensman@sheffield.ac.uk or panagiotis.papastamoulis@manchester.ac.uk or Magnus.Rattray@manchester.ac.uk Supplementary information: Supplementary data are available at Bioinformatics online
Accurate ω-ψ Spectral Solution of the Singular Driven Cavity Problem
NASA Astrophysics Data System (ADS)
Auteri, F.; Quartapelle, L.; Vigevano, L.
2002-08-01
This article provides accurate spectral solutions of the driven cavity problem, calculated in the vorticity-stream function representation without smoothing the corner singularities—a prima facie impossible task. As in a recent benchmark spectral calculation by primitive variables of Botella and Peyret, closed-form contributions of the singular solution for both zero and finite Reynolds numbers are subtracted from the unknown of the problem tackled here numerically in biharmonic form. The method employed is based on a split approach to the vorticity and stream function equations, a Galerkin-Legendre approximation of the problem for the perturbation, and an evaluation of the nonlinear terms by Gauss-Legendre numerical integration. Results computed for Re=0, 100, and 1000 compare well with the benchmark steady solutions provided by the aforementioned collocation-Chebyshev projection method. The validity of the proposed singularity subtraction scheme for computing time-dependent solutions is also established.
Efficient yet accurate approximations for ab initio calculations of alcohol cluster thermochemistry
NASA Astrophysics Data System (ADS)
Umer, Muhammad; Kopp, Wassja A.; Leonhard, Kai
2015-12-01
We have calculated the binding enthalpies and entropies of gas phase alcohol clusters from ethanol to 1-decanol. In addition to the monomers, we have investigated dimers, tetramers, and pentamers. Geometries have been obtained at the B3LYP/TZVP level and single point energy calculations have been performed with the Resolution of the Identity-MP2 (RIMP2) method and basis set limit extrapolation using aug-cc-pVTZ and aug-cc-pVQZ basis sets. Thermochemistry is calculated with decoupled hindered rotor treatment for large amplitude motions. The results show three points: First, it is more accurate to transfer the rigid-rotor harmonic oscillator entropies from propanol to longer alcohols than to compute them with an ultra-fine grid and tight geometry convergence criteria. Second, the computational effort can be reduced considerably by using dimerization energies of longer alcohols at density functional theory (B3LYP) level plus a RIMP2 correction obtained from 1-propanol. This approximation yields results almost with the same accuracy as RIMP2 — both methods differ for 1-decanol only 0.4 kJ/mol. Third, the entropy of dimerization including the hindered rotation contribution is converged at 1-propanol with respect to chain length. This allows for a transfer of hindered rotation contributions from smaller alcohols to longer ones which reduces the required computational and man power considerably.
Development of highly accurate approximate scheme for computing the charge transfer integral.
Pershin, Anton; Szalay, Péter G
2015-08-21
The charge transfer integral is a key parameter required by various theoretical models to describe charge transport properties, e.g., in organic semiconductors. The accuracy of this important property depends on several factors, which include the level of electronic structure theory and internal simplifications of the applied formalism. The goal of this paper is to identify the performance of various approximate approaches of the latter category, while using the high level equation-of-motion coupled cluster theory for the electronic structure. The calculations have been performed on the ethylene dimer as one of the simplest model systems. By studying different spatial perturbations, it was shown that while both energy split in dimer and fragment charge difference methods are equivalent with the exact formulation for symmetrical displacements, they are less efficient when describing transfer integral along the asymmetric alteration coordinate. Since the "exact" scheme was found computationally expensive, we examine the possibility to obtain the asymmetric fluctuation of the transfer integral by a Taylor expansion along the coordinate space. By exploring the efficiency of this novel approach, we show that the Taylor expansion scheme represents an attractive alternative to the "exact" calculations due to a substantial reduction of computational costs, when a considerably large region of the potential energy surface is of interest. Moreover, we show that the Taylor expansion scheme, irrespective of the dimer symmetry, is very accurate for the entire range of geometry fluctuations that cover the space the molecule accesses at room temperature. PMID:26298117
Development of highly accurate approximate scheme for computing the charge transfer integral
Pershin, Anton; Szalay, Péter G.
2015-08-21
The charge transfer integral is a key parameter required by various theoretical models to describe charge transport properties, e.g., in organic semiconductors. The accuracy of this important property depends on several factors, which include the level of electronic structure theory and internal simplifications of the applied formalism. The goal of this paper is to identify the performance of various approximate approaches of the latter category, while using the high level equation-of-motion coupled cluster theory for the electronic structure. The calculations have been performed on the ethylene dimer as one of the simplest model systems. By studying different spatial perturbations, it was shown that while both energy split in dimer and fragment charge difference methods are equivalent with the exact formulation for symmetrical displacements, they are less efficient when describing transfer integral along the asymmetric alteration coordinate. Since the “exact” scheme was found computationally expensive, we examine the possibility to obtain the asymmetric fluctuation of the transfer integral by a Taylor expansion along the coordinate space. By exploring the efficiency of this novel approach, we show that the Taylor expansion scheme represents an attractive alternative to the “exact” calculations due to a substantial reduction of computational costs, when a considerably large region of the potential energy surface is of interest. Moreover, we show that the Taylor expansion scheme, irrespective of the dimer symmetry, is very accurate for the entire range of geometry fluctuations that cover the space the molecule accesses at room temperature.
Efficient yet accurate approximations for ab initio calculations of alcohol cluster thermochemistry.
Umer, Muhammad; Kopp, Wassja A; Leonhard, Kai
2015-12-01
We have calculated the binding enthalpies and entropies of gas phase alcohol clusters from ethanol to 1-decanol. In addition to the monomers, we have investigated dimers, tetramers, and pentamers. Geometries have been obtained at the B3LYP/TZVP level and single point energy calculations have been performed with the Resolution of the Identity-MP2 (RIMP2) method and basis set limit extrapolation using aug-cc-pVTZ and aug-cc-pVQZ basis sets. Thermochemistry is calculated with decoupled hindered rotor treatment for large amplitude motions. The results show three points: First, it is more accurate to transfer the rigid-rotor harmonic oscillator entropies from propanol to longer alcohols than to compute them with an ultra-fine grid and tight geometry convergence criteria. Second, the computational effort can be reduced considerably by using dimerization energies of longer alcohols at density functional theory (B3LYP) level plus a RIMP2 correction obtained from 1-propanol. This approximation yields results almost with the same accuracy as RIMP2 - both methods differ for 1-decanol only 0.4 kJ/mol. Third, the entropy of dimerization including the hindered rotation contribution is converged at 1-propanol with respect to chain length. This allows for a transfer of hindered rotation contributions from smaller alcohols to longer ones which reduces the required computational and man power considerably. PMID:26646881
Accurate deterministic solutions for the classic Boltzmann shock profile
NASA Astrophysics Data System (ADS)
Yue, Yubei
The Boltzmann equation or Boltzmann transport equation is a classical kinetic equation devised by Ludwig Boltzmann in 1872. It is regarded as a fundamental law in rarefied gas dynamics. Rather than using macroscopic quantities such as density, temperature, and pressure to describe the underlying physics, the Boltzmann equation uses a distribution function in phase space to describe the physical system, and all the macroscopic quantities are weighted averages of the distribution function. The information contained in the Boltzmann equation is surprisingly rich, and the Euler and Navier-Stokes equations of fluid dynamics can be derived from it using series expansions. Moreover, the Boltzmann equation can reach regimes far from the capabilities of fluid dynamical equations, such as the realm of rarefied gases---the topic of this thesis. Although the Boltzmann equation is very powerful, it is extremely difficult to solve in most situations. Thus the only hope is to solve it numerically. But soon one finds that even a numerical simulation of the equation is extremely difficult, due to both the complex and high-dimensional integral in the collision operator, and the hyperbolic phase-space advection terms. For this reason, until few years ago most numerical simulations had to rely on Monte Carlo techniques. In this thesis I will present a new and robust numerical scheme to compute direct deterministic solutions of the Boltzmann equation, and I will use it to explore some classical gas-dynamical problems. In particular, I will study in detail one of the most famous and intrinsically nonlinear problems in rarefied gas dynamics, namely the accurate determination of the Boltzmann shock profile for a gas of hard spheres.
Are Quasi-Steady-State Approximated Models Suitable for Quantifying Intrinsic Noise Accurately?
Sengupta, Dola; Kar, Sandip
2015-01-01
Large gene regulatory networks (GRN) are often modeled with quasi-steady-state approximation (QSSA) to reduce the huge computational time required for intrinsic noise quantification using Gillespie stochastic simulation algorithm (SSA). However, the question still remains whether the stochastic QSSA model measures the intrinsic noise as accurately as the SSA performed for a detailed mechanistic model or not? To address this issue, we have constructed mechanistic and QSSA models for few frequently observed GRNs exhibiting switching behavior and performed stochastic simulations with them. Our results strongly suggest that the performance of a stochastic QSSA model in comparison to SSA performed for a mechanistic model critically relies on the absolute values of the mRNA and protein half-lives involved in the corresponding GRN. The extent of accuracy level achieved by the stochastic QSSA model calculations will depend on the level of bursting frequency generated due to the absolute value of the half-life of either mRNA or protein or for both the species. For the GRNs considered, the stochastic QSSA quantifies the intrinsic noise at the protein level with greater accuracy and for larger combinations of half-life values of mRNA and protein, whereas in case of mRNA the satisfactory accuracy level can only be reached for limited combinations of absolute values of half-lives. Further, we have clearly demonstrated that the abundance levels of mRNA and protein hardly matter for such comparison between QSSA and mechanistic models. Based on our findings, we conclude that QSSA model can be a good choice for evaluating intrinsic noise for other GRNs as well, provided we make a rational choice based on experimental half-life values available in literature. PMID:26327626
Accurate thermal imaging of low-emissivity surfaces using approximate blackbody cavities
NASA Astrophysics Data System (ADS)
Turner, S. Fiona; Metcalfe, Stuart F.; Mellor, Andrew; Willmott, Jon; Drögmöller, Peter
2012-06-01
Remote temperature sensing and thermal imaging can be invaluable tools for process control and optimization. Their utilization is limited within the metal processing industries, however, as bright metal surfaces are highly reflective, with low emissivity that can vary critically with oxide thickness and alloy composition. Any infrared temperature measurement is vulnerable to background reflection and limited to the uncertainty in the emissivity. An enclosure or cavity made of any material offers an approximation to blackbody radiation, as both emitted and reflected radiation are collected within the cavity, and background radiation is excluded by the geometry. By exploiting natural cavities formed during processing, emissivity-independent measurements can be made. This paper presents thermal imaging data from an aluminum rolling application. Data was gathered using Land's FTI-E imaging system. Based on an uncooled amorphous silicon array, the system provides measurement in the range 200°C to 600°C to an accuracy of +/-1°C. The 320 x 240 pixels each have field of view 570:1, providing a total viewing angle of 32° by 24°. Data was processed by Land's LIPS ASPS software, which features a patented algorithm for identifying the area of true temperature measurement within the cavity. The software automatically locates the wedge as the strip is coiled, and tracks its position as the coil increases in size. Successive profile graphs are collated to form a '2D map' of the whole strip. The results demonstrate that accurate, emissivity-independent temperature measurements can be obtained from the wedge-shaped cavity formed where the sheet aluminum joins the roll.
NASA Astrophysics Data System (ADS)
Zhang, Gang; Zhou, Di; Mortari, Daniele
2012-12-01
A new approximate analytical method for the two-body impulsive orbit rendezvous problem with short range is presented. The classical analytical approach derives the initial relative velocity from the state transition matrix of linear relative motion equations. This paper proposes a different analytical approach based on the relative Lambert solutions. An approximate expression for the transfer time is obtained as a function of chaser's and target's semi-major axes difference. This results in first and second order estimates of the chaser's semi-major axis. Singularity points of rendezvous time for the classical and proposed new methods are both analyzed. As compared with the classical method, the new solution is simpler, more accurate, and has fewer singularity points. Moreover, the proposed method can be easily expanded to higher order solutions. A numerical example quantifies the accuracy gain for multiple-revolution cases.
Parametric study of the Orbiter rollout using an approximate solution
NASA Technical Reports Server (NTRS)
Garland, B. J.
1979-01-01
An approximate solution to the motion of the Orbiter during rollout is used to perform a parametric study of the rollout distance required by the Orbiter. The study considers the maximum expected dispersions in the landing speed and the touchdown point. These dispersions are assumed to be correlated so that a fast landing occurs before the nominal touchdown point. The maximum rollout distance is required by the maximum landing speed with a 10 knot tailwind and the center of mass at the forward limit of its longitudinal travel. The maximum weight that can be stopped within 15,000 feet on a hot day at Kennedy Space Center is 248,800 pounds. The energy absorbed by the brakes would exceed the limit for reuse of the brakes.
Approximate explicit analytic solution of the Elenbaas-Heller equation
NASA Astrophysics Data System (ADS)
Liao, Meng-Ran; Li, Hui; Xia, Wei-Dong
2016-08-01
The Elenbaas-Heller equation describing the temperature field of a cylindrically symmetrical non-radiative electric arc has been solved, and approximate explicit analytic solutions are obtained. The radial distributions of the heat-flux potential and the electrical conductivity have been figured out briefly by using some special simplification techniques. The relations between both the core heat-flux potential and the electric field with the total arc current have also been given in several easy explicit formulas. Besides, the special voltage-ampere characteristic of electric arcs is explained intuitionally by a simple expression involving the Lambert W-function. The analyses also provide a preliminary estimation of the Joule heating per unit length, which has been verified in previous investigations. Helium arc is used to examine the theories, and the results agree well with the numerical computations.
Reaction fronts in a porous medium. Approximation techniques versus numerical solution
Escobedo, F.; Viljoen, H.J.
1995-03-01
The combustion of a gas fuel inside a porous media, as found in radiant porous burners (PRB), is one of the most promising novel combustion concepts. The flame sheet approximation (FS) and a novel polynomial approximation technique (PA) are compared in terms of their capability to describe reaction fronts of highly exothermic reactions in a porous medium. A one-phase model and a two-phase model of a system with adiabatic walls and a radiant output (to approximate the case of a porous radiant burner) are included in the analysis. By matching the reaction zone solution found by either the FS or PA method with the solutions of the nonreacting zones, the temperature, conversion, and position of the reaction zone were determined. Numerical solutions for catalytic and noncatalytic oxidation reactions were used to compare the predictions of both approaches. It was found that although both techniques yielded good approximations to the solutions, the PA technique proved to be more accurate, producing results with 3.5% of the numerical results. Both methods can find useful application in the analysis of this class of problems.
Zaslawsky, M.; Kennedy, W.N.
1992-09-30
Mathematical solutions to the problem consisting of a partially-full waste tank subjected to seismic loading, embedded in soil, is classically difficult in that one has to address: soil-structure interaction, fluid-structure interaction, non-linear behavior of material, dynamic effects. Separating the problem and applying numerous assumptions will yield approximate solutions. This paper explores methods for generating these solutions accurately.
Gu, S
2016-08-01
Despite its low accuracy and consistency, growing degree days (GDD) has been widely used to approximate growing heat summation (GHS) for regional classification and phenological prediction. GDD is usually calculated from the mean of daily minimum and maximum temperatures (GDDmm) above a growing base temperature (T gb). To determine approximation errors and accuracy, daily and cumulative GDDmm was compared to GDD based on daily average temperature (GDDavg), growing degree hours (GDH) based on hourly temperatures, and growing degree minutes (GDM) based on minute-by-minute temperatures. Finite error, due to the difference between measured and true temperatures above T gb is large in GDDmm but is negligible in GDDavg, GDH, and GDM, depending only upon the number of measured temperatures used for daily approximation. Hidden negative error, due to the temperatures below T gb when being averaged for approximation intervals larger than measuring interval, is large in GDDmm and GDDavg but is negligible in GDH and GDM. Both GDH and GDM improve GHS approximation accuracy over GDDmm or GDDavg by summation of multiple integration rectangles to reduce both finite and hidden negative errors. GDH is proposed as the standardized GHS approximation protocol, providing adequate accuracy and high precision independent upon T gb while requiring simple data recording and processing. PMID:26589826
NASA Astrophysics Data System (ADS)
Gu, S.
2016-08-01
Despite its low accuracy and consistency, growing degree days (GDD) has been widely used to approximate growing heat summation (GHS) for regional classification and phenological prediction. GDD is usually calculated from the mean of daily minimum and maximum temperatures (GDDmm) above a growing base temperature ( T gb). To determine approximation errors and accuracy, daily and cumulative GDDmm was compared to GDD based on daily average temperature (GDDavg), growing degree hours (GDH) based on hourly temperatures, and growing degree minutes (GDM) based on minute-by-minute temperatures. Finite error, due to the difference between measured and true temperatures above T gb is large in GDDmm but is negligible in GDDavg, GDH, and GDM, depending only upon the number of measured temperatures used for daily approximation. Hidden negative error, due to the temperatures below T gb when being averaged for approximation intervals larger than measuring interval, is large in GDDmm and GDDavg but is negligible in GDH and GDM. Both GDH and GDM improve GHS approximation accuracy over GDDmm or GDDavg by summation of multiple integration rectangles to reduce both finite and hidden negative errors. GDH is proposed as the standardized GHS approximation protocol, providing adequate accuracy and high precision independent upon T gb while requiring simple data recording and processing.
NASA Astrophysics Data System (ADS)
Mathias, Simon A.; Moutsopoulos, Konstantinos N.
2016-07-01
Understanding the hydraulics around injection and production wells in unconfined aquifers associated with rainwater and reclaimed water aquifer storage schemes is an issue of increasing importance. Much work has been done previously to understand the mathematics associated with Darcy's law in this context. However, groundwater flow velocities around injection and production wells are likely to be sufficiently large such as to induce significant non-Darcy effects. This article presents a mathematical analysis to look at Forchheimer's equation in the context of water injection and water production in unconfined aquifers. Three different approximate solutions are derived using quasi-steady-state assumptions and the method of matched asymptotic expansion. The resulting approximate solutions are shown to be accurate for a wide range of practical scenarios by comparison with a finite difference solution to the full problem of concern. The approximate solutions have led to an improved understanding of the flow dynamics. They can also be used as verification tools for future numerical models in this context.
In dealing with problems related to land-based nuclear waste management, a number of analytical and approximate solutions were developed to quantify radionuclide transport through fractures contained in the porous formation. t has been reported that by treating the radioactive de...
Bonetto, Paola; Qi, Jinyi; Leahy, Richard M.
1999-10-01
We describe a method for computing linear observer statistics for maximum a posteriori (MAP) reconstructions of PET images. The method is based on a theoretical approximation for the mean and covariance of MAP reconstructions. In particular, we derive here a closed form for the channelized Hotelling observer (CHO) statistic applied to 2D MAP images. We show reasonably good correspondence between these theoretical results and Monte Carlo studies. The accuracy and low computational cost of the approximation allow us to analyze the observer performance over a wide range of operating conditions and parameter settings for the MAP reconstruction algorithm.
Accurate solution of the Dirac equation on Lagrange meshes.
Baye, Daniel; Filippin, Livio; Godefroid, Michel
2014-04-01
The Lagrange-mesh method is an approximate variational method taking the form of equations on a grid because of the use of a Gauss quadrature approximation. With a basis of Lagrange functions involving associated Laguerre polynomials related to the Gauss quadrature, the method is applied to the Dirac equation. The potential may possess a 1/r singularity. For hydrogenic atoms, numerically exact energies and wave functions are obtained with small numbers n+1 of mesh points, where n is the principal quantum number. Numerically exact mean values of powers -2 to 3 of the radial coordinate r can also be obtained with n+2 mesh points. For the Yukawa potential, a 15-digit agreement with benchmark energies of the literature is obtained with 50 or fewer mesh points. PMID:24827362
Accurate solution of the Dirac equation on Lagrange meshes
NASA Astrophysics Data System (ADS)
Baye, Daniel; Filippin, Livio; Godefroid, Michel
2014-04-01
The Lagrange-mesh method is an approximate variational method taking the form of equations on a grid because of the use of a Gauss quadrature approximation. With a basis of Lagrange functions involving associated Laguerre polynomials related to the Gauss quadrature, the method is applied to the Dirac equation. The potential may possess a 1/r singularity. For hydrogenic atoms, numerically exact energies and wave functions are obtained with small numbers n +1 of mesh points, where n is the principal quantum number. Numerically exact mean values of powers -2 to 3 of the radial coordinate r can also be obtained with n +2 mesh points. For the Yukawa potential, a 15-digit agreement with benchmark energies of the literature is obtained with 50 or fewer mesh points.
Highly accurate boronimeter assay of concentrated boric acid solutions
Ball, R.M. )
1992-01-01
The Random-Walk Boronimeter has successfully been used as an on-line indicator of boric acid concentration in an operating commercial pressurized water reactor. The principle has been adapted for measurement of discrete samples to high accuracy and to concentrations up to 6000 ppm natural boron in light water. Boric acid concentration in an aqueous solution is a necessary measurement in many nuclear power plants, particularly those that use boric acid dissolved in the reactor coolant as a reactivity control system. Other nuclear plants use a high-concentration boric acid solution as a backup shutdown system. Such a shutdown system depends on rapid injection of the solution and frequent surveillance of the fluid to ensure the presence of the neutron absorber. The two methods typically used to measure boric acid are the chemical and the physical methods. The chemical method uses titration to determine the ionic concentration of the BO[sub 3] ions and infers the boron concentration. The physical method uses the attenuation of neutrons by the solution and infers the boron concentration from the neutron absorption properties. This paper describes the Random-Walk Boronimeter configured to measure discrete samples to high accuracy and high concentration.
NASA Astrophysics Data System (ADS)
Wu, Dongmei; Wang, Zhongcheng
2006-03-01
, we present a new iteration algorithm to calculate the coefficients of the Fourier series. By using this new method, the iteration procedure starts with a(x)cos(ωx)+b(x)sin(ωx), and the accuracy may be improved gradually by determining new coefficients a,a,… will be produced automatically in an one-by-one manner. In all the stage of calculation, we need only to solve a cubic equation. Using this new algorithm, we develop a Mathematica program, which demonstrates following main advantages over the previous HB method: (1) it avoids solving a set of associate nonlinear equations; (2) it is easier to be implemented into a computer program, and produces a highly accurate solution with analytical expression efficiently. It is interesting to find that, generally, for a given set of parameters, a nonlinear Duffing equation can have three independent oscillation modes. For some sets of the parameters, it can have two modes with complex displacement and one with real displacement. But in some cases, it can have three modes, all of them having real displacement. Therefore, we can divide the parameters into two classes, according to the solution property: there is only one mode with real displacement and there are three modes with real displacement. This program should be useful to study the dynamically periodic behavior of a Duffing oscillator and can provide an approximate analytical solution with high-accuracy for testing the error behavior of newly developed numerical methods with a wide range of parameters. Program summaryTitle of program:AnalyDuffing.nb Catalogue identifier:ADWR_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADWR_v1_0 Program obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland Licensing provisions:none Computer for which the program is designed and others on which it has been tested:the program has been designed for a microcomputer and been tested on the microcomputer. Computers:IBM PC Installations:the address
Accurate integral equation theory for the central force model of liquid water and ionic solutions
NASA Astrophysics Data System (ADS)
Ichiye, Toshiko; Haymet, A. D. J.
1988-10-01
The atom-atom pair correlation functions and thermodynamics of the central force model of water, introduced by Lemberg, Stillinger, and Rahman, have been calculated accurately by an integral equation method which incorporates two new developments. First, a rapid new scheme has been used to solve the Ornstein-Zernike equation. This scheme combines the renormalization methods of Allnatt, and Rossky and Friedman with an extension of the trigonometric basis-set solution of Labik and co-workers. Second, by adding approximate ``bridge'' functions to the hypernetted-chain (HNC) integral equation, we have obtained predictions for liquid water in which the hydrogen bond length and number are in good agreement with ``exact'' computer simulations of the same model force laws. In addition, for dilute ionic solutions, the ion-oxygen and ion-hydrogen coordination numbers display both the physically correct stoichiometry and good agreement with earlier simulations. These results represent a measurable improvement over both a previous HNC solution of the central force model and the ex-RISM integral equation solutions for the TIPS and other rigid molecule models of water.
Homotopic Approximate Solutions for the Perturbed CKdV Equation with Variable Coefficients
Lu, Dianchen; Chen, Tingting
2014-01-01
This work concerns how to find the double periodic form of approximate solutions of the perturbed combined KdV (CKdV) equation with variable coefficients by using the homotopic mapping method. The obtained solutions may degenerate into the approximate solutions of hyperbolic function form and the approximate solutions of trigonometric function form in the limit cases. Moreover, the first order approximate solutions and the second order approximate solutions of the variable coefficients CKdV equation in perturbation εun are also induced. PMID:24737983
Homotopic approximate solutions for the perturbed CKdV equation with variable coefficients.
Lu, Dianchen; Chen, Tingting; Hong, Baojian
2014-01-01
This work concerns how to find the double periodic form of approximate solutions of the perturbed combined KdV (CKdV) equation with variable coefficients by using the homotopic mapping method. The obtained solutions may degenerate into the approximate solutions of hyperbolic function form and the approximate solutions of trigonometric function form in the limit cases. Moreover, the first order approximate solutions and the second order approximate solutions of the variable coefficients CKdV equation in perturbation εu (n) are also induced. PMID:24737983
Marelli, Damián; Baumgartner, Robert; Majdak, Piotr
2015-01-01
Head-related transfer functions (HRTFs) describe the acoustic filtering of incoming sounds by the human morphology and are essential for listeners to localize sound sources in virtual auditory displays. Since rendering complex virtual scenes is computationally demanding, we propose four algorithms for efficiently representing HRTFs in subbands, i.e., as an analysis filterbank (FB) followed by a transfer matrix and a synthesis FB. All four algorithms use sparse approximation procedures to minimize the computational complexity while maintaining perceptually relevant HRTF properties. The first two algorithms separately optimize the complexity of the transfer matrix associated to each HRTF for fixed FBs. The other two algorithms jointly optimize the FBs and transfer matrices for complete HRTF sets by two variants. The first variant aims at minimizing the complexity of the transfer matrices, while the second one does it for the FBs. Numerical experiments investigate the latency-complexity trade-off and show that the proposed methods offer significant computational savings when compared with other available approaches. Psychoacoustic localization experiments were modeled and conducted to find a reasonable approximation tolerance so that no significant localization performance degradation was introduced by the subband representation. PMID:26681930
Accurate solutions for transonic viscous flow over finite wings
NASA Technical Reports Server (NTRS)
Vatsa, V. N.
1986-01-01
An explicit multistage Runge-Kutta type time-stepping scheme is used for solving the three-dimensional, compressible, thin-layer Navier-Stokes equations. A finite-volume formulation is employed to facilitate treatment of complex grid topologies encountered in three-dimensional calculations. Convergence to steady state is expedited through usage of acceleration techniques. Further numerical efficiency is achieved through vectorization of the computer code. The accuracy of the overall scheme is evaluated by comparing the computed solutions with the experimental data for a finite wing under different test conditions in the transonic regime. A grid refinement study ir conducted to estimate the grid requirements for adequate resolution of salient features of such flows.
Kottmann, Jakob S; Höfener, Sebastian; Bischoff, Florian A
2015-12-21
In the present work, we report an efficient implementation of configuration interaction singles (CIS) excitation energies and oscillator strengths using the multi-resolution analysis (MRA) framework to address the basis-set convergence of excited state computations. In MRA (ground-state) orbitals, excited states are constructed adaptively guaranteeing an overall precision. Thus not only valence but also, in particular, low-lying Rydberg states can be computed with consistent quality at the basis set limit a priori, or without special treatments, which is demonstrated using a small test set of organic molecules, basis sets, and states. We find that the new implementation of MRA-CIS excitation energy calculations is competitive with conventional LCAO calculations when the basis-set limit of medium-sized molecules is sought, which requires large, diffuse basis sets. This becomes particularly important if accurate calculations of molecular electronic absorption spectra with respect to basis-set incompleteness are required, in which both valence as well as Rydberg excitations can contribute to the molecule's UV/VIS fingerprint. PMID:25913482
NASA Astrophysics Data System (ADS)
Chung, H. Y.; Guo, G. Y.; Chiang, H.-P.; Tsai, D. P.; Leung, P. T.
2010-10-01
The optical response of a multilayered spherical system of unlimited number of layers (a “matryushka”) in the long wavelength limit can be accounted for from the knowledge of the static multipole polarizability of the system to first-order accuracy. However, for systems of ultrasmall dimensions or systems with sizes not-too-small compared to the wavelength, this ordinary quasistatic long wavelength approximation (LWA) becomes inaccurate. Here we introduce two significant modifications of the LWA for such a nanomatryushka in each of the two limits: the nonlocal optical response for ultrasmall systems (<10nm) , and the “finite-wavelength corrections” for systems ˜100nm . This is accomplished by employing the previous work for a single-layer shell, in combination with a certain effective-medium approach formulated recently in the literature. Numerical calculations for the extinction cross sections for such a system of different dimensions are provided as illustrations for these effects. This formulation thus provides significant improvements on the ordinary LWA, yielding enough accuracy for the description of the optical response of these nanoshell systems over an appreciable range of sizes, without resorting to more involved quantum mechanical or fully electrodynamic calculations.
NASA Astrophysics Data System (ADS)
Ghanbarian, Behzad; Daigle, Hugh; Hunt, Allen G.; Ewing, Robert P.; Sahimi, Muhammad
2015-01-01
Understanding and accurate prediction of gas or liquid phase (solute) diffusion are essential to accurate prediction of contaminant transport in partially saturated porous media. In this study, we propose analytical equations, using concepts from percolation theory and the Effective Medium Approximation (EMA) to model the saturation dependence of both gas and solute diffusion in porous media. The predictions of our theoretical approach agree well with the results of nine lattice Boltzmann simulations. We find that the universal quadratic scaling predicted by percolation theory, combined with the universal linear scaling predicted by the EMA, describes diffusion in porous media with both relatively broad and extremely narrow pore size distributions.
An Approximate Analytical Solution for Backward-Facing Step Flow
NASA Astrophysics Data System (ADS)
Celik, Ismail; Parsons, Don; Karaismail, Ertan; Nanduri, Jagannath
2007-11-01
Flow past a backward facing step is a classical bench mark for both laminar and turbulent flow calculations. Due to the near-singular behavior arising from the presence of the sharp step, it is very difficult to predict the size of the recirculation region and the reattachment length. This difficulty, in turn, manifests itself as a significant discrepancy between predicted and measured velocity profiles. The aim of the current work is to formulate an analytical solution to the 2D, steady flow in question that satisfies the Navier-Stokes equations with a source term. The proposed solution is a superposition of two stream functions, one being a semi-potential solution that satisfies all the boundary conditions for real incompressible fluids, and another composed of rotational vortices (e.g. Rankine vortices) which enable flow separation. The location and distribution of the vortices is selected to emulate the Reynolds number dependence of the re-attachment length, while other parameters in the model are used to minimize the additional source term that is needed. The proposed solution can be primarily used in code-verification, and quantification of discretization errors in CFD (Computational Fluid Dynamics). It can also be used to assess modeling errors, by adding additional source terms that represent the spatial variations in turbulent-eddy viscosity, the key quantity used in Boussinesq-type turbulence models.
NASA Technical Reports Server (NTRS)
Hagstrom, Thomas; Hariharan, S. I.; Maccamy, R. C.
1993-01-01
We consider the solution of scattering problems for the wave equation using approximate boundary conditions at artificial boundaries. These conditions are explicitly viewed as approximations to an exact boundary condition satisfied by the solution on the unbounded domain. We study the short and long term behavior of the error. It is provided that, in two space dimensions, no local in time, constant coefficient boundary operator can lead to accurate results uniformly in time for the class of problems we consider. A variable coefficient operator is developed which attains better accuracy (uniformly in time) than is possible with constant coefficient approximations. The theory is illustrated by numerical examples. We also analyze the proposed boundary conditions using energy methods, leading to asymptotically correct error bounds.
Exact solutions of regular approximate relativistic wave equations for hydrogen-like atoms
NASA Astrophysics Data System (ADS)
van Leeuwen, R.; van Lenthe, E.; Baerends, E. J.; Snijders, J. G.
1994-07-01
Apart from relativistic effects originating from high kinetic energy of an electron in a flat potential, which are treated in first order by the Pauli Hamiltonian, there are relativistic effects even for low-energy electrons if they move in a strong Coulomb potential. The latter effects can be accurately treated already in the zeroth order of an expansion of the Foldy-Wouthuysen transformation, if the expansion is carefully chosen to be nondivergent for r→0 even for Coulomb potentials, as shown by Van Lenthe et al. [J. Chem. Phys. 99, 4597 (1993)] (cf. also Heully et al. [J. Phys. B 19, 2799 (1986)] and Chang et al. [Phys. Scr. 34, 394 (1986)]). In the present paper, it is shown that the solutions of the zeroth order of this two-component regular approximate (ZORA) equation for hydrogen-like atoms are simply scaled solutions of the large component of the Dirac wave function for this problem. The eigenvalues are related in a similar way. As a consequence, it is proven that under some restrictions, the ZORA Hamiltonian is bounded from below for Coulomb-like potentials. Also, an exact result for the first order regular approximate Hamiltonian is given. The method can also be used to obtain exact results for regular approximations of scalar relativistic equations, like the Klein-Gordon equation. The balance between relativistic effects originating from the Coulombic singularity in the potential (typically core penetrating s and p valence electrons in atoms and molecules) and from high kinetic energy (important for high-energy electrons in a flat potential and also for core-avoiding high angular momentum (d, f, and g states in atoms) are discussed.
NASA Astrophysics Data System (ADS)
Bause, Markus
2008-02-01
In this work we study mixed finite element approximations of Richards' equation for simulating variably saturated subsurface flow and simultaneous reactive solute transport. Whereas higher order schemes have proved their ability to approximate reliably reactive solute transport (cf., e.g. [Bause M, Knabner P. Numerical simulation of contaminant biodegradation by higher order methods and adaptive time stepping. Comput Visual Sci 7;2004:61-78]), the Raviart- Thomas mixed finite element method ( RT0) with a first order accurate flux approximation is popular for computing the underlying water flow field (cf. [Bause M, Knabner P. Computation of variably saturated subsurface flow by adaptive mixed hybrid finite element methods. Adv Water Resour 27;2004:565-581, Farthing MW, Kees CE, Miller CT. Mixed finite element methods and higher order temporal approximations for variably saturated groundwater flow. Adv Water Resour 26;2003:373-394, Starke G. Least-squares mixed finite element solution of variably saturated subsurface flow problems. SIAM J Sci Comput 21;2000:1869-1885, Younes A, Mosé R, Ackerer P, Chavent G. A new formulation of the mixed finite element method for solving elliptic and parabolic PDE with triangular elements. J Comp Phys 149;1999:148-167, Woodward CS, Dawson CN. Analysis of expanded mixed finite element methods for a nonlinear parabolic equation modeling flow into variably saturated porous media. SIAM J Numer Anal 37;2000:701-724]). This combination might be non-optimal. Higher order techniques could increase the accuracy of the flow field calculation and thereby improve the prediction of the solute transport. Here, we analyse the application of the Brezzi- Douglas- Marini element ( BDM1) with a second order accurate flux approximation to elliptic, parabolic and degenerate problems whose solutions lack the regularity that is assumed in optimal order error analyses. For the flow field calculation a superiority of the BDM1 approach to the RT0 one is
NASA Astrophysics Data System (ADS)
Afanas'ev, A. P.; Dzyuba, S. M.
2015-10-01
A method for constructing approximate analytic solutions of systems of ordinary differential equations with a polynomial right-hand side is proposed. The implementation of the method is based on the Picard method of successive approximations and a procedure of continuation of local solutions. As an application, the problem of constructing the minimal sets of the Lorenz system is considered.
On comparison of accuracy of approximate solutions of operator equations with noisy data
NASA Astrophysics Data System (ADS)
Hämarik, Uno
2016-06-01
We consider an operator equation with noisy data where the noise level is given. The case of noisy data is especially actual in ill-posed problems. We formulate a criterion for comparison of accuracy of two approximate solutions get by arbitrary (different) methods. This generalizes previous results about monotonicity of error of approximate solutions generated by the same method but using different parameters.
Tian, Lian; Henningsen, Joseph; Salick, Max R; Crone, Wendy C; Gunderson, McLean; Dailey, Seth H; Chesler, Naomi C
2015-07-01
The mechanical properties of vascular tissues affect hemodynamics and can alter disease progression. The uniaxial tensile test is a simple and effective method for determining the stress-strain relationship in arterial tissue ex vivo. To enable calculation of strain, stretch can be measured directly with image tracking of markers on the tissue or indirectly from the distance between the grips used to hold the specimen. While the imaging technique is generally considered more accurate, it also requires more analysis, and the grip distance method is more widely used. The purpose of this study is to compare the stretch of the testing specimen calculated from the grip distance method to that obtained from the imaging method for canine descending aortas and large proximal pulmonary arteries. Our results showed a significant difference in stretch between the two methods; however, this difference was consistently less than 2%. Therefore, the grip distance method is an accurate approximation of the stretch in large elastic arteries in the uniaxial tensile test. PMID:25881308
Tian, Lian; Henningsen, Joseph; Salick, Max R.; Crone, Wendy C.; Gunderson, McLean; Dailey, Seth H.; Chesler, Naomi C.
2015-01-01
The mechanical properties of vascular tissues affect hemodynamics and can alter disease progression. The uniaxial tensile test is a simple and effective method for determining the stress-strain relationship in arterial tissue ex vivo. To enable calculation of strain, stretch can be measured directly with image tracking of markers on the tissue or indirectly from the distance between the grips used to hold the specimen. While the imaging technique is generally considered more accurate, it also requires more analysis, and the grip distance method is more widely used. The purpose of this study is to compare the stretch of the testing specimen calculated from the grip distance method to that obtained from the imaging method for canine descending aortas and large proximal pulmonary arteries. Our results showed a significant difference in stretch between the two methods; however, this difference was consistently less than 2%. Therefore, the grip distance method is an accurate approximation of the stretch in large elastic arteries in the uniaxial tensile test. PMID:25881308
Sharapov, Vladimir A; Mandelshtam, Vladimir A
2007-10-18
We consider systems undergoing very-low-temperature solid-solid transitions associated with minima of similar energy but different symmetry, and separated by a high potential barrier. In such cases the well-known "broken-ergodicity" problem is often difficult to overcome, even using the most advanced Monte Carlo (MC) techniques, including the replica exchange method (REM). The methodology that we develop in this paper is suitable for the above specified cases and is numerically accurate and efficient. It is based on a new MC move implemented within the REM framework, in which trial points are generated analytically using an auxiliary harmonic superposition system that mimics well the true system at low temperatures. Due to the new move, the low-temperature random walks are able to frequently switch the relevant potential energy funnels leading to an efficient sampling. Numerically accurate results are obtained for a number of Lennard-Jones clusters, including those that have so far been treated only by the harmonic superposition approximation (HSA). The latter is believed to provide good estimates for low-temperature equilibrium properties but is manifestly uncontrollable and is difficult to validate. The present results provide a good test for the HSA and demonstrate its reliability, particularly for estimation of the solid-solid transition temperatures in most cases considered. PMID:17685597
NASA Astrophysics Data System (ADS)
Tao, Jianmin; Mo, Yuxiang; Tian, Guocai; Ruzsinszky, Adrienn
2016-08-01
Long-range van der Waals (vdW) interaction is critically important for intermolecular interactions in molecular complexes and solids. However, accurate modeling of vdW coefficients presents a great challenge for nanostructures, in particular for fullerene clusters, which have huge vdW coefficients but also display very strong nonadditivity. In this work, we calculate the coefficients between fullerenes, fullerene and sodium clusters, and fullerene and alkali atoms with the hollow-sphere model within the modified single-frequency approximation (MSFA). In the MSFA, we assume that the electron density is uniform in a molecule and that only valence electrons in the outmost subshell of atoms contribute. The input to the model is the static multipole polarizability, which provides a sharp cutoff for the plasmon contribution outside the effective vdW radius. We find that the model can generate C6 in excellent agreement with expensive wave-function-based ab initio calculations, with a mean absolute relative error of only 3 % , without suffering size-dependent error. We show that the nonadditivities of the coefficients C6 between fullerenes and C60 and sodium clusters Nan revealed by the model agree remarkably well with those based on the accurate reference values. The great flexibility, simplicity, and high accuracy make the model particularly suitable for the study of the nonadditivity of vdW coefficients between nanostructures, advancing the development of better vdW corrections to conventional density functional theory.
A third-order-accurate upwind scheme for Navier-Stokes solutions at high Reynolds numbers
NASA Astrophysics Data System (ADS)
Agarwal, R. K.
1981-01-01
A third-order-accurate upwind scheme is presented for solution of the steady two-dimensional Navier-Stokes equations in stream-function/vorticity form. The scheme is found to be accurate and stable at high Reynolds numbers. A series of test computations is performed on flows with large recirculating regions. In particular, highly accurate solutions are obtained for flow in a driven square cavity up to Reynolds numbers of 10,000. These computations are used to critically evaluate the accuracy of other existing first- and second-order-accurate upwind schemes. In addition, computations are carried out for flow in a channel with symmetric sudden expansion, flow in a channel with a symmetrically placed blunt base, and the flowfield of an impinging jet. Good agreement is obtained with the computations of other investigators as well as with the available experimental data.
NASA Astrophysics Data System (ADS)
Ngo-Cong, D.; Mohammed, F. J.; Strunin, D. V.; Skvortsov, A. T.; Mai-Duy, N.; Tran-Cong, T.
2015-06-01
The contaminant transport process governed by the advection-diffusion equation plays an important role in modelling industrial and environmental flows. In this article, our aim is to accurately reduce the 2-D advection-diffusion equation governing the dispersion of a contaminant in a turbulent open channel flow to its 1-D approximation. The 1-D model helps to quickly estimate the horizontal size of contaminant clouds based on the values of the model coefficients. We derive these coefficients analytically and investigate numerically the model convergence. The derivation is based on the centre manifold theory to obtain successively more accurate approximations in a consistent manner. Two types of the average velocity profile are considered: the classical logarithmic profile and the power profile. We further develop the one-dimensional integrated radial basis function network method as a numerical approach to obtain the numerical solutions to both the original 2-D equation and the approximate 1-D equations. We compare the solutions of the original models with their centre-manifold approximations at very large Reynolds numbers. The numerical results obtained from the approximate 1-D models are in good agreement with those of the original 2-D model for both the logarithmic and power velocity profiles.
Large deflection of clamped circular plate and accuracy of its approximate analytical solutions
NASA Astrophysics Data System (ADS)
Zhang, Yin
2016-02-01
A different set of governing equations on the large deflection of plates are derived by the principle of virtual work (PVW), which also leads to a different set of boundary conditions. Boundary conditions play an important role in determining the computation accuracy of the large deflection of plates. Our boundary conditions are shown to be more appropriate by analyzing their difference with the previous ones. The accuracy of approximate analytical solutions is important to the bulge/blister tests and the application of various sensors with the plate structure. Different approximate analytical solutions are presented and their accuracies are evaluated by comparing them with the numerical results. The error sources are also analyzed. A new approximate analytical solution is proposed and shown to have a better approximation. The approximate analytical solution offers a much simpler and more direct framework to study the plate-membrane transition behavior of deflection as compared with the previous approaches of complex numerical integration.
NASA Astrophysics Data System (ADS)
Barrett, Steven R. H.; Britter, Rex E.
Predicting long-term mean pollutant concentrations in the vicinity of airports, roads and other industrial sources are frequently of concern in regulatory and public health contexts. Many emissions are represented geometrically as ground-level line or area sources. Well developed modelling tools such as AERMOD and ADMS are able to model dispersion from finite (i.e. non-point) sources with considerable accuracy, drawing upon an up-to-date understanding of boundary layer behaviour. Due to mathematical difficulties associated with line and area sources, computationally expensive numerical integration schemes have been developed. For example, some models decompose area sources into a large number of line sources orthogonal to the mean wind direction, for which an analytical (Gaussian) solution exists. Models also employ a time-series approach, which involves computing mean pollutant concentrations for every hour over one or more years of meteorological data. This can give rise to computer runtimes of several days for assessment of a site. While this may be acceptable for assessment of a single industrial complex, airport, etc., this level of computational cost precludes national or international policy assessments at the level of detail available with dispersion modelling. In this paper, we extend previous work [S.R.H. Barrett, R.E. Britter, 2008. Development of algorithms and approximations for rapid operational air quality modelling. Atmospheric Environment 42 (2008) 8105-8111] to line and area sources. We introduce approximations which allow for the development of new analytical solutions for long-term mean dispersion from line and area sources, based on hypergeometric functions. We describe how these solutions can be parameterized from a single point source run from an existing advanced dispersion model, thereby accounting for all processes modelled in the more costly algorithms. The parameterization method combined with the analytical solutions for long-term mean
An Accurate Upper Bound Solution for Plane Strain Extrusion through a Wedge-Shaped Die
Mustafa, Yusof; Lyamina, Elena
2014-01-01
An upper bound method for the process of plane strain extrusion through a wedge-shaped die is derived. A technique for constructing a kinematically admissible velocity field satisfying the exact asymptotic singular behavior of real velocity fields in the vicinity of maximum friction surfaces (the friction stress at sliding is equal to the shear yield stress on such surfaces) is described. Two specific upper bound solutions are found using the method derived. The solutions are compared to an accurate slip-line solution and it is shown that the accuracy of the new method is very high. PMID:25101311
Techniques for correcting approximate finite difference solutions. [applied to transonic flow
NASA Technical Reports Server (NTRS)
Nixon, D.
1979-01-01
A method of correcting finite-difference solutions for the effect of truncation error or the use of an approximate basic equation is presented. Applications to transonic flow problems are described and examples given.
İbiş, Birol
2014-01-01
This paper aims to obtain the approximate solution of time-fractional advection-dispersion equation (FADE) involving Jumarie's modification of Riemann-Liouville derivative by the fractional variational iteration method (FVIM). FVIM provides an analytical approximate solution in the form of a convergent series. Some examples are given and the results indicate that the FVIM is of high accuracy, more efficient, and more convenient for solving time FADEs. PMID:24578662
Ibiş, Birol; Bayram, Mustafa
2014-01-01
This paper aims to obtain the approximate solution of time-fractional advection-dispersion equation (FADE) involving Jumarie's modification of Riemann-Liouville derivative by the fractional variational iteration method (FVIM). FVIM provides an analytical approximate solution in the form of a convergent series. Some examples are given and the results indicate that the FVIM is of high accuracy, more efficient, and more convenient for solving time FADEs. PMID:24578662
The convergence rate of approximate solutions for nonlinear scalar conservation laws
NASA Technical Reports Server (NTRS)
Nessyahu, Haim; Tadmor, Eitan
1991-01-01
The convergence rate is discussed of approximate solutions for the nonlinear scalar conservation law. The linear convergence theory is extended into a weak regime. The extension is based on the usual two ingredients of stability and consistency. On the one hand, the counterexamples show that one must strengthen the linearized L(sup 2)-stability requirement. It is assumed that the approximate solutions are Lip(sup +)-stable in the sense that they satisfy a one-sided Lipschitz condition, in agreement with Oleinik's E-condition for the entropy solution. On the other hand, the lack of smoothness requires to weaken the consistency requirement, which is measured in the Lip'-(semi)norm. It is proved for Lip(sup +)-stable approximate solutions, that their Lip'convergence rate to the entropy solution is of the same order as their Lip'-consistency. The Lip'-convergence rate is then converted into stronger L(sup p) convergence rate estimates.
The convergence rate of approximate solutions for nonlinear scalar conservation laws. Final Report
Nessyahu, HAIM; Tadmor, EITAN.
1991-07-01
The convergence rate is discussed of approximate solutions for the nonlinear scalar conservation law. The linear convergence theory is extended into a weak regime. The extension is based on the usual two ingredients of stability and consistency. On the one hand, the counterexamples show that one must strengthen the linearized L{sup 2}-stability requirement. It is assumed that the approximate solutions are Lip{sup +}-stable in the sense that they satisfy a one-sided Lipschitz condition, in agreement with Oleinik's E-condition for the entropy solution. On the other hand, the lack of smoothness requires to weaken the consistency requirement, which is measured in the Lip'-(semi)norm. It is proved for Lip{sup +}-stable approximate solutions, that their Lip'convergence rate to the entropy solution is of the same order as their Lip'-consistency. The Lip'-convergence rate is then converted into stronger L{sup p} convergence rate estimates.
A novel numerical technique to obtain an accurate solution to the Thomas-Fermi equation
NASA Astrophysics Data System (ADS)
Parand, Kourosh; Yousefi, Hossein; Delkhosh, Mehdi; Ghaderi, Amin
2016-07-01
In this paper, a new algorithm based on the fractional order of rational Euler functions (FRE) is introduced to study the Thomas-Fermi (TF) model which is a nonlinear singular ordinary differential equation on a semi-infinite interval. This problem, using the quasilinearization method (QLM), converts to the sequence of linear ordinary differential equations to obtain the solution. For the first time, the rational Euler (RE) and the FRE have been made based on Euler polynomials. In addition, the equation will be solved on a semi-infinite domain without truncating it to a finite domain by taking FRE as basic functions for the collocation method. This method reduces the solution of this problem to the solution of a system of algebraic equations. We demonstrated that the new proposed algorithm is efficient for obtaining the value of y'(0) , y(x) and y'(x) . Comparison with some numerical and analytical solutions shows that the present solution is highly accurate.
Fall with Linear Drag and Wien's Displacement Law: Approximate Solution and Lambert Function
ERIC Educational Resources Information Center
Vial, Alexandre
2012-01-01
We present an approximate solution for the downward time of travel in the case of a mass falling with a linear drag force. We show how a quasi-analytical solution implying the Lambert function can be found. We also show that solving the previous problem is equivalent to the search for Wien's displacement law. These results can be of interest for…
Approximate analytical solution to the Boussinesq equation with a sloping water-land boundary
NASA Astrophysics Data System (ADS)
Tang, Yuehao; Jiang, Qinghui; Zhou, Chuangbing
2016-04-01
An approximate solution is presented to the 1-D Boussinesq equation (BEQ) characterizing transient groundwater flow in an unconfined aquifer subject to a constant water variation at the sloping water-land boundary. The flow equation is decomposed to a linearized BEQ and a head correction equation. The linearized BEQ is solved using a Laplace transform. By means of the frozen-coefficient technique and Gauss function method, the approximate solution for the head correction equation can be obtained, which is further simplified to a closed-form expression under the condition of local energy equilibrium. The solutions of the linearized and head correction equations are discussed from physical concepts. Especially for the head correction equation, the well posedness of the approximate solution obtained by the frozen-coefficient method is verified to demonstrate its boundedness, which can be further embodied as the upper and lower error bounds to the exact solution of the head correction by statistical analysis. The advantage of this approximate solution is in its simplicity while preserving the inherent nonlinearity of the physical phenomenon. Comparisons between the analytical and numerical solutions of the BEQ validate that the approximation method can achieve desirable precisions, even in the cases with strong nonlinearity. The proposed approximate solution is applied to various hydrological problems, in which the algebraic expressions that quantify the water flow processes are derived from its basic solutions. The results are useful for the quantification of stream-aquifer exchange flow rates, aquifer response due to the sudden reservoir release, bank storage and depletion, and front position and propagation speed.
An approximate analytical solution for interlaminar stresses in angle-ply laminates
NASA Technical Reports Server (NTRS)
Rose, Cheryl A.; Herakovich, Carl T.
1991-01-01
An improved approximate analytical solution for interlaminar stresses in finite width, symmetric, angle-ply laminated coupons subjected to axial loading is presented. The solution is based upon statically admissible stress fields which take into consideration local property mismatch effects and global equilibrium requirements. Unknown constants in the admissible stress states are determined through minimization of the complementary energy. Typical results are presented for through-the-thickness and interlaminar stress distributions for angle-ply laminates. It is shown that the results represent an improved approximate analytical solution for interlaminar stresses.
Construction of approximate analytical solutions to a new class of non-linear oscillator equation
NASA Technical Reports Server (NTRS)
Mickens, R. E.; Oyedeji, K.
1985-01-01
The principle of harmonic balance is invoked in the development of an approximate analytic model for a class of nonlinear oscillators typified by a mass attached to a stretched wire. By assuming that harmonic balance will hold, solutions are devised for a steady state limit cycle and/or limit point motion. A method of slowly varying amplitudes then allows derivation of approximate solutions by determining the form of the exact solutions and substituting into them the lowest order terms of their respective Fourier expansions. The latter technique is actually a generalization of the method proposed by Kryloff and Bogoliuboff (1943).
Accurate solution of the proton-hydrogen three-body scattering problem
NASA Astrophysics Data System (ADS)
Abdurakhmanov, I. B.; Kadyrov, A. S.; Bray, I.
2016-02-01
An accurate solution to the fundamental three-body problem of proton-hydrogen scattering including direct scattering and ionization, electron capture and electron capture into the continuum (ECC) is presented. The problem has been addressed using a quantum-mechanical two-center convergent close-coupling approach. At each energy the internal consistency of the solution is demonstrated with the help of single-center calculations, with both approaches converging independently to the same electron-loss cross section. This is the sum of the electron capture, ECC and direct ionization cross sections, which are only obtainable separately in the solution of the problem using the two-center expansion. Agreement with experiment for the electron-capture cross section is excellent. However, for the ionization cross sections some discrepancy exists. Given the demonstrated internal consistency we remain confident in the provided theoretical solution.
ASYMPTOTICALLY OPTIMAL HIGH-ORDER ACCURATE ALGORITHMS FOR THE SOLUTION OF CERTAIN ELLIPTIC PDEs
Leonid Kunyansky, PhD
2008-11-26
The main goal of the project, "Asymptotically Optimal, High-Order Accurate Algorithms for the Solution of Certain Elliptic PDE's" (DE-FG02-03ER25577) was to develop fast, high-order algorithms for the solution of scattering problems and spectral problems of photonic crystals theory. The results we obtained lie in three areas: (1) asymptotically fast, high-order algorithms for the solution of eigenvalue problems of photonics, (2) fast, high-order algorithms for the solution of acoustic and electromagnetic scattering problems in the inhomogeneous media, and (3) inversion formulas and fast algorithms for the inverse source problem for the acoustic wave equation, with applications to thermo- and opto- acoustic tomography.
NASA Technical Reports Server (NTRS)
Batina, John T.
1992-01-01
A time-accurate approximate-factorization (AF) algorithm is described for solution of the three-dimensional unsteady transonic small-disturbance equation. The AF algorithm consists of a time-linearization procedure coupled with a subiteration technique. The algorithm is the basis for the Computational Aeroelasticity Program-Transonic Small Disturbance (CAP-TSD) computer code, which was developed for the analysis of unsteady aerodynamics and aeroelasticity of realistic aircraft configurations. The paper describes details on the governing flow equations and boundary conditions, with an emphasis on documenting the finite-difference formulas of the AF algorithm.
NASA Technical Reports Server (NTRS)
Weston, K. C.; Reynolds, A. C., Jr.; Alikhan, A.; Drago, D. W.
1974-01-01
Numerical solutions for radiative transport in a class of anisotropically scattering materials are presented. Conditions for convergence and divergence of the iterative method are given and supported by computed results. The relation of two flux theories to the equation of radiative transfer for isotropic scattering is discussed. The adequacy of the two flux approach for the reflectance, radiative flux and radiative flux divergence of highly scattering media is evaluated with respect to solutions of the radiative transfer equation.
Approximate solutions of non-linear circular orbit relative motion in curvilinear coordinates
NASA Astrophysics Data System (ADS)
Bombardelli, Claudio; Gonzalo, Juan Luis; Roa, Javier
2016-07-01
A compact, time-explicit, approximate solution of the highly non-linear relative motion in curvilinear coordinates is provided under the assumption of circular orbit for the chief spacecraft. The rather compact, three-dimensional solution is obtained by algebraic manipulation of the individual Keplerian motions in curvilinear, rather than Cartesian coordinates, and provides analytical expressions for the secular, constant and periodic terms of each coordinate as a function of the initial relative motion conditions or relative orbital elements. Numerical test cases are conducted to show that the approximate solution can be effectively employed to extend the classical linear Clohessy-Wiltshire solution to include non-linear relative motion without significant loss of accuracy up to a limit of 0.4-0.45 in eccentricity and 40-45° in relative inclination for the follower. A very simple, quadratic extension of the classical Clohessy-Wiltshire solution in curvilinear coordinates is also presented.
An approximate solution for one-dimensional absorption in unsaturated porous media
NASA Astrophysics Data System (ADS)
Zimmerman, Robert W.; Bodvarsson, Gudmundur S.
1989-06-01
The "boundary layer" or "integral" method is used to derive a closed-form approximate solution for one-dimensional absorption of water in an unsaturated porous medium whose characteristic curves are of the van Genuchten type. In this approach, an assumed saturation profile is substituted into the governing equation, and integrated from the boundary out to the wetting front. This yields closed-form solutions for the front location and the instantaneous liquid flux at the boundary. The accuracy of this solution in predicting the flux, as determined by comparison with numerical solutions utilizing a Boltzmann-type transformation, is always within 15%, for any value of the initial saturation. As an example of the use of this approximate solution, saturation profiles are calculated for absorption into the Topopah Spring volcanic tuff at Yucca Mountain, Nevada, the site of the proposed nuclear waste repository.
An Approximation to the Periodic Solution of a Differential Equation of Abel
NASA Astrophysics Data System (ADS)
Mickens, Ronald E.
2011-10-01
The Abel equation, in canonical form, is y^' = sint- y^3 (*) and corresponds to the singular (ɛ --> 0) limit of the nonlinear, forced oscillator ɛy^'' + y^' + y^3 = sint, ɛ-> 0. (**) Equation (*) has the property that it has a unique periodic solution defined on (-∞,∞). Further, as t increases, all solutions are attracted into the strip |y| < 1 and any two different solutions y1(t) and y2(t) satisfy the condition Lim [y1(t) - y2(t)] = 0, (***) t --> ∞ and for t negatively decreasing, each solution, except for the periodic solution, becomes unbounded.ootnotetextU. Elias, American Mathematical Monthly, vol.115, (Feb. 2008), pps. 147-149. Our purpose is to calculate an approximation to the unique periodic solution of Eq. (*) using the method of harmonic balance. We also determine an estimation for the blow-up time of the non-periodic solutions.
NASA Technical Reports Server (NTRS)
Ballhaus, W. F.; Jameson, A.; Albert, J.
1977-01-01
Implicit approximate-factorization algorithms (AF) are developed for the solution of steady-state transonic flow problems. The performance of the AF solution method is evaluated relative to that of the standard solution method for transonic flow problems, successive line over-relaxation (SLOR). Both methods are applied to the solution of the nonlinear, two-dimensional transonic small-disturbance equation. Results indicate that the AF method requires substantially less computer time than SLOR to solve the nonlinear finite-difference matrix equation for a transonic flow field. This increase in computational efficiency is achieved with no appreciable increase in computer storage or coding complexity.
SOLUTIONS APPROXIMATING SOLUTE TRANSPORT IN A LEAKY AQUIFER RECEIVING WASTEWATER INJECTION
A mathematical model amenable to analytical solution techniques is developed for the investigation of contaminant transport from an injection well into a leaky aquifer system, which comprises a pumped and an unpumped aquifer connected to each other by an aquitard. A steady state ...
SOLUTIONS APPROXIMATING SOLUTE TRANSPORT IN A LEAKY AQUIFER RECEIVING WASTEWATER INJECTION
A mathematical model amenable to analytical solution techniques is developed for the investigation of contaminant transport from an injection well into a leaky aquifer system, which comprises a pumped and an unpumped aquifer connected to each other by an aquitard. teady state gro...
Free Vibration of Simply Supported General Triangular Thin Plates: AN Accurate Simplified Solution
NASA Astrophysics Data System (ADS)
Saliba, H. T.
1996-09-01
In this paper, a highly accurate, simplified, and economical solution is provided for the free vibration problem of simply supported thin general triangular plates. The method is applicable to thin plates with linear boundaries regardless of there geometrical shapes. Results are compared with previously published reliable data for both the isosceles as well as the general triangles. Excellent agreements are reported. Eigenvalues are provided for a wide range of place aspect ratios. The first five mode shapes for some isosceles triangles are also provided for illustrative purposes. The advantages of the solution presented in the paper over previously published solutions are briefly discussed. Although the paper deals only with simple support conditions, it is mentioned that any combination of classical boundary conditions, with or without complicating factors, can easily be handled.
NASA Technical Reports Server (NTRS)
Ito, Kazufumi
1987-01-01
The linear quadratic optimal control problem on infinite time interval for linear time-invariant systems defined on Hilbert spaces is considered. The optimal control is given by a feedback form in terms of solution pi to the associated algebraic Riccati equation (ARE). A Ritz type approximation is used to obtain a sequence pi sup N of finite dimensional approximations of the solution to ARE. A sufficient condition that shows pi sup N converges strongly to pi is obtained. Under this condition, a formula is derived which can be used to obtain a rate of convergence of pi sup N to pi. The results of the Galerkin approximation is demonstrated and applied for parabolic systems and the averaging approximation for hereditary differential systems.
Explicit solutions of the radiative transport equation in the P{sub 3} approximation
Liemert, André Kienle, Alwin
2014-11-01
Purpose: Explicit solutions of the monoenergetic radiative transport equation in the P{sub 3} approximation have been derived which can be evaluated with nearly the same computational effort as needed for solving the standard diffusion equation (DE). In detail, the authors considered the important case of a semi-infinite medium which is illuminated by a collimated beam of light. Methods: A combination of the classic spherical harmonics method and the recently developed method of rotated reference frames is used for solving the P{sub 3} equations in closed form. Results: The derived solutions are illustrated and compared to exact solutions of the radiative transport equation obtained via the Monte Carlo (MC) method as well as with other approximated analytical solutions. It is shown that for the considered cases which are relevant for biomedical optics applications, the P{sub 3} approximation is close to the exact solution of the radiative transport equation. Conclusions: The authors derived exact analytical solutions of the P{sub 3} equations under consideration of boundary conditions for defining a semi-infinite medium. The good agreement to Monte Carlo simulations in the investigated domains, for example, in the steady-state and time domains, as well as the short evaluation time needed suggests that the derived equations can replace the often applied solutions of the diffusion equation for the homogeneous semi-infinite medium.
An approximate solution for the free vibrations of rotating uniform cantilever beams
NASA Technical Reports Server (NTRS)
Peters, D. A.
1973-01-01
Approximate solutions are obtained for the uncoupled frequencies and modes of rotating uniform cantilever beams. The frequency approximations for flab bending, lead-lag bending, and torsion are simple expressions having errors of less than a few percent over the entire frequency range. These expressions provide a simple way of determining the relations between mass and stiffness parameters and the resultant frequencies and mode shapes of rotating uniform beams.
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.
Alarcón, Tomás
2014-05-14
In this paper, we propose two methods to carry out the quasi-steady state approximation in stochastic models of enzyme catalytic regulation, based on WKB asymptotics of the chemical master equation or of the corresponding partial differential equation for the generating function. The first of the methods we propose involves the development of multiscale generalisation of a WKB approximation of the solution of the master equation, where the separation of time scales is made explicit which allows us to apply the quasi-steady state approximation in a straightforward manner. To the lowest order, the multi-scale WKB method provides a quasi-steady state, Gaussian approximation of the probability distribution. The second method is based on the Hamilton-Jacobi representation of the stochastic process where, as predicted by large deviation theory, the solution of the partial differential equation for the corresponding characteristic function is given in terms of an effective action functional. The optimal transition paths between two states are then given by those paths that maximise the effective action. Such paths are the solutions of the Hamilton equations for the Hamiltonian associated to the effective action functional. The quasi-steady state approximation is applied to the Hamilton equations thus providing an approximation to the optimal transition paths and the transition time between two states. Using this approximation we predict that, unlike the mean-field quasi-steady approximation result, the rate of enzyme catalysis depends explicitly on the initial number of enzyme molecules. The accuracy and validity of our approximated results as well as that of our predictions regarding the behaviour of the stochastic enzyme catalytic models are verified by direct simulation of the stochastic model using Gillespie stochastic simulation algorithm. PMID:24832255
Alarcón, Tomás
2014-05-14
In this paper, we propose two methods to carry out the quasi-steady state approximation in stochastic models of enzyme catalytic regulation, based on WKB asymptotics of the chemical master equation or of the corresponding partial differential equation for the generating function. The first of the methods we propose involves the development of multiscale generalisation of a WKB approximation of the solution of the master equation, where the separation of time scales is made explicit which allows us to apply the quasi-steady state approximation in a straightforward manner. To the lowest order, the multi-scale WKB method provides a quasi-steady state, Gaussian approximation of the probability distribution. The second method is based on the Hamilton-Jacobi representation of the stochastic process where, as predicted by large deviation theory, the solution of the partial differential equation for the corresponding characteristic function is given in terms of an effective action functional. The optimal transition paths between two states are then given by those paths that maximise the effective action. Such paths are the solutions of the Hamilton equations for the Hamiltonian associated to the effective action functional. The quasi-steady state approximation is applied to the Hamilton equations thus providing an approximation to the optimal transition paths and the transition time between two states. Using this approximation we predict that, unlike the mean-field quasi-steady approximation result, the rate of enzyme catalysis depends explicitly on the initial number of enzyme molecules. The accuracy and validity of our approximated results as well as that of our predictions regarding the behaviour of the stochastic enzyme catalytic models are verified by direct simulation of the stochastic model using Gillespie stochastic simulation algorithm.
Approximate analytic solutions to 3D unconfined groundwater flow within regional 2D models
NASA Astrophysics Data System (ADS)
Luther, K.; Haitjema, H. M.
2000-04-01
We present methods for finding approximate analytic solutions to three-dimensional (3D) unconfined steady state groundwater flow near partially penetrating and horizontal wells, and for combining those solutions with regional two-dimensional (2D) models. The 3D solutions use distributed singularities (analytic elements) to enforce boundary conditions on the phreatic surface and seepage faces at vertical wells, and to maintain fixed-head boundary conditions, obtained from the 2D model, at the perimeter of the 3D model. The approximate 3D solutions are analytic (continuous and differentiable) everywhere, including on the phreatic surface itself. While continuity of flow is satisfied exactly in the infinite 3D flow domain, water balance errors can occur across the phreatic surface.
Approximate solution for frequency synchronization in a finite-size Kuramoto model
NASA Astrophysics Data System (ADS)
Wang, Chengwei; Rubido, Nicolás; Grebogi, Celso; Baptista, Murilo S.
2015-12-01
Scientists have been considering the Kuramoto model to understand the mechanism behind the appearance of collective behavior, such as frequency synchronization (FS) as a paradigm, in real-world networks with a finite number of oscillators. A major current challenge is to obtain an analytical solution for the phase angles. Here, we provide an approximate analytical solution for this problem by deriving a master solution for the finite-size Kuramoto model, with arbitrary finite-variance distribution of the natural frequencies of the oscillators. The master solution embodies all particular solutions of the finite-size Kuramoto model for any frequency distribution and coupling strength larger than the critical one. Furthermore, we present a criterion to determine the stability of the FS solution. This allows one to analytically infer the relationship between the physical parameters and the stable behavior of networks.
NASA Astrophysics Data System (ADS)
Stecca, Guglielmo; Siviglia, Annunziato; Blom, Astrid
2016-07-01
We present an accurate numerical approximation to the Saint-Venant-Hirano model for mixed-sediment morphodynamics in one space dimension. Our solution procedure originates from the fully-unsteady matrix-vector formulation developed in [54]. The principal part of the problem is solved by an explicit Finite Volume upwind method of the path-conservative type, by which all the variables are updated simultaneously in a coupled fashion. The solution to the principal part is embedded into a splitting procedure for the treatment of frictional source terms. The numerical scheme is extended to second-order accuracy and includes a bookkeeping procedure for handling the evolution of size stratification in the substrate. We develop a concept of balancedness for the vertical mass flux between the substrate and active layer under bed degradation, which prevents the occurrence of non-physical oscillations in the grainsize distribution of the substrate. We suitably modify the numerical scheme to respect this principle. We finally verify the accuracy in our solution to the equations, and its ability to reproduce one-dimensional morphodynamics due to streamwise and vertical sorting, using three test cases. In detail, (i) we empirically assess the balancedness of vertical mass fluxes under degradation; (ii) we study the convergence to the analytical linearised solution for the propagation of infinitesimal-amplitude waves [54], which is here employed for the first time to assess a mixed-sediment model; (iii) we reproduce Ribberink's E8-E9 flume experiment [46].
Approximate solutions to the quantum problem of two opposite charges in a constant magnetic field
NASA Astrophysics Data System (ADS)
Ardenghi, J. S.; Gadella, M.; Negro, J.
2016-05-01
We consider two particles of equal mass and opposite charge in a plane subject to a perpendicular constant magnetic field. This system is integrable but not superintegrable. From the quantum point of view, the solution is given by two fourth degree Hill differential equations which involve the energy as well as a second constant of motion. There are two solvable approximations in relation to the value of a parameter. Starting from each of these approximations, a consistent perturbation theory can be applied to get approximate values of the energy levels and of the second constant of motion.
A spectrally accurate method for overlapping grid solution of incompressible Navier-Stokes equations
NASA Astrophysics Data System (ADS)
Merrill, Brandon E.; Peet, Yulia T.; Fischer, Paul F.; Lottes, James W.
2016-02-01
An overlapping mesh methodology that is spectrally accurate in space and up to third-order accurate in time is developed for solution of unsteady incompressible flow equations in three-dimensional domains. The ability to decompose a global domain into separate, but overlapping, subdomains eases mesh generation procedures and increases flexibility of modeling flows with complex geometries. The methodology employs implicit spectral element discretization of equations in each subdomain and explicit treatment of subdomain interfaces with spectrally-accurate spatial interpolation and high-order accurate temporal extrapolation, and requires few, if any, iterations, yet maintains the global accuracy and stability of the underlying flow solver. The overlapping mesh methodology is thoroughly validated using two-dimensional and three-dimensional benchmark problems in laminar and turbulent flows. The spatial and temporal convergence is documented and is in agreement with the nominal order of accuracy of the solver. The influence of long integration times, as well as inflow-outflow global boundary conditions on the performance of the overlapping grid solver is assessed. In a turbulent benchmark of fully-developed turbulent pipe flow, the turbulent statistics with the overlapping grids is validated against published available experimental and other computation data. Scaling tests are presented that show near linear strong scaling, even for moderately large processor counts.
NASA Astrophysics Data System (ADS)
Zingerman, K. M.; Shavyrin, D. A.
2016-06-01
The approximate analytical solution of a quasi-static plane problem of the theory of viscoelasticity is obtained under finite strains. This is the problem of the stress-strain state in an infinite body with circular viscoelastic inclusion. The perturbation technique, Laplace transform, and complex Kolosov-Muskhelishvili's potentials are used for the solution. The numerical results are presented. The nonlinear effects and the effects of viscosity are estimated.
Unique solution for accurate in-situ infrared profiling in reheat furnaces
NASA Astrophysics Data System (ADS)
Primhak, David; Wileman, Ben; Drögmöller, Peter
2010-05-01
As thermal imaging becomes a more accepted technology in industrial environments it can provide exciting new solutions to applications that have been previously dominated by single point pyrometers. The new development of an uncooled focal plane array thermal imager with a narrow band 3.9μm filter and background compensation processing enables measurements in industrial furnaces to provide temperature profiling of the product. This paper will show why the use of a 3.9μm camera with a borescope optic is the most accurate noncontact method for in-furnace temperature measurement. This will be done using the example of a reheat furnace where in a controlled trial using an instrumented billet the measurement from the IR device was shown to accurately track the thermocouple temperature during a variety of furnace operating conditions.
A new algorithm for generating highly accurate benchmark solutions to transport test problems
Azmy, Y.Y.
1997-06-01
We present a new algorithm for solving the neutron transport equation in its discrete-variable form. The new algorithm is based on computing the full matrix relating the scalar flux spatial moments in all cells to the fixed neutron source spatial moments, foregoing the need to compute the angular flux spatial moments, and thereby eliminating the need for sweeping the spatial mesh in each discrete-angular direction. The matrix equation is solved exactly in test cases, producing a solution vector that is free from iteration convergence error, and subject only to truncation and roundoff errors. Our algorithm is designed to provide method developers with a quick and simple solution scheme to test their new methods on difficult test problems without the need to develop sophisticated solution techniques, e.g. acceleration, before establishing the worthiness of their innovation. We demonstrate the utility of the new algorithm by applying it to the Arbitrarily High Order Transport Nodal (AHOT-N) method, and using it to solve two of Burre`s Suite of Test Problems (BSTP). Our results provide highly accurate benchmark solutions, that can be distributed electronically and used to verify the pointwise accuracy of other solution methods and algorithms.
Thin airfoil theory based on approximate solution of the transonic flow equation
NASA Technical Reports Server (NTRS)
Spreiter, John R; Alksne, Alberta Y
1957-01-01
A method is presented for the approximate solution of the nonlinear equations transonic flow theory. Solutions are found for two-dimensional flows at a Mach number of 1 and for purely subsonic and purely supersonic flows. Results are obtained in closed analytic form for a large and significant class of nonlifting airfoils. At a Mach number of 1 general expressions are given for the pressure distribution on an airfoil of specified geometry and for the shape of an airfoil having a prescribed pressure distribution. Extensive comparisons are made with available data, particularly for a Mach number of 1, and with existing solutions.
Thin airfoil theory based on approximate solution of the transonic flow equation
NASA Technical Reports Server (NTRS)
Spreiter, John R; Alksne, Alberta Y
1958-01-01
A method is presented for the approximate solution of the nonlinear equations of transonic flow theory. Solutions are found for two-dimensional flows at a Mach number of 1 and for purely subsonic and purely supersonic flows. Results are obtained in closed analytic form for a large and significant class of nonlifting airfoils. At a Mach number of 1 general expressions are given for the pressure distribution on an airfoil of specified geometry and for the shape of an airfoil having a prescribed pressure distribution. Extensive comparisons are made with available data, particularly for a Mach number of 1, and with existing solutions.
NASA Astrophysics Data System (ADS)
Belov, A. M.; Myasnikov, V. V.
2015-02-01
The paper presents a method of atmospheric correction of remote sensing hyperspectral images. The method based on approximate solution of MODTRAN transmittance equation using simultaneous analysis of remote sensing hyperspectral image and "ideal" hyperspectral image which is free from atmospheric distortions. Experimental results show that proposed method is applicable to perform atmospheric correction.
Giao, Pham Huy
2003-01-01
Some approximations of Theis' solution are reviewed and compared with the numerical evaluation of the exponential integral by means of a short FORTRAN code. A graphical technique of Theis' curve matching is proposed to make this procedure as easy as plotting a graph. PMID:12772832
Numerical solution of 2D-vector tomography problem using the method of approximate inverse
NASA Astrophysics Data System (ADS)
Svetov, Ivan; Maltseva, Svetlana; Polyakova, Anna
2016-08-01
We propose a numerical solution of reconstruction problem of a two-dimensional vector field in a unit disk from the known values of the longitudinal and transverse ray transforms. The algorithm is based on the method of approximate inverse. Numerical simulations confirm that the proposed method yields good results of reconstruction of vector fields.
SDP-based approximation of stabilising solutions for periodic matrix Riccati differential equations
NASA Astrophysics Data System (ADS)
Gusev, Sergei V.; Shiriaev, Anton S.; Freidovich, Leonid B.
2016-07-01
Numerically finding stabilising feedback control laws for linear systems of periodic differential equations is a nontrivial task with no known reliable solutions. The most successful method requires solving matrix differential Riccati equations with periodic coefficients. All previously proposed techniques for solving such equations involve numerical integration of unstable differential equations and consequently fail whenever the period is too large or the coefficients vary too much. Here, a new method for numerical computation of stabilising solutions for matrix differential Riccati equations with periodic coefficients is proposed. Our approach does not involve numerical solution of any differential equations. The approximation for a stabilising solution is found in the form of a trigonometric polynomial, matrix coefficients of which are found solving a specially constructed finite-dimensional semidefinite programming (SDP) problem. This problem is obtained using maximality property of the stabilising solution of the Riccati equation for the associated Riccati inequality and sampling technique. Our previously published numerical comparisons with other methods shows that for a class of problems only this technique provides a working solution. Asymptotic convergence of the computed approximations to the stabilising solution is proved below under the assumption that certain combinations of the key parameters are sufficiently large. Although the rate of convergence is not analysed, it appeared to be exponential in our numerical studies.
Approximate semi-analytical solutions for the steady-state expansion of a contactor plasma
NASA Astrophysics Data System (ADS)
Camporeale, E.; Hogan, E. A.; MacDonald, E. A.
2015-04-01
We study the steady-state expansion of a collisionless, electrostatic, quasi-neutral plasma plume into vacuum, with a fluid model. We analyze approximate semi-analytical solutions, that can be used in lieu of much more expensive numerical solutions. In particular, we focus on the earlier studies presented in Parks and Katz (1979 American Institute of Aeronautics, Astronautics Conf. vol 1), Korsun and Tverdokhlebova (1997 33rd Joint Prop. Conf. (Seattle, WA) AIAA-97-3065), and Ashkenazy and Fruchtman (2001 27th Int. Electric Propulsion Conf. (Pasadena, CA)). By calculating the error with respect to the numerical solution, we can judge the range of validity for each solution. Moreover, we introduce a generalization of earlier models that has a wider range of applicability, in terms of plasma injection profiles. We conclude by showing a straightforward way to extend the discussed solutions to the case of a plasma plume injected with non-null azimuthal velocity.
An approximate solution for interlaminar stresses in laminated composites: Applied mechanics program
NASA Technical Reports Server (NTRS)
Rose, Cheryl A.; Herakovich, Carl T.
1992-01-01
An approximate solution for interlaminar stresses in finite width, laminated composites subjected to uniform extensional, and bending loads is presented. The solution is based upon the principle of minimum complementary energy and an assumed, statically admissible stress state, derived by considering local material mismatch effects and global equilibrium requirements. The stresses in each layer are approximated by polynomial functions of the thickness coordinate, multiplied by combinations of exponential functions of the in-plane coordinate, expressed in terms of fourteen unknown decay parameters. Imposing the stationary condition of the laminate complementary energy with respect to the unknown variables yields a system of fourteen non-linear algebraic equations for the parameters. Newton's method is implemented to solve this system. Once the parameters are known, the stresses can be easily determined at any point in the laminate. Results are presented for through-thickness and interlaminar stress distributions for angle-ply, cross-ply (symmetric and unsymmetric laminates), and quasi-isotropic laminates subjected to uniform extension and bending. It is shown that the solution compares well with existing finite element solutions and represents an improved approximate solution for interlaminar stresses, primarily at interfaces where global equilibrium is satisfied by the in-plane stresses, but large local mismatch in properties requires the presence of interlaminar stresses.
Approximate Solution Methods for Spectral Radiative Transfer in High Refractive Index Layers
NASA Technical Reports Server (NTRS)
Siegel, R.; Spuckler, C. M.
1994-01-01
Some ceramic materials for high temperature applications are partially transparent for radiative transfer. The refractive indices of these materials can be substantially greater than one which influences internal radiative emission and reflections. Heat transfer behavior of single and laminated layers has been obtained in the literature by numerical solutions of the radiative transfer equations coupled with heat conduction and heating at the boundaries by convection and radiation. Two-flux and diffusion methods are investigated here to obtain approximate solutions using a simpler formulation than required for exact numerical solutions. Isotropic scattering is included. The two-flux method for a single layer yields excellent results for gray and two band spectral calculations. The diffusion method yields a good approximation for spectral behavior in laminated multiple layers if the overall optical thickness is larger than about ten. A hybrid spectral model is developed using the two-flux method in the optically thin bands, and radiative diffusion in bands that are optically thick.
First-order corrections to approximate solutions to two-point boundary-condition problems
NASA Technical Reports Server (NTRS)
Andrus, J. F.; Greenleaf, W. G.
1973-01-01
A method, applicable to real time guidance, is developed for accurate solution to exo-atmospheric space flight optimization problems. In principle the method is applicable to many other two-point boundary-condition (TPBC) problems. The first step of the method is the iterative solution (using a shooting method) of a TPBC problem with differential equations simplified so that they may be solved analytically by means of a single closed-form solution over each stage of the flight. The second step is the addition of a closed-form correction to the solution to the TPBC problem obtained in the first step. The correction accounts (to first-order accuracy) for the errors due to the aforementioned simplifications. Numerical results are given for several orbital injection problems.
NASA Astrophysics Data System (ADS)
Dwyer, G. S.; Vengosh, A.
2008-12-01
The negative thermal ionization mass spectrometry technique has become the major tool for investigating boron isotopes in the environment. The high sensitivity of BO2- ionization enables measurements of ng levels of boron. However, B isotope measurement by this technique suffers from two fundamental problems (1) fractionation induced by selective ionization of B isotopes in the mass spectrometer; and (2) CNO- interference on mass 42 that is often present in some load solutions (such as B-free seawater processed through ion-exchange resin). Here we report a potentially improved methodology using an alternative filament loading solution with a recently-installed Thermo Scientific TRITON thermal ionization mass spectrometer. Our initial results suggest that this solution -- prepared by combining high-purity single- element standard solutions of Ca, Mg, Na, and K in proportions similar to those in seawater in a 5% HCl matrix -- may offer significant improvement over some other commonly used load solutions. Total loading blank is around 15pg as determined by isotope dilution (NIST952). Replicate analyses of NIST SRM951 and modern seawater thus far have yielded 11B/10B ratios of 4.0057 (±0.0008, n=14) and 4.1645 (±0.0017, n=7; δ11B=39.6 permil), respectively. Replicate analyses of samples and SRM951 yield an average standard deviation (1 σ) of approximately 0.001 (0.25 permil). Fractionation during analysis (60-90 minutes) has thus far typically been less than 0.002 (0.5 permil). The load solution delivers ionization efficiency similar to directly-loaded seawater and has negligible signal at mass 26 (CN-), a proxy for the common interfering molecular ion (CNO-) on mass 42. Standards and samples loaded with the solution behave fairly predictably during filament heating and analysis, thus allowing for the possibility of fully automated data collection.
NASA Astrophysics Data System (ADS)
Lin, Yezhi; Liu, Yinping; Li, Zhibin
2013-01-01
The Adomian decomposition method (ADM) is one of the most effective methods to construct analytic approximate solutions for nonlinear differential equations. In this paper, based on the new definition of the Adomian polynomials, Rach (2008) [22], the Adomian decomposition method and the Padé approximants technique, a new algorithm is proposed to construct analytic approximate solutions for nonlinear fractional differential equations with initial or boundary conditions. Furthermore, a MAPLE software package is developed to implement this new algorithm, which is user-friendly and efficient. One only needs to input the system equation, initial or boundary conditions and several necessary parameters, then our package will automatically deliver the analytic approximate solutions within a few seconds. Several different types of examples are given to illustrate the scope and demonstrate the validity of our package, especially for non-smooth initial value problems. Our package provides a helpful and easy-to-use tool in science and engineering simulations. Program summaryProgram title: ADMP Catalogue identifier: AENE_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AENE_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 12011 No. of bytes in distributed program, including test data, etc.: 575551 Distribution format: tar.gz Programming language: MAPLE R15. Computer: PCs. Operating system: Windows XP/7. RAM: 2 Gbytes Classification: 4.3. Nature of problem: Constructing analytic approximate solutions of nonlinear fractional differential equations with initial or boundary conditions. Non-smooth initial value problems can be solved by this program. Solution method: Based on the new definition of the Adomian polynomials [1], the Adomian decomposition method and the Pad
NASA Technical Reports Server (NTRS)
Schlosser, Herbert; Ferrante, John
1989-01-01
An accurate analytic expression for the nonlinear change of the volume of a solid as a function of applied pressure is of great interest in high-pressure experimentation. It is found that a two-parameter analytic expression, fits the experimental volume-change data to within a few percent over the entire experimentally attainable pressure range. Results are presented for 24 different materials including metals, ceramic semiconductors, polymers, and ionic and rare-gas solids.
Approximate solutions to the initial value problem for some compressible flows
NASA Astrophysics Data System (ADS)
Colombeau, M.
2015-10-01
For the natural initial conditions L 1 in the density field (more generally a positive bounded Radon measure) and L ∞ in the velocity field, we obtain global approximate solutions to the Cauchy problem for the 3-D systems of isothermal and isentropic gases and the 2-D shallow water equations. We obtain a sequence of functions which are differentiable in time and continuous in space and tend to satisfy the equations in the sense of distributions in the space variables and in the strong sense in the time variable. The method of construction relies on the study of a specific family of two ODEs in a classical Banach space (one for the continuity equation and one for the Euler equation). Standard convergent numerical methods for the solution of these ODEs can be used to provide concrete approximate solutions. It has been checked in numerous cases in which the solutions of systems of fluid dynamics are known that our construction always gives back the known solutions. It is also proved that it gives the classical analytic solutions in the domain of application of the Cauchy-Kovalevskaya theorem.
Approximate Analytical Solutions for Primary Chatter in the Non-Linear Metal Cutting Model
NASA Astrophysics Data System (ADS)
Warmiński, J.; Litak, G.; Cartmell, M. P.; Khanin, R.; Wiercigroch, M.
2003-01-01
This paper considers an accepted model of the metal cutting process dynamics in the context of an approximate analysis of the resulting non-linear differential equations of motion. The process model is based upon the established mechanics of orthogonal cutting and results in a pair of non-linear ordinary differential equations which are then restated in a form suitable for approximate analytical solution. The chosen solution technique is the perturbation method of multiple time scales and approximate closed-form solutions are generated for the most important non-resonant case. Numerical data are then substituted into the analytical solutions and key results are obtained and presented. Some comparisons between the exact numerical calculations for the forces involved and their reduced and simplified analytical counterparts are given. It is shown that there is almost no discernible difference between the two thus confirming the validity of the excitation functions adopted in the analysis for the data sets used, these being chosen to represent a real orthogonal cutting process. In an attempt to provide guidance for the selection of technological parameters for the avoidance of primary chatter, this paper determines for the first time the stability regions in terms of the depth of cut and the cutting speed co-ordinates.
NASA Technical Reports Server (NTRS)
Daso, E. O.
1986-01-01
An implicit approximate factorization algorithm is employed to quantify the parametric effects of Courant number and artificial smoothing on numerical solutions of the unsteady 3-D Euler equations for a windmilling propeller (low speed) flow field. The results show that propeller global or performance chracteristics vary strongly with Courant number and artificial dissipation parameters, though the variation is such less severe at high Courant numbers. Candidate sets of Courant number and dissipation parameters could result in parameter-dependent solutions. Parameter-independent numerical solutions can be obtained if low values of the dissipation parameter-time step ratio are used in the computations. Furthermore, it is realized that too much artificial damping can degrade numerical stability. Finally, it is demonstrated that highly resolved meshes may, in some cases, delay convergence, thereby suggesting some optimum cell size for a given flow solution. It is suspected that improper boundary treatment may account for the cell size constraint.
Orio, Patricio; Soudry, Daniel
2012-01-01
Background The phenomena that emerge from the interaction of the stochastic opening and closing of ion channels (channel noise) with the non-linear neural dynamics are essential to our understanding of the operation of the nervous system. The effects that channel noise can have on neural dynamics are generally studied using numerical simulations of stochastic models. Algorithms based on discrete Markov Chains (MC) seem to be the most reliable and trustworthy, but even optimized algorithms come with a non-negligible computational cost. Diffusion Approximation (DA) methods use Stochastic Differential Equations (SDE) to approximate the behavior of a number of MCs, considerably speeding up simulation times. However, model comparisons have suggested that DA methods did not lead to the same results as in MC modeling in terms of channel noise statistics and effects on excitability. Recently, it was shown that the difference arose because MCs were modeled with coupled gating particles, while the DA was modeled using uncoupled gating particles. Implementations of DA with coupled particles, in the context of a specific kinetic scheme, yielded similar results to MC. However, it remained unclear how to generalize these implementations to different kinetic schemes, or whether they were faster than MC algorithms. Additionally, a steady state approximation was used for the stochastic terms, which, as we show here, can introduce significant inaccuracies. Main Contributions We derived the SDE explicitly for any given ion channel kinetic scheme. The resulting generic equations were surprisingly simple and interpretable – allowing an easy, transparent and efficient DA implementation, avoiding unnecessary approximations. The algorithm was tested in a voltage clamp simulation and in two different current clamp simulations, yielding the same results as MC modeling. Also, the simulation efficiency of this DA method demonstrated considerable superiority over MC methods, except when
Accurate and molecular-size-tolerant NMR quantitation of diverse components in solution
Okamura, Hideyasu; Nishimura, Hiroshi; Nagata, Takashi; Kigawa, Takanori; Watanabe, Takashi; Katahira, Masato
2016-01-01
Determining the amount of each component of interest in a mixture is a fundamental first step in characterizing the nature of the solution and to develop possible means of utilization of its components. Similarly, determining the composition of units in complex polymers, or polymer mixtures, is crucial. Although NMR is recognized as one of the most powerful methods to achieve this and is widely used in many fields, variation in the molecular sizes or the relative mobilities of components skews quantitation due to the size-dependent decay of magnetization. Here, a method to accurately determine the amount of each component by NMR was developed. This method was validated using a solution that contains biomass-related components in which the molecular sizes greatly differ. The method is also tolerant of other factors that skew quantitation such as variation in the one-bond C–H coupling constant. The developed method is the first and only way to reliably overcome the skewed quantitation caused by several different factors to provide basic information on the correct amount of each component in a solution. PMID:26883279
Approximation of weak solution for the problem of a pH-gradient creation in isoelectrofocusing
Sakharova, L. V.; Shiryaeva, E. V.; Zhukov, M. Yu.
2014-01-01
The mathematical model describing the stationary natural pH-gradient arising under the action of an electric field in an aqueous solution of ampholytes is constructed and investigated. The model is a part of a more general model of the isoelectrofocusing process. Investigation is based on the approximation of a weak solution by the piecewise continuous non-smooth functions. The method can be used for solving classes of problems for ordinary differential equations with a small parameter at the highest derivatives and the turning points. PMID:25383021
Spectral solution of the viscous blunt body problem. 2: Multidomain approximation
NASA Technical Reports Server (NTRS)
Kopriva, David A.
1994-01-01
We present steady solutions of high speed viscous flows over blunt bodies using a multidomain Chebyshev spectral collocation method. The region with the shock layer is divided into subdomains so that internal layers can be well-resolved. In the interiors of the subdomains, the solution is approximated by Chebyshev collocation. At interfaces between subdomains, the advective terms are upwinded and the viscous terms are treated by a penalty method. The method is applied to five flows, the Mach number range 5-25 and Reynolds number range 2,000 - 83,000, based on nose radius. Results are compared to experimental data and to a finite difference result.
Căruntu, Bogdan
2014-01-01
The paper presents the optimal homotopy perturbation method, which is a new method to find approximate analytical solutions for nonlinear partial differential equations. Based on the well-known homotopy perturbation method, the optimal homotopy perturbation method presents an accelerated convergence compared to the regular homotopy perturbation method. The applications presented emphasize the high accuracy of the method by means of a comparison with previous results. PMID:25003150
Comparison of exact solution with Eikonal approximation for elastic heavy ion scattering
NASA Technical Reports Server (NTRS)
Dubey, Rajendra R.; Khandelwal, Govind S.; Cucinotta, Francis A.; Maung, Khin Maung
1995-01-01
A first-order optical potential is used to calculate the total and absorption cross sections for nucleus-nucleus scattering. The differential cross section is calculated by using a partial-wave expansion of the Lippmann-Schwinger equation in momentum space. The results are compared with solutions in the Eikonal approximation for the equivalent potential and with experimental data in the energy range from 25A to 1000A MeV.
Using trees to compute approximate solutions to ordinary differential equations exactly
NASA Technical Reports Server (NTRS)
Grossman, Robert
1991-01-01
Some recent work is reviewed which relates families of trees to symbolic algorithms for the exact computation of series which approximate solutions of ordinary differential equations. It turns out that the vector space whose basis is the set of finite, rooted trees carries a natural multiplication related to the composition of differential operators, making the space of trees an algebra. This algebraic structure can be exploited to yield a variety of algorithms for manipulating vector fields and the series and algebras they generate.
Strong pairing approximation in comparison with the exact solutions to the pairing Hamiltonian
NASA Astrophysics Data System (ADS)
Lunyov, A. V.; Mikhajlov, V. M.
2016-01-01
Results of the Strong Pairing Approximation (SPA) as a method with the exact particle number conservation are compared with those of the quasiparticle method (QM). It is shown that SPA comes to the same equations as QM for the gap parameter, chemical potential and one- and two-quasiparticle states. Calculations are performed for 14864Gd84 as an example, and compared with the exact solutions to the pairing Hamiltonian.
Global collocation methods for approximation and the solution of partial differential equations
NASA Technical Reports Server (NTRS)
Solomonoff, A.; Turkel, E.
1986-01-01
Polynomial interpolation methods are applied both to the approximation of functions and to the numerical solutions of hyperbolic and elliptic partial differential equations. The derivative matrix for a general sequence of the collocation points is constructed. The approximate derivative is then found by a matrix times vector multiply. The effects of several factors on the performance of these methods including the effect of different collocation points are then explored. The resolution of the schemes for both smooth functions and functions with steep gradients or discontinuities in some derivative are also studied. The accuracy when the gradients occur both near the center of the region and in the vicinity of the boundary is investigated. The importance of the aliasing limit on the resolution of the approximation is investigated in detail. Also examined is the effect of boundary treatment on the stability and accuracy of the scheme.
Bilgen, M.; Rose, J.H. )
1994-11-01
An approximate analytic series solution is obtained for the effects of randomly rough surfaces on the time-dependent ultrasonic backscatter that are due to beam-microstructure interactions. The transmission of sound through the rough surface is modeled by scalar waves by use of the phase-screen and Fresnel approximations, whereas the transducer is assumed to produce a focused normally oriented Gaussian beam. The beam-microstructure interaction is described by a simple, generic model that attributes backscattering to inhomogeneities in the elastic constants of the sample; density variations are ignored. Key predictions of the approximate series solution are that (a) acoustic backscatter is relatively insensitive to surface roughness for unfocused probes, (b) roughness can dramatically reduce the backscatter noise seen by focused probes, (c) backscatter is increased at early times because of weak localization, and (d) backscatter is reduced at late times because of increased diffraction. The predictions of the series solution are briefly compared with available experiment. 32 refs., 10 figs.
Ohyanagi, Toshio; Sengoku, Yasuhito
2010-02-01
This article presents a new solution for measuring accurate reaction time (SMART) to visual stimuli. The SMART is a USB device realized with a Cypress Programmable System-on-Chip (PSoC) mixed-signal array programmable microcontroller. A brief overview of the hardware and firmware of the PSoC is provided, together with the results of three experiments. In Experiment 1, we investigated the timing accuracy of the SMART in measuring reaction time (RT) under different conditions of operating systems (OSs; Windows XP or Vista) and monitor displays (a CRT or an LCD). The results indicated that the timing error in measuring RT by the SMART was less than 2 msec, on average, under all combinations of OS and display and that the SMART was tolerant to jitter and noise. In Experiment 2, we tested the SMART with 8 participants. The results indicated that there was no significant difference among RTs obtained with the SMART under the different conditions of OS and display. In Experiment 3, we used Microsoft (MS) PowerPoint to present visual stimuli on the display. We found no significant difference in RTs obtained using MS DirectX technology versus using the PowerPoint file with the SMART. We are certain that the SMART is a simple and practical solution for measuring RTs accurately. Although there are some restrictions in using the SMART with RT paradigms, the SMART is capable of providing both researchers and health professionals working in clinical settings with new ways of using RT paradigms in their work. PMID:20160303
NASA Astrophysics Data System (ADS)
Frid, Hermano; Rendón, Leonardo
We prove the asymptotic stability of nonplanar two-states Riemann solutions in BGK approximations of a class of multidimensional systems of conservation laws. The latter consists of systems whose flux-functions in different directions share a common complete system of Riemann invariants, the level surfaces of which are hyperplanes. The asymptotic stability to which the main result refers is in the sense of the convergence as t→∞ in Lloc1 of the space of directions ζ=x/t. That is, the solution z(t,x,ξ) of the perturbed Cauchy problem for the corresponding BGK system satisfies ∫z(t,tζ,ξ) dμ(ξ)→R(ζ) as t→∞, in Lloc1(R), where R(ζ) is the self-similar entropy solution of the two-states nonplanar Riemann problem for the system of conservation laws.
S{sub N} Schemes, Linear Infinite-Medium Solutions, and the Diffusion Approximation
Larsen, E.W.
2001-06-17
It is standard practice to require an S{sub N} spatial discretization scheme to preserve the ''flat infinite-medium'' solution of the transport equation. This solution consists of a spatially independent source that gives rise to a spatially independent flux. However, there exist many other exact solutions of the transport equation that are typically not preserved by approximation schemes. Here, we discuss one of these: a source that is linear in space giving rise to an angular flux that is linear in space and angle. For one-group, planar-geometry S{sub N} problems, we show that (a) among the class of weighted-diamond schemes, only one - the diamond-difference scheme - preserves this exact ''linear'' solution; (b) consequently, only the diamond scheme preserves the correct Fick's Law; and (c) as a further consequence, nondiamond schemes can produce significant errors (not observed in the diamond solution) for diffusive problems with spatial cells that are not optically thin. These results demonstrate that it is advantageous for S{sub N} discretization schemes to preserve the ''flat'' and ''linear'' infinite-medium solutions.
Approximate N-Player Nonzero-Sum Game Solution for an Uncertain Continuous Nonlinear System.
Johnson, Marcus; Kamalapurkar, Rushikesh; Bhasin, Shubhendu; Dixon, Warren E
2015-08-01
An approximate online equilibrium solution is developed for an N -player nonzero-sum game subject to continuous-time nonlinear unknown dynamics and an infinite horizon quadratic cost. A novel actor-critic-identifier structure is used, wherein a robust dynamic neural network is used to asymptotically identify the uncertain system with additive disturbances, and a set of critic and actor NNs are used to approximate the value functions and equilibrium policies, respectively. The weight update laws for the actor neural networks (NNs) are generated using a gradient-descent method, and the critic NNs are generated by least square regression, which are both based on the modified Bellman error that is independent of the system dynamics. A Lyapunov-based stability analysis shows that uniformly ultimately bounded tracking is achieved, and a convergence analysis demonstrates that the approximate control policies converge to a neighborhood of the optimal solutions. The actor, critic, and identifier structures are implemented in real time continuously and simultaneously. Simulations on two and three player games illustrate the performance of the developed method. PMID:25312943
NASA Astrophysics Data System (ADS)
Neese, Frank; Wennmohs, Frank; Hansen, Andreas
2009-03-01
Coupled-electron pair approximations (CEPAs) and coupled-pair functionals (CPFs) have been popular in the 1970s and 1980s and have yielded excellent results for small molecules. Recently, interest in CEPA and CPF methods has been renewed. It has been shown that these methods lead to competitive thermochemical, kinetic, and structural predictions. They greatly surpass second order Møller-Plesset and popular density functional theory based approaches in accuracy and are intermediate in quality between CCSD and CCSD(T) in extended benchmark studies. In this work an efficient production level implementation of the closed shell CEPA and CPF methods is reported that can be applied to medium sized molecules in the range of 50-100 atoms and up to about 2000 basis functions. The internal space is spanned by localized internal orbitals. The external space is greatly compressed through the method of pair natural orbitals (PNOs) that was also introduced by the pioneers of the CEPA approaches. Our implementation also makes extended use of density fitting (or resolution of the identity) techniques in order to speed up the laborious integral transformations. The method is called local pair natural orbital CEPA (LPNO-CEPA) (LPNO-CPF). The implementation is centered around the concepts of electron pairs and matrix operations. Altogether three cutoff parameters are introduced that control the size of the significant pair list, the average number of PNOs per electron pair, and the number of contributing basis functions per PNO. With the conservatively chosen default values of these thresholds, the method recovers about 99.8% of the canonical correlation energy. This translates to absolute deviations from the canonical result of only a few kcal mol-1. Extended numerical test calculations demonstrate that LPNO-CEPA (LPNO-CPF) has essentially the same accuracy as parent CEPA (CPF) methods for thermochemistry, kinetics, weak interactions, and potential energy surfaces but is up to 500
NASA Astrophysics Data System (ADS)
Chardon, Gilles; Daudet, Laurent
2013-11-01
This paper extends the method of particular solutions (MPS) to the computation of eigenfrequencies and eigenmodes of thin plates, in the framework of the Kirchhoff-Love plate theory. Specific approximation schemes are developed, with plane waves (MPS-PW) or Fourier-Bessel functions (MPS-FB). This framework also requires a suitable formulation of the boundary conditions. Numerical tests, on two plates with various boundary conditions, demonstrate that the proposed approach provides competitive results with standard numerical schemes such as the finite element method, at reduced complexity, and with large flexibility in the implementation choices.
Numerical and approximate solution of the high Reynolds number small separation problem
NASA Technical Reports Server (NTRS)
Davis, R. T.
1976-01-01
Several possible methods of solving the small separation problem at high Reynolds number are investigated. In addition to using analytical methods, there are several numerical approaches which are used. High Reynolds number laminar two dimensional problems are used for simplicity. A brief discussion is given of the finite difference methods since these methods are discussed in detail. Most of the emphasis is placed on developing an approximate integral method. As a model problem the supersonic compression ramp problem is chosen since several numerical solutions along with experimental data are available. The techniques discussed are modified and applied to other similar type wall geometries.
NASA Technical Reports Server (NTRS)
Jameson, A.
1976-01-01
A review is presented of some recently developed numerical methods for the solution of nonlinear equations of mixed type. The methods considered use finite difference approximations to the differential equation. Central difference formulas are employed in the subsonic zone and upwind difference formulas are used in the supersonic zone. The relaxation method for the small disturbance equation is discussed and a description is given of difference schemes for the potential flow equation in quasi-linear form. Attention is also given to difference schemes for the potential flow equation in conservation form, the analysis of relaxation schemes by the time dependent analogy, the accelerated iterative method, and three-dimensional calculations.
An approximate solution to the stress and deformation states of functionally graded rotating disks
NASA Astrophysics Data System (ADS)
Sondhi, Lakshman; Sanyal, Shubhashis; Saha, Kashi Nath; Bhowmick, Shubhankar
2016-07-01
The present work employs variational principle to investigate the stress and deformation states and estimate the limit angular speed of functionally graded high-speed rotating annular disks of constant thickness. Assuming a series approximation following Galerkin's principle, the solution of the governing equation is obtained. In the present study, elasticity modulus and density of the disk material are taken as power function of radius with the gradient parameter ranging between 0.0 and 1.0. Results obtained from numerical solutions are validated with benchmark results and are found to be in good agreement. The results are reported in dimensional form and presented graphically. The results provide a substantial insight in understanding the behavior of FGM rotating disks with constant thickness and different gradient parameter. Furthermore, the stress and deformation state of the disk at constant angular speed and limit angular speed is investigated to explain the existence of optimum gradient parameters.
Cao, Youfang; Terebus, Anna; Liang, Jie
2016-04-01
truncation and error analysis methods developed here can be used to ensure accurate direct solutions to the dCME for a large number of stochastic networks. PMID:27105653
Cao, Youfang; Terebus, Anna; Liang, Jie
2016-01-01
truncation and error analysis methods developed here can be used to ensure accurate direct solutions to the dCME for a large number of stochastic networks. PMID:27105653
An approximate analytic solution for the radiation from a line-driven fluid-loaded plate
NASA Astrophysics Data System (ADS)
Diperna, Daniel T.; Feit, David
2001-12-01
In the analysis of a fluid loaded line-driven plate, the fields in the structure and the fluid are often expressed in terms of a Fourier transform. Once the boundary conditions are matched, the structural displacement can be expressed as an inverse transform, which can be evaluated using contour integration. The result is then a sum of propagating or decaying waves, each arising from poles in the complex plane, plus a branch cut integral. The branch cut is due to a square root in the transform of the acoustic impedance. The complex layer analysis (CLA) used here eliminates the branch cut singularity by approximating the square root with a rational function, causing the characteristic equation to become a polynomial in the transform variable. An approximate analytic solution to the characteristic equation is then found using a perturbation method. The result is four poles corresponding to the roots of the in vacuo plate, modified by the presence of the fluid, plus an infinity of poles located along the branch cut of the acoustic impedance. The solution is then found analytically using contour integration, with the integrand containing only simple poles.
Discrete-time nonlinear HJB solution using approximate dynamic programming: convergence proof.
Al-Tamimi, Asma; Lewis, Frank L; Abu-Khalaf, Murad
2008-08-01
Convergence of the value-iteration-based heuristic dynamic programming (HDP) algorithm is proven in the case of general nonlinear systems. That is, it is shown that HDP converges to the optimal control and the optimal value function that solves the Hamilton-Jacobi-Bellman equation appearing in infinite-horizon discrete-time (DT) nonlinear optimal control. It is assumed that, at each iteration, the value and action update equations can be exactly solved. The following two standard neural networks (NN) are used: a critic NN is used to approximate the value function, whereas an action network is used to approximate the optimal control policy. It is stressed that this approach allows the implementation of HDP without knowing the internal dynamics of the system. The exact solution assumption holds for some classes of nonlinear systems and, specifically, in the specific case of the DT linear quadratic regulator (LQR), where the action is linear and the value quadratic in the states and NNs have zero approximation error. It is stressed that, for the LQR, HDP may be implemented without knowing the system A matrix by using two NNs. This fact is not generally appreciated in the folklore of HDP for the DT LQR, where only one critic NN is generally used. PMID:18632382
High-Order Accurate Solutions to the Helmholtz Equation in the Presence of Boundary Singularities
NASA Astrophysics Data System (ADS)
Britt, Darrell Steven, Jr.
Problems of time-harmonic wave propagation arise in important fields of study such as geological surveying, radar detection/evasion, and aircraft design. These often involve highfrequency waves, which demand high-order methods to mitigate the dispersion error. We propose a high-order method for computing solutions to the variable-coefficient inhomogeneous Helmholtz equation in two dimensions on domains bounded by piecewise smooth curves of arbitrary shape with a finite number of boundary singularities at known locations. We utilize compact finite difference (FD) schemes on regular structured grids to achieve highorder accuracy due to their efficiency and simplicity, as well as the capability to approximate variable-coefficient differential operators. In this work, a 4th-order compact FD scheme for the variable-coefficient Helmholtz equation on a Cartesian grid in 2D is derived and tested. The well known limitation of finite differences is that they lose accuracy when the boundary curve does not coincide with the discretization grid, which is a severe restriction on the geometry of the computational domain. Therefore, the algorithm presented in this work combines high-order FD schemes with the method of difference potentials (DP), which retains the efficiency of FD while allowing for boundary shapes that are not aligned with the grid without sacrificing the accuracy of the FD scheme. Additionally, the theory of DP allows for the universal treatment of the boundary conditions. One of the significant contributions of this work is the development of an implementation that accommodates general boundary conditions (BCs). In particular, Robin BCs with discontinuous coefficients are studied, for which we introduce a piecewise parameterization of the boundary curve. Problems with discontinuities in the boundary data itself are also studied. We observe that the design convergence rate suffers whenever the solution loses regularity due to the boundary conditions. This is
Subsonic-Sonic Limit of Approximate Solutions to Multidimensional Steady Euler Equations
NASA Astrophysics Data System (ADS)
Chen, Gui-Qiang; Huang, Fei-Min; Wang, Tian-Yi
2016-02-01
A compactness framework is established for approximate solutions to subsonic-sonic flows governed by the steady full Euler equations for compressible fluids in arbitrary dimension. The existing compactness frameworks for the two-dimensional irrotational case do not directly apply for the steady full Euler equations in higher dimensions. The new compactness framework we develop applies for both non-homentropic and rotational flows. One of our main observations is that the compactness can be achieved by using only natural weak estimates for the mass balance and the vorticity, along with the Bernoulli law and the entropy relation, through a more delicate analysis on the phase space. As direct applications, we establish two existence theorems for multidimensional subsonic-sonic full Euler flows through infinitely long nozzles.
Variational superposed Gaussian approximation for time-dependent solutions of Langevin equations
NASA Astrophysics Data System (ADS)
Hasegawa, Yoshihiko
2015-04-01
We propose a variational superposed Gaussian approximation (VSGA) for dynamical solutions of Langevin equations subject to applied signals, determining time-dependent parameters of superposed Gaussian distributions by the variational principle. We apply the proposed VSGA to systems driven by a chaotic signal, where the conventional Fourier method cannot be adopted, and calculate the time evolution of probability density functions (PDFs) and moments. Both white and colored Gaussian noises terms are included to describe fluctuations. Our calculations show that time-dependent PDFs obtained by VSGA agree excellently with those obtained by Monte Carlo simulations. The correlation between the chaotic input signal and the mean response are also calculated as a function of the noise intensity, which confirms the occurrence of aperiodic stochastic resonance with both white and colored noises.
Slender-Body Theory Based On Approximate Solution of the Transonic Flow Equation
NASA Technical Reports Server (NTRS)
Spreiter, John R.; Alksne, Alberta Y.
1959-01-01
Approximate solution of the nonlinear equations of the small disturbance theory of transonic flow are found for the pressure distribution on pointed slender bodies of revolution for flows with free-stream, Mach number 1, and for flows that are either purely subsonic or purely supersonic. These results are obtained by application of a method based on local linearization that was introduced recently in the analysis of similar problems in two-dimensional flows. The theory is developed for bodies of arbitrary shape, and specific results are given for cone-cylinders and for parabolic-arc bodies at zero angle of attack. All results are compared either with existing theoretical results or with experimental data.
Approximate Solutions for a Self-Folding Problem of Carbon Nanotubes
Y Mikata
2006-08-22
This paper treats approximate solutions for a self-folding problem of carbon nanotubes. It has been observed in the molecular dynamics calculations [1] that a carbon nanotube with a large aspect ratio can self-fold due to van der Waals force between the parts of the same carbon nanotube. The main issue in the self-folding problem is to determine the minimum threshold length of the carbon nanotube at which it becomes possible for the carbon nanotube to self-fold due to the van der Waals force. An approximate mathematical model based on the force method is constructed for the self-folding problem of carbon nanotubes, and it is solved exactly as an elastica problem using elliptic functions. Additionally, three other mathematical models are constructed based on the energy method. As a particular example, the lower and upper estimates for the critical threshold (minimum) length are determined based on both methods for the (5,5) armchair carbon nanotube.
Least squares solutions of the HJB equation with neural network value-function approximators.
Tassa, Yuval; Erez, Tom
2007-07-01
In this paper, we present an empirical study of iterative least squares minimization of the Hamilton-Jacobi-Bellman (HJB) residual with a neural network (NN) approximation of the value function. Although the nonlinearities in the optimal control problem and NN approximator preclude theoretical guarantees and raise concerns of numerical instabilities, we present two simple methods for promoting convergence, the effectiveness of which is presented in a series of experiments. The first method involves the gradual increase of the horizon time scale, with a corresponding gradual increase in value function complexity. The second method involves the assumption of stochastic dynamics which introduces a regularizing second derivative term to the HJB equation. A gradual reduction of this term provides further stabilization of the convergence. We demonstrate the solution of several problems, including the 4-D inverted-pendulum system with bounded control. Our approach requires no initial stabilizing policy or any restrictive assumptions on the plant or cost function, only knowledge of the plant dynamics. In the Appendix, we provide the equations for first- and second-order differential backpropagation. PMID:17668659
Coelho, P.J.
2009-05-15
An analysis of the relevance of turbulence-radiation interaction in the numerical simulation of turbulent reactive flows is presented. A semi-causal stochastic model was used to generate a time-series of turbulent scalar fluctuations along optical paths of Sandia flame D, a widely studied piloted turbulent jet nonpremixed flame. The radiative transfer equation was integrated along these paths for every realization using a grid resolution typical of a direct numerical simulation. The correlated k-distribution method was employed to compute the radiative properties of the medium. The results were used to determine the ensemble average, as well as the extreme values, of quantities that indicate the importance of the turbulence-radiation interaction. Several approximate methods are then proposed to solve the filtered radiative transfer equation in the framework of large eddy simulations. The proposed methods are applicable along with combustion models that either assume the filtered probability density function of a conserved scalar or solve a transport equation for a joint scalar or joint scalar/velocity filtered density function. It is concluded that the errors resulting from neglecting the turbulence-radiation interaction in large eddy simulations are much lower than those found in Reynolds-averaged Navier-Stokes calculations. The optically thin fluctuation approximation may be extended to large eddy simulations yielding predictions in excellent agreement with the reference solution. If the turbulence-radiation interaction is accounted for using this approximation, the average relative error of the filtered total radiation intensity is generally below 0.3% for the studied flame. (author)
A deterministic annealing algorithm for approximating a solution of the min-bisection problem.
Dang, Chuangyin; Ma, Wei; Liang, Jiye
2009-01-01
The min-bisection problem is an NP-hard combinatorial optimization problem. In this paper an equivalent linearly constrained continuous optimization problem is formulated and an algorithm is proposed for approximating its solution. The algorithm is derived from the introduction of a logarithmic-cosine barrier function, where the barrier parameter behaves as temperature in an annealing procedure and decreases from a sufficiently large positive number to zero. The algorithm searches for a better solution in a feasible descent direction, which has a desired property that lower and upper bounds are always satisfied automatically if the step length is a number between zero and one. We prove that the algorithm converges to at least a local minimum point of the problem if a local minimum point of the barrier problem is generated for a sequence of descending values of the barrier parameter with a limit of zero. Numerical results show that the algorithm is much more efficient than two of the best existing heuristic methods for the min-bisection problem, Kernighan-Lin method with multiple starting points (MSKL) and multilevel graph partitioning scheme (MLGP). PMID:18995985
Sanders, C R; Schwonek, J P
1993-01-01
An empirical model of a liquid crystalline (L alpha phase) phosphatidylcholine (PC) bilayer interface is presented along with a function which calculates the position-dependent energy of associated solutes. The model approximates the interface as a gradual two-step transition, the first step being from an aqueous phase to a phase of reduced polarity, but which maintains a high enough concentration of water and/or polar head group moieties to satisfy the hydrogen bond-forming potential of the solute. The second transition is from the hydrogen bonding/low polarity region to an effectively anhydrous hydrocarbon phase. The "interfacial energies" of solutes within this variable medium are calculated based upon atomic positions and atomic parameters describing general polarity and hydrogen bond donor/acceptor propensities. This function was tested for its ability to reproduce experimental water-solvent partitioning energies and water-bilayer partitioning data. In both cases, the experimental data was reproduced fairly well. Energy minimizations carried out on beta-hexyl glucopyranoside led to identification of a global minimum for the interface-associated glycolipid which exhibited glycosidic torsion angles in agreement with prior results (Hare, B.J., K.P. Howard, and J.H. Prestegard. 1993. Biophys. J. 64:392-398). Molecular dynamics simulations carried out upon this same molecule within the simulated interface led to results which were consistent with a number of experimentally based conclusions from previous work, but failed to quantitatively reproduce an available NMR quadrupolar/dipolar coupling data set (Sanders, C.R., and J.H. Prestegard. 1991. J. Am. Chem. Soc. 113:1987-1996). The proposed model and functions are readily incorporated into computational energy modeling algorithms and may prove useful in future studies of membrane-associated molecules. PMID:8241401
ERIC Educational Resources Information Center
Johannessen, Kim
2010-01-01
An analytic approximation of the solution to the differential equation describing the oscillations of a simple pendulum at large angles and with initial velocity is discussed. In the derivation, a sinusoidal approximation has been applied, and an analytic formula for the large-angle period of the simple pendulum is obtained, which also includes…
NASA Astrophysics Data System (ADS)
Zhang, Ji; Ding, Mingyue; Yuchi, Ming; Hou, Wenguang; Ye, Huashan; Qiu, Wu
2010-03-01
Factor analysis is an efficient technique to the analysis of dynamic structures in medical image sequences and recently has been used in contrast-enhanced ultrasound (CEUS) of hepatic perfusion. Time-intensity curves (TICs) extracted by factor analysis can provide much more diagnostic information for radiologists and improve the diagnostic rate of focal liver lesions (FLLs). However, one of the major drawbacks of factor analysis of dynamic structures (FADS) is nonuniqueness of the result when only the non-negativity criterion is used. In this paper, we propose a new method of replace-approximation based on apex-seeking for ambiguous FADS solutions. Due to a partial overlap of different structures, factor curves are assumed to be approximately replaced by the curves existing in medical image sequences. Therefore, how to find optimal curves is the key point of the technique. No matter how many structures are assumed, our method always starts to seek apexes from one-dimensional space where the original high-dimensional data is mapped. By finding two stable apexes from one dimensional space, the method can ascertain the third one. The process can be continued until all structures are found. This technique were tested on two phantoms of blood perfusion and compared to the two variants of apex-seeking method. The results showed that the technique outperformed two variants in comparison of region of interest measurements from phantom data. It can be applied to the estimation of TICs derived from CEUS images and separation of different physiological regions in hepatic perfusion.
Bu, Sunyoung; Huang, Jingfang; Boyer, Treavor H.; Miller, Cass T.
2010-01-01
The focus of this work is on the modeling of an ion exchange process that occurs in drinking water treatment applications. The model formulation consists of a two-scale model in which a set of microscale diffusion equations representing ion exchange resin particles that vary in size and age are coupled through a boundary condition with a macroscopic ordinary differential equation (ODE), which represents the concentration of a species in a well-mixed reactor. We introduce a new age-averaged model (AAM) that averages all ion exchange particle ages for a given size particle to avoid the expensive Monte-Carlo simulation associated with previous modeling applications. We discuss two different numerical schemes to approximate both the original Monte Carlo algorithm and the new AAM for this two-scale problem. The first scheme is based on the finite element formulation in space coupled with an existing backward-difference-formula-based ODE solver in time. The second scheme uses an integral equation based Krylov deferred correction (KDC) method and a fast elliptic solver (FES) for the resulting elliptic equations. Numerical results are presented to validate the new AAM algorithm, which is also shown to be more computationally efficient than the original Monte Carlo algorithm. We also demonstrate that the higher order KDC scheme is more efficient than the traditional finite element solution approach and this advantage becomes increasingly important as the desired accuracy of the solution increases. We also discuss issues of smoothness, which affect the efficiency of the KDC-FES approach, and outline additional algorithmic changes that would further improve the efficiency of these developing methods for a wide range of applications. PMID:20577570
NASA Technical Reports Server (NTRS)
Nemeth, Michael P.
2013-01-01
Nondimensional linear-bifurcation buckling equations for balanced, symmetrically laminated cylinders with negligible shell-wall anisotropies and subjected to uniform axial compression loads are presented. These equations are solved exactly for the practical case of simply supported ends. Nondimensional quantities are used to characterize the buckling behavior that consist of a stiffness-weighted length-to-radius parameter, a stiffness-weighted shell-thinness parameter, a shell-wall nonhomogeneity parameter, two orthotropy parameters, and a nondimensional buckling load. Ranges for the nondimensional parameters are established that encompass a wide range of laminated-wall constructions and numerous generic plots of nondimensional buckling load versus a stiffness-weighted length-to-radius ratio are presented for various combinations of the other parameters. These plots are expected to include many practical cases of interest to designers. Additionally, these plots show how the parameter values affect the distribution and size of the festoons forming each response curve and how they affect the attenuation of each response curve to the corresponding solution for an infinitely long cylinder. To aid in preliminary design studies, approximate formulas for the nondimensional buckling load are derived, and validated against the corresponding exact solution, that give the attenuated buckling response of an infinitely long cylinder in terms of the nondimensional parameters presented herein. A relatively small number of "master curves" are identified that give a nondimensional measure of the buckling load of an infinitely long cylinder as a function of the orthotropy and wall inhomogeneity parameters. These curves reduce greatly the complexity of the design-variable space as compared to representations that use dimensional quantities as design variables. As a result of their inherent simplicity, these master curves are anticipated to be useful in the ongoing development of
NASA Astrophysics Data System (ADS)
Bu, Sunyoung; Huang, Jingfang; Boyer, Treavor H.; Miller, Cass T.
2010-07-01
The focus of this work is on the modeling of an ion exchange process that occurs in drinking water treatment applications. The model formulation consists of a two-scale model in which a set of microscale diffusion equations representing ion exchange resin particles that vary in size and age are coupled through a boundary condition with a macroscopic ordinary differential equation (ODE), which represents the concentration of a species in a well-mixed reactor. We introduce a new age-averaged model (AAM) that averages all ion exchange particle ages for a given size particle to avoid the expensive Monte-Carlo simulation associated with previous modeling applications. We discuss two different numerical schemes to approximate both the original Monte-Carlo algorithm and the new AAM for this two-scale problem. The first scheme is based on the finite element formulation in space coupled with an existing backward difference formula-based ODE solver in time. The second scheme uses an integral equation based Krylov deferred correction (KDC) method and a fast elliptic solver (FES) for the resulting elliptic equations. Numerical results are presented to validate the new AAM algorithm, which is also shown to be more computationally efficient than the original Monte-Carlo algorithm. We also demonstrate that the higher order KDC scheme is more efficient than the traditional finite element solution approach and this advantage becomes increasingly important as the desired accuracy of the solution increases. We also discuss issues of smoothness, which affect the efficiency of the KDC-FES approach, and outline additional algorithmic changes that would further improve the efficiency of these developing methods for a wide range of applications.
NASA Astrophysics Data System (ADS)
Lajohn, L. A.; Pratt, R. H.
2015-05-01
There is no simple parameter that can be used to predict when impulse approximation (IA) can yield accurate Compton scattering doubly differential cross sections (DDCS) in relativistic regimes. When Z is low, a small value of the parameter /q (where is the average initial electron momentum and q is the momentum transfer) suffices. For small Z the photon electron kinematic contribution described in relativistic S-matrix (SM) theory reduces to an expression, Xrel, which is present in the relativistic impulse approximation (RIA) formula for Compton DDCS. When Z is high, the S-Matrix photon electron kinematics no longer reduces to Xrel, and this along with the error characterized by the magnitude of /q contribute to the RIA error Δ. We demonstrate and illustrate in the form of contour plots that there are regimes of incident photon energy ωi and scattering angle θ in which the two types of errors at least partially cancel. Our calculations show that when θ is about 65° for Uranium K-shell scattering, Δ is less than 1% over an ωi range of 300 to 900 keV.
Are 0. 1%-accurate gamma-ray assays possible for /sup 235/U solutions
Parker, J.L.
1983-01-01
The factors influencing the accuracy of passive gamma-ray assay of uniform, homogeneous solution samples have been studied in some detail, particularly for the assay of /sup 235/U in uranium solutions. Factors considered are the overall long-term electronic stability, the information losses caused by the rate-related electronic processes of pulse pileup and dead-time, and the self-attenuation of gamma rays within the samples. Both experimental and computational studies indicate that gamma-ray assay procedures for solution samples of moderate size (from approx. 10 to perhaps a few hundred milliliters) are now capable of accuracies approaching 0.1% in many practical cases.
NASA Astrophysics Data System (ADS)
Kupriyanov, N. A.; Simankin, F. A.; Manabaev, K. K.
2016-04-01
A new approximate algorithm for calculating a stress-strain state of viscoelastic bodies is used. The algorithm is based on the derivation of the expressions of time-effective modules. These modules are obtained by iterative changes, compressing the fork of Voigt-Reuss. As an example the analytic solution about the action of a concentrated force on the viscoelastic half-space is compared with the approximate solution. Numerical calculations are performed for a wide range of relaxation characteristics of a viscoelastic half-space. Results of the comparison of stresses and displacements with the analytic solution give coincidence within 3... 15%.
NASA Astrophysics Data System (ADS)
Kafri, H. Q.; Khuri, S. A.; Sayfy, A.
2016-03-01
In this paper, a novel approach is introduced for the solution of the non-linear Troesch's boundary value problem. The underlying strategy is based on Green's functions and fixed-point iterations, including Picard's and Krasnoselskii-Mann's schemes. The resulting numerical solutions are compared with both the analytical solutions and numerical solutions that exist in the literature. Convergence of the iterative schemes is proved via manipulation of the contraction principle. It is observed that the method handles the boundary layer very efficiently, reduces lengthy calculations, provides rapid convergence, and yields accurate results particularly for large eigenvalues. Indeed, to our knowledge, this is the first time that this problem is solved successfully for very large eigenvalues, actually the rate of convergence increases as the magnitude of the eigenvalues increases.
Accurate and inexpensive prediction of the color optical properties of anthocyanins in solution.
Ge, Xiaochuan; Timrov, Iurii; Binnie, Simon; Biancardi, Alessandro; Calzolari, Arrigo; Baroni, Stefano
2015-04-23
The simulation of the color optical properties of molecular dyes in liquid solution requires the calculation of time evolution of the solute absorption spectra fluctuating in the solvent at finite temperature. Time-averaged spectra can be directly evaluated by combining ab initio Car-Parrinello molecular dynamics and time-dependent density functional theory calculations. The inclusion of hybrid exchange-correlation functionals, necessary for the prediction of the correct transition frequencies, prevents one from using these techniques for the simulation of the optical properties of large realistic systems. Here we present an alternative approach for the prediction of the color of natural dyes in solution with a low computational cost. We applied this approach to representative anthocyanin dyes: the excellent agreement between the simulated and the experimental colors makes this method a straightforward and inexpensive tool for the high-throughput prediction of colors of molecules in liquid solvents. PMID:25830823
Toward Accurate Modeling of the Effect of Ion-Pair Formation on Solute Redox Potential.
Qu, Xiaohui; Persson, Kristin A
2016-09-13
A scheme to model the dependence of a solute redox potential on the supporting electrolyte is proposed, and the results are compared to experimental observations and other reported theoretical models. An improved agreement with experiment is exhibited if the effect of the supporting electrolyte on the redox potential is modeled through a concentration change induced via ion pair formation with the salt, rather than by only considering the direct impact on the redox potential of the solute itself. To exemplify the approach, the scheme is applied to the concentration-dependent redox potential of select molecules proposed for nonaqueous flow batteries. However, the methodology is general and enables rational computational electrolyte design through tuning of the operating window of electrochemical systems by shifting the redox potential of its solutes; including potentially both salts as well as redox active molecules. PMID:27500744
Direct Coupling Method for Time-Accurate Solution of Incompressible Navier-Stokes Equations
NASA Technical Reports Server (NTRS)
Soh, Woo Y.
1992-01-01
A noniterative finite difference numerical method is presented for the solution of the incompressible Navier-Stokes equations with second order accuracy in time and space. Explicit treatment of convection and diffusion terms and implicit treatment of the pressure gradient give a single pressure Poisson equation when the discretized momentum and continuity equations are combined. A pressure boundary condition is not needed on solid boundaries in the staggered mesh system. The solution of the pressure Poisson equation is obtained directly by Gaussian elimination. This method is tested on flow problems in a driven cavity and a curved duct.
Highly Accurate Beam Torsion Solutions Using the p-Version Finite Element Method
NASA Technical Reports Server (NTRS)
Smith, James P.
1996-01-01
A new treatment of the classical beam torsion boundary value problem is applied. Using the p-version finite element method with shape functions based on Legendre polynomials, torsion solutions for generic cross-sections comprised of isotropic materials are developed. Element shape functions for quadrilateral and triangular elements are discussed, and numerical examples are provided.
NASA Technical Reports Server (NTRS)
Hyer, M. W.; Cooper, D. E.; Cohen, D.
1985-01-01
The effects of a uniform temperature change on the stresses and deformations of composite tubes are investigated. The accuracy of an approximate solution based on the principle of complementary virtual work is determined. Interest centers on tube response away from the ends and so a planar elasticity approach is used. For the approximate solution a piecewise linear variation of stresses with the radial coordinate is assumed. The results from the approximate solution are compared with the elasticity solution. The stress predictions agree well, particularly peak interlaminar stresses. Surprisingly, the axial deformations also agree well. This, despite the fact that the deformations predicted by the approximate solution do not satisfy the interface displacement continuity conditions required by the elasticity solution. The study shows that the axial thermal expansion coefficient of tubes with a specific number of axial and circumferential layers depends on the stacking sequence. This is in contrast to classical lamination theory which predicts the expansion to be independent of the stacking arrangement. As expected, the sign and magnitude of the peak interlaminar stresses depends on stacking sequence.
NASA Astrophysics Data System (ADS)
Maggio, Emanuele; Kresse, Georg
2016-06-01
The correlation energy of the homogeneous electron gas is evaluated by solving the Bethe-Salpeter equation (BSE) beyond the Tamm-Dancoff approximation for the electronic polarization propagator. The BSE is expected to improve on the random-phase approximation, owing to the inclusion of exchange diagrams. For instance, since the BSE reduces in second order to Møller-Plesset perturbation theory, it is self-interaction free in second order. Results for the correlation energy are compared with quantum Monte Carlo benchmarks and excellent agreement is observed. For low densities, however, we find imaginary eigenmodes in the polarization propagator. To avoid the occurrence of imaginary eigenmodes, an approximation to the BSE kernel is proposed that allows us to completely remove this issue in the low-electron-density region. We refer to this approximation as the random-phase approximation with screened exchange (RPAsX). We show that this approximation even slightly improves upon the standard BSE kernel.
NASA Astrophysics Data System (ADS)
Jiang, Shidong; Luo, Li-Shi
2016-07-01
The integral equation for the flow velocity u (x ; k) in the steady Couette flow derived from the linearized Bhatnagar-Gross-Krook-Welander kinetic equation is studied in detail both theoretically and numerically in a wide range of the Knudsen number k between 0.003 and 100.0. First, it is shown that the integral equation is a Fredholm equation of the second kind in which the norm of the compact integral operator is less than 1 on Lp for any 1 ≤ p ≤ ∞ and thus there exists a unique solution to the integral equation via the Neumann series. Second, it is shown that the solution is logarithmically singular at the endpoints. More precisely, if x = 0 is an endpoint, then the solution can be expanded as a double power series of the form ∑n=0∞∑m=0∞cn,mxn(xln x) m about x = 0 on a small interval x ∈ (0 , a) for some a > 0. And third, a high-order adaptive numerical algorithm is designed to compute the solution numerically to high precision. The solutions for the flow velocity u (x ; k), the stress Pxy (k), and the half-channel mass flow rate Q (k) are obtained in a wide range of the Knudsen number 0.003 ≤ k ≤ 100.0; and these solutions are accurate for at least twelve significant digits or better, thus they can be used as benchmark solutions.
The accurate solution of Poisson's equation by expansion in Chebyshev polynomials
NASA Technical Reports Server (NTRS)
Haidvogel, D. B.; Zang, T.
1979-01-01
A Chebyshev expansion technique is applied to Poisson's equation on a square with homogeneous Dirichlet boundary conditions. The spectral equations are solved in two ways - by alternating direction and by matrix diagonalization methods. Solutions are sought to both oscillatory and mildly singular problems. The accuracy and efficiency of the Chebyshev approach compare favorably with those of standard second- and fourth-order finite-difference methods.
Zeinali-Rafsanjani, B.; Mosleh-Shirazi, M. A.; Faghihi, R.; Karbasi, S.; Mosalaei, A.
2015-01-01
To accurately recompute dose distributions in chest-wall radiotherapy with 120 kVp kilovoltage X-rays, an MCNP4C Monte Carlo model is presented using a fast method that obviates the need to fully model the tube components. To validate the model, half-value layer (HVL), percentage depth doses (PDDs) and beam profiles were measured. Dose measurements were performed for a more complex situation using thermoluminescence dosimeters (TLDs) placed within a Rando phantom. The measured and computed first and second HVLs were 3.8, 10.3 mm Al and 3.8, 10.6 mm Al, respectively. The differences between measured and calculated PDDs and beam profiles in water were within 2 mm/2% for all data points. In the Rando phantom, differences for majority of data points were within 2%. The proposed model offered an approximately 9500-fold reduced run time compared to the conventional full simulation. The acceptable agreement, based on international criteria, between the simulations and the measurements validates the accuracy of the model for its use in treatment planning and radiobiological modeling studies of superficial therapies including chest-wall irradiation using kilovoltage beam. PMID:26170553
Zeinali-Rafsanjani, B; Mosleh-Shirazi, M A; Faghihi, R; Karbasi, S; Mosalaei, A
2015-01-01
To accurately recompute dose distributions in chest-wall radiotherapy with 120 kVp kilovoltage X-rays, an MCNP4C Monte Carlo model is presented using a fast method that obviates the need to fully model the tube components. To validate the model, half-value layer (HVL), percentage depth doses (PDDs) and beam profiles were measured. Dose measurements were performed for a more complex situation using thermoluminescence dosimeters (TLDs) placed within a Rando phantom. The measured and computed first and second HVLs were 3.8, 10.3 mm Al and 3.8, 10.6 mm Al, respectively. The differences between measured and calculated PDDs and beam profiles in water were within 2 mm/2% for all data points. In the Rando phantom, differences for majority of data points were within 2%. The proposed model offered an approximately 9500-fold reduced run time compared to the conventional full simulation. The acceptable agreement, based on international criteria, between the simulations and the measurements validates the accuracy of the model for its use in treatment planning and radiobiological modeling studies of superficial therapies including chest-wall irradiation using kilovoltage beam. PMID:26170553
NASA Astrophysics Data System (ADS)
Haven, Emmanuel
2005-11-01
Analytical solutions to the backward Kolmogorov PDE are very dependent on the functional form of b(y,t) and a(y,t). We suggest one solution technique for obtaining analytical solutions via the use of an adiabatic approximation to the Schrödinger PDE. This approximation takes the specific form of a so-called WKB (W D Wentzel [G. Wentzel, Eine Verallgemeinerung der Quantenbedingungen für die Zwecke der Wellenmechanik, Z. Phys. 38 (1926) 518-529], K D Kramers [H. Kramers, Wellenmechanik und halbzahlige Quantisierung, Z. Phys. 39 (1926) 828-840], B D Brillouin [L. Brillouin, La mécanique ondulatoire de Schrödinger: une méthode générale de résolution par approximations successives, C. R. Acad. Sci. 183 (1926) 24-26]) approximation. We provide for two examples, in financial option pricing, where we show how the proposed approximation could be of use.
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
Thin film flow of an Oldroyd 6-constant fluid over a moving belt: an analytic approximate solution
NASA Astrophysics Data System (ADS)
Ene, Remus-Daniel; Marinca, Vasile; Marinca, Valentin Bogdan
2016-01-01
In this paper the thin film flow of an Oldroyd 6-constant fluid on a vertically moving belt is investigated. The basic equation of a non-Newtonian fluid in a container with a wide moving belt which passes through the container moving vertically upward with constant velocity, is reduced to an ordinary nonlinear differential equation. This equation is solved approximately by means of the Optimal Homotopy Asymptotic Method (OHAM). The solutions take into account the behavior of Newtonian and non-Newtonian fluids. Our procedure intended for solving nonlinear problems does not need small parameters in the equation and provides a convenient way to control the convergence of the approximate solutions.
Alemgadmi, Khaled I. K. Suparmi; Cari; Deta, U. A.
2015-09-30
The approximate analytical solution of Schrodinger equation for Q-Deformed Rosen-Morse potential was investigated using Supersymmetry Quantum Mechanics (SUSY QM) method. The approximate bound state energy is given in the closed form and the corresponding approximate wave function for arbitrary l-state given for ground state wave function. The first excited state obtained using upper operator and ground state wave function. The special case is given for the ground state in various number of q. The existence of Rosen-Morse potential reduce energy spectra of system. The larger value of q, the smaller energy spectra of system.
Dworak, D; Loskiewicz, J; Janik, M
2001-05-01
The diffusion approximation solution for neutron transport has been used in well-logging geophysics for calculating tool responses in boreholes, sometimes with success. The problem of the dimension of different materials to which it can be applied with success is important for the borehole environment. The results obtained show that the diffusion approximation can be used for distances greater than a few millimetre in some rock types. For iron, barium, and other highly absorbing media the use of the diffusion approximation is inappropriate even for large distances. PMID:11258535
Ene, Remus-Daniel; Marinca, Vasile; Marinca, Bogdan
2016-01-01
Analytic approximate solutions using Optimal Homotopy Perturbation Method (OHPM) are given for steady boundary layer flow over a nonlinearly stretching wall in presence of partial slip at the boundary. The governing equations are reduced to nonlinear ordinary differential equation by means of similarity transformations. Some examples are considered and the effects of different parameters are shown. OHPM is a very efficient procedure, ensuring a very rapid convergence of the solutions after only two iterations. PMID:27031232
Ene, Remus-Daniel; Marinca, Vasile; Marinca, Bogdan
2016-01-01
Analytic approximate solutions using Optimal Homotopy Perturbation Method (OHPM) are given for steady boundary layer flow over a nonlinearly stretching wall in presence of partial slip at the boundary. The governing equations are reduced to nonlinear ordinary differential equation by means of similarity transformations. Some examples are considered and the effects of different parameters are shown. OHPM is a very efficient procedure, ensuring a very rapid convergence of the solutions after only two iterations. PMID:27031232
Xiang, Yanhui; Jiang, Yiqi; Chao, Xiaomei; Wu, Qihan; Mo, Lei
2016-01-01
Approximate strategies are crucial in daily human life. The studies on the "difficulty effect" seen in approximate complex arithmetic have long been neglected. Here, we aimed to explore the brain mechanisms related to this difficulty effect in the case of complex addition, using event-related potential-based methods. Following previous path-finding studies, we used the inequality paradigm and different split sizes to induce the use of two approximate strategies for different difficulty levels. By comparing dependent variables from the medium- and large-split conditions, we anticipated being able to dissociate the effects of task difficulty based on approximate strategy in electrical components. In the fronto-central region, early P2 (150-250 ms) and an N400-like wave (250-700 ms) were significantly different between different difficulty levels. Differences in P2 correlated with the difficulty of separation of the approximate strategy from the early physical stimulus discrimination process, which is dominant before 200 ms, and differences in the putative N400 correlated with different difficulties of approximate strategy execution. Moreover, this difference may be linked to speech processing. In addition, differences were found in the fronto-central region, which may reflect the regulatory role of this part of the cortex in approximate strategy execution when solving complex arithmetic problems. PMID:27072753
Xiang, Yanhui; Jiang, Yiqi; Chao, Xiaomei; Wu, Qihan; Mo, Lei
2016-01-01
Approximate strategies are crucial in daily human life. The studies on the “difficulty effect” seen in approximate complex arithmetic have long been neglected. Here, we aimed to explore the brain mechanisms related to this difficulty effect in the case of complex addition, using event-related potential-based methods. Following previous path-finding studies, we used the inequality paradigm and different split sizes to induce the use of two approximate strategies for different difficulty levels. By comparing dependent variables from the medium- and large-split conditions, we anticipated being able to dissociate the effects of task difficulty based on approximate strategy in electrical components. In the fronto−central region, early P2 (150–250 ms) and an N400-like wave (250–700 ms) were significantly different between different difficulty levels. Differences in P2 correlated with the difficulty of separation of the approximate strategy from the early physical stimulus discrimination process, which is dominant before 200 ms, and differences in the putative N400 correlated with different difficulties of approximate strategy execution. Moreover, this difference may be linked to speech processing. In addition, differences were found in the fronto-central region, which may reflect the regulatory role of this part of the cortex in approximate strategy execution when solving complex arithmetic problems. PMID:27072753
Design and Construction Solutions in the Accurate Realization of NCSX Magnetic Fields
Heitzenroeder, P.; Dudek, Lawrence E.; Brooks, Arthur W.; Viola, Michael E.; Brown, Thomas; Neilson, George H.; Zarnstorff, Michael C.; Rej, Donald; Cole,Michael J.; Freudenberg, Kevin D.; Harris J. H.; McGinnis, Gary
2008-09-29
The National Compact Stellarator Experiment, NCSX, is being constructed at the Princeton Plasma Physics Laboratory (PPPL) in partnership with the Oak Ridge national Laboratory. The goal of NCSX is to provide the understanding necessary to develop an attractive, disruption free, steady state compact stellaratorbased reactor design. This paper describes the recently revised designs of the critical interfaces between the modular coils, the construction solutions developed to meet assembly tolerances, and the recently revised trim coil system that provides the required compensation to correct for the “as built” conditions and to allow flexibility in the disposition of as-built conditions. In May, 2008, the sponsor decided to terminate the NCSX project due to growth in the project’s cost and schedule estimates. However significant technical challenges in design and construction were overcome, greatly reducing the risk in the remaining work to complete the project.
Mutelet, Fabrice; Jaubert, Jean-Noël
2006-01-13
Activity coefficients at infinite dilution of 29 organic compounds in two room temperature ionic liquids were determined using inverse gas chromatography. The measurements were carried out at different temperatures between 323.15 and 343.15K. To establish the influence of concurrent retention mechanisms on the accuracy of activity coefficients at infinite dilution for 1-butyl-3-methylimidazolium octyl sulfate and 1-ethyl-3-methylimidazolium tosylate, phase loading studies of the net retention volume per gram of packing as a function of the percent phase loading were used. It is shown that most of the solutes are retained largely by partition with a small contribution from adsorption on 1-butyl-3-methylimidazolium octyl sulfate and that the n-alkanes are retained predominantly by interfacial adsorption on 1-ethyl-3-methylimidazolium tosylate. PMID:16310203
Hong Xinguo; Hao Quan
2009-01-15
In this paper, we report a method of precise in situ x-ray scattering measurements on protein solutions using small stationary sample cells. Although reduction in the radiation damage induced by intense synchrotron radiation sources is indispensable for the correct interpretation of scattering data, there is still a lack of effective methods to overcome radiation-induced aggregation and extract scattering profiles free from chemical or structural damage. It is found that radiation-induced aggregation mainly begins on the surface of the sample cell and grows along the beam path; the diameter of the damaged region is comparable to the x-ray beam size. Radiation-induced aggregation can be effectively avoided by using a two-dimensional scan (2D mode), with an interval as small as 1.5 times the beam size, at low temperature (e.g., 4 deg. C). A radiation sensitive protein, bovine hemoglobin, was used to test the method. A standard deviation of less than 5% in the small angle region was observed from a series of nine spectra recorded in 2D mode, in contrast to the intensity variation seen using the conventional stationary technique, which can exceed 100%. Wide-angle x-ray scattering data were collected at a standard macromolecular diffraction station using the same data collection protocol and showed a good signal/noise ratio (better than the reported data on the same protein using a flow cell). The results indicate that this method is an effective approach for obtaining precise measurements of protein solution scattering.
An Exact Solution for Geophysical Edge Waves in the {β}-Plane Approximation
NASA Astrophysics Data System (ADS)
Ionescu-Kruse, Delia
2015-12-01
By taking into account the {β}-plane effects, we provide an exact nonlinear solution to the geophysical edge-wave problem within the Lagrangian framework. This solution describes trapped waves propagating eastward or westward along a sloping beach with the shoreline parallel to the Equator.
NASA Astrophysics Data System (ADS)
Gorpas, Dimitris; Andersson-Engels, Stefan
2012-03-01
The solution of the forward problem in fluorescence molecular imaging is among the most important premises for the successful confrontation of the inverse reconstruction problem. To date, the most typical approach has been the application of the diffusion approximation as the forward model. This model is basically a first order angular approximation for the radiative transfer equation, and thus it presents certain limitations. The scope of this manuscript is to present the dual coupled radiative transfer equation and diffusion approximation model for the solution of the forward problem in fluorescence molecular imaging. The integro-differential equations of its weak formalism were solved via the finite elements method. Algorithmic blocks with cubature rules and analytical solutions of the multiple integrals have been constructed for the solution. Furthermore, specialized mapping matrices have been developed to assembly the finite elements matrix. As a radiative transfer equation based model, the integration over the angular discretization was implemented analytically, while quadrature rules were applied whenever required. Finally, this model was evaluated on numerous virtual phantoms and its relative accuracy, with respect to the radiative transfer equation, was over 95%, when the widely applied diffusion approximation presented almost 85% corresponding relative accuracy for the fluorescence emission.
NASA Astrophysics Data System (ADS)
Lee, Dongwook
2013-06-01
In this paper, we extend the unsplit staggered mesh scheme (USM) for 2D magnetohydrodynamics (MHD) [D. Lee, A.E. Deane, An unsplit staggered mesh scheme for multidimensional magnetohydrodynamics, J. Comput. Phys. 228 (2009) 952-975] to a full 3D MHD scheme. The scheme is a finite-volume Godunov method consisting of a constrained transport (CT) method and an efficient and accurate single-step, directionally unsplit multidimensional data reconstruction-evolution algorithm, which extends Colella's original 2D corner transport upwind (CTU) method [P. Colella, Multidimensional upwind methods for hyperbolic conservation laws, J. Comput. Phys. 87 (1990) 446-466]. We present two types of data reconstruction-evolution algorithms for 3D: (1) a reduced CTU scheme and (2) a full CTU scheme. The reduced 3D CTU scheme is a variant of a simple 3D extension of Collela's 2D CTU method and is considered as a direct extension from the 2D USM scheme. The full 3D CTU scheme is our primary 3D solver which includes all multidimensional cross-derivative terms for stability. The latter method is logically analogous to the 3D unsplit CTU method by Saltzman [J. Saltzman, An unsplit 3D upwind method for hyperbolic conservation laws, J. Comput. Phys. 115 (1994) 153-168]. The major novelties in our algorithms are twofold. First, we extend the reduced CTU scheme to the full CTU scheme which is able to run with CFL numbers close to unity. Both methods utilize the transverse update technique developed in the 2D USM algorithm to account for transverse fluxes without solving intermediate Riemann problems, which in turn gives cost-effective 3D methods by reducing the total number of Riemann solves. The proposed algorithms are simple and efficient especially when including multidimensional MHD terms that maintain in-plane magnetic field dynamics. Second, we introduce a new CT scheme that makes use of proper upwind information in taking averages of electric fields. Our 3D USM schemes can be easily
Energy expenditure during level human walking: seeking a simple and accurate predictive solution.
Ludlow, Lindsay W; Weyand, Peter G
2016-03-01
Accurate prediction of the metabolic energy that walking requires can inform numerous health, bodily status, and fitness outcomes. We adopted a two-step approach to identifying a concise, generalized equation for predicting level human walking metabolism. Using literature-aggregated values we compared 1) the predictive accuracy of three literature equations: American College of Sports Medicine (ACSM), Pandolf et al., and Height-Weight-Speed (HWS); and 2) the goodness-of-fit possible from one- vs. two-component descriptions of walking metabolism. Literature metabolic rate values (n = 127; speed range = 0.4 to 1.9 m/s) were aggregated from 25 subject populations (n = 5-42) whose means spanned a 1.8-fold range of heights and a 4.2-fold range of weights. Population-specific resting metabolic rates (V̇o2 rest) were determined using standardized equations. Our first finding was that the ACSM and Pandolf et al. equations underpredicted nearly all 127 literature-aggregated values. Consequently, their standard errors of estimate (SEE) were nearly four times greater than those of the HWS equation (4.51 and 4.39 vs. 1.13 ml O2·kg(-1)·min(-1), respectively). For our second comparison, empirical best-fit relationships for walking metabolism were derived from the data set in one- and two-component forms for three V̇o2-speed model types: linear (∝V(1.0)), exponential (∝V(2.0)), and exponential/height (∝V(2.0)/Ht). We found that the proportion of variance (R(2)) accounted for, when averaged across the three model types, was substantially lower for one- vs. two-component versions (0.63 ± 0.1 vs. 0.90 ± 0.03) and the predictive errors were nearly twice as great (SEE = 2.22 vs. 1.21 ml O2·kg(-1)·min(-1)). Our final analysis identified the following concise, generalized equation for predicting level human walking metabolism: V̇o2 total = V̇o2 rest + 3.85 + 5.97·V(2)/Ht (where V is measured in m/s, Ht in meters, and V̇o2 in ml O2·kg(-1)·min(-1)). PMID:26679617
Accurate solutions, parameter studies and comparisons for the Euler and potential flow equations
NASA Technical Reports Server (NTRS)
Anderson, W. Kyle; Batina, John T.
1988-01-01
Parameter studies are conducted using the Euler and potential flow equation models for steady and unsteady flows in both two and three dimensions. The Euler code is an implicit, upwind, finite volume code which uses the Van Leer method of flux vector splitting which has been recently extended for use on dynamic meshes and maintain all the properties of the original splitting. The potential flow code is an implicit, finite difference method for solving the transonic small disturbance equations and incorporates both entropy and vorticity corrections into the solution procedures thereby extending its applicability into regimes where shock strength normally precludes its use. Parameter studies resulting in benchmark type calculations include the effects of spatial and temporal refinement, spatial order of accuracy, far field boundary conditions for steady flow, frequency of oscillation, and the use of subiterations at each time step to reduce linearization and factorization errors. Comparisons between Euler and potential flow results are made, as well as with experimental data where available.
Accurate solutions, parameter studies and comparisons for the Euler and potential flow equations
NASA Technical Reports Server (NTRS)
Anderson, W. Kyle; Batina, John T.
1988-01-01
Parameter studies are conducted using the Euler and potential flow equation models for unsteady and steady flows in both two and three dimensions. The Euler code is an implicit, upwind, finite volume code which uses the Van Leer method of flux-vector-splitting which has been recently extended for use on dynamic meshes and maintain all the properties of the original splitting. The potential flow code is an implicit, finite difference method for solving the transonic small disturbance equations and incorporates both entropy and vorticity corrections into the solution procedures thereby extending its applicability into regimes where shock strength normally precludes its use. Parameter studies resulting in benchmark type calculations include the effects of spatial and temporal refinement, spatial order of accuracy, far field boundary conditions for steady flow, frequency of oscillation, and the use of subiterations at each time step to reduce linearization and factorization errors. Comparisons between Euler and potential flows results are made as well as with experimental data where available.
NASA Technical Reports Server (NTRS)
Tal-Ezer, Hillel
1987-01-01
During the process of solving a mathematical model numerically, there is often a need to operate on a vector v by an operator which can be expressed as f(A) while A is NxN matrix (ex: exp(A), sin(A), A sup -1). Except for very simple matrices, it is impractical to construct the matrix f(A) explicitly. Usually an approximation to it is used. In the present research, an algorithm is developed which uses a polynomial approximation to f(A). It is reduced to a problem of approximating f(z) by a polynomial in z while z belongs to the domain D in the complex plane which includes all the eigenvalues of A. This problem of approximation is approached by interpolating the function f(z) in a certain set of points which is known to have some maximal properties. The approximation thus achieved is almost best. Implementing the algorithm to some practical problem is described. Since a solution to a linear system Ax = b is x= A sup -1 b, an iterative solution to it can be regarded as a polynomial approximation to f(A) = A sup -1. Implementing the algorithm in this case is also described.
NASA Astrophysics Data System (ADS)
Elizarov, A. M.; Kalimullina, A. N.
2009-03-01
The lift/drag ratio of an airfoil placed in an incompressible attached flow is maximized taking into account the viscosity in the boundary-layer approximation. An exact solution is constructed. The situation when the resulting solutions are not in the admissible class of univalent flows is discussed. A procedure is proposed for determining physically feasible airfoils (with a univalent flow region) with a high lift/drag ratio. For this purpose, a class of airfoils is constructed that are determined by a twoparameter function approximating the found exact solution to the variational problem. For this class, the ranges of free parameters leading to physically feasible flows are found. The results are verified by computing a turbulent boundary layer using Eppler’s method, and airfoils with a high lift/drag ratio in an attached flow are detected.
Simulating higher-dimensional geometries in GADRAS using approximate one-dimensional solutions.
Thoreson, Gregory G.; Mitchell, Dean James; Harding, Lee T.
2013-02-01
The Gamma Detector Response and Analysis Software (GADRAS) software package is capable of simulating the radiation transport physics for one-dimensional models. Spherical shells are naturally one-dimensional, and have been the focus of development and benchmarking. However, some objects are not spherical in shape, such as cylinders and boxes. These are not one-dimensional. Simulating the radiation transport in two or three dimensions is unattractive because of the extra computation time required. To maintain computational efficiency, higher-dimensional geometries require approximations to simulate them in one-dimension. This report summarizes the theory behind these approximations, tests the theory against other simulations, and compares the results to experimental data. Based on the results, it is recommended that GADRAS users always attempt to approximate reality using spherical shells. However, if fissile material is present, it is imperative that the shape of the one-dimensional model matches the fissile material, including the use of slab and cylinder geometry.
Approximation of traveling wave solutions in wall-bounded flows using resolvent modes
NASA Astrophysics Data System (ADS)
McKeon, Beverley; Graham, Michael; Moarref, Rashad; Park, Jae Sung; Sharma, Ati; Willis, Ashley
2014-11-01
Significant recent attention has been devoted to computing and understanding exact traveling wave solutions of the Navier-Stokes equations. These solutions can be interpreted as the state-space skeleton of turbulence and are attractive benchmarks for studying low-order models of wall turbulence. Here, we project such solutions onto the velocity response (or resolvent) modes supplied by the gain-based resolvent analysis outlined by McKeon & Sharma (JFM, 2010). We demonstrate that in both pipe (Pringle et al., Phil. Trans. R. Soc. A, 2009) and channel (Waleffe, JFM, 2001) flows, the solutions can be well-described by a small number of resolvent modes. Analysis of the nonlinear forcing modes sustaining these solutions reveals the importance of small amplitude forcing, consistent with the large amplifications admitted by the resolvent operator. We investigate the use of resolvent modes as computationally cheap ``seeds'' for the identification of further traveling wave solutions. The support of AFOSR under Grants FA9550-09-1-0701, FA9550-12-1-0469, FA9550-11-1-0094 and FA9550-14-1-0042 (program managers Rengasamy Ponnappan, Doug Smith and Gregg Abate) is gratefully acknowledged.
NASA Astrophysics Data System (ADS)
Gráf, Lukáš; Čížek, Martin
2014-09-01
A two dimensional model for the electron interaction with molecular vibrations in molecular junctions is proposed. Alternatively the model can be applied to tunneling through a cylindrical nano-structure. The transmission function is calculated accurately numerically. The exact results are then compared with various approximations: (1) completely frozen vibrations for very light molecule, (2) Chase approximation for very heavy molecule, and (3) discrete-state-in-continuum model in resonant regime. The validity of these approximations is discussed in terms of the characteristic time-scales and coupling strengths. The excitation of the vibrational degree of freedom and the emergence of prominent threshold structures in the strong coupling regime are discussed in more details.
Loglinear Approximate Solutions to Real-Business-Cycle Models: Some Observations
ERIC Educational Resources Information Center
Lau, Sau-Him Paul; Ng, Philip Hoi-Tak
2007-01-01
Following the analytical approach suggested in Campbell, the authors consider a baseline real-business-cycle (RBC) model with endogenous labor supply. They observe that the coefficients in the loglinear approximation of the dynamic equations characterizing the equilibrium are related to the fundamental parameters in a relatively simple manner.…
An approximate method for solution to variable moment of inertia problems
NASA Technical Reports Server (NTRS)
Beans, E. W.
1981-01-01
An approximation method is presented for reducing a nonlinear differential equation (for the 'weather vaning' motion of a wind turbine) to an equivalent constant moment of inertia problem. The integrated average of the moment of inertia is determined. Cycle time was found to be the equivalent cycle time if the rotating speed is 4 times greater than the system's minimum natural frequency.
Duan, Qianqian; Yang, Genke; Xu, Guanglin; Pan, Changchun
2014-01-01
This paper is devoted to develop an approximation method for scheduling refinery crude oil operations by taking into consideration the demand uncertainty. In the stochastic model the demand uncertainty is modeled as random variables which follow a joint multivariate distribution with a specific correlation structure. Compared to deterministic models in existing works, the stochastic model can be more practical for optimizing crude oil operations. Using joint chance constraints, the demand uncertainty is treated by specifying proximity level on the satisfaction of product demands. However, the joint chance constraints usually hold strong nonlinearity and consequently, it is still hard to handle it directly. In this paper, an approximation method combines a relax-and-tight technique to approximately transform the joint chance constraints to a serial of parameterized linear constraints so that the complicated problem can be attacked iteratively. The basic idea behind this approach is to approximate, as much as possible, nonlinear constraints by a lot of easily handled linear constraints which will lead to a well balance between the problem complexity and tractability. Case studies are conducted to demonstrate the proposed methods. Results show that the operation cost can be reduced effectively compared with the case without considering the demand correlation. PMID:24757433
NASA Astrophysics Data System (ADS)
Pazzona, Federico G.; Demontis, Pierfranco; Suffritti, Giuseppe B.
2014-08-01
The adsorption isotherm for the recently proposed parallel Kawasaki (PK) lattice-gas model [Phys. Rev. E 88, 062144 (2013), 10.1103/PhysRevE.88.062144] is calculated exactly in one dimension. To do so, a third-order difference equation for the grand-canonical partition function is derived and solved analytically. In the present version of the PK model, the attraction and repulsion effects between two neighboring particles and between a particle and a neighboring empty site are ruled, respectively, by the dimensionless parameters ϕ and θ. We discuss the inflections induced in the isotherms by situations of high repulsion, the role played by finite lattice sizes in the emergence of substeps, and the adequacy of the two most widely used mean-field approximations in lattice gases, namely, the Bragg-Williams and the Bethe-Peierls approximations.
Heat flux solutions of the 13-moment approximation transport equations in a multispecies gas
Jian Wu; Taieb, C.
1993-09-01
The authors study steady state heat flux equations by means of the 13-moment approximation for situations applicable to aeronomy and space plasmas. They compare their results with Fourier`s law applied to similar problems, to test validity conditions for it. They look at the flux of oxygen and hydrogen ions in the high-latitude ionosphere, and compare calculations with observations from EISCAT radar measurements. These plasma components are observed to have strongly non-Maxwellian distributions.
Technology Transfer Automated Retrieval System (TEKTRAN)
The system of governing equations of a simplified slab model of the uniformly-mixed, purely convective, diurnal atmospheric boundary layer (ABL) is shown to allow immediate solutions for the potential temperature and specific humidity as functions of the ABL height and net radiation when expressed i...
Guo, Chengan; Yang, Qingshan
2015-07-01
Finding the optimal solution to the constrained l0 -norm minimization problems in the recovery of compressive sensed signals is an NP-hard problem and it usually requires intractable combinatorial searching operations for getting the global optimal solution, unless using other objective functions (e.g., the l1 norm or lp norm) for approximate solutions or using greedy search methods for locally optimal solutions (e.g., the orthogonal matching pursuit type algorithms). In this paper, a neurodynamic optimization method is proposed to solve the l0 -norm minimization problems for obtaining the global optimum using a recurrent neural network (RNN) model. For the RNN model, a group of modified Gaussian functions are constructed and their sum is taken as the objective function for approximating the l0 norm and for optimization. The constructed objective function sets up a convexity condition under which the neurodynamic system is guaranteed to obtain the globally convergent optimal solution. An adaptive adjustment scheme is developed for improving the performance of the optimization algorithm further. Extensive experiments are conducted to test the proposed approach in this paper and the output results validate the effectiveness of the new method. PMID:25122603
Ori, Amos
2010-11-15
Callan, Giddings, Harvey, and Strominger (CGHS) previously introduced a two-dimensional semiclassical model of gravity coupled to a dilaton and to matter fields. Their model yields a system of field equations which may describe the formation of a black hole in gravitational collapse as well as its subsequent evaporation. Here we present an approximate analytical solution to the semiclassical CGHS field equations. This solution is constructed using the recently introduced formalism of flux-conserving hyperbolic systems. We also explore the asymptotic behavior at the horizon of the evaporating black hole.
NASA Astrophysics Data System (ADS)
Tinoco Arenas, A.; González Bolívar, M.; Medina Covarrubias, R.; Raga, A. C.
2015-10-01
We present analytic models for a photoionized region in pressure equilibrium with the surrounding, neutral material. The models are based on the assumption of a linear relation between the H ionization fraction and the square of the sound speed of the gas. We show that under these assumptions the "grey" radiative transfer equation has analytic solutions that provide the ionization structure and the density of the nebula as a function of radius.
Kim, S.
1994-12-31
Parallel iterative procedures based on domain decomposition techniques are defined and analyzed for the numerical solution of wave propagation by finite element and finite difference methods. For finite element methods, in a Lagrangian framework, an efficient way for choosing the algorithm parameter as well as the algorithm convergence are indicated. Some heuristic arguments for finding the algorithm parameter for finite difference schemes are addressed. Numerical results are presented to indicate the effectiveness of the methods.
Approximation methods for the solution of inverse problems in lake and sea sediment core analysis
NASA Technical Reports Server (NTRS)
Banks, H. T.; Rosen, I. G.
1985-01-01
A theoretical model employing one-dimensional (depth) transport equations to describe vertical redistribution of ocean-floor and lake-floor sediment (particulates, volcanic ash, microtektites, or radioactive tracers) by episodic and nonepisodic events including bioturbation is developed analytically and demonstrated. The principles underlying the model are explained; the model equations are derived; the inverse problem of identifying the depth-dependent bioturbation coefficient is addressed; two approximation theorems are presented; and numerical results for two sample problems are presented graphically. It is suggested that compatification, porosity effects, and depth-dependent sedimentation be taken into account when formulating future models.
Delfino, A.; Silva, J.B.; Malheiro, M.
2006-03-15
We study nuclear matter, at the mean-field approximation, by considering as equal the values of the scalar and the vector density in the Walecka model, which is a very reasonable approximation up to the nuclear matter saturation density. It turns out that the model has an analytical solution for the scalar and vector couplings as functions only of the nuclear matter density and binding energy. The nuclear matter properties are very close to the original version of the model. This solution allows us to show that the correlation between the binding energy and the saturation density is Coester line like. The liquid-gas phase transition is also studied and the critical and flash temperatures are again very similar to the original ones.
NASA Astrophysics Data System (ADS)
Ikot, Akpan N.; Maghsoodi, Elham; Hassanabadi, Hassan; Obu, Joseph A.
2014-05-01
In this paper, we obtain the approximate analytical bound-state solutions of the Dirac particle with the generalized Yukawa potential within the framework of spin and pseudospin symmetries for the arbitrary к state with a generalized tensor interaction. The generalized parametric Nikiforov-Uvarov method is used to obtain the energy eigenvalues and the corresponding wave functions in closed form. We also report some numerical results and present figures to show the effect of the tensor interaction.
Krause, Katharina; Bauer, Mirko; Klopper, Wim
2016-06-14
Theoretical description of phosphorescence lifetimes in the condensed phase requires a method that takes into account both spin-orbit coupling and solvent-solute interactions. To obtain such a method, we have coupled our recently developed two-component coupled-cluster method with singles and approximated doubles to a polarizable environment. With this new method, we investigate how different solvents effect the electronic phosphorescence energies and lifetimes of 4H-pyran-4-thione. PMID:27158835
Traytak, Sergey D.
2014-06-14
The anisotropic 3D equation describing the pointlike particles diffusion in slender impermeable tubes of revolution with cross section smoothly depending on the longitudinal coordinate is the object of our study. We use singular perturbations approach to find the rigorous asymptotic expression for the local particles concentration as an expansion in the ratio of the characteristic transversal and longitudinal diffusion relaxation times. The corresponding leading-term approximation is a generalization of well-known Fick-Jacobs approximation. This result allowed us to delineate the conditions on temporal and spatial scales under which the Fick-Jacobs approximation is valid. A striking analogy between solution of our problem and the method of inner-outer expansions for low Knudsen numbers gas kinetic theory is established. With the aid of this analogy we clarify the physical and mathematical meaning of the obtained results.
Approximate Solution to the Fractional Second-Type Lane-Emden Equation
NASA Astrophysics Data System (ADS)
Abdel-Salam, E. A.-B.; Nouh, M. I.
2016-09-01
The spherical isothermal Lane-Emden equation is a second-order nonlinear differential equation that models many configurations in astrophysics. Using the fractal index technique and the power series expansion, we solve the fractional Lane-Emden equation involving the modified Riemann-Liouville derivative. The results indicate that the series converges over the range of radii 0≤ x< 2200 for a wide spread of values for the fractional parameter α. Comparison with the numerical solution reveals good agreement with a maximum relative error of 0.05.
NASA Technical Reports Server (NTRS)
Gunzburger, M. D.; Nicolaides, R. A.
1986-01-01
Substructuring methods are in common use in mechanics problems where typically the associated linear systems of algebraic equations are positive definite. Here these methods are extended to problems which lead to nonpositive definite, nonsymmetric matrices. The extension is based on an algorithm which carries out the block Gauss elimination procedure without the need for interchanges even when a pivot matrix is singular. Examples are provided wherein the method is used in connection with finite element solutions of the stationary Stokes equations and the Helmholtz equation, and dual methods for second-order elliptic equations.
An approximate closed-form solution for lead lag damping of rotor blades in hover
NASA Technical Reports Server (NTRS)
Peters, D. A.
1975-01-01
Simple stability methods are used to derive an approximate, closed-form expression for the lead-lag damping of rotor blades in hover. Destabilizing terms are shown to be a result of two dynamic mechanisms. First, the destabilizing aerodynamic forces that can occur when blade lift is higher than a critical value are maximized when the blade motion is in a straight line equidistant from the blade chord and the average direction of the air flow velocity. This condition occurs when the Coriolis terms vanish and when the elastic coupling terms align the blade motion with this least stable direction. Second, the nonconservative stiffness terms that result from pitch-flap or pitch-lag coupling can add or subtract energy from the system depending upon whether the motion of the blade tip is clockwise or counterclockwise.
NASA Technical Reports Server (NTRS)
White, C. W.
1981-01-01
The computational efficiency of the impedance type loads prediction method was studied. Three goals were addressed: devise a method to make the impedance method operate more efficiently in the computer; assess the accuracy and convenience of the method for determining the effect of design changes; and investigate the use of the method to identify design changes for reduction of payload loads. The method is suitable for calculation of dynamic response in either the frequency or time domain. It is concluded that: the choice of an orthogonal coordinate system will allow the impedance method to operate more efficiently in the computer; the approximate mode impedance technique is adequate for determining the effect of design changes, and is applicable for both statically determinate and statically indeterminate payload attachments; and beneficial design changes to reduce payload loads can be identified by the combined application of impedance techniques and energy distribution review techniques.
Dahlen, Nils Erik; van Leeuwen, Robert
2005-04-22
We have calculated the self-consistent Green's function for a number of atoms and diatomic molecules. This Green's function is obtained from a conserving self-energy approximation, which implies that the observables calculated from the Green's functions agree with the macroscopic conservation laws for particle number, momentum, and energy. As a further consequence, the kinetic and potential energies agree with the virial theorem, and the many possible methods for calculating the total energy all give the same result. In these calculations we use the finite temperature formalism and calculate the Green's function on the imaginary time axis. This allows for a simple extension to nonequilibrium systems. We have compared the energies from self-consistent Green's functions to those of nonselfconsistent schemes and also calculated ionization potentials from the Green's functions by using the extended Koopmans' theorem. PMID:15945667
Atta-Fynn, Raymond; Biswas, Parthapratim
2009-07-01
Localized basis ab initio molecular dynamics simulation within the density functional framework has been used to generate realistic configurations of amorphous silicon carbide (a-SiC). Our approach consists of constructing a set of smart initial configurations that conform to essential geometrical and structural aspects of the materials obtained from experimental data, which is subsequently driven via a first-principles force field to obtain the best solution in a reduced solution space. A combination of a priori information (primarily structural and topological) along with the ab initio optimization of the total energy makes it possible to model a large system size (1000 atoms) without compromising the quantum mechanical accuracy of the force field to describe the complex bonding chemistry of Si and C. The structural, electronic and vibrational properties of the models have been studied and compared to existing theoretical models and available data from experiments. We demonstrate that the approach is capable of producing large, realistic configurations of a-SiC from first-principles simulation that display its excellent structural and electronic properties. Our study reveals the presence of predominant short range order in the material originating from heteronuclear Si-C bonds with a coordination defect concentration as small as 5% and a chemical disorder parameter of about 8%. PMID:21828477
Cobb, J.W.
1995-02-01
There is an increasing need for more accurate numerical methods for large-scale nonlinear magneto-fluid turbulence calculations. These methods should not only increase the current state of the art in terms of accuracy, but should also continue to optimize other desired properties such as simplicity, minimized computation, minimized memory requirements, and robust stability. This includes the ability to stably solve stiff problems with long time-steps. This work discusses a general methodology for deriving higher-order numerical methods. It also discusses how the selection of various choices can affect the desired properties. The explicit discussion focuses on third-order Runge-Kutta methods, including general solutions and five examples. The study investigates the linear numerical analysis of these methods, including their accuracy, general stability, and stiff stability. Additional appendices discuss linear multistep methods, discuss directions for further work, and exhibit numerical analysis results for some other commonly used lower-order methods.
Modeling in nuclear waste isolation: Approximate solutions for flow in unsaturated porous media
Martinez, M.J.; McTigue, D.F.
1996-12-31
Mathematical modeling plays a key role in the design and licensing of repositories for radioactive waste. Because safe isolation of nuclear waste involves extremely long time scales, and there exists very little engineering experience upon which to draw, modeling takes on a particularly crucial role. An example of a model problem motivated by hydrological issues in high-level waste isolation is presented. A repository concept involving storage in rock above the water table requires models for the flow of groundwater in unsaturated, porous media. Such flow is governed by an extremely nonlinear diffusion equation, and poses some difficult numerical challenges. A special form of the hydraulic conductivity function however, results in a linear field equation for steady-state problems, for which a boundary integral method yields very fast solutions.
Solution of nonlinear time-dependent PDEs through componentwise approximation of matrix functions
NASA Astrophysics Data System (ADS)
Cibotarica, Alexandru; Lambers, James V.; Palchak, Elisabeth M.
2016-09-01
Exponential propagation iterative (EPI) methods provide an efficient approach to the solution of large stiff systems of ODEs, compared to standard integrators. However, the bulk of the computational effort in these methods is due to products of matrix functions and vectors, which can become very costly at high resolution due to an increase in the number of Krylov projection steps needed to maintain accuracy. In this paper, it is proposed to modify EPI methods by using Krylov subspace spectral (KSS) methods, instead of standard Krylov projection methods, to compute products of matrix functions and vectors. Numerical experiments demonstrate that this modification causes the number of Krylov projection steps to become bounded independently of the grid size, thus dramatically improving efficiency and scalability. As a result, for each test problem featured, as the total number of grid points increases, the growth in computation time is just below linear, while other methods achieved this only on selected test problems or not at all.
NASA Astrophysics Data System (ADS)
Mondescu, Radu P.; Muthukumar, M.
1999-06-01
We have considered the stationary state of a one-dimensional Gaussian polymer chain of length Nl (N is number of segments, l is the Kuhn length) subjected to a one-parameter class of exactly solvable, symmetric, repulsive potentials with multiple (two or three) minima. We have calculated analytically and numerically the exact Green's function G(R|R';L) and the mean-square end-to-end distance <(R-R')2>, respectively, of the polymer chain. The instanton approximation is translated in polymer physics language and used to analyze the conformation of the polymer chain in its ground state, by evaluating the average number of folds that connect the potential minima. All quantities are functions of the barrier height and of the separation between wells. Our results show that for a given length N, the polymer expands with increasing barrier width until a maximum value of <(R-R')2> is reached. Afterwards, the polymer apparently collapses in one of the wells. This behavior defines a critical length N* and may offer the possibility of applications in separation and pattern recognition processes.
Zhang, X.C.; Xu, B.S.; Wang, H.D.; Wu, Y.X.
2005-09-01
Stoney's equation and subsequent modifications and some approximations are widely used to evaluate the macrostress within a film on a substrate, though some of these solutions are only applicable for thin films. The purpose of this paper is to review the considerable efforts devoted to the analysis of residual stresses in a single-layer film in the last century and recent years and to estimate the errors involved in using these formulas. The following are some of the important results that can be obtained. (1) The exact solution for the residual stress can be expressed in terms of Stoney's equation [Proc. R. Soc. London A82, 172 (1909)] and a correction factor (1+{sigma}{eta}{sup 3})/(1+{eta}), where {sigma},{eta} are the ratios of the elastic modulus and the thickness of the film to those of the substrate, respectively. (2) When the thickness ratio of the film and the substrate is less than 0.1, Stoney's equation and Roell's approximation [J. Appl. Phys. 47, 3224 (1976)] do not cause serious errors. (3) The approximation proposed by Vilms and Kerps [J. Appl. Phys. 53, 1536 (1982)] is an improved modification for Stoney's equation and can be applicable when {eta}{<=}0.3. (4) The approximations proposed by Brenner and Senderoff [J. Res. Natl. Bur. Stand. 42, 105 (1949)] and Teixeira [Thin Solid Films 392, 276 (2001)] can lead to serious errors and should be avoided. (5) The approximation based on the assumption of constant elastic modulus is only applicable for a ratio of {eta}{<=}0.01 and can be very misleading.
NASA Astrophysics Data System (ADS)
Chae, Kyu-Hyun
2002-04-01
Fourier series solutions to the deflection and magnification by a family of three-dimensional cusped two-power-law ellipsoidal mass distributions are presented. The cusped two-power-law ellipsoidal mass distributions are characterized by inner and outer power-law radial indices and a break (or transition) radius. The model family includes mass models mimicking Jaffe, Hernquist, and η models and dark matter halo profiles from numerical simulations. The Fourier series solutions for the cusped two-power-law mass distributions are relatively simple and allow a very fast calculation, even for a chosen small fractional calculational error (e.g., 10-5). These results will be particularly useful for studying lensed systems that provide a number of accurate lensing constraints and for systematic analyses of large numbers of lenses. Subroutines employing these results for the two-power-law model and the results by Chae, Khersonsky, & Turnshek for the generalized single-power-law mass model are made publicly available.
NASA Astrophysics Data System (ADS)
Chen, Jui-Sheng; Jang, Cheng-Shin; Cheng, Chung-Ting; Liu, Chen-Wuing
2010-09-01
SummaryThis study presents a novel mathematical model for describing the transport of the remedial reagent in a vertical circulation flow field in an anisotropic aquifer. To develop the mathematical model, the radial and vertical components of the pore water velocity are calculated first by using an analytical solution for steady-state drawdown distribution near a vertical circulation well. Next, the obtained radial and vertical components of the pore water velocity are then incorporated into a three-dimensional axisymmetrical advection-dispersion equation in cylindrical coordinates from which to build the reagent transport equation. The Laplace transform finite difference technique is applied to solve the three-dimensional axisymmetrical advection-dispersion equation with spatial variable-dependent coefficients. The developed mathematical model is used to investigate the effects of various parameters such as hydraulic conductivity anisotropy, longitudinal and transverse dispersivities, the placement of the extraction and injection screened intervals of the vertical circulation well and the injection modes on the transport regime of the remedial reagent. Results show that those parameters have different degrees of impacts on the distribution of the remedial reagent. The mathematical model provides an effective tool for designing and operating an enhanced groundwater remediation in an anisotropic aquifer using the vertical circulation well technology.
NASA Astrophysics Data System (ADS)
Stukel, Michael R.; Landry, Michael R.; Ohman, Mark D.; Goericke, Ralf; Samo, Ty; Benitez-Nelson, Claudia R.
2012-03-01
Despite the increasing use of linear inverse modeling techniques to elucidate fluxes in undersampled marine ecosystems, the accuracy with which they estimate food web flows has not been resolved. New Markov Chain Monte Carlo (MCMC) solution methods have also called into question the biases of the commonly used L2 minimum norm (L 2MN) solution technique. Here, we test the abilities of MCMC and L 2MN methods to recover field-measured ecosystem rates that are sequentially excluded from the model input. For data, we use experimental measurements from process cruises of the California Current Ecosystem (CCE-LTER) Program that include rate estimates of phytoplankton and bacterial production, micro- and mesozooplankton grazing, and carbon export from eight study sites varying from rich coastal upwelling to offshore oligotrophic conditions. Both the MCMC and L 2MN methods predicted well-constrained rates of protozoan and mesozooplankton grazing with reasonable accuracy, but the MCMC method overestimated primary production. The MCMC method more accurately predicted the poorly constrained rate of vertical carbon export than the L 2MN method, which consistently overestimated export. Results involving DOC and bacterial production were equivocal. Overall, when primary production is provided as model input, the MCMC method gives a robust depiction of ecosystem processes. Uncertainty in inverse ecosystem models is large and arises primarily from solution under-determinacy. We thus suggest that experimental programs focusing on food web fluxes expand the range of experimental measurements to include the nature and fate of detrital pools, which play large roles in the model.
NASA Astrophysics Data System (ADS)
Din, Alif
2016-08-01
The theory of positive-ion collection by a probe immersed in a low-pressure plasma was reviewed and extended by Allen et al. [Proc. Phys. Soc. 70, 297 (1957)]. The numerical computations for cylindrical and spherical probes in a sheath region were presented by F. F. Chen [J. Nucl. Energy C 7, 41 (1965)]. Here, in this paper, the sheath and presheath solutions for a cylindrical probe are matched through a numerical matching procedure to yield "matched" potential profile or "M solution." The solution based on the Bohm criterion approach "B solution" is discussed for this particular problem. The comparison of cylindrical probe characteristics obtained from the correct potential profile (M solution) and the approximated Bohm-criterion approach are different. This raises questions about the correctness of cylindrical probe theories relying only on the Bohm-criterion approach. Also the comparison between theoretical and experimental ion current characteristics shows that in an argon plasma the ions motion towards the probe is almost radial.
NASA Astrophysics Data System (ADS)
Weber, F.; Distl, H.
2015-11-01
This paper derives an approximate collocated control solution for the mitigation of multi-mode cable vibration by semi-active damping with negative stiffness based on the control force characteristics of clipped linear quadratic regulator (LQR). The control parameters are derived from optimal modal viscous damping and corrected in order to guarantee that both the equivalent viscous damping coefficient and the equivalent stiffness coefficient of the semi-active cable damper force are equal to their desired counterparts. The collocated control solution with corrected control parameters is numerically validated by free decay tests of the first four cable modes and combinations of these modes. The results of the single-harmonic tests demonstrate that the novel approach yields 1.86 times more cable damping than optimal modal viscous damping and 1.87 to 2.33 times more damping compared to a passive oil damper whose viscous damper coefficient is optimally tuned to the targeted mode range of the first four modes. The improvement in case of the multi-harmonic vibration tests, i.e. when modes 1 and 3 and modes 2 and 4 are vibrating at the same time, is between 1.55 and 3.81. The results also show that these improvements are obtained almost independent of the cable anti-node amplitude. Thus, the proposed approximate real-time applicable collocated semi-active control solution which can be realized by magnetorheological dampers represents a promising tool for the efficient mitigation of stay cable vibrations.
Dodin, Amro; Tscherbul, Timur V; Brumer, Paul
2016-06-28
Closed-form analytic solutions to non-secular Bloch-Redfield master equations for quantum dynamics of a V-type system driven by weak coupling to a thermal bath, relevant to light harvesting processes, are obtained and discussed. We focus on noise-induced Fano coherences among the excited states induced by incoherent driving of the V-system initially in the ground state. For suddenly turned-on incoherent driving, the time evolution of the coherences is determined by the damping parameter ζ=12(γ1+γ2)/Δp, where γi are the radiative decay rates of the excited levels i = 1, 2, and Δp=Δ(2)+(1-p(2))γ1γ2 depends on the excited-state level splitting Δ > 0 and the angle between the transition dipole moments in the energy basis. The coherences oscillate as a function of time in the underdamped limit (ζ ≫ 1), approach a long-lived quasi-steady state in the overdamped limit (ζ ≪ 1), and display an intermediate behavior at critical damping (ζ = 1). The sudden incoherent turn-on is shown to generate a mixture of excited eigenstates |e1〉 and |e2〉 and their in-phase coherent superposition |ϕ+〉=1r1+r2(r1|e1〉+r2|e2〉), which is remarkably long-lived in the overdamped limit (where r1 and r2 are the incoherent pumping rates). Formation of this coherent superposition enhances the decay rate from the excited states to the ground state. In the strongly asymmetric V-system where the coupling strengths between the ground state and the excited states differ significantly, additional asymptotic quasistationary coherences are identified, which arise due to slow equilibration of one of the excited states. Finally, we demonstrate that noise-induced Fano coherences are maximized with respect to populations when r1 = r2 and the transition dipole moments are fully aligned. PMID:27369498
NASA Astrophysics Data System (ADS)
Dodin, Amro; Tscherbul, Timur V.; Brumer, Paul
2016-06-01
Closed-form analytic solutions to non-secular Bloch-Redfield master equations for quantum dynamics of a V-type system driven by weak coupling to a thermal bath, relevant to light harvesting processes, are obtained and discussed. We focus on noise-induced Fano coherences among the excited states induced by incoherent driving of the V-system initially in the ground state. For suddenly turned-on incoherent driving, the time evolution of the coherences is determined by the damping parameter ζ = /1 2 ( γ 1 + γ 2) / Δ p , where γi are the radiative decay rates of the excited levels i = 1, 2, and Δ p = √{ Δ 2 + ( 1 - p 2) γ 1 γ 2 } depends on the excited-state level splitting Δ > 0 and the angle between the transition dipole moments in the energy basis. The coherences oscillate as a function of time in the underdamped limit (ζ ≫ 1), approach a long-lived quasi-steady state in the overdamped limit (ζ ≪ 1), and display an intermediate behavior at critical damping (ζ = 1). The sudden incoherent turn-on is shown to generate a mixture of excited eigenstates |e1> and |e2> and their in-phase coherent superposition | ϕ + > = /1 √{ r 1 + r 2 } ( √{ r 1 } | e 1 > + √{ r 2 } | e 2 >) , which is remarkably long-lived in the overdamped limit (where r1 and r2 are the incoherent pumping rates). Formation of this coherent superposition enhances the decay rate from the excited states to the ground state. In the strongly asymmetric V-system where the coupling strengths between the ground state and the excited states differ significantly, additional asymptotic quasistationary coherences are identified, which arise due to slow equilibration of one of the excited states. Finally, we demonstrate that noise-induced Fano coherences are maximized with respect to populations when r1 = r2 and the transition dipole moments are fully aligned.
NASA Technical Reports Server (NTRS)
Ratkiewicz, Romana E.; Scherer, Klaus; Fahr, Hans J.; Cuzzi, Jeffrey N. (Technical Monitor)
1994-01-01
The solar system is in relative motion with respect to the ambient interstellar medium. The supersonic solar wind is expected to pass through the termination shock, thus the solar wind plasma eventually has to enter into an asymptotic outflow geometry appropriately adopted to this counterflow situation. Many attempts have been done to simulate the interaction between the solar wind and the LISM numerically. In this paper we generalize a Parker type analytical solution of the counterflow. The idea is to introduce a special kind of compressibility of the solar wind flow. With the assumption that only a transversal component of the density gradient normal to the flow lines exists we are able to calculate a full set of hydrodynamical quantities describing the circumsolar flow field of a Sun moving through the LISM. The equations governing the velocity and density fields lead to analytical solutions which can be taken as good approximations to the more general case of compressible plasma flows.
NASA Astrophysics Data System (ADS)
Trujillo Bueno, J.; Fabiani Bendicho, P.
1995-12-01
Iterative schemes based on Gauss-Seidel (G-S) and optimal successive over-relaxation (SOR) iteration are shown to provide a dramatic increase in the speed with which non-LTE radiation transfer (RT) problems can be solved. The convergence rates of these new RT methods are identical to those of upper triangular nonlocal approximate operator splitting techniques, but the computing time per iteration and the memory requirements are similar to those of a local operator splitting method. In addition to these properties, both methods are particularly suitable for multidimensional geometry, since they neither require the actual construction of nonlocal approximate operators nor the application of any matrix inversion procedure. Compared with the currently used Jacobi technique, which is based on the optimal local approximate operator (see Olson, Auer, & Buchler 1986), the G-S method presented here is faster by a factor 2. It gives excellent smoothing of the high-frequency error components, which makes it the iterative scheme of choice for multigrid radiative transfer. This G-S method can also be suitably combined with standard acceleration techniques to achieve even higher performance. Although the convergence rate of the optimal SOR scheme developed here for solving non-LTE RT problems is much higher than G-S, the computing time per iteration is also minimal, i.e., virtually identical to that of a local operator splitting method. While the conventional optimal local operator scheme provides the converged solution after a total CPU time (measured in arbitrary units) approximately equal to the number n of points per decade of optical depth, the time needed by this new method based on the optimal SOR iterations is only √n/2√2. This method is competitive with those that result from combining the above-mentioned Jacobi and G-S schemes with the best acceleration techniques. Contrary to what happens with the local operator splitting strategy currently in use, these novel
NASA Astrophysics Data System (ADS)
Boyd, John P.
2011-02-01
Radial basis function (RBF) interpolants have become popular in computer graphics, neural networks and for solving partial differential equations in many fields of science and engineering. In this article, we compare five different species of RBFs: Gaussians, hyperbolic secant (sech's), inverse quadratics, multiquadrics and inverse multiquadrics. We show that the corresponding cardinal functions for a uniform, unbounded grid are all approximated by the same function: C(X) ∼ (1/(ρ)) sin (πX)/sinh (πX/ρ) for some constant ρ(α) which depends on the inverse width parameter (“shape parameter”) α of the RBF and also on the RBF species. The error in this approximation is exponentially small in 1/α for sech's and inverse quadratics and exponentially small in 1/α2 for Gaussians; the error is proportional to α4 for multiquadrics and inverse multiquadrics. The error in all cases is small even for α ∼ O(1). These results generalize to higher dimensions. The Gaussian RBF cardinal functions in any number of dimensions d are, without approximation, the tensor product of one dimensional Gaussian cardinal functions: Cd(x1,x2…,xd)=∏j=1dC(xj). For other RBF species, we show that the two-dimensional cardinal functions are well approximated by the products of one-dimensional cardinal functions; again the error goes to zero as α → 0. The near-identity of the cardinal functions implies that all five species of RBF interpolants are (almost) the same, despite the great differences in the RBF ϕ's themselves.
NASA Astrophysics Data System (ADS)
Louarroudi, E.; Pintelon, R.; Lataire, J.
2014-10-01
Time-periodic (TP) phenomena occurring, for instance, in wind turbines, helicopters, anisotropic shaft-bearing systems, and cardiovascular/respiratory systems, are often not addressed when classical frequency response function (FRF) measurements are performed. As the traditional FRF concept is based on the linear time-invariant (LTI) system theory, it is only approximately valid for systems with varying dynamics. Accordingly, the quantification of any deviation from this ideal LTI framework is more than welcome. The “measure of deviation” allows us to define the notion of the best LTI (BLTI) approximation, which yields the best - in mean square sense - LTI description of a linear time-periodic LTP system. By taking into consideration the TP effects, it is shown in this paper that the variability of the BLTI measurement can be reduced significantly compared with that of classical FRF estimators. From a single experiment, the proposed identification methods can handle (non-)linear time-periodic [(N)LTP] systems in open-loop with a quantification of (i) the noise and/or the NL distortions, (ii) the TP distortions and (iii) the transient (leakage) errors. Besides, a geometrical interpretation of the BLTI approximation is provided, leading to a framework called vector FRF analysis. The theory presented is supported by numerical simulations as well as real measurements mimicking the well-known mechanical Mathieu oscillator.
NASA Astrophysics Data System (ADS)
Saliba, H. T.
1995-06-01
The practical engineering problem of right angled triangular plates with combinations of clamped and simply supported boundary conditions is dealt with in this paper. A highly accurate, economical and practical solution is used for the transverse free vibration analysis of these plates. The solution is based on a modified superposition method. The accuracy of the solution is discussed. Numerical results are compared with previously published reliable data. The advantages of the solution used in the paper over previously published solutions are discussed. Eigenvalues, mode shapes and contour plots are provided for a large number of plates.
Bouallègue, Fayçal Ben; Crouzet, Jean-François; Comtat, Claude; Fourcade, Marjolaine; Mohammadi, Bijan; Mariano-Goulart, Denis
2007-07-01
This paper presents an extended 3-D exact rebinning formula in the Fourier space that leads to an iterative reprojection algorithm (iterative FOREPROJ), which enables the estimation of unmeasured oblique projection data on the basis of the whole set of measured data. In first approximation, this analytical formula also leads to an extended Fourier rebinning equation that is the basis for an approximate reprojection algorithm (extended FORE). These algorithms were evaluated on numerically simulated 3-D positron emission tomography (PET) data for the solution of the truncation problem, i.e., the estimation of the missing portions in the oblique projection data, before the application of algorithms that require complete projection data such as some rebinning methods (FOREX) or 3-D reconstruction algorithms (3DRP or direct Fourier methods). By taking advantage of all the 3-D data statistics, the iterative FOREPROJ reprojection provides a reliable alternative to the classical FOREPROJ method, which only exploits the low-statistics nonoblique data. It significantly improves the quality of the external reconstructed slices without loss of spatial resolution. As for the approximate extended FORE algorithm, it clearly exhibits limitations due to axial interpolations, but will require clinical studies with more realistic measured data in order to decide on its pertinence. PMID:17649913
NASA Astrophysics Data System (ADS)
Zhang, Yongfang; Wu, Peng; Guo, Bo; Lü, Yanjun; Liu, Fuxi; Yu, Yingtian
2015-01-01
The instability of the rotor dynamic system supported by oil journal bearing is encountered frequently, such as the half-speed whirl of the rotor, which is caused by oil film lubricant with nonlinearity. Currently, more attention is paid to the physical characteristics of oil film due to an oil-lubricated journal bearing being the important supporting component of the bearing-rotor systems and its nonlinear nature. In order to analyze the lubrication characteristics of journal bearings efficiently and save computational efforts, an approximate solution of nonlinear oil film forces of a finite length turbulent journal bearing with couple stress flow is proposed based on Sommerfeld and Ocvirk numbers. Reynolds equation in lubrication of a finite length turbulent journal bearing is solved based on multi-parametric principle. Load-carrying capacity of nonlinear oil film is obtained, and the results obtained by different methods are compared. The validation of the proposed method is verified, meanwhile, the relationships of load-carrying capacity versus eccentricity ratio and width-to-diameter ratio under turbulent and couple stress working conditions are analyzed. The numerical results show that both couple stress flow and eccentricity ratio have obvious influence on oil film pressure distribution, and the proposed method approximates the load-carrying capacity of turbulent journal bearings efficiently with various width-to-diameter ratios. This research proposes an approximate solution of oil film load-carrying capacity of turbulent journal bearings with different width-to-diameter ratios, which are suitable for high eccentricity ratios and heavy loads.
Precise and accurate measurement of U and Th isotopes via ICP-MS using a single solution
NASA Astrophysics Data System (ADS)
Mertz-Kraus, R.; Sharp, W. D.; Ludwig, K. R.
2012-04-01
, allowing the sample's 238U/235U ratio to be measured. In step 3, we monitor peak-tails at half-mass positions (229.5, 231.5, 234.5) and on mass 237 while aspirating sample solution. Tail measurement requires a distinct cup configuration to maintain 238U in the cups; however, no sample is consumed during automated cup reconfiguration. We monitor the accuracy of 234U/238U ratios using CRM 145, which gives a weighted mean atom ratio of (5.2846 ± 0.0029) - 10-5 (all errors 2σ), consistent with published and reference values. The reproducibility of 230Th/238U ratios is monitored using the Schwartzwalder Mine secular-equilibrium standard (SM). We detect no bias in 230Th/238U or 234U/238U ratios measured for SM at beam intensities ranging over a factor of four, consistent with accurate correction for IC yields. Aladdin's cave coral (AC-1) was analyzed to check our ICP-MS method (and the preceding purification by ion exchange) on a carbonate and yields a mean age of 125.43 ± 0.38 ka, in agreement with published values. We are currently applying the method to corals, speleothems, pedogenic coatings, and tufas.
NASA Astrophysics Data System (ADS)
Mourenas, D.; Artemyev, A. V.; Agapitov, O. V.; Krasnoselskikh, V.; Li, W.
2014-12-01
The distribution of trapped energetic electrons inside the Earth's radiation belts is the focus of intense studies aiming at better describing the evolution of the space environment in the presence of various disturbances induced by the solar wind or by an enhanced lightning activity. Such studies are usually performed by means of comparisons with full numerical simulations solving the Fokker-Planck quasi-linear diffusion equation for the particle distribution function. Here we present for the first time approximate but realistic analytical solutions for the electron distribution, which are shown to be in good agreement with exact numerical solutions in situations where resonant scattering of energetic electrons by whistler mode hiss, lightning-generated or chorus waves, is the dominant process. Quiet time distributions are well recovered, as well as the evolution of energized relativistic electron distributions during disturbed geomagnetic conditions. It is further shown that careful comparisons between the analytical solutions and measured distributions may allow to infer important bounce- and drift-averaged wave characteristics (such as wave amplitude). It could also help to improve the global understanding of underlying physical phenomena.
NASA Astrophysics Data System (ADS)
Alcoba, D. R.; Valdemoro, C.; Tel, L. M.; Pérez-Romero, E.
The equation obtained by mapping the matrix representation of the Schrödinger equation with the 2nd-order correlation transition matrix elements into the 2-body space is the so called correlation contracted Schrödinger equation (CCSE) (Alcoba, Phys Rev A 2002, 65, 032519). As shown by Alcoba (Phys Rev A 2002, 65, 032519) the solution of the CCSE coincides with that of the Schrödinger equation. Here the attention is focused in the vanishing hypervirial of the correlation operator (GHV), which can be identified with the anti-Hermitian part of the CCSE. A comparative analysis of the GHV and the anti-Hermitian part of the contracted Schrödinger equation (ACSE) indicates that the former is a stronger stationarity condition than the latter. By applying a Heisenberg-like unitary transformation to the G-particle-hole operator (Valdemoro et al., Phys Rev A 2000, 61, 032507), a good approximation of the expectation value of this operator as well as of the GHV is obtained. The method is illustrated for the case of the Beryllium isoelectronic series as well as for the Li2 and BeH2 molecules. The correlation energies obtained are within 98.80-100.09% of the full-configuration interaction ones. The convergence of these calculations was faster when using the GHV than with the ACSE.
NASA Astrophysics Data System (ADS)
Oluwadare, O. J.; Oyewumi, K. J.; Akoshile, C. O.; Babalola, O. A.
2012-09-01
By employing the Pekeris-type (or a new improved approximation) to deal with the (pseudo or) centrifugal term, we solve the Klein-Gordon and Dirac equations with equally mixed scalar and vector Deng-Fan molecular potentials for all values of l (orbital the angular momentum quantum number) and κ (spin-orbit coupling quantum number), respectively. Using the formalism of the Nikiforov-Uvarov method, the approximate analytical bound state energy equations and the associated two-component spinors corresponding to the two relativistic equations are obtained. Also, special cases including the non-relativistic limits of the relativistic equation are obtained.
NASA Astrophysics Data System (ADS)
Gutiérrez, César; Jiménez, Bienvenido; Novo, Vicente
2005-10-01
In this work we obtain a chain rule for the approximate subdifferential considering a vector-valued proper convex function and its post-composition with a proper convex function of several variables nondecreasing in the sense of the Pareto order. We derive an interesting formula for the conjugate of a composition in the same framework and we prove the chain rule using this formula. To get the results, we require qualification conditions since, in the composition, the initial function is extended vector-valued. This chain rule extends analogous well-known calculus rules obtained when the functions involved are finite and it gives a complementary simple expression for other chain rules proved without assuming any qualification condition. As application we deduce the well-known calculus rule for the addition and we extend the formula for the maximum of functions. Finally, we use them and a scalarization process to obtain Kuhn-Tucker type necessary and sufficient conditions for approximate solutions in convex Pareto problems. These conditions extend other obtained in scalar optimization problems.
NASA Astrophysics Data System (ADS)
Guevara, Carlos; Graf, Thomas
2013-04-01
Subsurface water systems are endangered due to salt water intrusion in coastal aquifers, leachate infiltration from waste disposal sites and salt transport in agricultural sites. This leads to the situation where more dense fluid overlies a less dense fluid creating a density gradient. Under certain conditions this density gradient produces instabilities in form dense plume fingers that move downwards. This free convection increases solute transport over large distances and shorter times. In cases where a significantly larger density gradient exists, the effect of free convection on transport is non-negligible. The assumption of a constant density distribution in space and time is no longer valid. Therefore variable-density flow must be considered. The flow equation and the transport equation govern the numerical modeling of variable-density flow and solute transport. Computer simulation programs mathematically describe variable-density flow using the Oberbeck-Boussinesq Approximation (OBA). Three levels of simplifications can de considered, which are denoted by OB1, OB2 and OB3. OB1 is the usually applied simplification where variable density is taken into account in the hydraulic potential. In OB2 variable density is considered in the flow equation and in OB3 variable density is additionally considered in the transport equation. Using the results from a laboratory-scale experiment of variable-density flow and solute transport (Simmons et al., Transp. Porous Medium, 2002) it is investigated which level of mathematical accuracy is required to represent the physical experiment the most accurate. Differences between the physical and mathematical model are evaluated using qualitative indicators (e.g. mass fluxes, Nusselt number). Results show that OB1 is required for small density gradients and OB3 is required for large density gradients.
Pernal, Katarzyna; Chatterjee, Koushik; Kowalski, Piotr H.
2014-01-07
Performance of the antisymmetrized product of strongly orthogonal geminal (APSG) ansatz in describing ground states of molecules has been extensively explored in the recent years. Not much is known, however, about possibilities of obtaining excitation energies from methods that would rely on the APSG ansatz. In the paper we investigate the recently proposed extended random phase approximations, ERPA and ERPA2, that employ APSG reduced density matrices. We also propose a time-dependent linear response APSG method (TD-APSG). Its relation to the recently proposed phase including natural orbital theory is elucidated. The methods are applied to Li{sub 2}, BH, H{sub 2}O, and CH{sub 2}O molecules at equilibrium geometries and in the dissociating limits. It is shown that ERPA2 and TD-APSG perform better in describing double excitations than ERPA due to inclusion of the so-called diagonal double elements. Analysis of the potential energy curves of Li{sub 2}, BH, and H{sub 2}O reveals that ERPA2 and TD-APSG describe correctly excitation energies of dissociating molecules if orbitals involved in breaking bonds are involved. For single excitations of molecules at equilibrium geometries the accuracy of the APSG-based methods approaches that of the time-dependent Hartree-Fock method with the increase of the system size. A possibility of improving the accuracy of the TD-APSG method for single excitations by splitting the electron-electron interaction operator into the long- and short-range terms and employing density functionals to treat the latter is presented.
NASA Astrophysics Data System (ADS)
Pernal, Katarzyna; Chatterjee, Koushik; Kowalski, Piotr H.
2014-01-01
Performance of the antisymmetrized product of strongly orthogonal geminal (APSG) ansatz in describing ground states of molecules has been extensively explored in the recent years. Not much is known, however, about possibilities of obtaining excitation energies from methods that would rely on the APSG ansatz. In the paper we investigate the recently proposed extended random phase approximations, ERPA and ERPA2, that employ APSG reduced density matrices. We also propose a time-dependent linear response APSG method (TD-APSG). Its relation to the recently proposed phase including natural orbital theory is elucidated. The methods are applied to Li2, BH, H2O, and CH2O molecules at equilibrium geometries and in the dissociating limits. It is shown that ERPA2 and TD-APSG perform better in describing double excitations than ERPA due to inclusion of the so-called diagonal double elements. Analysis of the potential energy curves of Li2, BH, and H2O reveals that ERPA2 and TD-APSG describe correctly excitation energies of dissociating molecules if orbitals involved in breaking bonds are involved. For single excitations of molecules at equilibrium geometries the accuracy of the APSG-based methods approaches that of the time-dependent Hartree-Fock method with the increase of the system size. A possibility of improving the accuracy of the TD-APSG method for single excitations by splitting the electron-electron interaction operator into the long- and short-range terms and employing density functionals to treat the latter is presented.
NASA Astrophysics Data System (ADS)
Shen, Jianhe; Han, Maoan
2014-08-01
This paper considers the existence and uniformly valid asymptotic approximation of canard solutions in a second-order nonlinear singularly perturbed boundary value problem with a turning point. We get the main results by constructing the asymptotic solution first and then defining a couple of upper and lower solutions suitably on the basis of the asymptotic solution. Two examples are carried out to illustrate and verify the theoretical results.
NASA Technical Reports Server (NTRS)
Savin, Raymond C.
1958-01-01
The flow about slender flat-top wing-body configurations traveling at high supersonic speeds and small angles of attack is investigated analytically. In the case of conical configurations, approximate algebraic solutions to the flow field are obtained. In the case of configurations which are conical at the vertex but curved in the stream direction, these solutions are combined with a slender-body approximation to the generalized shock-expansion method to obtain the flow downstream of the vertex. Surface pressures were obtained experimentally at Mach numbers from 3.0 to 6.0 and angles of attack up to 6 deg for several flat-top wing-body configurations. These configurations consisted of half-bodies of revolution mounted beneath thin highly swept wings. Three different bodies were employed. The two conical bodies consisted of one-half of a fineness-ratio-5 cone and one-half of a fineness-ratio-2-1/2 cone. The body of the third configuration consisted of one-half of a fineness-ratio-5 ogive. For the ogive configuration, the leading edges of the wing were curved and designed to just maintain the theoretically determined bow shock along the leading edge at a Mach number of 5.0 and an angle of attack of 3 deg. The predictions of the conical flow theory of this paper for the surface pressures are found to be in good agreement with experiment at Mach numbers of 5.0 and 6.0 up to angles of attack of approximately 3 deg. Estimated lift, drag, and pitching-moment coefficients, as well as maximum lift-drag ratio, are also in good agreement with existing experimental data at a Mach number of 5.0 for a conical configuration having an arrow plan-form wing. It is also found that the generalized shock-expansion method yields reasonable good agreement with experiment for the surface pressures on the half-ogive configuration at a Mach number of 5.0 and an angle of attack of 3 deg.
NASA Astrophysics Data System (ADS)
Sarwar, S.; Rashidi, M. M.
2016-07-01
This paper deals with the investigation of the analytical approximate solutions for two-term fractional-order diffusion, wave-diffusion, and telegraph equations. The fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals [0,1], (1,2), and [1,2], respectively. In this paper, we extended optimal homotopy asymptotic method (OHAM) for two-term fractional-order wave-diffusion equations. Highly approximate solution is obtained in series form using this extended method. Approximate solution obtained by OHAM is compared with the exact solution. It is observed that OHAM is a prevailing and convergent method for the solutions of nonlinear-fractional-order time-dependent partial differential problems. The numerical results rendering that the applied method is explicit, effective, and easy to use, for handling more general fractional-order wave diffusion, diffusion, and telegraph problems.
Frolov, Andrey I
2015-05-12
Accurate calculation of solvation free energies (SFEs) is a fundamental problem of theoretical chemistry. In this work we perform a careful validation of the theory of solutions in energy representation (ER method) developed by Matubayasi et al. [J. Chem. Phys. 2000, 113, 6070-6081] for SFE calculations in supercritical solvents. This method can be seen as a bridge between the molecular simulations and the classical (not quantum) density functional theory (DFT) formulated in energy representation. We performed extensive calculations of SFEs of organic molecules of different chemical natures in pure supercritical CO2 (sc-CO2) and in sc-CO2 with addition of 6 mol % of ethanol, acetone, and n-hexane as cosolvents. We show that the ER method reproduces SFE data calculated by a method free of theoretical approximations (the Bennett's acceptance ratio) with the mean absolute error of only 0.05 kcal/mol. However, the ER method requires by an order less computational resources. Also, we show that the quality of ER calculations should be carefully monitored since the lack of sampling can result into a considerable bias in predictions. The present calculations reproduce the trends in the cosolvent-induced solubility enhancement factors observed in experimental data. Thus, we think that molecular simulations coupled with the ER method can be used for quick calculations of the effect of variation of temperature, pressure, and cosolvent concentration on SFE and hence solubility of bioactive compounds in supercritical fluids. This should dramatically reduce the burden of experimental work on optimizing solvency of supercritical solvents. PMID:26574423
NASA Astrophysics Data System (ADS)
Chen, Chang-Yuan; Lu, Fa-Lin; Sun, Dong-Sheng
2008-12-01
In this paper, using the exponential function transformation approach along with an approximation for the centrifugal potential, the radial Klein-Gordon equation with the vector and scalar Hulthén potential is transformed to a hypergeometric differential equation. The approximate analytical solutions of t-waves scattering states are presented. The normalized wave functions expressed in terms of hypergeometric functions of scattering states on the “ k/2π scale” and the calculation formula of phase shifts are given. The physical meaning of the approximate analytical solution is discussed.
NASA Astrophysics Data System (ADS)
Chen, Chang-Yuan; Lu, Fa-Lin; Sun, Dong-Sheng
2008-12-01
In this paper, using the exponential function transformation approach along with an approximation for the centrifugal potential, the radial Klein-Gordon equation with the vector and scalar Hulthén potential is transformed to a hypergeometric differential equation. The approximate analytical solutions of t-waves scattering states are presented. The normalized wave functions expressed in terms of hypergeometric functions of scattering states on the "k/2π scale" and the calculation formula of phase shifts are given. The physical meaning of the approximate analytical solution is discussed.
NASA Astrophysics Data System (ADS)
Jasinski, Jerzy
2015-05-01
In the paper propagation of axially-symmetric (1+2)D beam in nonlinear medium with dual-power nonlinearity is analyzed. The ordinary differential equation for transverse stationary profile of the propagating field is derived and solved using a perturbation technique. The simple analytical formulas for the three lowest order solutions are obtained. They describe fields of algebraic profiles. The zero order solution satisfies exactly the nonlinear Schrödinger equation in (1+2)D case. Higher order solutions are determined by propagation constant and describe fields of different initial amplitude. The accuracy of approximation and stability of the obtained solutions are discussed.
Drabina, A; Dwora, D; Loskiewicz, J; Zorski, T
2001-05-01
Czubek has obtained a practically successful solution of a two-layer borehole geometry using the neutron diffusion approximation. Czubek and Woznicka solved the more realistic three-layer borehole model, which includes a middle layer consisting of mud-cake or steel tubing, in this approximation. The comparison with experimental data for 15-mm thick steel tubes shows some discrepancies. By using an exact Monte Carlo solution of the Boltzmann equation for calculating the migration length, measurements agree more closely with the "general calibration curve". This opens opportunities for cased borehole logging. PMID:11258536
NASA Astrophysics Data System (ADS)
Huang, C.-S.; Yang, S.-Y.; Yeh, H.-D.
2015-06-01
An aquifer consisting of a skin zone and a formation zone is considered as a two-zone aquifer. Existing solutions for the problem of constant-flux pumping in a two-zone confined aquifer involve laborious calculation. This study develops a new approximate solution for the problem based on a mathematical model describing steady-state radial and vertical flows in a two-zone aquifer. Hydraulic parameters in these two zones can be different but are assumed homogeneous in each zone. A partially penetrating well may be treated as the Neumann condition with a known flux along the screened part and zero flux along the unscreened part. The aquifer domain is finite with an outer circle boundary treated as the Dirichlet condition. The steady-state drawdown solution of the model is derived by the finite Fourier cosine transform. Then, an approximate transient solution is developed by replacing the radius of the aquifer domain in the steady-state solution with an analytical expression for a dimensionless time-dependent radius of influence. The approximate solution is capable of predicting good temporal drawdown distributions over the whole pumping period except at the early stage. A quantitative criterion for the validity of neglecting the vertical flow due to a partially penetrating well is also provided. Conventional models considering radial flow without the vertical component for the constant-flux pumping have good accuracy if satisfying the criterion.
NASA Astrophysics Data System (ADS)
Huang, C.-S.; Yang, S.-Y.; Yeh, H.-D.
2015-03-01
An aquifer consisting of a skin zone and a formation zone is considered as a two-zone aquifer. Existing solutions for the problem of constant-flux pumping (CFP) in a two-zone confined aquifer involve laborious calculation. This study develops a new approximate solution for the problem based on a mathematical model including two steady-state flow equations with different hydraulic parameters for the skin and formation zones. A partially penetrating well may be treated as the Neumann condition with a known flux along the screened part and zero flux along the unscreened part. The aquifer domain is finite with an outer circle boundary treated as the Dirichlet condition. The steady-state drawdown solution of the model is derived by the finite Fourier cosine transform. Then, an approximate transient solution is developed by replacing the radius of the boundary in the steady-state solution with an analytical expression for a dimensionless time-dependent radius of influence. The approximate solution is capable of predicting good temporal drawdown distributions over the whole pumping period except at the early stage. A quantitative criterion for the validity of neglecting the vertical flow component due to a partially penetrating well is also provided. Conventional models considering radial flow without the vertical component for the CFP have good accuracy if satisfying the criterion.
Forgeron, Michelle A.; Wasylishen, Roderick E.
2006-06-21
Solid-state 95Mo NMR spectroscopy is shown to be an efficient and effective tool for analyzing the diamagnetic octacyanomolybdate(IV) anions, Mo(CN)8 4-, of approximate dodecahedral, D2d, and square antiprismatic, D4d, symmetry. The sensitivity of the Mo magnetic shielding (?) and electric field gradient (EFG) tensors to small changes in the local structure of these anions allows the approximate D2d and D4d Mo(CN)8 4- anions to be readily distinguished. The use of high applied magnetic fields, 11.75, 17.63 and 21.1 T, amplifies the overall sensitivity of the NMR experiment and enables more accurate characterization of the Mo ? and EFG tensors. Although the magnitudes of the Mo ? and EFG interactions are comparable for the D2d and D4d Mo(CN)8 4- anions, the relative values and orientations of the principal components of the Mo ? and EFG tensors give rise to 95Mo NMR line shapes that are significantly different at the fields utilized here. Quantum chemical calculations of the Mo ? and EFG tensors, using zeroth-order regular approximation density functional theory (ZORA DFT) and restricted Hartree-Fock (RHF) methods, have also been carried out and are in good agreement with experiment. The most significant and surprising result from the DFT and RHF calculations is a significant EFG at Mo for an isolated Mo(CN)8 4- anion possessing an ideal square antiprismatic structure; this is contrary to the point-charge approximation, PCA, which predicts a zero EFG at Mo for this structure.
NASA Astrophysics Data System (ADS)
Kawashima, I.; Toh, H.; Satake, K.
2013-12-01
A seafloor geomagnetic observatory in the northwest Pacific detected clear electromagnetic (EM) variations associated with tsunami passage from two earthquakes that occurred along the Kuril Trench (Toh et al., 2011). Previous seismological analyses indicated that the M8.3 earthquake on 15 November 2006 was an underthrust type on the landward slope of the trench, while the M8.1 earthquake on 13 January 2007 was a normal fault type on the seaward side (Ammon et al., 2008). Here we report the simulation results on the frequency dependence of those tsunami-induced EM signals observed at the seafloor, using a three-dimensional (3-D) non-uniform thin-sheet approximation by Dawson and Weaver (1979) and McKirdy, Weaver, & Dawson (1985), which can accommodate not only the inducing non-uniform source fields caused by particle motions of conducting seawater at the time of tsunami passage but also the self-induction effect within the ocean and its conductive substrata. Horizontal particle motions were calculated by Fujii and Satake (2008) with two types of hydrodynamic approximation, viz., the Boussinesq approximation and the long-wave approximation. Because the dispersion effect of each tsunami was more remarkable in the 2007 event, the observed EM variations were expected to be more compatible with the simulated EM signals using the Boussinesq approximation than the long-wave approximation. We calculated EM variations after we confirmed that synthetic plane waves in a flat ocean produced theoretically predicted harmonic EM variations well. In both approximations, the calculated EM variations associated with the initial wave of the tsunami at the time of the 2006 event are consistent with the observed ones, but the agreement became worse for the subsequent tsunami phases. As for the 2007 event, the calculated EM variations were less consistent compared with the 2006 event irrespective to the hydrodynamic approximations used. This can be due to the current limitation of thin
NASA Astrophysics Data System (ADS)
Artrith, Nongnuch; Kolpak, Alexie
2014-03-01
The shape, size, and composition of catalyst nanoparticles can have a significant influence on their catalytic activity. Understanding such structure-reactivity relationships is crucial for the optimization of industrial catalysts and the design of novel catalysts with enhanced properties. In this work, we investigate the equilibrium shape and surface structure/composition of Au/Cu nanoparticles in solution, which have recently been shown to be stable and efficient catalysts for CO2 reduction. Using a combination of density functional theory calculations and large-scale Monte-Carlo and molecular dynamics simulations with reactive atomistic potentials, we determine how the nanoparticle shape, surface structure, and surface stoichiometry (i.e., fraction of Au at the surface relative to overall composition), evolve as a function of varying catalytic conditions. We discuss the effects of these changes on the surface electronic structure and binding energies of CO2, H2, and CH3OH. Our results emphasize the important relationships between catalytic environment (e.g., solvent effects), catalyst structure, and catalytic activity. We thank the Schlumberger Foundation Faculty for the Future for financial support. Computing time at XSEDE and NERSC clusters are gratefully acknowledged.
NASA Technical Reports Server (NTRS)
Oden, J. T.
1976-01-01
Some results of studies of convergence and accuracy of finite element approximations of certain nonlinear problems encountered in finite elasticity are presented. A general technique for obtaining error bounds is also described together with an existence theorem. Numerical results obtained by solving a representative problem are also included.
NASA Astrophysics Data System (ADS)
Wang, Chun-Xiao; Liu, Mo-Lin; Liu, Hong-Ya
2008-10-01
As one exact candidate of the higher dimensional black hole, the 5D Ricci Qat Schwarzschild-de Sitter black string space presents something interesting. In this paper, we give a numerical solution to the real scalar field around the Nariai black hole by the polynomial approximation. Unlike the previous tangent approximation, this fitting function makes a perfect match in the leading intermediate region and gives a good description near both the event and the cosmological horizons. We can read from our results that the wave is close to a harmonic one with the tortoise coordinate. Furthermore, with the actual radial coordinate the waves pile up almost equally near the both horizons.
NASA Technical Reports Server (NTRS)
Schlesinger, Robert E.
1990-01-01
Results are presented from a linear Lagrangian entraining parcel model of an overshooting thunderstorm cloud top. The model, which is similar to that of Adler and Mack (1986), gives analytic exact solutions for vertical velocity and temperature by representing mixing with Rayleigh damping instead of nonlinearly. Model results are presented for various combinations of stratospheric lapse rate, drag intensity, and mixing strength. The results are compared to those of Adler and Mack.
Kristóf, T; Boda, D; Szalai, I
2012-08-22
An analytic formula is derived for the magnetization of a two-dimensional dipolar hard disk fluid using a variational functional series expansion of the free energy as a function of the orientational distribution function. The excess term expressing the effect of the intermolecular forces is calculated on the basis of the mean spherical approximation. Comparison with our own Monte Carlo simulation data shows excellent agreement for large external fields and for the zero-field susceptibility. At intermediate field strengths, the agreement is satisfactory for moderate dipole moments and densities. PMID:22810162
NASA Technical Reports Server (NTRS)
Box, M. A.; Deepak, A.
1981-01-01
The propagation of photons in a medium with strongly anisotropic scattering is a problem with a considerable history. Like the propagation of electrons in metal foils, it may be solved in the small-angle scattering approximation by the use of Fourier-transform techniques. In certain limiting cases, one may even obtain analytic expressions. This paper presents some of these results in a model-independent form and also illustrates them by the use of four different phase-function models. Sample calculations are provided for comparison purposes
Zhou Qi-huang
1988-12-01
Starting with the general expression of a static state axisymmetric metric and using the principle of equivalence and the Maccullagh formula, the Einstein--Maxwell equations of a charged axisymmetric celestial body are obtained. Next, using the method of undetermined coefficients these equations are solved up to fourth-order approximate. These sets of solutions are generally appropriate for all kinds of charged axisymmetric celestial bodies.
NASA Technical Reports Server (NTRS)
Copper, G. K.
1980-01-01
The implementation of the approximate factorization algorithm and its ability to efficiently and accurately describe transonic flow about an NACA 64A010 airfoil section is examined. The approximate factorization algorithm is developed from the nondimensional, conservative, vectorized Navier-Stokes equations expressed in curvilinear coordinates. Equations of state and transport coefficient relations appropriate to atmospheric air are appended to close the system of partial differential equations. An algebraic turbulence model is also incorporated into the equation set. This algorithm was verified by investigating the flow about an NACA 64A010 airfoil at 0, 2, and 3.5 deg angle of attack for free-stream conditions of 2,000,000 Reynolds number and 0.8 Mach number. Overall results were in good qualitative agreement with wind tunnel data sets. However, while nondimensional times of six were attained, numerical difficulties prevented any case from reaching a true steady state.
DiFrancesco, D; Noble, D
1980-01-01
1. Regular perturbation theory was used to obtain analytical solutions for the time course of membrane current decay following voltage-clamp depolarizing pulses when both time-dependent K conductance mechanisms and the process of K accumulation in extracellular spaces are present. These solutions apply when the current and K concentration changes are small enough for linear relations to be assumed between current and K concentration. 2. In the case of a single Hodgkin-Huxley type conductance variable with time constant tau chi the presence of an accumulation process which, by itself, would produce a current decay with time constant tau alpha, induces the appearance of two infinite sets of components with decreasing time constants (1/(n+1/tau chi) and 1/(1/tau alpha + n/tau chi), where n is integer), and decreasing magnitudes. 3. The analytical solutions are used to investigate the range of conditions over which semi-exponential (curve-stripping) analysis of current decay tails may give useful information on the kinetics of current change. It is shown that, except at very large decay tail amplitudes, the method may give a good estimate of the true time constants of conductance decay even when the currents are assumed to be strongly dependent on external K concentration. 4. The method introduces error in current amplitude, but over the range in which curve-stripping gives useful results, the direct distortion of activation curves by variations in external K concentration is fairly small. However, as the current decay becomes grossly distorted in its time course by accumulation, so does the activation curve. The effects are very similar both to those obtained using numerical computation without linearization, and to those obtained experimentally. 5. Even with a large dependence of current on external K concentration the linear model does not reproduce i chi, fast as a perturbation of i chi, slow by K accumulation. PMID:7463358
Świrniak, Grzegorz; Mroczka, Janusz
2016-04-01
When a plane electromagnetic wave is scattered by an optically transparent object, whose size is much larger than the wavelength, a series of bright and dark fringes forms the primary rainbow, which is one of the most splendid phenomena in nature. In this work, an optical technique is discussed for simultaneous measurement of the diameter and refractive index of an axisymmetric and dielectric fiber by studying some rainbow features. This noncontact optical technique uses a beam of light exhibiting low temporal coherence, which enabled us to reduce the detrimental sensitivity of the rainbow features to variations of the fiber properties, thus allowing for high-precision estimates. Approximate mathematical formulas for the diameter and refractive index measurements were derived from the lowest-order complex angular momentum correction to Airy theory of rainbow. Furthermore, sensitivity of the measurement data to small deformation of the fiber's cross section into an ellipse was discussed. Preliminary empirical results provide a qualitative verification. PMID:27140778
Serdyuk, Vladimir; Rudnitsky, Anton
2015-05-01
We present an approximate 2D asymptotic analytic theory of light field excitation in a plane thin dielectric layer under conditions of frustrated total internal reflection, when an inclined Gaussian beam, falling from a triangular prism, excites a decaying field in air spacing between a prism and a plane dielectric. Ignoring the radiation scattering on the sharp edges of a prism, we have obtained the formulas that allow us to compute spatial structures of an electromagnetic field in every point of space and to estimate the integral efficiency of waveguide mode excitation in a plane dielectric layer and the total energy of a reflected beam. It is shown that the width of an initial Gaussian beam has an effect on waveguide mode intensity. PMID:26366908
Ruas, Alexandre; Simonin, Jean-Pierre; Turq, Pierre; Moisy, Philippe
2005-12-01
This work is aimed at a description of the thermodynamic properties of actinide salt solutions at high concentration. The predictive capability of the binding mean spherical approximation (BIMSA) theory to describe the thermodynamic properties of electrolytes is assessed in the case of aqueous solutions of lanthanide(III) nitrate and chloride salts. Osmotic coefficients of cerium(III) nitrate and chloride were calculated from other lanthanide(III) salts properties. In parallel, concentrated binary solutions of cerium nitrate were prepared in order to measure experimentally its water activity and density as a function of concentration, at 25 degrees C. Water activities of several binary solutions of cerium chloride were also measured to check existing data on this salt. Then, the properties of cerium chloride and cerium nitrate solutions were compared within the BIMSA model. Osmotic coefficient values for promethium nitrate and promethium chloride given by this theory are proposed. Finally, water activity measurements were made to examine the fact that the ternary system Ce(NO3)3/HNO3/H2O and the quaternary system Ce(NO3)3/HNO3/N2H5NO3/H2O may be regarded as "simple solutions" (in the sense of Zdanovskii and Mikulin). PMID:16854002
Samuha, Shmuel; Mugnaioli, Enrico; Grushko, Benjamin; Kolb, Ute; Meshi, Louisa
2014-12-01
The crystal structure of the novel Al77Rh15Ru8 phase (which is an approximant of decagonal quasicrystals) was determined using modern direct methods (MDM) applied to automated electron diffraction tomography (ADT) data. The Al77Rh15Ru8 E-phase is orthorhombic [Pbma, a = 23.40 (5), b = 16.20 (4) and c = 20.00 (5) Å] and has one of the most complicated intermetallic structures solved solely by electron diffraction methods. Its structural model consists of 78 unique atomic positions in the unit cell (19 Rh/Ru and 59 Al). Precession electron diffraction (PED) patterns and high-resolution electron microscopy (HRTEM) images were used for the validation of the proposed atomic model. The structure of the E-phase is described using hierarchical packing of polyhedra and a single type of tiling in the form of a parallelogram. Based on this description, the structure of the E-phase is compared with that of the ε6-phase formed in Al-Rh-Ru at close compositions. PMID:25449623
Frantti, Johannes; Fujioka, Y; Puretzky, Alexander A; Xie, Y; Glazer, A
2013-01-01
Lead titanate (PbTiO3) is a classical example of a ferroelectric perovskite oxide illustrating a displacive phase transition accompanied by a softening of a symmetry-breaking mode. The underlying assumption justifying the soft-mode theory is that the crystal is macroscopically sufficiently uniform so that a meaningful free energy function can be formed. In contrast to PbTiO3, experimental studies show that the phase transition behaviour of lead-zirconate-titanate solid solution (PZT) is far more subtle. Most of the studies on the PZT system have been dedicated to ceramic or powder samples, in which case an unambiguous soft-mode study is not possible, as modes with different symmetries appear together. Our Raman scattering study on titanium-rich PZT single crystal shows that the phase transitions in PZT cannot be described by a simple soft-mode theory. In strong contrast to PbTiO3, splitting of transverse E-symmetry modes reveals that there are different locally-ordered regions. The role of crystal defects, random distribution of Ti and Zr at the B- cation site and Pb ions shifted away from their ideal positions, dictates the phase transition mechanism. A statistical model explaining the observed peak splitting and phase transformation to a complex state with spatially varying local order in the vicinity of the morphotropic phase boundary is given.
Accurate quantum chemical calculations
NASA Technical Reports Server (NTRS)
Bauschlicher, Charles W., Jr.; Langhoff, Stephen R.; Taylor, Peter R.
1989-01-01
An important goal of quantum chemical calculations is to provide an understanding of chemical bonding and molecular electronic structure. A second goal, the prediction of energy differences to chemical accuracy, has been much harder to attain. First, the computational resources required to achieve such accuracy are very large, and second, it is not straightforward to demonstrate that an apparently accurate result, in terms of agreement with experiment, does not result from a cancellation of errors. Recent advances in electronic structure methodology, coupled with the power of vector supercomputers, have made it possible to solve a number of electronic structure problems exactly using the full configuration interaction (FCI) method within a subspace of the complete Hilbert space. These exact results can be used to benchmark approximate techniques that are applicable to a wider range of chemical and physical problems. The methodology of many-electron quantum chemistry is reviewed. Methods are considered in detail for performing FCI calculations. The application of FCI methods to several three-electron problems in molecular physics are discussed. A number of benchmark applications of FCI wave functions are described. Atomic basis sets and the development of improved methods for handling very large basis sets are discussed: these are then applied to a number of chemical and spectroscopic problems; to transition metals; and to problems involving potential energy surfaces. Although the experiences described give considerable grounds for optimism about the general ability to perform accurate calculations, there are several problems that have proved less tractable, at least with current computer resources, and these and possible solutions are discussed.
NASA Astrophysics Data System (ADS)
Majer, G.; Zick, K.
2015-04-01
A pulsed field gradient spin-echo nuclear magnetic resonance (NMR) sequence with solvent suppression (PGSE-WATERGATE) was applied to accurately measure the diffusion coefficients of Rhodamine 6G (Rh6G) in low-concentration aqueous solutions. Three samples with Rh6G concentrations of CRh6G = 1, 4.5, and 25 μM were investigated. The precise determination of the diffusion coefficients in this low-concentration range was made possible by using a cryogenically cooled NMR probe and by the effective solvent suppression of the PGSE-WATERGATE sequence. The present results bridge the gap between diffusion data measured by fluorescence correlation spectroscopy in the single molecule limit and diffusivities obtained by pulsed field gradient NMR (PFG-NMR) without solvent suppression at higher concentrations. To further extend the concentration range, the diffusion coefficient of Rh6G was also measured on a sample with CRh6G = 410 μM by PFG-NMR. The overall concentration dependence of the Rh6G diffusion at 25 °C is discussed in terms of dimerization of the Rh6G molecules. The concentration-dependent monomer/dimer proportion is deduced from the diffusion data.
Majer, G.; Zick, K.
2015-04-28
A pulsed field gradient spin-echo nuclear magnetic resonance (NMR) sequence with solvent suppression (PGSE-WATERGATE) was applied to accurately measure the diffusion coefficients of Rhodamine 6G (Rh6G) in low-concentration aqueous solutions. Three samples with Rh6G concentrations of C{sub Rh6G} = 1, 4.5, and 25 μM were investigated. The precise determination of the diffusion coefficients in this low-concentration range was made possible by using a cryogenically cooled NMR probe and by the effective solvent suppression of the PGSE-WATERGATE sequence. The present results bridge the gap between diffusion data measured by fluorescence correlation spectroscopy in the single molecule limit and diffusivities obtained by pulsed field gradient NMR (PFG-NMR) without solvent suppression at higher concentrations. To further extend the concentration range, the diffusion coefficient of Rh6G was also measured on a sample with C{sub Rh6G} = 410 μM by PFG-NMR. The overall concentration dependence of the Rh6G diffusion at 25 °C is discussed in terms of dimerization of the Rh6G molecules. The concentration-dependent monomer/dimer proportion is deduced from the diffusion data.
Approximate Bayesian multibody tracking.
Lanz, Oswald
2006-09-01
Visual tracking of multiple targets is a challenging problem, especially when efficiency is an issue. Occlusions, if not properly handled, are a major source of failure. Solutions supporting principled occlusion reasoning have been proposed but are yet unpractical for online applications. This paper presents a new solution which effectively manages the trade-off between reliable modeling and computational efficiency. The Hybrid Joint-Separable (HJS) filter is derived from a joint Bayesian formulation of the problem, and shown to be efficient while optimal in terms of compact belief representation. Computational efficiency is achieved by employing a Markov random field approximation to joint dynamics and an incremental algorithm for posterior update with an appearance likelihood that implements a physically-based model of the occlusion process. A particle filter implementation is proposed which achieves accurate tracking during partial occlusions, while in cases of complete occlusion, tracking hypotheses are bound to estimated occlusion volumes. Experiments show that the proposed algorithm is efficient, robust, and able to resolve long-term occlusions between targets with identical appearance. PMID:16929730
Legendre-tau approximations for functional differential equations
NASA Technical Reports Server (NTRS)
Ito, K.; Teglas, R.
1986-01-01
The numerical approximation of solutions to linear retarded functional differential equations are considered using the so-called Legendre-tau method. The functional differential equation is first reformulated as a partial differential equation with a nonlocal boundary condition involving time-differentiation. The approximate solution is then represented as a truncated Legendre series with time-varying coefficients which satisfy a certain system of ordinary differential equations. The method is very easy to code and yields very accurate approximations. Convergence is established, various numerical examples are presented, and comparison between the latter and cubic spline approximation is made.
Legendre-Tau approximations for functional differential equations
NASA Technical Reports Server (NTRS)
Ito, K.; Teglas, R.
1983-01-01
The numerical approximation of solutions to linear functional differential equations are considered using the so called Legendre tau method. The functional differential equation is first reformulated as a partial differential equation with a nonlocal boundary condition involving time differentiation. The approximate solution is then represented as a truncated Legendre series with time varying coefficients which satisfy a certain system of ordinary differential equations. The method is very easy to code and yields very accurate approximations. Convergence is established, various numerical examples are presented, and comparison between the latter and cubic spline approximations is made.
Simple analytic approximations for the Blasius problem
NASA Astrophysics Data System (ADS)
Iacono, R.; Boyd, John P.
2015-08-01
The classical boundary layer problem formulated by Heinrich Blasius more than a century ago is revisited, with the purpose of deriving simple and accurate analytical approximations to its solution. This is achieved through the combined use of a generalized Padé approach and of an integral iteration scheme devised by Hermann Weyl. The iteration scheme is also used to derive very accurate bounds for the value of the second derivative of the Blasius function at the origin, which plays a crucial role in this problem.
On the Accuracy of the MINC approximation
Lai, C.H.; Pruess, K.; Bodvarsson, G.S.
1986-02-01
The method of ''multiple interacting continua'' is based on the assumption that changes in thermodynamic conditions of rock matrix blocks are primarily controlled by the distance from the nearest fracture. The accuracy of this assumption was evaluated for regularly shaped (cubic and rectangular) rock blocks with uniform initial conditions, which are subjected to a step change in boundary conditions on the surface. Our results show that pressures (or temperatures) predicted from the MINC approximation may deviate from the exact solutions by as much as 10 to 15% at certain points within the blocks. However, when fluid (or heat) flow rates are integrated over the entire block surface, MINC-approximation and exact solution agree to better than 1%. This indicates that the MINC approximation can accurately represent transient inter-porosity flow in fractured porous media, provided that matrix blocks are indeed subjected to nearly uniform boundary conditions at all times.
NASA Technical Reports Server (NTRS)
Schwenke, David W.
1993-01-01
We report the results of a series of calculations of state-to-state integral cross sections for collisions between O and nonvibrating H2O in the gas phase on a model nonreactive potential energy surface. The dynamical methods used include converged quantum mechanical scattering calculations, the j(z) conserving centrifugal sudden (j(z)-CCS) approximation, and quasi-classical trajectory (QCT) calculations. We consider three total energies 0.001, 0.002, and 0.005 E(h) and the nine initial states with rotational angular momentum less than or equal to 2 (h/2 pi). The j(z)-CCS approximation gives good results, while the QCT method can be quite unreliable for transitions to specific rotational sublevels. However, the QCT cross sections summed over final sublevels and averaged over initial sublevels are in better agreement with the quantum results.
An approximation based global optimization strategy for structural synthesis
NASA Technical Reports Server (NTRS)
Sepulveda, A. E.; Schmit, L. A.
1991-01-01
A global optimization strategy for structural synthesis based on approximation concepts is presented. The methodology involves the solution of a sequence of highly accurate approximate problems using a global optimization algorithm. The global optimization algorithm implemented consists of a branch and bound strategy based on the interval evaluation of the objective function and constraint functions, combined with a local feasible directions algorithm. The approximate design optimization problems are constructed using first order approximations of selected intermediate response quantities in terms of intermediate design variables. Some numerical results for example problems are presented to illustrate the efficacy of the design procedure setforth.
An adiabatic approximation for grain alignment theory
NASA Astrophysics Data System (ADS)
Roberge, W. G.
1997-10-01
The alignment of interstellar dust grains is described by the joint distribution function for certain `internal' and `external' variables, where the former describe the orientation of the axes of a grain with respect to its angular momentum, J, and the latter describe the orientation of J relative to the interstellar magnetic field. I show how the large disparity between the dynamical time-scales of the internal and external variables - which is typically 2-3 orders of magnitude - can be exploited to simplify calculations of the required distribution greatly. The method is based on an `adiabatic approximation' which closely resembles the Born-Oppenheimer approximation in quantum mechanics. The adiabatic approximation prescribes an analytic distribution function for the `fast' dynamical variables and a simplified Fokker-Planck equation for the `slow' variables which can be solved straightforwardly using various techniques. These solutions are accurate to O(epsilon), where epsilon is the ratio of the fast and slow dynamical time-scales. As a simple illustration of the method, I derive an analytic solution for the joint distribution established when Barnett relaxation acts in concert with gas damping. The statistics of the analytic solution agree with the results of laborious numerical calculations which do not exploit the adiabatic approximation.
An Adiabatic Approximation for Grain Alignment Theory
NASA Astrophysics Data System (ADS)
Roberge, W. G.
1997-12-01
The alignment of interstellar dust grains is described by the joint distribution function for certain ``internal'' and ``external'' variables, where the former describe the orientation of a grain's axes with respect to its angular momentum, J, and the latter describe the orientation of J relative to the interstellar magnetic field. I show how the large disparity between the dynamical timescales of the internal and external variables--- which is typically 2--3 orders of magnitude--- can be exploited to greatly simplify calculations of the required distribution. The method is based on an ``adiabatic approximation'' which closely resembles the Born-Oppenheimer approximation in quantum mechanics. The adiabatic approximation prescribes an analytic distribution function for the ``fast'' dynamical variables and a simplified Fokker-Planck equation for the ``slow'' variables which can be solved straightforwardly using various techniques. These solutions are accurate to cal {O}(epsilon ), where epsilon is the ratio of the fast and slow dynamical timescales. As a simple illustration of the method, I derive an analytic solution for the joint distribution established when Barnett relaxation acts in concert with gas damping. The statistics of the analytic solution agree with the results of laborious numerical calculations which do not exploit the adiabatic approximation.
Approximations for the period of the simple pendulum based on the arithmetic-geometric mean
NASA Astrophysics Data System (ADS)
Carvalhaes, Claudio G.; Suppes, Patrick
2008-12-01
We use the arithmetic-geometric mean to derive approximate solutions for the period of the simple pendulum. The fast convergence of the arithmetic-geometric mean yields accurate solutions. We also discuss the invention of the pendulum clock by Christiaan Huygens in 1656-1657.
Multicriteria approximation through decomposition
Burch, C. |; Krumke, S.; Marathe, M.; Phillips, C.; Sundberg, E. |
1997-12-01
The authors propose a general technique called solution decomposition to devise approximation algorithms with provable performance guarantees. The technique is applicable to a large class of combinatorial optimization problems that can be formulated as integer linear programs. Two key ingredients of the technique involve finding a decomposition of a fractional solution into a convex combination of feasible integral solutions and devising generic approximation algorithms based on calls to such decompositions as oracles. The technique is closely related to randomized rounding. The method yields as corollaries unified solutions to a number of well studied problems and it provides the first approximation algorithms with provable guarantees for a number of new problems. The particular results obtained in this paper include the following: (1) The authors demonstrate how the technique can be used to provide more understanding of previous results and new algorithms for classical problems such as Multicriteria Spanning Trees, and Suitcase Packing. (2) They show how the ideas can be extended to apply to multicriteria optimization problems, in which they wish to minimize a certain objective function subject to one or more budget constraints. As corollaries they obtain first non-trivial multicriteria approximation algorithms for problems including the k-Hurdle and the Network Inhibition problems.
Multicriteria approximation through decomposition
Burch, C.; Krumke, S.; Marathe, M.; Phillips, C.; Sundberg, E.
1998-06-01
The authors propose a general technique called solution decomposition to devise approximation algorithms with provable performance guarantees. The technique is applicable to a large class of combinatorial optimization problems that can be formulated as integer linear programs. Two key ingredients of their technique involve finding a decomposition of a fractional solution into a convex combination of feasible integral solutions and devising generic approximation algorithms based on calls to such decompositions as oracles. The technique is closely related to randomized rounding. Their method yields as corollaries unified solutions to a number of well studied problems and it provides the first approximation algorithms with provable guarantees for a number of new problems. The particular results obtained in this paper include the following: (1) the authors demonstrate how the technique can be used to provide more understanding of previous results and new algorithms for classical problems such as Multicriteria Spanning Trees, and Suitcase Packing; (2) they also show how the ideas can be extended to apply to multicriteria optimization problems, in which they wish to minimize a certain objective function subject to one or more budget constraints. As corollaries they obtain first non-trivial multicriteria approximation algorithms for problems including the k-Hurdle and the Network Inhibition problems.
Vergeynst, Leendert; Van Langenhove, Herman; Joos, Pieter; Demeestere, Kristof
2013-07-30
Uniform guidelines for the data processing and validation of qualitative and quantitative multi-residue analysis using full-spectrum high-resolution mass spectrometry are scarce. Through systematic research, optimal mass accuracy and sensitivity are obtained after refining the post-processing of the HRMS data. For qualitative analysis, transforming the raw profile spectra to centroid spectra is recommended resulting in a 2.3 fold improved precision on the accurate mass determination of spectrum peaks. However, processing centroid data for quantitative purposes could lead to signal interruption when too narrow mass windows are applied for the construction of extracted ion chromatograms. Therefore, peak integration on the raw profile data is recommended. An optimal width of the mass window of 50 ppm, which is a trade-off between sensitivity and selectivity, was obtained for a TOF instrument providing a resolving power of 20,000 at full width at half maximum (FWHM). For the validation of HRMS analytical methods, widespread concepts such as the signal-to-noise ratios for the determination of decision limits and detection capabilities have shown to be not always applicable because in some cases almost no noise can be detected anymore. A statistical methodology providing a reliable alternative is extended and applied. PMID:23856232
NASA Astrophysics Data System (ADS)
Russell, John
2003-11-01
I will use the abbreviation BMHR equation (for uc(Bussman,) Münz, Hughes & Reid) to denote the modified uc(Orr-Sommerfeld) equation that applies when the mean flow is the asymptotic suction boundary layer. uc(Dieter) Grohne (1950) and uc(Paul) Baldwin (1970) found several integral representations of exact solutions of the uc(Orr-Sommerfeld) and BMHR equations, respectively. In the mean time uc(W.D.) Lakin & W.H. Reid (1982) derived uniformly-valid asymptotic solutions of the BMHR equation. Following uc(W.) Wasow (1953) they found it useful to define a family of seven canonical solutions (one of which is well-balanced, three of which are balanced and three of which are dominant-recessive). As uc(Baldwin) pointed out in a 1985 paper the family of integral representations found by uc(Grohne) and uc(Baldwin,) though large enough to represent the general solution, does not immediately permit a one-to-one correlation with seven solutions of uc(Lakin) & Reid. A careful correlation must enlarge the family of exact solutions given by uc(Grohne) and uc(Baldwin) and must, moreover, include suitable conformal transfomations between the planes of the uc(Langer) variable used by uc(Lakin) & Reid and the independent variable that arises most naturally in the exact integral representations. The present work addresses these problems.
Accurate upwind-monotone (nonoscillatory) methods for conservation laws
NASA Technical Reports Server (NTRS)
Huynh, Hung T.
1992-01-01
The well known MUSCL scheme of Van Leer is constructed using a piecewise linear approximation. The MUSCL scheme is second order accurate at the smooth part of the solution except at extrema where the accuracy degenerates to first order due to the monotonicity constraint. To construct accurate schemes which are free from oscillations, the author introduces the concept of upwind monotonicity. Several classes of schemes, which are upwind monotone and of uniform second or third order accuracy are then presented. Results for advection with constant speed are shown. It is also shown that the new scheme compares favorably with state of the art methods.
A simple approximation for the current-voltage characteristics of high-power, relativistic diodes
NASA Astrophysics Data System (ADS)
Ekdahl, Carl
2016-06-01
A simple approximation for the current-voltage characteristics of a relativistic electron diode is presented. The approximation is accurate from non-relativistic through relativistic electron energies. Although it is empirically developed, it has many of the fundamental properties of the exact diode solutions. The approximation is simple enough to be remembered and worked on almost any pocket calculator, so it has proven to be quite useful on the laboratory floor.
A simple approximation for the current-voltage characteristics of high-power, relativistic diodes
Ekdahl, Carl
2016-06-01
A simple approximation for the current-voltage characteristics of a relativistic electron diode is presented. The approximation is accurate from non-relativistic through relativistic electron energies. Although it is empirically developed, it has many of the fundamental properties of the exact diode solutions. Lastly, the approximation is simple enough to be remembered and worked on almost any pocket calculator, so it has proven to be quite useful on the laboratory floor.
A simple approximation for the current-voltage characteristics of high-power, relativistic diodes.
Ekdahl, Carl
2016-06-01
A simple approximation for the current-voltage characteristics of a relativistic electron diode is presented. The approximation is accurate from non-relativistic through relativistic electron energies. Although it is empirically developed, it has many of the fundamental properties of the exact diode solutions. The approximation is simple enough to be remembered and worked on almost any pocket calculator, so it has proven to be quite useful on the laboratory floor. PMID:27370443
Impact of inflow transport approximation on light water reactor analysis
NASA Astrophysics Data System (ADS)
Choi, Sooyoung; Smith, Kord; Lee, Hyun Chul; Lee, Deokjung
2015-10-01
The impact of the inflow transport approximation on light water reactor analysis is investigated, and it is verified that the inflow transport approximation significantly improves the accuracy of the transport and transport/diffusion solutions. A methodology for an inflow transport approximation is implemented in order to generate an accurate transport cross section. The inflow transport approximation is compared to the conventional methods, which are the consistent-PN and the outflow transport approximations. The three transport approximations are implemented in the lattice physics code STREAM, and verification is performed for various verification problems in order to investigate their effects and accuracy. From the verification, it is noted that the consistent-PN and the outflow transport approximations cause significant error in calculating the eigenvalue and the power distribution. The inflow transport approximation shows very accurate and precise results for the verification problems. The inflow transport approximation shows significant improvements not only for the high leakage problem but also for practical large core problem analyses.
NASA Astrophysics Data System (ADS)
Reineker, P.; Kühne, R.
1980-03-01
Starting from the stochastic Liouville equation of the full Haken-Strobl model, describing the coupled coherent and incoherent motion of excitons in molecular crystals, the Nakajima-Zwanzig generalized master equation (GME) for the probability of finding an exciton at a specific lattice site is derived by an exact straightforward evaluation of its memory function. Various recently derived generalized master equations describing the excition motion are obtained as limiting cases and the Born approximation is discussed. It is shown that, even in the case of nearest-neighbor interaction in the stochastic Liouville equation, in the GME generalized time-dependent transition rates evolve between non-nearest neighbors and that their time behavior shows damped oscillations. Applying the Born approximation to the GME, the range of the generalized transition rates reduces to that of the interaction in the stochastic Liouville equation. Furthermore in this approximation the transition rates show a purely exponential decay with increasing time. Taking into account the interaction with an arbitrary number of neighbors, the mean square displacement of the exciton motion is calculated exactly from the GME. Finally the GME is solved exactly in the general case and several limiting expressions are discussed.
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.
NASA Astrophysics Data System (ADS)
Okita, Taishi; Takagi, Toshiyuki
2010-01-01
We analytically derive the solutions for electromagnetic fields of electric current dipole moment, which is placed in the exterior of the spherical homogeneous conductor, and is pointed along the radial direction. The dipole moment is driven in the low frequency f = 1 kHz and high frequency f = 1 GHz regimes. The electrical properties of the conductor are appropriately chosen in each frequency. Electromagnetic fields are rigorously formulated at an arbitrary point in a spherical geometry, in which the magnetic vector potential is straightforwardly given by the Biot-Savart formula, and the scalar potential is expanded with the Legendre polynomials, taking into account the appropriate boundary conditions at the spherical surface of the conductor. The induced electric fields are numerically calculated along the several paths in the low and high frequeny excitation. The self-consistent solutions obtained in this work will be of much importance in a wide region of electromagnetic induction problems.
Fast and accurate propagation of coherent light
Lewis, R. D.; Beylkin, G.; Monzón, L.
2013-01-01
We describe a fast algorithm to propagate, for any user-specified accuracy, a time-harmonic electromagnetic field between two parallel planes separated by a linear, isotropic and homogeneous medium. The analytical formulation of this problem (ca 1897) requires the evaluation of the so-called Rayleigh–Sommerfeld integral. If the distance between the planes is small, this integral can be accurately evaluated in the Fourier domain; if the distance is very large, it can be accurately approximated by asymptotic methods. In the large intermediate region of practical interest, where the oscillatory Rayleigh–Sommerfeld kernel must be applied directly, current numerical methods can be highly inaccurate without indicating this fact to the user. In our approach, for any user-specified accuracy ϵ>0, we approximate the kernel by a short sum of Gaussians with complex-valued exponents, and then efficiently apply the result to the input data using the unequally spaced fast Fourier transform. The resulting algorithm has computational complexity , where we evaluate the solution on an N×N grid of output points given an M×M grid of input samples. Our algorithm maintains its accuracy throughout the computational domain. PMID:24204184
NASA Astrophysics Data System (ADS)
Truzzolillo, D.; Bordi, F.; Sciortino, F.; Sennato, S.
2010-07-01
We study the effective interaction between differently charged polyelectrolyte-colloid complexes in electrolyte solutions via Monte Carlo simulations. These complexes are formed when short and flexible polyelectrolyte chains adsorb onto oppositely charged colloidal spheres, dispersed in an electrolyte solution. In our simulations the bending energy between adjacent monomers is small compared to the electrostatic energy, and the chains, once adsorbed, do not exchange with the solution, although they rearrange on the particles surface to accommodate further adsorbing chains or due to the electrostatic interaction with neighbor complexes. Rather unexpectedly, when two interacting particles approach each other, the rearrangement of the surface charge distribution invariably produces antiparallel dipolar doublets that invert their orientation at the isoelectric point. These findings clearly rule out a contribution of dipole-dipole interactions to the observed attractive interaction between the complexes, pointing out that such suspensions cannot be considered dipolar fluids. On varying the ionic strength of the electrolyte, we find that a screening length κ-1, short compared with the size of the colloidal particles, is required in order to observe the attraction between like-charged complexes due to the nonuniform distribution of the electric charge on their surface ("patch attraction"). On the other hand, by changing the polyelectrolyte/particle charge ratio ξs, the interaction between like-charged polyelectrolyte-decorated particles, at short separations, evolves from purely repulsive to strongly attractive. Hence, the effective interaction between the complexes is characterized by a potential barrier, whose height depends on the net charge and on the nonuniformity of their surface charge distribution.
NASA Astrophysics Data System (ADS)
Ghoumaid, A.; Benamira, F.; Guechi, L.
2016-02-01
It is shown that the application of the Nikiforov-Uvarov method by Ikhdair for solving the Dirac equation with the radial Rosen-Morse potential plus the spin-orbit centrifugal term is inadequate because the required conditions are not satisfied. The energy spectra given is incorrect and the wave functions are not physically acceptable. We clarify the problem and prove that the spinor wave functions are expressed in terms of the generalized hypergeometric functions 2F1(a, b, c; z). The energy eigenvalues for the bound states are given by the solution of a transcendental equation involving the hypergeometric function.
Ginovska, Bojana; Camaioni, Donald M; Dupuis, Michel; Schwerdtfeger, Christine A; Gilcrease, Quinn
2008-10-23
Dielectric continuum solvation models are widely used because they are a computationally efficacious way to simulate equilibrium properties of solutes. With advances that allow for molecular-shaped cavities, they have reached a high level of accuracy, in particular for neutral solutes. However, benchmark tests show that existing schemes for defining cavities are unable to consistently predict accurately the effects of solvation on ions, especially anions. This work involves the further development of a protocol put forth earlier for defining the cavities of aqueous solutes, with resulting advances that are most striking for anions. Molecular cavities are defined as interlocked spheres around atoms or groups of atoms in the solute, but the sphere radii are determined by simple empirically-based expressions involving the effective atomic charges of the solute atoms (derived from molecular electrostatic potential) and base radii. Both of these terms are optimized for the different types of atoms or functional groups in a training set of neutral and charged solutes. Parameters in these expressions for radii were fitted by minimizing residuals between calculated and measured standard free energies of solvation (ΔG_{s}*), weighted by the uncertainty in the measured value. The calculations were performed using density functional theory with the B3LYP functional and the 6-311+G** basis set and the COnductor-like Screening MOdel (COSMO). The optimized radii definitions reproduce ΔG_{s}* of neutral solutes and singly-charged ions in the training set to within experimental uncertainty and, more importantly, accurately predict ΔG_{s}* of compounds outside the training set, in particular anions. Inherent to this approach, the cavity definitions reflect the strength of specific solute-water interactions. We surmise that this feature underlies the success of the model, referred to as the CD-COSMO model for Charge-Dependent (also Camaioni-Dupuis) COSMO
NASA Astrophysics Data System (ADS)
Asgharzadeh, Hafez; Borazjani, Iman
2014-11-01
Time step-size restrictions and low convergence rates are major bottle necks for implicit solution of the Navier-Stokes in simulations involving complex geometries with moving boundaries. Newton-Krylov method (NKM) is a combination of a Newton-type method for super-linearly convergent solution of nonlinear equations and Krylov subspace methods for solving the Newton correction equations, which can theoretically address both bottle necks. The efficiency of this method vastly depends on the Jacobian forming scheme e.g. automatic differentiation is very expensive and Jacobian-free methods slow down as the mesh is refined. A novel, computationally efficient analytical Jacobian for NKM was developed to solve unsteady incompressible Navier-Stokes momentum equations on staggered curvilinear grids with immersed boundaries. The NKM was validated and verified against Taylor-Green vortex and pulsatile flow in a 90 degree bend and efficiently handles complex geometries such as an intracranial aneurysm with multiple overset grids, pulsatile inlet flow and immersed boundaries. The NKM method is shown to be more efficient than the semi-implicit Runge-Kutta methods and Jabobian-free Newton-Krylov methods. We believe NKM can be applied to many CFD techniques to decrease the computational cost. This work was supported partly by the NIH Grant R03EB014860, and the computational resources were partly provided by Center for Computational Research (CCR) at University at Buffalo.
Green-Ampt approximations: A comprehensive analysis
NASA Astrophysics Data System (ADS)
Ali, Shakir; Islam, Adlul; Mishra, P. K.; Sikka, Alok K.
2016-04-01
Green-Ampt (GA) model and its modifications are widely used for simulating infiltration process. Several explicit approximate solutions to the implicit GA model have been developed with varying degree of accuracy. In this study, performance of nine explicit approximations to the GA model is compared with the implicit GA model using the published data for broad range of soil classes and infiltration time. The explicit GA models considered are Li et al. (1976) (LI), Stone et al. (1994) (ST), Salvucci and Entekhabi (1994) (SE), Parlange et al. (2002) (PA), Barry et al. (2005) (BA), Swamee et al. (2012) (SW), Ali et al. (2013) (AL), Almedeij and Esen (2014) (AE), and Vatankhah (2015) (VA). Six statistical indicators (e.g., percent relative error, maximum absolute percent relative error, average absolute percent relative errors, percent bias, index of agreement, and Nash-Sutcliffe efficiency) and relative computer computation time are used for assessing the model performance. Models are ranked based on the overall performance index (OPI). The BA model is found to be the most accurate followed by the PA and VA models for variety of soil classes and infiltration periods. The AE, SW, SE, and LI model also performed comparatively better. Based on the overall performance index, the explicit models are ranked as BA > PA > VA > LI > AE > SE > SW > ST > AL. Results of this study will be helpful in selection of accurate and simple explicit approximate GA models for solving variety of hydrological problems.
Yamada, Atsushi; Kojima, Hidekazu; Okazaki, Susumu
2014-08-28
In order to investigate proton transfer reaction in solution, mixed quantum-classical molecular dynamics calculations have been carried out based on our previously proposed quantum equation of motion for the reacting system [A. Yamada and S. Okazaki, J. Chem. Phys. 128, 044507 (2008)]. Surface hopping method was applied to describe forces acting on the solvent classical degrees of freedom. In a series of our studies, quantum and solvent effects on the reaction dynamics in solutions have been analysed in detail. Here, we report our mixed quantum-classical molecular dynamics calculations for intramolecular proton transfer of malonaldehyde in water. Thermally activated proton transfer process, i.e., vibrational excitation in the reactant state followed by transition to the product state and vibrational relaxation in the product state, as well as tunneling reaction can be described by solving the equation of motion. Zero point energy is, of course, included, too. The quantum simulation in water has been compared with the fully classical one and the wave packet calculation in vacuum. The calculated quantum reaction rate in water was 0.70 ps{sup −1}, which is about 2.5 times faster than that in vacuum, 0.27 ps{sup −1}. This indicates that the solvent water accelerates the reaction. Further, the quantum calculation resulted in the reaction rate about 2 times faster than the fully classical calculation, which indicates that quantum effect enhances the reaction rate, too. Contribution from three reaction mechanisms, i.e., tunneling, thermal activation, and barrier vanishing reactions, is 33:46:21 in the mixed quantum-classical calculations. This clearly shows that the tunneling effect is important in the reaction.
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.
Approximate formula for the escape function for nearly conservative scattering
NASA Astrophysics Data System (ADS)
Yanovitskij, E. G.
2002-02-01
The escape function u(μ) (i.e., the boundary solution of the Milne problem for a semi-infinite atmosphere) is considered. It is presented in the form u(μ) = u0 (μ ) + √ {1 - λ}u1(μ) + (1-λ)u2(μ) + ldots, where λ is the single-scattering albedo. A rather accurate approximate formula for a the function u0 (μ) is obtained for not highly elongated phase function. An approximate expression for the function u2 (μ) is also derived, it is exact in the case of the most simple anisotropic scattering.
Accurate theoretical chemistry with coupled pair models.
Neese, Frank; Hansen, Andreas; Wennmohs, Frank; Grimme, Stefan
2009-05-19
Quantum chemistry has found its way into the everyday work of many experimental chemists. Calculations can predict the outcome of chemical reactions, afford insight into reaction mechanisms, and be used to interpret structure and bonding in molecules. Thus, contemporary theory offers tremendous opportunities in experimental chemical research. However, even with present-day computers and algorithms, we cannot solve the many particle Schrodinger equation exactly; inevitably some error is introduced in approximating the solutions of this equation. Thus, the accuracy of quantum chemical calculations is of critical importance. The affordable accuracy depends on molecular size and particularly on the total number of atoms: for orientation, ethanol has 9 atoms, aspirin 21 atoms, morphine 40 atoms, sildenafil 63 atoms, paclitaxel 113 atoms, insulin nearly 800 atoms, and quaternary hemoglobin almost 12,000 atoms. Currently, molecules with up to approximately 10 atoms can be very accurately studied by coupled cluster (CC) theory, approximately 100 atoms with second-order Møller-Plesset perturbation theory (MP2), approximately 1000 atoms with density functional theory (DFT), and beyond that number with semiempirical quantum chemistry and force-field methods. The overwhelming majority of present-day calculations in the 100-atom range use DFT. Although these methods have been very successful in quantum chemistry, they do not offer a well-defined hierarchy of calculations that allows one to systematically converge to the correct answer. Recently a number of rather spectacular failures of DFT methods have been found-even for seemingly simple systems such as hydrocarbons, fueling renewed interest in wave function-based methods that incorporate the relevant physics of electron correlation in a more systematic way. Thus, it would be highly desirable to fill the gap between 10 and 100 atoms with highly correlated ab initio methods. We have found that one of the earliest (and now
Approximate factorization with source terms
NASA Technical Reports Server (NTRS)
Shih, T. I.-P.; Chyu, W. J.
1991-01-01
A comparative evaluation is made of three methodologies with a view to that which offers the best approximate factorization error. While two of these methods are found to lead to more efficient algorithms in cases where factors which do not contain source terms can be diagonalized, the third method used generates the lowest approximate factorization error. This method may be preferred when the norms of source terms are large, and transient solutions are of interest.
Approximating random quantum optimization problems
NASA Astrophysics Data System (ADS)
Hsu, B.; Laumann, C. R.; Läuchli, A. M.; Moessner, R.; Sondhi, S. L.
2013-06-01
We report a cluster of results regarding the difficulty of finding approximate ground states to typical instances of the quantum satisfiability problem k-body quantum satisfiability (k-QSAT) on large random graphs. As an approximation strategy, we optimize the solution space over “classical” product states, which in turn introduces a novel autonomous classical optimization problem, PSAT, over a space of continuous degrees of freedom rather than discrete bits. Our central results are (i) the derivation of a set of bounds and approximations in various limits of the problem, several of which we believe may be amenable to a rigorous treatment; (ii) a demonstration that an approximation based on a greedy algorithm borrowed from the study of frustrated magnetism performs well over a wide range in parameter space, and its performance reflects the structure of the solution space of random k-QSAT. Simulated annealing exhibits metastability in similar “hard” regions of parameter space; and (iii) a generalization of belief propagation algorithms introduced for classical problems to the case of continuous spins. This yields both approximate solutions, as well as insights into the free energy “landscape” of the approximation problem, including a so-called dynamical transition near the satisfiability threshold. Taken together, these results allow us to elucidate the phase diagram of random k-QSAT in a two-dimensional energy-density-clause-density space.
Approximate two layer (inviscid/viscous) methods to model aerothermodynamic environments
NASA Technical Reports Server (NTRS)
Dejarnette, Fred R.
1992-01-01
Approximate inviscid and boundary layer techniques for aerodynamic heating calculations are discussed. An inviscid flowfield solution is needed to provide surface pressures and boundary-layer edge properties. Modified Newtonian pressures coupled with an approximate shock shape will suffice for relatively simple shapes like sphere-cones with cone half-angles between 15 and 45 deg. More accurate approximate methods have been developed which make use of modified Maslen techniques. Slender and large angle sphere-cones and more complex shapes generally require an Euler code, like HALIS, to provide that information. The boundary-layer solution is reduced significantly by using the axisymmetric analog and approximate heating relations developed by Zoby, et al. (1981). Analysis is presented for the calculation of inviscid surface streamlines and metrics. Entropy-layer swallowing effects require coupling the inviscid and boundary-layer solutions.
Structural optimization with approximate sensitivities
NASA Technical Reports Server (NTRS)
Patnaik, S. N.; Hopkins, D. A.; Coroneos, R.
1994-01-01
Computational efficiency in structural optimization can be enhanced if the intensive computations associated with the calculation of the sensitivities, that is, gradients of the behavior constraints, are reduced. Approximation to gradients of the behavior constraints that can be generated with small amount of numerical calculations is proposed. Structural optimization with these approximate sensitivities produced correct optimum solution. Approximate gradients performed well for different nonlinear programming methods, such as the sequence of unconstrained minimization technique, method of feasible directions, sequence of quadratic programming, and sequence of linear programming. Structural optimization with approximate gradients can reduce by one third the CPU time that would otherwise be required to solve the problem with explicit closed-form gradients. The proposed gradient approximation shows potential to reduce intensive computation that has been associated with traditional structural optimization.
On the dynamics of approximating schemes for dissipative nonlinear equations
NASA Technical Reports Server (NTRS)
Jones, Donald A.
1993-01-01
Since one can rarely write down the analytical solutions to nonlinear dissipative partial differential equations (PDE's), it is important to understand whether, and in what sense, the behavior of approximating schemes to these equations reflects the true dynamics of the original equations. Further, because standard error estimates between approximations of the true solutions coming from spectral methods - finite difference or finite element schemes, for example - and the exact solutions grow exponentially in time, this analysis provides little value in understanding the infinite time behavior of a given approximating scheme. The notion of the global attractor has been useful in quantifying the infinite time behavior of dissipative PDEs, such as the Navier-Stokes equations. Loosely speaking, the global attractor is all that remains of a sufficiently large bounded set in phase space mapped infinitely forward in time under the evolution of the PDE. Though the attractor has been shown to have some nice properties - it is compact, connected, and finite dimensional, for example - it is in general quite complicated. Nevertheless, the global attractor gives a way to understand how the infinite time behavior of approximating schemes such as the ones coming from a finite difference, finite element, or spectral method relates to that of the original PDE. Indeed, one can often show that such approximations also have a global attractor. We therefore only need to understand how the structure of the attractor for the PDE behaves under approximation. This is by no means a trivial task. Several interesting results have been obtained in this direction. However, we will not go into the details. We mention here that approximations generally lose information about the system no matter how accurate they are. There are examples that show certain parts of the attractor may be lost by arbitrary small perturbations of the original equations.
Altitude-adjusted corrected geomagnetic coordinates: Definition and functional approximations
NASA Astrophysics Data System (ADS)
Shepherd, S. G.
2014-09-01
Analysis of the functional approximations used to transform between geographic and Altitude-Adjusted Corrected Geomagnetic (AACGM) coordinates reveals that errors of >50 km can occur in the auroral and polar regions. These errors are the result of efforts to better approximate AACGM coordinates near the magnetic equator and the South Atlantic Anomaly. In these regions AACGM coordinates are not defined and alternate coordinates have been used. This augmentation and emphasis on the solution in regions near the equator result in spherical harmonic approximating functions that are less accurate than need be in the auroral and polar regions. In response, a new set of spherical harmonic coefficients have been derived that better represent AACGM coordinates in these regions. These new AACGM coefficients are limited to below 2000 km in altitude in order to ensure accuracy. For altitudes above 2000 km, a magnetic field-line tracing solution is recommended. A software package developed to take advantage of the new AACGM coefficients provides the capability of tracing magnetic field lines at any altitude, for improved accuracy. In addition, linear interpolation between 5 year epochs is used to produce coordinates that vary smoothly over the entire period from 1965 to present. The intent of this work is to provide a more accurate procedure for determining AACGM coordinates in the auroral and polar regions for the study of magnetospheric and ionospheric processes.
Diagonal Pade approximations for initial value problems
Reusch, M.F.; Ratzan, L.; Pomphrey, N.; Park, W.
1987-06-01
Diagonal Pade approximations to the time evolution operator for initial value problems are applied in a novel way to the numerical solution of these problems by explicitly factoring the polynomials of the approximation. A remarkable gain over conventional methods in efficiency and accuracy of solution is obtained. 20 refs., 3 figs., 1 tab.
Approximation for nonresonant beam target fusion reactivities
Mikkelsen, D.R.
1988-11-01
The beam target fusion reactivity for a monoenergetic beam in a Maxwellian target is approximately evaluated for nonresonant reactions. The approximation is accurate for the DD and TT fusion reactions to better than 4% for all beam energies up to 300 keV and all ion temperatures up to 2/3 of the beam energy. 12 refs., 1 fig., 1 tab.
Approximate simulation of CO[sub 2] and H[sub 2]S absorption into aqueous alkanolamines
Glasscock, D.A.; Rochelle, G.T. . Dept. of Chemical Engineering)
1993-08-01
Rigorous and approximate methods are compared for the simulation of CO[sub 2] absorption into aqueous alkanolamine mixtures of methyldiethanolamine and diethanolamine. In addition, data for the mixtures containing monoethanolamine and the simultaneous absorption of CO[sub 2] and H[sub 2]S are presented. For the rigorous approach, the simplified eddy diffusivity theory is used to simulate the liquid-phase hydrodynamic characteristics. The approximation methods examined are the pseudo-first-order approximation, the interpolation approximation of Wellek et al. (1978), the algebraic combined flux (ACFLUX) approximation and the modified combined flux (MCFLUX) approximation. The latter approximation utilizes the reaction zone concept to determine the kinetic preference of the absorbing gas at the gas-liquid interface. Under the range of conditions studied, the MCFLUX approximation predicts very accurately the CO[sub 2] and H[sub 2]S flux rates in mixed amine systems, as compared with the rigorous solution of the differential equations.
Mathematical algorithms for approximate reasoning
NASA Technical Reports Server (NTRS)
Murphy, John H.; Chay, Seung C.; Downs, Mary M.
1988-01-01
Most state of the art expert system environments contain a single and often ad hoc strategy for approximate reasoning. Some environments provide facilities to program the approximate reasoning algorithms. However, the next generation of expert systems should have an environment which contain a choice of several mathematical algorithms for approximate reasoning. To meet the need for validatable and verifiable coding, the expert system environment must no longer depend upon ad hoc reasoning techniques but instead must include mathematically rigorous techniques for approximate reasoning. Popular approximate reasoning techniques are reviewed, including: certainty factors, belief measures, Bayesian probabilities, fuzzy logic, and Shafer-Dempster techniques for reasoning. A group of mathematically rigorous algorithms for approximate reasoning are focused on that could form the basis of a next generation expert system environment. These algorithms are based upon the axioms of set theory and probability theory. To separate these algorithms for approximate reasoning various conditions of mutual exclusivity and independence are imposed upon the assertions. Approximate reasoning algorithms presented include: reasoning with statistically independent assertions, reasoning with mutually exclusive assertions, reasoning with assertions that exhibit minimum overlay within the state space, reasoning with assertions that exhibit maximum overlay within the state space (i.e. fuzzy logic), pessimistic reasoning (i.e. worst case analysis), optimistic reasoning (i.e. best case analysis), and reasoning with assertions with absolutely no knowledge of the possible dependency among the assertions. A robust environment for expert system construction should include the two modes of inference: modus ponens and modus tollens. Modus ponens inference is based upon reasoning towards the conclusion in a statement of logical implication, whereas modus tollens inference is based upon reasoning away
ERIC Educational Resources Information Center
Rom, Mark Carl
2011-01-01
Grades matter. College grading systems, however, are often ad hoc and prone to mistakes. This essay focuses on one factor that contributes to high-quality grading systems: grading accuracy (or "efficiency"). I proceed in several steps. First, I discuss the elements of "efficient" (i.e., accurate) grading. Next, I present analytical results…
An explicit series approximation to the optimal exercise boundary of American put options
NASA Astrophysics Data System (ADS)
Cheng, Jun; Zhu, Song-Ping; Liao, Shi-Jun
2010-05-01
This paper derives an explicit series approximation solution for the optimal exercise boundary of an American put option by means of a new analytical method for strongly nonlinear problems, namely the homotopy analysis method (HAM). The Black-Sholes equation subject to the moving boundary conditions for an American put option is transferred into an infinite number of linear sub-problems in a fixed domain through the deformation equations. Different from perturbation/asymptotic approximations, the HAM approximation can be applicable for options with much longer expiry. Accuracy tests are made in comparison with numerical solutions. It is found that the current approximation is as accurate as many numerical methods. Considering its explicit form of expression, it can bring great convenience to the market practitioners.
An approximation technique for jet impingement flow
Najafi, Mahmoud; Fincher, Donald; Rahni, Taeibi; Javadi, KH.; Massah, H.
2015-03-10
The analytical approximate solution of a non-linear jet impingement flow model will be demonstrated. We will show that this is an improvement over the series approximation obtained via the Adomian decomposition method, which is itself, a powerful method for analysing non-linear differential equations. The results of these approximations will be compared to the Runge-Kutta approximation in order to demonstrate their validity.
Multivariate Padé Approximations For Solving Nonlinear Diffusion Equations
NASA Astrophysics Data System (ADS)
Turut, V.
2015-11-01
In this paper, multivariate Padé approximation is applied to power series solutions of nonlinear diffusion equations. As it is seen from tables, multivariate Padé approximation (MPA) gives reliable solutions and numerical results.
Rasin, A.
1994-04-01
We discuss the idea of approximate flavor symmetries. Relations between approximate flavor symmetries and natural flavor conservation and democracy models is explored. Implications for neutrino physics are also discussed.
Abrupt PN junctions: Analytical solutions under equilibrium and non-equilibrium
NASA Astrophysics Data System (ADS)
Khorasani, Sina
2016-08-01
We present an explicit solution of carrier and field distributions in abrupt PN junctions under equilibrium. An accurate logarithmic numerical method is implemented and results are compared to the analytical solutions. Analysis of results shows reasonable agreement with numerical solution as well as the depletion layer approximation. We discuss extensions to the asymmetric junctions. Approximate relations for differential capacitance C-V and current-voltage I-V characteristics are also found under non-zero external bias.
Accurate monotone cubic interpolation
NASA Technical Reports Server (NTRS)
Huynh, Hung T.
1991-01-01
Monotone piecewise cubic interpolants are simple and effective. They are generally third-order accurate, except near strict local extrema where accuracy degenerates to second-order due to the monotonicity constraint. Algorithms for piecewise cubic interpolants, which preserve monotonicity as well as uniform third and fourth-order accuracy are presented. The gain of accuracy is obtained by relaxing the monotonicity constraint in a geometric framework in which the median function plays a crucial role.
Accurate Finite Difference Algorithms
NASA Technical Reports Server (NTRS)
Goodrich, John W.
1996-01-01
Two families of finite difference algorithms for computational aeroacoustics are presented and compared. All of the algorithms are single step explicit methods, they have the same order of accuracy in both space and time, with examples up to eleventh order, and they have multidimensional extensions. One of the algorithm families has spectral like high resolution. Propagation with high order and high resolution algorithms can produce accurate results after O(10(exp 6)) periods of propagation with eight grid points per wavelength.
An optimum approximation of n-point correlation functions of random heterogeneous material systems
Baniassadi, M.; Garmestani, H.; Ahzi, S.; Remond, Y.
2014-02-21
An approximate solution for n-point correlation functions is developed in this study. In the approximate solution, weight functions are used to connect subsets of (n-1)-point correlation functions to estimate the full set of n-point correlation functions. In previous related studies, simple weight functions were introduced for the approximation of three and four-point correlation functions. In this work, the general framework of the weight functions is extended and derived to achieve optimum accuracy for approximate n-point correlation functions. Such approximation can be utilized to construct global n-point correlation functions for a system when there exist limited information about these functions in a subset of space. To verify its accuracy, the new formulation is used to approximate numerically three-point correlation functions from the set of two-point functions directly evaluated from a virtually generated isotropic heterogeneous microstructure representing a particulate composite system. Similarly, three-point functions are approximated for an anisotropic glass fiber/epoxy composite system and compared to their corresponding reference values calculated from an experimental dataset acquired by computational tomography. Results from both virtual and experimental studies confirm the accuracy of the new approximation. The new formulation can be utilized to attain a more accurate approximation to global n-point correlation functions for heterogeneous material systems with a hierarchy of length scales.
An optimum approximation of n-point correlation functions of random heterogeneous material systems
NASA Astrophysics Data System (ADS)
Baniassadi, M.; Safdari, M.; Garmestani, H.; Ahzi, S.; Geubelle, P. H.; Remond, Y.
2014-02-01
An approximate solution for n-point correlation functions is developed in this study. In the approximate solution, weight functions are used to connect subsets of (n-1)-point correlation functions to estimate the full set of n-point correlation functions. In previous related studies, simple weight functions were introduced for the approximation of three and four-point correlation functions. In this work, the general framework of the weight functions is extended and derived to achieve optimum accuracy for approximate n-point correlation functions. Such approximation can be utilized to construct global n-point correlation functions for a system when there exist limited information about these functions in a subset of space. To verify its accuracy, the new formulation is used to approximate numerically three-point correlation functions from the set of two-point functions directly evaluated from a virtually generated isotropic heterogeneous microstructure representing a particulate composite system. Similarly, three-point functions are approximated for an anisotropic glass fiber/epoxy composite system and compared to their corresponding reference values calculated from an experimental dataset acquired by computational tomography. Results from both virtual and experimental studies confirm the accuracy of the new approximation. The new formulation can be utilized to attain a more accurate approximation to global n-point correlation functions for heterogeneous material systems with a hierarchy of length scales.
An approximate model for pulsar navigation simulation
NASA Astrophysics Data System (ADS)
Jovanovic, Ilija; Enright, John
2016-02-01
This paper presents an approximate model for the simulation of pulsar aided navigation systems. High fidelity simulations of these systems are computationally intensive and impractical for simulating periods of a day or more. Simulation of yearlong missions is done by abstracting navigation errors as periodic Gaussian noise injections. This paper presents an intermediary approximate model to simulate position errors for periods of several weeks, useful for building more accurate Gaussian error models. This is done by abstracting photon detection and binning, replacing it with a simple deterministic process. The approximate model enables faster computation of error injection models, allowing the error model to be inexpensively updated throughout a simulation. Testing of the approximate model revealed an optimistic performance prediction for non-millisecond pulsars with more accurate predictions for pulsars in the millisecond spectrum. This performance gap was attributed to noise which is not present in the approximate model but can be predicted and added to improve accuracy.
Generalized Gradient Approximation Made Simple
Perdew, J.P.; Burke, K.; Ernzerhof, M.
1996-10-01
Generalized gradient approximations (GGA{close_quote}s) for the exchange-correlation energy improve upon the local spin density (LSD) description of atoms, molecules, and solids. We present a simple derivation of a simple GGA, in which all parameters (other than those in LSD) are fundamental constants. Only general features of the detailed construction underlying the Perdew-Wang 1991 (PW91) GGA are invoked. Improvements over PW91 include an accurate description of the linear response of the uniform electron gas, correct behavior under uniform scaling, and a smoother potential. {copyright} {ital 1996 The American Physical Society.}
An n log n Generalized Born Approximation.
Anandakrishnan, Ramu; Daga, Mayank; Onufriev, Alexey V
2011-03-01
Molecular dynamics (MD) simulations based on the generalized Born (GB) model of implicit solvation offer a number of important advantages over the traditional explicit solvent based simulations. Yet, in MD simulations, the GB model has not been able to reach its full potential partly due to its computational cost, which scales as ∼n(2), where n is the number of solute atoms. We present here an ∼n log n approximation for the generalized Born (GB) implicit solvent model. The approximation is based on the hierarchical charge partitioning (HCP) method (Anandakrishnan and Onufriev J. Comput. Chem. 2010 , 31 , 691 - 706 ) previously developed and tested for electrostatic computations in gas-phase and distant dependent dielectric models. The HCP uses the natural organization of biomolecular structures to partition the structures into multiple hierarchical levels of components. The charge distribution for each of these components is approximated by a much smaller number of charges. The approximate charges are then used for computing electrostatic interactions with distant components, while the full set of atomic charges are used for nearby components. To apply the HCP concept to the GB model, we define the equivalent of the effective Born radius for components. The component effective Born radius is then used in GB computations for points that are distant from the component. This HCP approximation for GB (HCP-GB) is implemented in the open source MD software, NAB in AmberTools, and tested on a set of representative biomolecular structures ranging in size from 632 atoms to ∼3 million atoms. For this set of test structures, the HCP-GB method is 1.1-390 times faster than the GB computation without additional approximations (the reference GB computation), depending on the size of the structure. Similar to the spherical cutoff method with GB (cutoff-GB), which also scales as ∼n log n, the HCP-GB is relatively simple. However, for the structures considered here, we show
Approximation by hinge functions
Faber, V.
1997-05-01
Breiman has defined {open_quotes}hinge functions{close_quotes} for use as basis functions in least squares approximations to data. A hinge function is the max (or min) function of two linear functions. In this paper, the author assumes the existence of smooth function f(x) and a set of samples of the form (x, f(x)) drawn from a probability distribution {rho}(x). The author hopes to find the best fitting hinge function h(x) in the least squares sense. There are two problems with this plan. First, Breiman has suggested an algorithm to perform this fit. The author shows that this algorithm is not robust and also shows how to create examples on which the algorithm diverges. Second, if the author tries to use the data to minimize the fit in the usual discrete least squares sense, the functional that must be minimized is continuous in the variables, but has a derivative which jumps at the data. This paper takes a different approach. This approach is an example of a method that the author has developed called {open_quotes}Monte Carlo Regression{close_quotes}. (A paper on the general theory is in preparation.) The author shall show that since the function f is continuous, the analytic form of the least squares equation is continuously differentiable. A local minimum is solved for by using Newton`s method, where the entries of the Hessian are estimated directly from the data by Monte Carlo. The algorithm has the desirable properties that it is quadratically convergent from any starting guess sufficiently close to a solution and that each iteration requires only a linear system solve.
NASA Technical Reports Server (NTRS)
Hannah, S. R.; Palazotto, A. N.
1978-01-01
A new trigonometric approach to the finite difference calculus was applied to the problem of beam buckling as represented by virtual work and equilibrium equations. The trigonometric functions were varied by adjusting a wavelength parameter in the approximating Fourier series. Values of the critical force obtained from the modified approach for beams with a variety of boundary conditions were compared to results using the conventional finite difference method. The trigonometric approach produced significantly more accurate approximations for the critical force than the conventional approach for a relatively wide range in values of the wavelength parameter; and the optimizing value of the wavelength parameter corresponded to the half-wavelength of the buckled mode shape. It was found from a modal analysis that the most accurate solutions are obtained when the approximating function closely represents the actual displacement function and matches the actual boundary conditions.
2D/1D approximations to the 3D neutron transport equation. I: Theory
Kelley, B. W.; Larsen, E. W.
2013-07-01
A new class of '2D/1D' approximations is proposed for the 3D linear Boltzmann equation. These approximate equations preserve the exact transport physics in the radial directions x and y and diffusion physics in the axial direction z. Thus, the 2D/1D equations are more accurate approximations of the 3D Boltzmann equation than the conventional 3D diffusion equation. The 2D/1D equations can be systematically discretized, to yield accurate simulation methods for 3D reactor core problems. The resulting solutions will be more accurate than 3D diffusion solutions, and less expensive to generate than standard 3D transport solutions. In this paper, we (i) show that the simplest 2D/1D equation has certain desirable properties, (ii) systematically discretize this equation, and (iii) derive a stable iteration scheme for solving the discrete system of equations. In a companion paper [1], we give numerical results that confirm the theoretical predictions of accuracy and iterative stability. (authors)
NASA Astrophysics Data System (ADS)
Itano, Wayne M.; Ramsey, Norman F.
1993-07-01
The paper discusses current methods for accurate measurements of time by conventional atomic clocks, with particular attention given to the principles of operation of atomic-beam frequency standards, atomic hydrogen masers, and atomic fountain and to the potential use of strings of trapped mercury ions as a time device more stable than conventional atomic clocks. The areas of application of the ultraprecise and ultrastable time-measuring devices that tax the capacity of modern atomic clocks include radio astronomy and tests of relativity. The paper also discusses practical applications of ultraprecise clocks, such as navigation of space vehicles and pinpointing the exact position of ships and other objects on earth using the GPS.
NASA Technical Reports Server (NTRS)
Dutta, Soumitra
1988-01-01
A model for approximate spatial reasoning using fuzzy logic to represent the uncertainty in the environment is presented. Algorithms are developed which can be used to reason about spatial information expressed in the form of approximate linguistic descriptions similar to the kind of spatial information processed by humans. Particular attention is given to static spatial reasoning.
NASA Astrophysics Data System (ADS)
Shigeta, Takemi; Young, D. L.; Liu, Chein-Shan
2012-08-01
The mixed boundary value problem of the Laplace equation is considered. The method of fundamental solutions (MFS) approximates the exact solution to the Laplace equation by a linear combination of independent fundamental solutions with different source points. The accuracy of the numerical solution depends on the distribution of source points. In this paper, a weighted greedy QR decomposition (GQRD) is proposed to choose significant source points by introducing a weighting parameter. An index called an average degree of approximation is defined to show the efficiency of the proposed method. From numerical experiments, it is concluded that the numerical solution tends to be more accurate when the average degree of approximation is larger, and that the proposed method can yield more accurate solutions with a less number of source points than the conventional GQRD.
Adaptive approximation of higher order posterior statistics
Lee, Wonjung
2014-02-01
Filtering is an approach for incorporating observed data into time-evolving systems. Instead of a family of Dirac delta masses that is widely used in Monte Carlo methods, we here use the Wiener chaos expansion for the parametrization of the conditioned probability distribution to solve the nonlinear filtering problem. The Wiener chaos expansion is not the best method for uncertainty propagation without observations. Nevertheless, the projection of the system variables in a fixed polynomial basis spanning the probability space might be a competitive representation in the presence of relatively frequent observations because the Wiener chaos approach not only leads to an accurate and efficient prediction for short time uncertainty quantification, but it also allows to apply several data assimilation methods that can be used to yield a better approximate filtering solution. The aim of the present paper is to investigate this hypothesis. We answer in the affirmative for the (stochastic) Lorenz-63 system based on numerical simulations in which the uncertainty quantification method and the data assimilation method are adaptively selected by whether the dynamics is driven by Brownian motion and the near-Gaussianity of the measure to be updated, respectively.
A High-Order Accurate Parallel Solver for Maxwell's Equations on Overlapping Grids
Henshaw, W D
2005-09-23
A scheme for the solution of the time dependent Maxwell's equations on composite overlapping grids is described. The method uses high-order accurate approximations in space and time for Maxwell's equations written as a second-order vector wave equation. High-order accurate symmetric difference approximations to the generalized Laplace operator are constructed for curvilinear component grids. The modified equation approach is used to develop high-order accurate approximations that only use three time levels and have the same time-stepping restriction as the second-order scheme. Discrete boundary conditions for perfect electrical conductors and for material interfaces are developed and analyzed. The implementation is optimized for component grids that are Cartesian, resulting in a fast and efficient method. The solver runs on parallel machines with each component grid distributed across one or more processors. Numerical results in two- and three-dimensions are presented for the fourth-order accurate version of the method. These results demonstrate the accuracy and efficiency of the approach.
A New 2D-Transport, 1D-Diffusion Approximation of the Boltzmann Transport equation
Larsen, Edward
2013-06-17
The work performed in this project consisted of the derivation, implementation, and testing of a new, computationally advantageous approximation to the 3D Boltz- mann transport equation. The solution of the Boltzmann equation is the neutron flux in nuclear reactor cores and shields, but solving this equation is difficult and costly. The new “2D/1D” approximation takes advantage of a special geometric feature of typical 3D reactors to approximate the neutron transport physics in a specific (ax- ial) direction, but not in the other two (radial) directions. The resulting equation is much less expensive to solve computationally, and its solutions are expected to be sufficiently accurate for many practical problems. In this project we formulated the new equation, discretized it using standard methods, developed a stable itera- tion scheme for solving the equation, implemented the new numerical scheme in the MPACT code, and tested the method on several realistic problems. All the hoped- for features of this new approximation were seen. For large, difficult problems, the resulting 2D/1D solution is highly accurate, and is calculated about 100 times faster than a 3D discrete ordinates simulation.
Computer Experiments for Function Approximations
Chang, A; Izmailov, I; Rizzo, S; Wynter, S; Alexandrov, O; Tong, C
2007-10-15
This research project falls in the domain of response surface methodology, which seeks cost-effective ways to accurately fit an approximate function to experimental data. Modeling and computer simulation are essential tools in modern science and engineering. A computer simulation can be viewed as a function that receives input from a given parameter space and produces an output. Running the simulation repeatedly amounts to an equivalent number of function evaluations, and for complex models, such function evaluations can be very time-consuming. It is then of paramount importance to intelligently choose a relatively small set of sample points in the parameter space at which to evaluate the given function, and then use this information to construct a surrogate function that is close to the original function and takes little time to evaluate. This study was divided into two parts. The first part consisted of comparing four sampling methods and two function approximation methods in terms of efficiency and accuracy for simple test functions. The sampling methods used were Monte Carlo, Quasi-Random LP{sub {tau}}, Maximin Latin Hypercubes, and Orthogonal-Array-Based Latin Hypercubes. The function approximation methods utilized were Multivariate Adaptive Regression Splines (MARS) and Support Vector Machines (SVM). The second part of the study concerned adaptive sampling methods with a focus on creating useful sets of sample points specifically for monotonic functions, functions with a single minimum and functions with a bounded first derivative.
On the Applicability of High-frequency Approximations to Lilley's Equation
NASA Technical Reports Server (NTRS)
Wundrow, David W.; Khavaran, Abbas; Bridges, James (Technical Monitor)
2003-01-01
Three forms of the high-frequency asymptotic Green's function for Lilley's equation are reviewed and compared to the exact solution over wide range of Strouhal numbers. The asymmetric approximation, which applies to sources away form the jet axis, and the quasi-symmetric approximation, which is arrived at by making a near-axis source assumption, are both obtained for parallel round jets from a formal Fourier-transform solution. The ray-theory solution, which is the only high-frequency approximation that can be applied to more general mean flows, follows from a WKB ansatz and is shown to be closely related to the asymmetric approximation. The comparisons show that the best overall prediction of the exact Green's function is given by the asymmetric approximation which remains accurate down to a Strouhal number of 1/2. The close relationship between the asymmetric and ray-theory approximations suggests that the high-frequency asymptotic Green's function for more general mean flows would be similarly successful.
Development and application of accurate analytical models for single active electron potentials
NASA Astrophysics Data System (ADS)
Miller, Michelle; Jaron-Becker, Agnieszka; Becker, Andreas
2015-05-01
The single active electron (SAE) approximation is a theoretical model frequently employed to study scenarios in which inner-shell electrons may productively be treated as frozen spectators to a physical process of interest, and accurate analytical approximations for these potentials are sought as a useful simulation tool. Density function theory is often used to construct a SAE potential, requiring that a further approximation for the exchange correlation functional be enacted. In this study, we employ the Krieger, Li, and Iafrate (KLI) modification to the optimized-effective-potential (OEP) method to reduce the complexity of the problem to the straightforward solution of a system of linear equations through simple arguments regarding the behavior of the exchange-correlation potential in regions where a single orbital dominates. We employ this method for the solution of atomic and molecular potentials, and use the resultant curve to devise a systematic construction for highly accurate and useful analytical approximations for several systems. Supported by the U.S. Department of Energy (Grant No. DE-FG02-09ER16103), and the U.S. National Science Foundation (Graduate Research Fellowship, Grants No. PHY-1125844 and No. PHY-1068706).
Calculator Function Approximation.
ERIC Educational Resources Information Center
Schelin, Charles W.
1983-01-01
The general algorithm used in most hand calculators to approximate elementary functions is discussed. Comments on tabular function values and on computer function evaluation are given first; then the CORDIC (Coordinate Rotation Digital Computer) scheme is described. (MNS)
An approximation function for frequency constrained structural optimization
NASA Technical Reports Server (NTRS)
Canfield, R. A.
1989-01-01
The purpose is to examine a function for approximating natural frequency constraints during structural optimization. The nonlinearity of frequencies has posed a barrier to constructing approximations for frequency constraints of high enough quality to facilitate efficient solutions. A new function to represent frequency constraints, called the Rayleigh Quotient Approximation (RQA), is presented. Its ability to represent the actual frequency constraint results in stable convergence with effectively no move limits. The objective of the optimization problem is to minimize structural weight subject to some minimum (or maximum) allowable frequency and perhaps subject to other constraints such as stress, displacement, and gage size, as well. A reason for constraining natural frequencies during design might be to avoid potential resonant frequencies due to machinery or actuators on the structure. Another reason might be to satisy requirements of an aircraft or spacecraft's control law. Whatever the structure supports may be sensitive to a frequency band that must be avoided. Any of these situations or others may require the designer to insure the satisfaction of frequency constraints. A further motivation for considering accurate approximations of natural frequencies is that they are fundamental to dynamic response constraints.
Approximating a Giving Up Smoking Dynamic on Adolescent Nicotine Dependence in Fractional Order
2016-01-01
In this work, we consider giving up smoking dynamic on adolescent nicotine dependence. First, we use the Caputo derivative to develop the model in fractional order. Then we apply two different numerical methods to compute accurate approximate solutions of this new model in fractional order and compare their results. In order to do this, we consider the generalized Euler method (GEM) and multi-step generalized differential transform method (MSGDTM). We also show the unique positive solution for this model and present numerical results graphically. PMID:27105426
Approximating a Giving Up Smoking Dynamic on Adolescent Nicotine Dependence in Fractional Order.
Zeb, Anwar; Zaman, Gul; Erturk, Vedat Suat; Alzalg, Baha; Yousafzai, Faisal; Khan, Madad
2016-01-01
In this work, we consider giving up smoking dynamic on adolescent nicotine dependence. First, we use the Caputo derivative to develop the model in fractional order. Then we apply two different numerical methods to compute accurate approximate solutions of this new model in fractional order and compare their results. In order to do this, we consider the generalized Euler method (GEM) and multi-step generalized differential transform method (MSGDTM). We also show the unique positive solution for this model and present numerical results graphically. PMID:27105426
NASA Technical Reports Server (NTRS)
Dutta, Soumitra
1988-01-01
Much of human reasoning is approximate in nature. Formal models of reasoning traditionally try to be precise and reject the fuzziness of concepts in natural use and replace them with non-fuzzy scientific explicata by a process of precisiation. As an alternate to this approach, it has been suggested that rather than regard human reasoning processes as themselves approximating to some more refined and exact logical process that can be carried out with mathematical precision, the essence and power of human reasoning is in its capability to grasp and use inexact concepts directly. This view is supported by the widespread fuzziness of simple everyday terms (e.g., near tall) and the complexity of ordinary tasks (e.g., cleaning a room). Spatial reasoning is an area where humans consistently reason approximately with demonstrably good results. Consider the case of crossing a traffic intersection. We have only an approximate idea of the locations and speeds of various obstacles (e.g., persons and vehicles), but we nevertheless manage to cross such traffic intersections without any harm. The details of our mental processes which enable us to carry out such intricate tasks in such apparently simple manner are not well understood. However, it is that we try to incorporate such approximate reasoning techniques in our computer systems. Approximate spatial reasoning is very important for intelligent mobile agents (e.g., robots), specially for those operating in uncertain or unknown or dynamic domains.
Approximate kernel competitive learning.
Wu, Jian-Sheng; Zheng, Wei-Shi; Lai, Jian-Huang
2015-03-01
Kernel competitive learning has been successfully used to achieve robust clustering. However, kernel competitive learning (KCL) is not scalable for large scale data processing, because (1) it has to calculate and store the full kernel matrix that is too large to be calculated and kept in the memory and (2) it cannot be computed in parallel. In this paper we develop a framework of approximate kernel competitive learning for processing large scale dataset. The proposed framework consists of two parts. First, it derives an approximate kernel competitive learning (AKCL), which learns kernel competitive learning in a subspace via sampling. We provide solid theoretical analysis on why the proposed approximation modelling would work for kernel competitive learning, and furthermore, we show that the computational complexity of AKCL is largely reduced. Second, we propose a pseudo-parallelled approximate kernel competitive learning (PAKCL) based on a set-based kernel competitive learning strategy, which overcomes the obstacle of using parallel programming in kernel competitive learning and significantly accelerates the approximate kernel competitive learning for large scale clustering. The empirical evaluation on publicly available datasets shows that the proposed AKCL and PAKCL can perform comparably as KCL, with a large reduction on computational cost. Also, the proposed methods achieve more effective clustering performance in terms of clustering precision against related approximate clustering approaches. PMID:25528318
Leng, Wei; Ju, Lili; Gunzburger, Max; Price, Stephen; Ringler, Todd
2012-01-01
The numerical modeling of glacier and ice sheet evolution is a subject of growing interest, in part because of the potential for models to inform estimates of global sea level change. This paper focuses on the development of a numerical model that determines the velocity and pressure fields within an ice sheet. Our numerical model features a high-fidelity mathematical model involving the nonlinear Stokes system and combinations of no-sliding and sliding basal boundary conditions, high-order accurate finite element discretizations based on variable resolution grids, and highly scalable parallel solution strategies, all of which contribute to a numerical model that can achieve accurate velocity and pressure approximations in a highly efficient manner. We demonstrate the accuracy and efficiency of our model by analytical solution tests, established ice sheet benchmark experiments, and comparisons with other well-established ice sheet models.
On the Accuracy of Double Scattering Approximation for Atmospheric Polarization Computations
NASA Technical Reports Server (NTRS)
Korkin, Sergey V.; Lyapustin, Alexei I.; Marshak, Alexander L.
2011-01-01
Interpretation of multi-angle spectro-polarimetric data in remote sensing of atmospheric aerosols require fast and accurate methods of solving the vector radiative transfer equation (VRTE). The single and double scattering approximations could provide an analytical framework for the inversion algorithms and are relatively fast, however accuracy assessments of these approximations for the aerosol atmospheres in the atmospheric window channels have been missing. This paper provides such analysis for a vertically homogeneous aerosol atmosphere with weak and strong asymmetry of scattering. In both cases, the double scattering approximation gives a high accuracy result (relative error approximately 0.2%) only for the low optical path - 10(sup -2) As the error rapidly grows with optical thickness, a full VRTE solution is required for the practical remote sensing analysis. It is shown that the scattering anisotropy is not important at low optical thicknesses neither for reflected nor for transmitted polarization components of radiation.
Numerical study of superfluid turbulence in the self-induction approximation
Buttke, T.F.
1988-06-01
Two stable numerical methods are presented to solve the self-induction equation of vortex theory. These numerical methods are validated by comparison with known exact solutions. A new self-similar solution of the self-induction equation is presented and the approximate solutions are shown to converge to the exact solution for the self-similar solution. The numerical method is then generalized to solve the equations of motion of a superfluid vortex in the self-induction approximation where reconnection is allowed. A careful numerical study shows that the mesh spacing of the method must be restricted so that the approximate solutions are accurate. The line length density of a system of superfluid vortices is calculated. Contrary to earlier results it is found that the line length density produced does not scale as the velocity squared and therefore is not characteristic of homogeneous turbulence. It is concluded that the model equationn used is inadequate to describe superfluid turbulence. copyright 1988 Academic Press, Inc.
Covariant approximation averaging
NASA Astrophysics Data System (ADS)
Shintani, Eigo; Arthur, Rudy; Blum, Thomas; Izubuchi, Taku; Jung, Chulwoo; Lehner, Christoph
2015-06-01
We present a new class of statistical error reduction techniques for Monte Carlo simulations. Using covariant symmetries, we show that correlation functions can be constructed from inexpensive approximations without introducing any systematic bias in the final result. We introduce a new class of covariant approximation averaging techniques, known as all-mode averaging (AMA), in which the approximation takes account of contributions of all eigenmodes through the inverse of the Dirac operator computed from the conjugate gradient method with a relaxed stopping condition. In this paper we compare the performance and computational cost of our new method with traditional methods using correlation functions and masses of the pion, nucleon, and vector meson in Nf=2 +1 lattice QCD using domain-wall fermions. This comparison indicates that AMA significantly reduces statistical errors in Monte Carlo calculations over conventional methods for the same cost.
Fast approximate motif statistics.
Nicodème, P
2001-01-01
We present in this article a fast approximate method for computing the statistics of a number of non-self-overlapping matches of motifs in a random text in the nonuniform Bernoulli model. This method is well suited for protein motifs where the probability of self-overlap of motifs is small. For 96% of the PROSITE motifs, the expectations of occurrences of the motifs in a 7-million-amino-acids random database are computed by the approximate method with less than 1% error when compared with the exact method. Processing of the whole PROSITE takes about 30 seconds with the approximate method. We apply this new method to a comparison of the C. elegans and S. cerevisiae proteomes. PMID:11535175
The Guiding Center Approximation
NASA Astrophysics Data System (ADS)
Pedersen, Thomas Sunn
The guiding center approximation for charged particles in strong magnetic fields is introduced here. This approximation is very useful in situations where the charged particles are very well magnetized, such that the gyration (Larmor) radius is small compared to relevant length scales of the confinement device, and the gyration is fast relative to relevant timescales in an experiment. The basics of motion in a straight, uniform, static magnetic field are reviewed, and are used as a starting point for analyzing more complicated situations where more forces are present, as well as inhomogeneities in the magnetic field -- magnetic curvature as well as gradients in the magnetic field strength. The first and second adiabatic invariant are introduced, and slowly time-varying fields are also covered. As an example of the use of the guiding center approximation, the confinement concept of the cylindrical magnetic mirror is analyzed.
Monotone Boolean approximation
Hulme, B.L.
1982-12-01
This report presents a theory of approximation of arbitrary Boolean functions by simpler, monotone functions. Monotone increasing functions can be expressed without the use of complements. Nonconstant monotone increasing functions are important in their own right since they model a special class of systems known as coherent systems. It is shown here that when Boolean expressions for noncoherent systems become too large to treat exactly, then monotone approximations are easily defined. The algorithms proposed here not only provide simpler formulas but also produce best possible upper and lower monotone bounds for any Boolean function. This theory has practical application for the analysis of noncoherent fault trees and event tree sequences.
NASA Technical Reports Server (NTRS)
Graves, R. A., Jr.
1975-01-01
The previously obtained second-order-accurate partial implicitization numerical technique used in the solution of fluid dynamic problems was modified with little complication to achieve fourth-order accuracy. The Von Neumann stability analysis demonstrated the unconditional linear stability of the technique. The order of the truncation error was deduced from the Taylor series expansions of the linearized difference equations and was verified by numerical solutions to Burger's equation. For comparison, results were also obtained for Burger's equation using a second-order-accurate partial-implicitization scheme, as well as the fourth-order scheme of Kreiss.
NASA Astrophysics Data System (ADS)
Probe, A.; Macomber, B.; Kim, D.; Woollands, R.; Junkins, J.
2014-09-01
Modified Chebyshev Picard Iteration (MCPI) is a numerical method for approximating solutions of Ordinary Differential Equations (ODEs). MCPI uses Picard Iteration with Orthogonal Chebyshev Polynomial basis functions to recursively update approximate time histories of system states. Unlike stepping numerical integrators, such as explicit Runge-Kutta methods, MCPI approximates large segments of the trajectory by evaluating the forcing function at multiple nodes along the current approximation during each iteration. Importantly, the Picard sequence theoretically converges to the solution over large time intervals if the forces are continuous and once differentiable. Orthogonality of the basis functions and a vector-matrix formulation allow for low overhead cost, efficient iterations, and parallel evaluation of the forcing function. Despite these advantages MCPI only achieves a geometric rate of convergence. Depending on the quality of the starting approximation, MCPI sometimes requires more function evaluations than competing methods; for parallel applications, this is not a serious drawback, but may be for some serial applications. To improve efficiency, the Terminal Convergence Approximation Modified Chebyshev Picard Iteration (TCA-MCPI) was developed. TCA-MCPI takes advantage of the property that once moderate accuracy of the approximating trajectory has been achieved, the subsequent displacement of nodes asymptotically approaches zero. Applying judicious approximation methods to the force function at each node in the terminal convergence iterations is shown to dramatically reduce the computational cost to achieve accurate convergence. To illustrate this approach we consider high-order spherical-harmonic gravity for high accuracy orbital propagation. When combined with a starting approximation from the 2-body solution TCA-MCPI, is shown to outperform 2 current state-of-practice integration methods for astrodynamics. This paper presents the development of TCA
Time-dependent perturbation and the Born-Oppenheimer approximation
NASA Astrophysics Data System (ADS)
Jilcott, Steven Wayne, Jr.
2000-12-01
We discuss the physical problem of a molecule interacting with an electromagnetic field pulse and model the problem using a time-dependent perturbation of the Born- Oppenheimer approximation to the Schrödinger equation. Using previous results that develop asymptotic series solutions in the Born-Oppenheimer parameter ɛ, we derive a formal Dyson series expansion in the perturbation parameter μ, which is proportional to the electromagnetic field strength. We then prove that this series is asymptotically accurate in both parameters, provided that the Hamiltonian for the electrons has purely discrete spectrum. Under more general hypotheses, we show that the series is accurate to first order in μ, and that it is accurate to one higher order if we place conditions on the abruptness of the EM pulse. We also show how this series development provides a justification for the Franck-Condon factors in the case of a diatomic molecule.* *This work was supported by the Cunningham Research Fellowship provided by Virginia Tech.
Approximating Integrals Using Probability
ERIC Educational Resources Information Center
Maruszewski, Richard F., Jr.; Caudle, Kyle A.
2005-01-01
As part of a discussion on Monte Carlo methods, which outlines how to use probability expectations to approximate the value of a definite integral. The purpose of this paper is to elaborate on this technique and then to show several examples using visual basic as a programming tool. It is an interesting method because it combines two branches of…
Optimizing the Zeldovich approximation
NASA Technical Reports Server (NTRS)
Melott, Adrian L.; Pellman, Todd F.; Shandarin, Sergei F.
1994-01-01
We have recently learned that the Zeldovich approximation can be successfully used for a far wider range of gravitational instability scenarios than formerly proposed; we study here how to extend this range. In previous work (Coles, Melott and Shandarin 1993, hereafter CMS) we studied the accuracy of several analytic approximations to gravitational clustering in the mildly nonlinear regime. We found that what we called the 'truncated Zeldovich approximation' (TZA) was better than any other (except in one case the ordinary Zeldovich approximation) over a wide range from linear to mildly nonlinear (sigma approximately 3) regimes. TZA was specified by setting Fourier amplitudes equal to zero for all wavenumbers greater than k(sub nl), where k(sub nl) marks the transition to the nonlinear regime. Here, we study the cross correlation of generalized TZA with a group of n-body simulations for three shapes of window function: sharp k-truncation (as in CMS), a tophat in coordinate space, or a Gaussian. We also study the variation in the crosscorrelation as a function of initial truncation scale within each type. We find that k-truncation, which was so much better than other things tried in CMS, is the worst of these three window shapes. We find that a Gaussian window e(exp(-k(exp 2)/2k(exp 2, sub G))) applied to the initial Fourier amplitudes is the best choice. It produces a greatly improved crosscorrelation in those cases which most needed improvement, e.g. those with more small-scale power in the initial conditions. The optimum choice of kG for the Gaussian window is (a somewhat spectrum-dependent) 1 to 1.5 times k(sub nl). Although all three windows produce similar power spectra and density distribution functions after application of the Zeldovich approximation, the agreement of the phases of the Fourier components with the n-body simulation is better for the Gaussian window. We therefore ascribe the success of the best-choice Gaussian window to its superior treatment
NASA Technical Reports Server (NTRS)
Cheatwood, F. M.; Dejarnette, F. R.
1992-01-01
An approximate axisymmetric method has been developed which can reliably calculate nonequilibrium 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. This procedure is significantly faster than the parabolized Navier-Stokes and VSL solvers and would be useful in a preliminary design environment. Solutions have been generated for air flows over several analytic body shapes. Surface heat transfer and pressure predictions are comparable to VSL results. 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.
Difference equation state approximations for nonlinear hereditary control problems
NASA Technical Reports Server (NTRS)
Rosen, I. G.
1984-01-01
Discrete approximation schemes for the solution of nonlinear hereditary control problems are constructed. The methods involve approximation by a sequence of optimal control problems in which the original infinite dimensional state equation has been approximated by a finite dimensional discrete difference equation. Convergence of the state approximations is argued using linear semigroup theory and is then used to demonstrate that solutions to the approximating optimal control problems in some sense approximate solutions to the original control problem. Two schemes, one based upon piecewise constant approximation, and the other involving spline functions are discussed. Numerical results are presented, analyzed and used to compare the schemes to other available approximation methods for the solution of hereditary control problems. Previously announced in STAR as N83-33589
Difference equation state approximations for nonlinear hereditary control problems
NASA Technical Reports Server (NTRS)
Rosen, I. G.
1982-01-01
Discrete approximation schemes for the solution of nonlinear hereditary control problems are constructed. The methods involve approximation by a sequence of optimal control problems in which the original infinite dimensional state equation has been approximated by a finite dimensional discrete difference equation. Convergence of the state approximations is argued using linear semigroup theory and is then used to demonstrate that solutions to the approximating optimal control problems in some sense approximate solutions to the original control problem. Two schemes, one based upon piecewise constant approximation, and the other involving spline functions are discussed. Numerical results are presented, analyzed and used to compare the schemes to other available approximation methods for the solution of hereditary control problems.
NASA Technical Reports Server (NTRS)
Meador, W. E.; Weaver, W. R.
1980-01-01
Existing two-stream approximations to radiative transfer theory for particulate media are shown to be represented by identical forms of coupled differential equations if the intensity is replaced by integrals of the intensity over hemispheres. One set of solutions thus suffices for all methods and provides convenient analytical comparisons. The equations also suggest modifications of the standard techniques so as to duplicate exact solutions for thin atmospheres and thus permit accurate determinations of the effects of typical aerosol layers. Numerical results for the plane albedos of plane-parallel atmospheres are given for conventional and modified Eddington approximations, conventional and modified two-point quadrature schemes, the hemispheric-constant method and the delta-function method, all for comparison with accurate discrete-ordinate solutions. A new two-stream approximation is introduced that reduces to the modified Eddington approximation in the limit of isotropic phase functions and to the exact solution in the limit of extreme anisotropic scattering. Comparisons of plane albedos and transmittances show the new method to be generally superior over a wide range of atmospheric conditions (including cloud and aerosol layers), especially in the case of nonconservative scattering.
Weber's gravitational force as static weak field approximation
NASA Astrophysics Data System (ADS)
Tiandho, Yuant
2016-02-01
Weber's gravitational force (WGF) is one of gravitational model that can accommodate a non-static system because it depends not only on the distance but also on the velocity and the acceleration. Unlike Newton's law of gravitation, WGF can predict the anomalous of Mercury and gravitational bending of light near massive object very well. Then, some researchers use WGF as an alternative model of gravitation and propose a new mechanics theory namely the relational mechanics theory. However, currently we have known that the theory of general relativity which proposed by Einstein can explain gravity with very accurate. Through the static weak field approximation for the non-relativistic object, we also have known that the theory of general relativity will reduce to Newton's law of gravity. In this work, we expand the static weak field approximation that compatible with relativistic object and we obtain a force equation which correspond to WGF. Therefore, WGF is more precise than Newton's gravitational law. The static-weak gravitational field that we used is a solution of the Einstein's equation in the vacuum that satisfy the linear field approximation. The expression of WGF with ξ = 1 and satisfy the requirement of energy conservation are obtained after resolving the geodesic equation. By this result, we can conclude that WGF can be derived from the general relativity.
Approximate probability distributions of the master equation
NASA Astrophysics Data System (ADS)
Thomas, Philipp; Grima, Ramon
2015-07-01
Master equations are common descriptions of mesoscopic systems. Analytical solutions to these equations can rarely be obtained. We here derive an analytical approximation of the time-dependent probability distribution of the master equation using orthogonal polynomials. The solution is given in two alternative formulations: a series with continuous and a series with discrete support, both of which can be systematically truncated. While both approximations satisfy the system size expansion of the master equation, the continuous distribution approximations become increasingly negative and tend to oscillations with increasing truncation order. In contrast, the discrete approximations rapidly converge to the underlying non-Gaussian distributions. The theory is shown to lead to particularly simple analytical expressions for the probability distributions of molecule numbers in metabolic reactions and gene expression systems.
Accurate computation of Stokes flow driven by an open immersed interface
NASA Astrophysics Data System (ADS)
Li, Yi; Layton, Anita T.
2012-06-01
We present numerical methods for computing two-dimensional Stokes flow driven by forces singularly supported along an open, immersed interface. Two second-order accurate methods are developed: one for accurately evaluating boundary integral solutions at a point, and another for computing Stokes solution values on a rectangular mesh. We first describe a method for computing singular or nearly singular integrals, such as a double layer potential due to sources on a curve in the plane, evaluated at a point on or near the curve. To improve accuracy of the numerical quadrature, we add corrections for the errors arising from discretization, which are found by asymptotic analysis. When used to solve the Stokes equations with sources on an open, immersed interface, the method generates second-order approximations, for both the pressure and the velocity, and preserves the jumps in the solutions and their derivatives across the boundary. We then combine the method with a mesh-based solver to yield a hybrid method for computing Stokes solutions at N2 grid points on a rectangular grid. Numerical results are presented which exhibit second-order accuracy. To demonstrate the applicability of the method, we use the method to simulate fluid dynamics induced by the beating motion of a cilium. The method preserves the sharp jumps in the Stokes solution and their derivatives across the immersed boundary. Model results illustrate the distinct hydrodynamic effects generated by the effective stroke and by the recovery stroke of the ciliary beat cycle.
Chalasani, P.; Saias, I.; Jha, S.
1996-04-08
As increasingly large volumes of sophisticated options (called derivative securities) are traded in world financial markets, determining a fair price for these options has become an important and difficult computational problem. Many valuation codes use the binomial pricing model, in which the stock price is driven by a random walk. In this model, the value of an n-period option on a stock is the expected time-discounted value of the future cash flow on an n-period stock price path. Path-dependent options are particularly difficult to value since the future cash flow depends on the entire stock price path rather than on just the final stock price. Currently such options are approximately priced by Monte carlo methods with error bounds that hold only with high probability and which are reduced by increasing the number of simulation runs. In this paper the authors show that pricing an arbitrary path-dependent option is {number_sign}-P hard. They show that certain types f path-dependent options can be valued exactly in polynomial time. Asian options are path-dependent options that are particularly hard to price, and for these they design deterministic polynomial-time approximate algorithms. They show that the value of a perpetual American put option (which can be computed in constant time) is in many cases a good approximation to the value of an otherwise identical n-period American put option. In contrast to Monte Carlo methods, the algorithms have guaranteed error bounds that are polynormally small (and in some cases exponentially small) in the maturity n. For the error analysis they derive large-deviation results for random walks that may be of independent interest.
Beyond the Kirchhoff approximation
NASA Technical Reports Server (NTRS)
Rodriguez, Ernesto
1989-01-01
The three most successful models for describing scattering from random rough surfaces are the Kirchhoff approximation (KA), the small-perturbation method (SPM), and the two-scale-roughness (or composite roughness) surface-scattering (TSR) models. In this paper it is shown how these three models can be derived rigorously from one perturbation expansion based on the extinction theorem for scalar waves scattering from perfectly rigid surface. It is also shown how corrections to the KA proportional to the surface curvature and higher-order derivatives may be obtained. Using these results, the scattering cross section is derived for various surface models.
NASA Astrophysics Data System (ADS)
Ikot, Akpan N.; Hassanabadi, Hassan; Obong, Hillary Patrick; Mehraban, H.; Yazarloo, Bentol Hoda
2015-07-01
The effects of Coulomb-like tensor (CLT), Yukawa-like tensor (YLT) and generalized tensor (GLT) interactions are investigated in the Dirac theory with Schiöberg and Manning-Rosen potentials within the framework of spin and pseudospin symmetries using the Nikiforov-Uvarov method. The bound state energy spectra and the radial wave functions have been approximately obtained in the case of spin and pseudospin symmetries. We have also reported some numerical results and figures to show the effects these tensor interactions.
NASA Astrophysics Data System (ADS)
Enright, W. H.
2016-06-01
In recent years we have developed a class of reliable order p methods for the approximate solution of general systems of initial value problems (IVPs) and delay differential equations (DDEs). In the theoretical analysis of these methods we have identified several trade-offs that do arise and have to be addressed when applying these methods to problems that exhibit special structure. Similar trade-offs also arise when one is concerned with investigating other important properties of the solutions. We will give examples of such trade-offs that arise when investigating the sensitivities of the solutions, and when very accurate approximate solutions are required.
Wavelet Approximation in Data Assimilation
NASA Technical Reports Server (NTRS)
Tangborn, Andrew; Atlas, Robert (Technical Monitor)
2002-01-01
Estimation of the state of the atmosphere with the Kalman filter remains a distant goal because of high computational cost of evolving the error covariance for both linear and nonlinear systems. Wavelet approximation is presented here as a possible solution that efficiently compresses both global and local covariance information. We demonstrate the compression characteristics on the the error correlation field from a global two-dimensional chemical constituent assimilation, and implement an adaptive wavelet approximation scheme on the assimilation of the one-dimensional Burger's equation. In the former problem, we show that 99%, of the error correlation can be represented by just 3% of the wavelet coefficients, with good representation of localized features. In the Burger's equation assimilation, the discrete linearized equations (tangent linear model) and analysis covariance are projected onto a wavelet basis and truncated to just 6%, of the coefficients. A nearly optimal forecast is achieved and we show that errors due to truncation of the dynamics are no greater than the errors due to covariance truncation.
'LTE-diffusion approximation' for arc calculations
NASA Astrophysics Data System (ADS)
Lowke, J. J.; Tanaka, M.
2006-08-01
This paper proposes the use of the 'LTE-diffusion approximation' for predicting the properties of electric arcs. Under this approximation, local thermodynamic equilibrium (LTE) is assumed, with a particular mesh size near the electrodes chosen to be equal to the 'diffusion length', based on De/W, where De is the electron diffusion coefficient and W is the electron drift velocity. This approximation overcomes the problem that the equilibrium electrical conductivity in the arc near the electrodes is almost zero, which makes accurate calculations using LTE impossible in the limit of small mesh size, as then voltages would tend towards infinity. Use of the LTE-diffusion approximation for a 200 A arc with a thermionic cathode gives predictions of total arc voltage, electrode temperatures, arc temperatures and radial profiles of heat flux density and current density at the anode that are in approximate agreement with more accurate calculations which include an account of the diffusion of electric charges to the electrodes, and also with experimental results. Calculations, which include diffusion of charges, agree with experimental results of current and heat flux density as a function of radius if the Milne boundary condition is used at the anode surface rather than imposing zero charge density at the anode.
NASA Astrophysics Data System (ADS)
Pau, George Shu Heng; Shen, Chaopeng; Riley, William J.; Liu, Yaning
2016-02-01
The topography, and the biotic and abiotic parameters are typically upscaled to make watershed-scale hydrologic-biogeochemical models computationally tractable. However, upscaling procedure can produce biases when nonlinear interactions between different processes are not fully captured at coarse resolutions. Here we applied the Proper Orthogonal Decomposition Mapping Method (PODMM) to downscale the field solutions from a coarse (7 km) resolution grid to a fine (220 m) resolution grid. PODMM trains a reduced-order model (ROM) with coarse-resolution and fine-resolution solutions, here obtained using PAWS+CLM, a quasi-3-D watershed processes model that has been validated for many temperate watersheds. Subsequent fine-resolution solutions were approximated based only on coarse-resolution solutions and the ROM. The approximation errors were efficiently quantified using an error estimator. By jointly estimating correlated variables and temporally varying the ROM parameters, we further reduced the approximation errors by up to 20%. We also improved the method's robustness by constructing multiple ROMs using different set of variables, and selecting the best approximation based on the error estimator. The ROMs produced accurate downscaling of soil moisture, latent heat flux, and net primary production with O(1000) reduction in computational cost. The subgrid distributions were also nearly indistinguishable from the ones obtained using the fine-resolution model. Compared to coarse-resolution solutions, biases in upscaled ROM solutions were reduced by up to 80%. This method has the potential to help address the long-standing spatial scaling problem in hydrology and enable long-time integration, parameter estimation, and stochastic uncertainty analysis while accurately representing the heterogeneities.
Accuracy of the non-relativistic approximation for momentum diffusion
NASA Astrophysics Data System (ADS)
Liang, Shiuan-Ni; Lan, Boon Leong
2016-06-01
The accuracy of the non-relativistic approximation, which is calculated using the same parameter and the same initial ensemble of trajectories, to relativistic momentum diffusion at low speed is studied numerically for a prototypical nonlinear Hamiltonian system -the periodically delta-kicked particle. We find that if the initial ensemble is a non-localized semi-uniform ensemble, the non-relativistic approximation to the relativistic mean square momentum displacement is always accurate. However, if the initial ensemble is a localized Gaussian, the non-relativistic approximation may not always be accurate and the approximation can break down rapidly.
Practical aspects of spatially high accurate methods
NASA Technical Reports Server (NTRS)
Godfrey, Andrew G.; Mitchell, Curtis R.; Walters, Robert W.
1992-01-01
The computational qualities of high order spatially accurate methods for the finite volume solution of the Euler equations are presented. Two dimensional essentially non-oscillatory (ENO), k-exact, and 'dimension by dimension' ENO reconstruction operators are discussed and compared in terms of reconstruction and solution accuracy, computational cost and oscillatory behavior in supersonic flows with shocks. Inherent steady state convergence difficulties are demonstrated for adaptive stencil algorithms. An exact solution to the heat equation is used to determine reconstruction error, and the computational intensity is reflected in operation counts. Standard MUSCL differencing is included for comparison. Numerical experiments presented include the Ringleb flow for numerical accuracy and a shock reflection problem. A vortex-shock interaction demonstrates the ability of the ENO scheme to excel in simulating unsteady high-frequency flow physics.
Approximate inference on planar graphs using loop calculus and belief progagation
Chertkov, Michael; Gomez, Vicenc; Kappen, Hilbert
2009-01-01
We introduce novel results for approximate inference on planar graphical models using the loop calculus framework. The loop calculus (Chertkov and Chernyak, 2006b) allows to express the exact partition function Z of a graphical model as a finite sum of terms that can be evaluated once the belief propagation (BP) solution is known. In general, full summation over all correction terms is intractable. We develop an algorithm for the approach presented in Chertkov et al. (2008) which represents an efficient truncation scheme on planar graphs and a new representation of the series in terms of Pfaffians of matrices. We analyze in detail both the loop series and the Pfaffian series for models with binary variables and pairwise interactions, and show that the first term of the Pfaffian series can provide very accurate approximations. The algorithm outperforms previous truncation schemes of the loop series and is competitive with other state-of-the-art methods for approximate inference.
Fast solvers for finite difference approximations for the Stokes and Navier-Stokes equations
Shin, D.
1992-01-01
The authors consider several methods for solving the linear equations arising from finite difference discretizations of the Stokes equations. The pressure equation method presented here for the first time, apparently, and the method, presented by Bramble and Pasciak, are shown to have computational effort that grows slowly with the number of grid points. The methods work with second-order accurate discretizations. Computational results are shown for both the Stokes and incompressible Navier-Stokes at low Reynolds number. The inf-sup conditions resulting from three finite difference approximations of the Stokes equations are proven. These conditions are used to prove that the Schur complement Q[sub h] of the linear system generated by each of these approximations is bounded uniformly away from zero. For the pressure equation method, this guarantees that the conjugate gradient method applied to Q[sub h] converges in a finite number of iterations which is independent of mesh size. The fact that Q[sub h] is bounded below is used to prove convergence estimates for the solutions generated by these finite difference approximations. One of the estimates is for a staggered grid and the estimate of the scheme shows that both the pressure and the velocity parts of the solution are second-order accurate. Iterative methods are compared by the use of the regularized central differencing introduced by Strikwerda. Several finite difference approximations of the Stokes equations by the SOR method are compared and the excellence of the approximations by the regularized central differencing over the other finite difference approximation is mentioned. This difference gives rise to a linear equation with a matrix which is slightly non-symmetric. The convergence of the typical steepest descent method and conjugate gradient method, which is almost as same as the typical conjugate gradient method, applied to slightly non-symmetric positive definite matrices are proven.
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.
Approximations for crossing two nearby spin resonances
NASA Astrophysics Data System (ADS)
Ranjbar, V. H.
2015-01-01
Solutions to the Thomas-Bargmann-Michel-Telegdi spin equation for spin 1 /2 particles have to date been confined to the single-resonance crossing. However, in reality, most cases of interest concern the overlapping of several resonances. While there have been several serious studies of this problem, a good analytical solution or even an approximation has eluded the community. We show that this system can be transformed into a Hill-like equation. In this representation, we show that, while the single-resonance crossing represents the solution to the parabolic cylinder equation, the overlapping case becomes a parametric type of resonance.
Shear viscosity in the postquasistatic approximation
Peralta, C.; Rosales, L.; Rodriguez-Mueller, B.; Barreto, W.
2010-05-15
We apply the postquasistatic approximation, an iterative method for the evolution of self-gravitating spheres of matter, to study the evolution of anisotropic nonadiabatic radiating and dissipative distributions in general relativity. Dissipation is described by viscosity and free-streaming radiation, assuming an equation of state to model anisotropy induced by the shear viscosity. We match the interior solution, in noncomoving coordinates, with the Vaidya exterior solution. Two simple models are presented, based on the Schwarzschild and Tolman VI solutions, in the nonadiabatic and adiabatic limit. In both cases, the eventual collapse or expansion of the distribution is mainly controlled by the anisotropy induced by the viscosity.
Approximate analysis of electromagnetically coupled microstrip dipoles
NASA Astrophysics Data System (ADS)
Kominami, M.; Yakuwa, N.; Kusaka, H.
1990-10-01
A new dynamic analysis model for analyzing electromagnetically coupled (EMC) microstrip dipoles is proposed. The formulation is based on an approximate treatment of the dielectric substrate. Calculations of the equivalent impedance of two different EMC dipole configurations are compared with measured data and full-wave solutions. The agreement is very good.
NNLOPS accurate associated HW production
NASA Astrophysics Data System (ADS)
Astill, William; Bizon, Wojciech; Re, Emanuele; Zanderighi, Giulia
2016-06-01
We present a next-to-next-to-leading order accurate description of associated HW production consistently matched to a parton shower. The method is based on reweighting events obtained with the HW plus one jet NLO accurate calculation implemented in POWHEG, extended with the MiNLO procedure, to reproduce NNLO accurate Born distributions. Since the Born kinematics is more complex than the cases treated before, we use a parametrization of the Collins-Soper angles to reduce the number of variables required for the reweighting. We present phenomenological results at 13 TeV, with cuts suggested by the Higgs Cross section Working Group.
Countably QC-Approximating Posets
Mao, Xuxin; Xu, Luoshan
2014-01-01
As a generalization of countably C-approximating posets, the concept of countably QC-approximating posets is introduced. With the countably QC-approximating property, some characterizations of generalized completely distributive lattices and generalized countably approximating posets are given. The main results are as follows: (1) a complete lattice is generalized completely distributive if and only if it is countably QC-approximating and weakly generalized countably approximating; (2) a poset L having countably directed joins is generalized countably approximating if and only if the lattice σc(L)op of all σ-Scott-closed subsets of L is weakly generalized countably approximating. PMID:25165730
How to accurately bypass damage
Broyde, Suse; Patel, Dinshaw J.
2016-01-01
Ultraviolet radiation can cause cancer through DNA damage — specifically, by linking adjacent thymine bases. Crystal structures show how the enzyme DNA polymerase η accurately bypasses such lesions, offering protection. PMID:20577203
Accurate Evaluation of Quantum Integrals
NASA Technical Reports Server (NTRS)
Galant, David C.; Goorvitch, D.
1994-01-01
Combining an appropriate finite difference method with Richardson's extrapolation results in a simple, highly accurate numerical method for solving a Schr\\"{o}dinger's equation. Important results are that error estimates are provided, and that one can extrapolate expectation values rather than the wavefunctions to obtain highly accurate expectation values. We discuss the eigenvalues, the error growth in repeated Richardson's extrapolation, and show that the expectation values calculated on a crude mesh can be extrapolated to obtain expectation values of high accuracy.
Kojima, H; Yamada, A; Okazaki, S
2015-05-01
The intramolecular proton transfer reaction of malonaldehyde in neon solvent has been investigated by mixed quantum-classical molecular dynamics (QCMD) calculations and fully classical molecular dynamics (FCMD) calculations. Comparing these calculated results with those for malonaldehyde in water reported in Part I [A. Yamada, H. Kojima, and S. Okazaki, J. Chem. Phys. 141, 084509 (2014)], the solvent dependence of the reaction rate, the reaction mechanism involved, and the quantum effect therein have been investigated. With FCMD, the reaction rate in weakly interacting neon is lower than that in strongly interacting water. However, with QCMD, the order of the reaction rates is reversed. To investigate the mechanisms in detail, the reactions were categorized into three mechanisms: tunneling, thermal activation, and barrier vanishing. Then, the quantum and solvent effects were analyzed from the viewpoint of the reaction mechanism focusing on the shape of potential energy curve and its fluctuations. The higher reaction rate that was found for neon in QCMD compared with that found for water solvent arises from the tunneling reactions because of the nearly symmetric double-well shape of the potential curve in neon. The thermal activation and barrier vanishing reactions were also accelerated by the zero-point energy. The number of reactions based on these two mechanisms in water was greater than that in neon in both QCMD and FCMD because these reactions are dominated by the strength of solute-solvent interactions. PMID:25956108
NASA Astrophysics Data System (ADS)
Kojima, H.; Yamada, A.; Okazaki, S.
2015-05-01
The intramolecular proton transfer reaction of malonaldehyde in neon solvent has been investigated by mixed quantum-classical molecular dynamics (QCMD) calculations and fully classical molecular dynamics (FCMD) calculations. Comparing these calculated results with those for malonaldehyde in water reported in Part I [A. Yamada, H. Kojima, and S. Okazaki, J. Chem. Phys. 141, 084509 (2014)], the solvent dependence of the reaction rate, the reaction mechanism involved, and the quantum effect therein have been investigated. With FCMD, the reaction rate in weakly interacting neon is lower than that in strongly interacting water. However, with QCMD, the order of the reaction rates is reversed. To investigate the mechanisms in detail, the reactions were categorized into three mechanisms: tunneling, thermal activation, and barrier vanishing. Then, the quantum and solvent effects were analyzed from the viewpoint of the reaction mechanism focusing on the shape of potential energy curve and its fluctuations. The higher reaction rate that was found for neon in QCMD compared with that found for water solvent arises from the tunneling reactions because of the nearly symmetric double-well shape of the potential curve in neon. The thermal activation and barrier vanishing reactions were also accelerated by the zero-point energy. The number of reactions based on these two mechanisms in water was greater than that in neon in both QCMD and FCMD because these reactions are dominated by the strength of solute-solvent interactions.
Second-order accurate nonoscillatory schemes for scalar conservation laws
NASA Technical Reports Server (NTRS)
Huynh, Hung T.
1989-01-01
Explicit finite difference schemes for the computation of weak solutions of nonlinear scalar conservation laws is presented and analyzed. These schemes are uniformly second-order accurate and nonoscillatory in the sense that the number of extrema of the discrete solution is not increasing in time.
Approximation methods in gravitational-radiation theory
NASA Technical Reports Server (NTRS)
Will, C. M.
1986-01-01
The observation of gravitational-radiation damping in the binary pulsar PSR 1913 + 16 and the ongoing experimental search for gravitational waves of extraterrestrial origin have made the theory of gravitational radiation an active branch of classical general relativity. In calculations of gravitational radiation, approximation methods play a crucial role. Recent developments are summarized in two areas in which approximations are important: (a) the quadrupole approxiamtion, which determines the energy flux and the radiation reaction forces in weak-field, slow-motion, source-within-the-near-zone systems such as the binary pulsar; and (b) the normal modes of oscillation of black holes, where the Wentzel-Kramers-Brillouin approximation gives accurate estimates of the complex frequencies of the modes.
Numerical study of superfluid turbulence in the self-induction approximation
Buttke, T.F.
1986-08-01
A numerical method is presented that determines the evolution of a superfluid vortex, and it is used to calculate some properties of turbulent superfluid vortex systems. A stable finite difference method is derived for solving the self-induction equation in vortex dynamics. The self-induction equation is equivalent to a non-linear Schroedinger equation. A numerical method is designed that preserves some of the invariants of the Schroedinger equations; this leads to stability. The method is written in terms of the tangent field of the vortex lines. It is shown that the approximate solutions exist for all times and for all initial conditions. This is the first stable numerical method for solving the self-induction equation. A new exact self-similar solution of the self-induction equation is found and this solution is used along with the previously known soliton solutions to validate the method. The method is second order accurate in space and time. The equations which govern the evolution of a superfluid vortex are viewed as a perturbation of the self-induction equation and a method is developed for determining the evolution of a superfluid vortex. The reconnection ansatz of Schwarz is incorporated into the method. The method is validated by comparison with known exact solutions and by a careful convergence check for cases where the exact solution is not known. If the spatial step in the approximation is chosen too large the approximate solutions became inaccurate. Following the work of Schwarz experiments on vortex systems are performed to find the line length density of a turbulent superfluid. The turbulence produced is not homogeneous, and line length densities and critical properties disagree with earlier results. 45 refs., 42 figs., 3 tabs.
Accurate basis set truncation for wavefunction embedding
NASA Astrophysics Data System (ADS)
Barnes, Taylor A.; Goodpaster, Jason D.; Manby, Frederick R.; Miller, Thomas F.
2013-07-01
Density functional theory (DFT) provides a formally exact framework for performing embedded subsystem electronic structure calculations, including DFT-in-DFT and wavefunction theory-in-DFT descriptions. In the interest of efficiency, it is desirable to truncate the atomic orbital basis set in which the subsystem calculation is performed, thus avoiding high-order scaling with respect to the size of the MO virtual space. In this study, we extend a recently introduced projection-based embedding method [F. R. Manby, M. Stella, J. D. Goodpaster, and T. F. Miller III, J. Chem. Theory Comput. 8, 2564 (2012)], 10.1021/ct300544e to allow for the systematic and accurate truncation of the embedded subsystem basis set. The approach is applied to both covalently and non-covalently bound test cases, including water clusters and polypeptide chains, and it is demonstrated that errors associated with basis set truncation are controllable to well within chemical accuracy. Furthermore, we show that this approach allows for switching between accurate projection-based embedding and DFT embedding with approximate kinetic energy (KE) functionals; in this sense, the approach provides a means of systematically improving upon the use of approximate KE functionals in DFT embedding.
Mean-Field Approximation to the Hydrophobic Hydration in the Liquid-Vapor Interface of Water.
Abe, Kiharu; Sumi, Tomonari; Koga, Kenichiro
2016-03-01
A mean-field approximation to the solvation of nonpolar solutes in the liquid-vapor interface of aqueous solutions is proposed. It is first remarked with a numerical illustration that the solvation of a methane-like solute in bulk liquid water is accurately described by the mean-field theory of liquids, the main idea of which is that the probability (Pcav) of finding a cavity in the solvent that can accommodate the solute molecule and the attractive interaction energy (uatt) that the solute would feel if it is inserted in such a cavity are both functions of the solvent density alone. It is then assumed that the basic idea is still valid in the liquid-vapor interface, but Pcav and uatt are separately functions of different coarse-grained local densities, not functions of a common local density. Validity of the assumptions is confirmed for the solvation of the methane-like particle in the interface of model water at temperatures between 253 and 613 K. With the mean-field approximation extended to the inhomogeneous system the local solubility profiles across the interface at various temperatures are calculated from Pcav and uatt obtained at a single temperature. The predicted profiles are in excellent agreement with those obtained by the direct calculation of the excess chemical potential over an interfacial region where the solvent local density varies most rapidly. PMID:26595441
An approximate viscous shock layer approach to calculating hypersonic flows about blunt-nosed bodies
NASA Technical Reports Server (NTRS)
Cheatwood, F. MCN.; Dejarnette, F. R.
1991-01-01
An approximate axisymmetric method has been 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. Since the method is fully viscous, the problems associated with coupling 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. Surface heat transfer and pressure predictions are comparable to both VSL results and 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.
Numerical approximation of head and flux covariances in three dimensions using mixed finite elements
NASA Astrophysics Data System (ADS)
James, Andrew I.; Graham, Wendy D.
A numerical method is developed for accurately approximating head and flux covariances and cross-covariances in finite two- and three-dimensional domains using the mixed finite element method. The method is useful for determining head and flux covariances for non-stationary flow fields, for example those induced by injection or extraction wells, impermeable subsurface barriers, or non-stationary hydraulic conductivity fields. Because the numerical approximations to the flux covariances are obtained directly from the solution to the coupled problem rather than having to differentiate head covariances, the approximations are in general more accurate than those obtained from conventional finite difference or finite element methods. Results for uniform flow example problems are consistent with results from previously published finite domain analyses and demonstrate that head variances and covariances are quite sensitive to boundary conditions and the size of the bounded domain. Flux variances and covariances are less sensitive to boundary conditions and domain size. Results comparing approximations from lower-order Raviart-Thomas-Nedelec and higher order Brezzi-Douglas-Marini [9] finite element spaces indicate that higher order element space improve the estimate of the flux covariances, but do not significantly affect the estimate of the head covariances.
ERIC Educational Resources Information Center
Bellera, Carine A.; Julien, Marilyse; Hanley, James A.
2010-01-01
The Wilcoxon statistics are usually taught as nonparametric alternatives for the 1- and 2-sample Student-"t" statistics in situations where the data appear to arise from non-normal distributions, or where sample sizes are so small that we cannot check whether they do. In the past, critical values, based on exact tail areas, were presented in…
Approximate but accurate quantum dynamics from the Mori formalism: I. Nonequilibrium dynamics.
Montoya-Castillo, Andrés; Reichman, David R
2016-05-14
We present a formalism that explicitly unifies the commonly used Nakajima-Zwanzig approach for reduced density matrix dynamics with the more versatile Mori theory in the context of nonequilibrium dynamics. Employing a Dyson-type expansion to circumvent the difficulty of projected dynamics, we obtain a self-consistent equation for the memory kernel which requires only knowledge of normally evolved auxiliary kernels. To illustrate the properties of the current approach, we focus on the spin-boson model and limit our attention to the use of a simple and inexpensive quasi-classical dynamics, given by the Ehrenfest method, for the calculation of the auxiliary kernels. For the first time, we provide a detailed analysis of the dependence of the properties of the memory kernels obtained via different projection operators, namely, the thermal (Redfield-type) and population based (NIBA-type) projection operators. We further elucidate the conditions that lead to short-lived memory kernels and the regions of parameter space to which this program is best suited. Via a thorough analysis of the different closures available for the auxiliary kernels and the convergence properties of the self-consistently extracted memory kernel, we identify the mechanisms whereby the current approach leads to a significant improvement over the direct usage of standard semi- and quasi-classical dynamics. PMID:27179468
Approximate but accurate quantum dynamics from the Mori formalism: I. Nonequilibrium dynamics
NASA Astrophysics Data System (ADS)
Montoya-Castillo, Andrés; Reichman, David R.
2016-05-01
We present a formalism that explicitly unifies the commonly used Nakajima-Zwanzig approach for reduced density matrix dynamics with the more versatile Mori theory in the context of nonequilibrium dynamics. Employing a Dyson-type expansion to circumvent the difficulty of projected dynamics, we obtain a self-consistent equation for the memory kernel which requires only knowledge of normally evolved auxiliary kernels. To illustrate the properties of the current approach, we focus on the spin-boson model and limit our attention to the use of a simple and inexpensive quasi-classical dynamics, given by the Ehrenfest method, for the calculation of the auxiliary kernels. For the first time, we provide a detailed analysis of the dependence of the properties of the memory kernels obtained via different projection operators, namely, the thermal (Redfield-type) and population based (NIBA-type) projection operators. We further elucidate the conditions that lead to short-lived memory kernels and the regions of parameter space to which this program is best suited. Via a thorough analysis of the different closures available for the auxiliary kernels and the convergence properties of the self-consistently extracted memory kernel, we identify the mechanisms whereby the current approach leads to a significant improvement over the direct usage of standard semi- and quasi-classical dynamics.
High-order-harmonic spectra from atoms in intense laser fields: Exact versus approximate methods
NASA Astrophysics Data System (ADS)
Pugliese, S. N.; Simonsen, A. S.; Førre, M.; Hansen, J. P.
2015-08-01
We compare harmonic spectra from hydrogen based on the numerical solution of the time-dependent Schrödinger equation and three approximate models: (i) the strong field approximation (SFA), (ii) the Coulomb-Volkov modified strong field approximation (CVA), and (iii) the strong field approximation with the stationary phase approximation applied to the momentum integrals (SPSFA). At laser intensities in the range of (1 -3 ) ×1014W/cm 2 we find good agreement when comparing the SFA and CVA with exact results. In general the CVA displays an overall better agreement with ab initio results, which reflects the role of the Coulomb field in the ionization as well as in the recombination process. Furthermore, it is found that the widely used SPSFA breaks down for low-order harmonic generation; i.e., the approximation turns out to be accurate only in the outer part of the harmonic plateau region as well as in the cutoff region. We trace this deficiency to the singularity of the SPSFA associated with short trajectories, i.e., short return times. When removing these, we obtain a version of the SPSFA which works rather well for the entire harmonic spectrum.
Approximate methods for building extreme mass ratio inspiral waveforms
NASA Astrophysics Data System (ADS)
Hughes, Scott
2007-04-01
The ``extreme mass ratio inspiral'' (or EMRI) problem has captured much attention in recent years. This is due to its relevance at describing a potentially important gravitational-wave source, and to the elegance of techniques which are being developed to solve it. A complete, self-consistent solution to this problem will require detailed knowledge of the self-interaction of a small body orbiting a Kerr black hole, taken (at least in part) to second order. This challenge will consume much time and effort. In the meantime, there is an exigent need for waveforms which, though not correct in all details, are sufficiently reliable that they can be used to understand how to measure these waves with space-based gravitational-wave antennae. I will describe in this talk results from a crude but surprisingly effective ``kludge'' approximation. The kludge produces waves which match well with available strong-field results, requiring only a fraction of the computational effort. Motivated by how the kludge operates, I will argue that a good medium between the kludge and the full solution is a ``hybrid'' approach to waveform generation. This hybrid combines the best features of both time and frequency domain approaches to black hole perturbation theory, using them to make EMRI waves that are as accurate as is possible without incorporating self-force information.
Quantum Calisthenics: Gaussians, The Path Integral and Guided Numerical Approximations
Weinstein, Marvin; /SLAC
2009-02-12
It is apparent to anyone who thinks about it that, to a large degree, the basic concepts of Newtonian physics are quite intuitive, but quantum mechanics is not. My purpose in this talk is to introduce you to a new, much more intuitive way to understand how quantum mechanics works. I begin with an incredibly easy way to derive the time evolution of a Gaussian wave-packet for the case free and harmonic motion without any need to know the eigenstates of the Hamiltonian. This discussion is completely analytic and I will later use it to relate the solution for the behavior of the Gaussian packet to the Feynman path-integral and stationary phase approximation. It will be clear that using the information about the evolution of the Gaussian in this way goes far beyond what the stationary phase approximation tells us. Next, I introduce the concept of the bucket brigade approach to dealing with problems that cannot be handled totally analytically. This approach combines the intuition obtained in the initial discussion, as well as the intuition obtained from the path-integral, with simple numerical tools. My goal is to show that, for any specific process, there is a simple Hilbert space interpretation of the stationary phase approximation. I will then argue that, from the point of view of numerical approximations, the trajectory obtained from my generalization of the stationary phase approximation specifies that subspace of the full Hilbert space that is needed to compute the time evolution of the particular state under the full Hamiltonian. The prescription I will give is totally non-perturbative and we will see, by the grace of Maple animations computed for the case of the anharmonic oscillator Hamiltonian, that this approach allows surprisingly accurate computations to be performed with very little work. I think of this approach to the path-integral as defining what I call a guided numerical approximation scheme. After the discussion of the anharmonic oscillator I will
Approximate convective heating equations for hypersonic flows
NASA Technical Reports Server (NTRS)
Zoby, E. V.; Moss, J. N.; Sutton, K.
1979-01-01
Laminar and turbulent heating-rate equations appropriate for engineering predictions of the convective heating rates about blunt reentry spacecraft at hypersonic conditions are developed. The approximate methods are applicable to both nonreacting and reacting gas mixtures for either constant or variable-entropy edge conditions. A procedure which accounts for variable-entropy effects and is not based on mass balancing is presented. Results of the approximate heating methods are in good agreement with existing experimental results as well as boundary-layer and viscous-shock-layer solutions.
Fast and Provably Accurate Bilateral Filtering
NASA Astrophysics Data System (ADS)
Chaudhury, Kunal N.; Dabhade, Swapnil D.
2016-06-01
The bilateral filter is a non-linear filter that uses a range filter along with a spatial filter to perform edge-preserving smoothing of images. A direct computation of the bilateral filter requires $O(S)$ operations per pixel, where $S$ is the size of the support of the spatial filter. In this paper, we present a fast and provably accurate algorithm for approximating the bilateral filter when the range kernel is Gaussian. In particular, for box and Gaussian spatial filters, the proposed algorithm can cut down the complexity to $O(1)$ per pixel for any arbitrary $S$. The algorithm has a simple implementation involving $N+1$ spatial filterings, where $N$ is the approximation order. We give a detailed analysis of the filtering accuracy that can be achieved by the proposed approximation in relation to the target bilateral filter. This allows us to to estimate the order $N$ required to obtain a given accuracy. We also present comprehensive numerical results to demonstrate that the proposed algorithm is competitive with state-of-the-art methods in terms of speed and accuracy.
Fast and Provably Accurate Bilateral Filtering.
Chaudhury, Kunal N; Dabhade, Swapnil D
2016-06-01
The bilateral filter is a non-linear filter that uses a range filter along with a spatial filter to perform edge-preserving smoothing of images. A direct computation of the bilateral filter requires O(S) operations per pixel, where S is the size of the support of the spatial filter. In this paper, we present a fast and provably accurate algorithm for approximating the bilateral filter when the range kernel is Gaussian. In particular, for box and Gaussian spatial filters, the proposed algorithm can cut down the complexity to O(1) per pixel for any arbitrary S . The algorithm has a simple implementation involving N+1 spatial filterings, where N is the approximation order. We give a detailed analysis of the filtering accuracy that can be achieved by the proposed approximation in relation to the target bilateral filter. This allows us to estimate the order N required to obtain a given accuracy. We also present comprehensive numerical results to demonstrate that the proposed algorithm is competitive with the state-of-the-art methods in terms of speed and accuracy. PMID:27093722
Accurate Molecular Polarizabilities Based on Continuum Electrostatics
Truchon, Jean-François; Nicholls, Anthony; Iftimie, Radu I.; Roux, Benoît; Bayly, Christopher I.
2013-01-01
A novel approach for representing the intramolecular polarizability as a continuum dielectric is introduced to account for molecular electronic polarization. It is shown, using a finite-difference solution to the Poisson equation, that the Electronic Polarization from Internal Continuum (EPIC) model yields accurate gas-phase molecular polarizability tensors for a test set of 98 challenging molecules composed of heteroaromatics, alkanes and diatomics. The electronic polarization originates from a high intramolecular dielectric that produces polarizabilities consistent with B3LYP/aug-cc-pVTZ and experimental values when surrounded by vacuum dielectric. In contrast to other approaches to model electronic polarization, this simple model avoids the polarizability catastrophe and accurately calculates molecular anisotropy with the use of very few fitted parameters and without resorting to auxiliary sites or anisotropic atomic centers. On average, the unsigned error in the average polarizability and anisotropy compared to B3LYP are 2% and 5%, respectively. The correlation between the polarizability components from B3LYP and this approach lead to a R2 of 0.990 and a slope of 0.999. Even the F2 anisotropy, shown to be a difficult case for existing polarizability models, can be reproduced within 2% error. In addition to providing new parameters for a rapid method directly applicable to the calculation of polarizabilities, this work extends the widely used Poisson equation to areas where accurate molecular polarizabilities matter. PMID:23646034
NASA Astrophysics Data System (ADS)
Sultan, Cornel
2010-10-01
The design of vector second-order linear systems for accurate proportional damping approximation is addressed. For this purpose an error system is defined using the difference between the generalized coordinates of the non-proportionally damped system and its proportionally damped approximation in modal space. The accuracy of the approximation is characterized using the energy gain of the error system and the design problem is formulated as selecting parameters of the non-proportionally damped system to ensure that this gain is sufficiently small. An efficient algorithm that combines linear matrix inequalities and simultaneous perturbation stochastic approximation is developed to solve the problem and examples of its application to tensegrity structures design are presented.
An asymptotic homogenized neutron diffusion approximation. II. Numerical comparisons
Trahan, T. J.; Larsen, E. W.
2012-07-01
In a companion paper, a monoenergetic, homogenized, anisotropic diffusion equation is derived asymptotically for large, 3-D, multiplying systems with a periodic lattice structure [1]. In the present paper, this approximation is briefly compared to several other well known diffusion approximations. Although the derivation is different, the asymptotic diffusion approximation matches that proposed by Deniz and Gelbard, and is closely related to those proposed by Benoist. The focus of this paper, however, is a numerical comparison of the various methods for simple reactor analysis problems in 1-D. The comparisons show that the asymptotic diffusion approximation provides a more accurate estimate of the eigenvalue than the Benoist diffusion approximations. However, the Benoist diffusion approximations and the asymptotic diffusion approximation provide very similar estimates of the neutron flux. The asymptotic method and the Benoist methods both outperform the standard homogenized diffusion approximation, with flux weighted cross sections. (authors)
An approximation theory for the identification of linear thermoelastic systems
NASA Technical Reports Server (NTRS)
Rosen, I. G.; Su, Chien-Hua Frank
1990-01-01
An abstract approximation framework and convergence theory for the identification of thermoelastic systems is developed. Starting from an abstract operator formulation consisting of a coupled second order hyperbolic equation of elasticity and first order parabolic equation for heat conduction, well-posedness is established using linear semigroup theory in Hilbert space, and a class of parameter estimation problems is then defined involving mild solutions. The approximation framework is based upon generic Galerkin approximation of the mild solutions, and convergence of solutions of the resulting sequence of approximating finite dimensional parameter identification problems to a solution of the original infinite dimensional inverse problem is established using approximation results for operator semigroups. An example involving the basic equations of one dimensional linear thermoelasticity and a linear spline based scheme are discussed. Numerical results indicate how the approach might be used in a study of damping mechanisms in flexible structures.
Rapidly converging series approximation to Kepler's equation
NASA Astrophysics Data System (ADS)
Peters, R. D.
1984-08-01
A power series solution in eccentricity e and normalized mean anomaly f has been developed for elliptic orbits. Expansion through the fourth order yields approximate errors about an order of magnitude smaller than the corresponding Lagrange series. For large e, a particular algorithm is shown to be superior to published initializers for Newton iteration solutions. The normalized variable f varies between zero and one on each of two separately defined intervals: 0 to x = (pi/2-e) and x to pi. The expansion coefficients are polynomials based on a one-time evaluation of sine and cosine terms in f.
B-term approximation using tree-structured Haar transforms
NASA Astrophysics Data System (ADS)
Ho, Hsin-Han; Egiazarian, Karen O.; Mitra, Sanjit K.
2009-02-01
We present a heuristic solution for B-term approximation using Tree-Structured Haar (TSH) transforms. Our solution consists of two main stages: best basis selection and greedy approximation. In addition, when approximating the same signal with different B constraint or error metric, our solution also provides the flexibility of having less overall running time at expense of more storage space. We adopted lattice structure to index basis vectors, so that one index value can fully specify a basis vector. Based on the concept of fast computation of TSH transform by butterfly network, we also developed an algorithm for directly deriving butterfly parameters and incorporated it into our solution. Results show that, when the error metric is normalized l1-norm and normalized l2-norm, our solution has comparable (sometimes better) approximation quality with prior data synopsis algorithms.
NASA Astrophysics Data System (ADS)
Liu, Y.; Li, T.; Zhu, C.; Zhang, R.; Wu, Y.
2015-12-01
Three-dimensional (3-D) electromagnetic (EM) forward modelling and inversion continues to be an important issue for the correct interpretation of EM data.To this end,approximate solutions have been developed that allow the construction of relatively fast forward modelling and inversion schemes.We have developed an improved quasi-linear approximation which is more appropriate in solving the linear equation for greatly shortening calculation time.We achieved this by using green's function properties.Then we introduced the improved quasi-linear approximation to spectral induced polarization (SIP) to tackle the problem of the resolution and the efficiency.The localized quasi-linear (LQL) approximation theory is appropriate for multisource array-type surveys assuming that the normal field is slowly varying within the inhomogeneity domain.However,the normal field of attenuates severely which dose not satisfy the assumption of the LQL approximation.As a consenquence,the imaginary part is not accurate when LQL approximation is adopted for the simulation.The improved quasi-linear approximation provide a new approach with the same resolution of QL approximation and much less calculation time.We have also constructed three-dimensional SIP forward modeling based on improved quasi-linear approximation method.It only takes 0.8s for forward modeling when inhomogeneity domain is divided into 2000 blocks.Beyond that, we have introduced the Cole-Cole model to the algorithm and complete the three-dimensional complex resistivity conjugate gradient inversion with parameter restraint.The model trial results show that this method can obtain good inversion results in physical parameters such as zero frequency resistivity, polarization.The results demonstrate the stability and the efficiency of the improved quasi-linear approximation and the method may be a practical solution for3-D EM forward modelling and inversion of SIP.
NASA Technical Reports Server (NTRS)
Robertson, J. S.; Siegman, W. L.; Jacobson, M. J.
1989-01-01
There is substantial interest in the analytical and numerical modeling of low-frequency, long-range atmospheric acoustic propagation. Ray-based models, because of frequency limitations, do not always give an adequate prediction of quantities such as sound pressure or intensity levels. However, the parabolic approximation method, widely used in ocean acoustics, and often more accurate than ray models for lower frequencies of interest, can be applied to acoustic propagation in the atmosphere. Modifications of an existing implicit finite-difference implementation for computing solutions to the parabolic approximation are discussed. A locally-reacting boundary is used together with a one-parameter impedance model. Intensity calculations are performed for a number of flow resistivity values in both quiescent and windy atmospheres. Variations in the value of this parameter are shown to have substantial effects on the spatial variation of the acoustic signal.
Pre-equilibrium approximation in chemical and photophysical kinetics
NASA Astrophysics Data System (ADS)
Rae, Margaret; Berberan-Santos, Mário N.
2002-07-01
For most mechanisms of chemical reactions and molecular photophysical processes the time evolution of the concentration of the intervening species cannot be obtained analytically. The pre-equilibrium approximation is one of several useful approximation methods that allow the derivation of explicit solutions and simplify numerical solutions. In this work, a general view of the pre-equilibrium approximation is presented, along with the respective analytical solution. It is also shown that the kinetic behavior of systems subject to pre-equilibration can be obtained by the application of perturbation theory. Several photophysical systems are discussed, including excimer formation, thermally activated delayed fluorescence, and external-heavy atom quenching of luminescence.
Approximate analytical solutions for excitation and propagation in cardiac tissue.
Greene, D'Artagnan; Shiferaw, Yohannes
2015-04-01
It is well known that a variety of cardiac arrhythmias are initiated by a focal excitation in heart tissue. At the single cell level these currents are typically induced by intracellular processes such as spontaneous calcium release (SCR). However, it is not understood how the size and morphology of these focal excitations are related to the electrophysiological properties of cardiac cells. In this paper a detailed physiologically based ionic model is analyzed by projecting the excitation dynamics to a reduced one-dimensional parameter space. Based on this analysis we show that the inward current required for an excitation to occur is largely dictated by the voltage dependence of the inward rectifier potassium current (I(K1)), and is insensitive to the detailed properties of the sodium current. We derive an analytical expression relating the size of a stimulus and the critical current required to induce a propagating action potential (AP), and argue that this relationship determines the necessary number of cells that must undergo SCR in order to induce ectopic activity in cardiac tissue. Finally, we show that, once a focal excitation begins to propagate, its propagation characteristics, such as the conduction velocity and the critical radius for propagation, are largely determined by the sodium and gap junction currents with a substantially lesser effect due to repolarizing potassium currents. These results reveal the relationship between ion channel properties and important tissue scale processes such as excitation and propagation. PMID:25974539
Approximate analytical solutions for excitation and propagation in cardiac tissue
NASA Astrophysics Data System (ADS)
Greene, D'Artagnan; Shiferaw, Yohannes
2015-04-01
It is well known that a variety of cardiac arrhythmias are initiated by a focal excitation in heart tissue. At the single cell level these currents are typically induced by intracellular processes such as spontaneous calcium release (SCR). However, it is not understood how the size and morphology of these focal excitations are related to the electrophysiological properties of cardiac cells. In this paper a detailed physiologically based ionic model is analyzed by projecting the excitation dynamics to a reduced one-dimensional parameter space. Based on this analysis we show that the inward current required for an excitation to occur is largely dictated by the voltage dependence of the inward rectifier potassium current (IK 1) , and is insensitive to the detailed properties of the sodium current. We derive an analytical expression relating the size of a stimulus and the critical current required to induce a propagating action potential (AP), and argue that this relationship determines the necessary number of cells that must undergo SCR in order to induce ectopic activity in cardiac tissue. Finally, we show that, once a focal excitation begins to propagate, its propagation characteristics, such as the conduction velocity and the critical radius for propagation, are largely determined by the sodium and gap junction currents with a substantially lesser effect due to repolarizing potassium currents. These results reveal the relationship between ion channel properties and important tissue scale processes such as excitation and propagation.
Revisiting Twomey's approximation for peak supersaturation
NASA Astrophysics Data System (ADS)
Shipway, B. J.
2015-04-01
Twomey's seminal 1959 paper provided lower and upper bound approximations to the estimation of peak supersaturation within an updraft and thus provides the first closed expression for the number of nucleated cloud droplets. The form of this approximation is simple, but provides a surprisingly good estimate and has subsequently been employed in more sophisticated treatments of nucleation parametrization. In the current paper, we revisit the lower bound approximation of Twomey and make a small adjustment that can be used to obtain a more accurate calculation of peak supersaturation under all potential aerosol loadings and thermodynamic conditions. In order to make full use of this improved approximation, the underlying integro-differential equation for supersaturation evolution and the condition for calculating peak supersaturation are examined. A simple rearrangement of the algebra allows for an expression to be written down that can then be solved with a single lookup table with only one independent variable for an underlying lognormal aerosol population. While multimodal aerosol with N different dispersion characteristics requires 2N+1 inputs to calculate the activation fraction, only N of these one-dimensional lookup tables are needed. No additional information is required in the lookup table to deal with additional chemical, physical or thermodynamic properties. The resulting implementation provides a relatively simple, yet computationally cheap, physically based parametrization of droplet nucleation for use in climate and Numerical Weather Prediction models.
Exponential Approximations Using Fourier Series Partial Sums
NASA Technical Reports Server (NTRS)
Banerjee, Nana S.; Geer, James F.
1997-01-01
The problem of accurately reconstructing a piece-wise smooth, 2(pi)-periodic function f and its first few derivatives, given only a truncated Fourier series representation of f, is studied and solved. The reconstruction process is divided into two steps. In the first step, the first 2N + 1 Fourier coefficients of f are used to approximate the locations and magnitudes of the discontinuities in f and its first M derivatives. This is accomplished by first finding initial estimates of these quantities based on certain properties of Gibbs phenomenon, and then refining these estimates by fitting the asymptotic form of the Fourier coefficients to the given coefficients using a least-squares approach. It is conjectured that the locations of the singularities are approximated to within O(N(sup -M-2), and the associated jump of the k(sup th) derivative of f is approximated to within O(N(sup -M-l+k), as N approaches infinity, and the method is robust. These estimates are then used with a class of singular basis functions, which have certain 'built-in' singularities, to construct a new sequence of approximations to f. Each of these new approximations is the sum of a piecewise smooth function and a new Fourier series partial sum. When N is proportional to M, it is shown that these new approximations, and their derivatives, converge exponentially in the maximum norm to f, and its corresponding derivatives, except in the union of a finite number of small open intervals containing the points of singularity of f. The total measure of these intervals decreases exponentially to zero as M approaches infinity. The technique is illustrated with several examples.
Exact PDF equations and closure approximations for advective-reactive transport
Venturi, D.; Tartakovsky, Daniel M.; Tartakovsky, Alexandre M.; Karniadakis, George E.
2013-06-01
Mathematical models of advection–reaction phenomena rely on advective flow velocity and (bio) chemical reaction rates that are notoriously random. By using functional integral methods, we derive exact evolution equations for the probability density function (PDF) of the state variables of the advection–reaction system in the presence of random transport velocity and random reaction rates with rather arbitrary distributions. These PDF equations are solved analytically for transport with deterministic flow velocity and a linear reaction rate represented mathematically by a heterog eneous and strongly-correlated random field. Our analytical solution is then used to investigate the accuracy and robustness of the recently proposed large-eddy diffusivity (LED) closure approximation [1]. We find that the solution to the LED-based PDF equation, which is exact for uncorrelated reaction rates, is accurate even in the presence of strong correlations and it provides an upper bound of predictive uncertainty.
HNC solution for the central force model for liquid water
NASA Astrophysics Data System (ADS)
Thuraisingham, Ranjit A.; Friedman, Harold L.
1983-05-01
Beginning with the central force model for water introduced by Lemberg, Stillinger, and Rahman, the HNC approximation method has been used to calculate the atom-atom pair correlation functions of a state of liquid water. Although a stable and accurate solution to the HNC equation for the model is obtained using the Rossky-Dale algorithm, the structure and thermodynamics agree only crudely with the published molecular dynamics results for the same model and the same N, V, T state.
Numerical Stability and Convergence of Approximate Methods for Conservation Laws
NASA Astrophysics Data System (ADS)
Galkin, V. A.
We present the new approach to background of approximate methods convergence based on functional solutions theory for conservation laws. The applications to physical kinetics, gas and fluid dynamics are considered.
Approximate inverse preconditioning of iterative methods for nonsymmetric linear systems
Benzi, M.; Tuma, M.
1996-12-31
A method for computing an incomplete factorization of the inverse of a nonsymmetric matrix A is presented. The resulting factorized sparse approximate inverse is used as a preconditioner in the iterative solution of Ax = b by Krylov subspace methods.
A test of the adhesion approximation for gravitational clustering
NASA Technical Reports Server (NTRS)
Melott, Adrian L.; Shandarin, Sergei; Weinberg, David H.
1993-01-01
We quantitatively compare a particle implementation of the adhesion approximation to fully non-linear, numerical 'N-body' simulations. Our primary tool, cross-correlation of N-body simulations with the adhesion approximation, indicates good agreement, better than that found by the same test performed with the Zel-dovich approximation (hereafter ZA). However, the cross-correlation is not as good as that of the truncated Zel-dovich approximation (TZA), obtained by applying the Zel'dovich approximation after smoothing the initial density field with a Gaussian filter. We confirm that the adhesion approximation produces an excessively filamentary distribution. Relative to the N-body results, we also find that: (a) the power spectrum obtained from the adhesion approximation is more accurate than that from ZA or TZA, (b) the error in the phase angle of Fourier components is worse than that from TZA, and (c) the mass distribution function is more accurate than that from ZA or TZA. It appears that adhesion performs well statistically, but that TZA is more accurate dynamically, in the sense of moving mass to the right place.
Dombrovsky, Leonid; Randrianalisoa, Jaona; Baillis, Dominique
2006-01-01
A modified two-flux approximation is suggested for calculating the hemispherical transmittance and reflectance of a refracting, absorbing, and scattering medium in the case of collimated irradiation of the sample along the normal to the interface. The Fresnel reflection is taken into account in this approach. It is shown that the new approximation is rather accurate for the model transport scattering function. For an arbitrary scattering medium, the error of the modified two-flux approximation is estimated by comparison with the exact numerical calculations for the Henyey-Greenstein scattering function in a wide range of albedos and optical thicknesses. Possible applications of the derived analytical solution to identification problems are discussed. PMID:16478064
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.
Cavity approximation for graphical models.
Rizzo, T; Wemmenhove, B; Kappen, H J
2007-07-01
We reformulate the cavity approximation (CA), a class of algorithms recently introduced for improving the Bethe approximation estimates of marginals in graphical models. In our formulation, which allows for the treatment of multivalued variables, a further generalization to factor graphs with arbitrary order of interaction factors is explicitly carried out, and a message passing algorithm that implements the first order correction to the Bethe approximation is described. Furthermore, we investigate an implementation of the CA for pairwise interactions. In all cases considered we could confirm that CA[k] with increasing k provides a sequence of approximations of markedly increasing precision. Furthermore, in some cases we could also confirm the general expectation that the approximation of order k , whose computational complexity is O(N(k+1)) has an error that scales as 1/N(k+1) with the size of the system. We discuss the relation between this approach and some recent developments in the field. PMID:17677405
Approximate circuits for increased reliability
Hamlet, Jason R.; Mayo, Jackson R.
2015-08-18
Embodiments of the invention describe a Boolean circuit having a voter circuit and a plurality of approximate circuits each based, at least in part, on a reference circuit. The approximate circuits are each to generate one or more output signals based on values of received input signals. The voter circuit is to receive the one or more output signals generated by each of the approximate circuits, and is to output one or more signals corresponding to a majority value of the received signals. At least some of the approximate circuits are to generate an output value different than the reference circuit for one or more input signal values; however, for each possible input signal value, the majority values of the one or more output signals generated by the approximate circuits and received by the voter circuit correspond to output signal result values of the reference circuit.
Approximate circuits for increased reliability
Hamlet, Jason R.; Mayo, Jackson R.
2015-12-22
Embodiments of the invention describe a Boolean circuit having a voter circuit and a plurality of approximate circuits each based, at least in part, on a reference circuit. The approximate circuits are each to generate one or more output signals based on values of received input signals. The voter circuit is to receive the one or more output signals generated by each of the approximate circuits, and is to output one or more signals corresponding to a majority value of the received signals. At least some of the approximate circuits are to generate an output value different than the reference circuit for one or more input signal values; however, for each possible input signal value, the majority values of the one or more output signals generated by the approximate circuits and received by the voter circuit correspond to output signal result values of the reference circuit.
NASA Astrophysics Data System (ADS)
Ganapol, Barry D.; Townsend, Lawrence W.; Wilson, John W.
1989-03-01
Nontrivial benchmark solutions are developed for the galactic ion transport (GIT) equations in the straight-ahead approximation. These equations are used to predict potential radiation hazards in the upper atmosphere and in space. Two levels of difficulty are considered: (1) energy independent, and (2) spatially independent. The analysis emphasizes analytical methods never before applied to the GIT equations. Most of the representations derived have been numerically implemented and compared to more approximate calculations. Accurate ion fluxes are obtained (3 to 5 digits) for nontrivial sources. For monoenergetic beams, both accurate doses and fluxes are found. The benchmarks presented are useful in assessing the accuracy of transport algorithms designed to accommodate more complex radiation protection problems. In addition, these solutions can provide fast and accurate assessments of relatively simple shield configurations.
NASA Technical Reports Server (NTRS)
Ganapol, Barry D.; Townsend, Lawrence W.; Wilson, John W.
1989-01-01
Nontrivial benchmark solutions are developed for the galactic ion transport (GIT) equations in the straight-ahead approximation. These equations are used to predict potential radiation hazards in the upper atmosphere and in space. Two levels of difficulty are considered: (1) energy independent, and (2) spatially independent. The analysis emphasizes analytical methods never before applied to the GIT equations. Most of the representations derived have been numerically implemented and compared to more approximate calculations. Accurate ion fluxes are obtained (3 to 5 digits) for nontrivial sources. For monoenergetic beams, both accurate doses and fluxes are found. The benchmarks presented are useful in assessing the accuracy of transport algorithms designed to accommodate more complex radiation protection problems. In addition, these solutions can provide fast and accurate assessments of relatively simple shield configurations.
Perturbation approximation for orbits in axially symmetric funnels
NASA Astrophysics Data System (ADS)
Nauenberg, Michael
2014-11-01
A perturbation method that can be traced back to Isaac Newton is applied to obtain approximate analytic solutions for objects sliding in axially symmetric funnels in near circular orbits. Some experimental observations are presented for balls rolling in inverted cones with different opening angles, and in a funnel with a hyperbolic surface that approximately simulates the gravitational force.
Numerical solution of a diffusion problem by exponentially fitted finite difference methods.
D'Ambrosio, Raffaele; Paternoster, Beatrice
2014-01-01
This paper is focused on the accurate and efficient solution of partial differential differential equations modelling a diffusion problem by means of exponentially fitted finite difference numerical methods. After constructing and analysing special purpose finite differences for the approximation of second order partial derivatives, we employed them in the numerical solution of a diffusion equation with mixed boundary conditions. Numerical experiments reveal that a special purpose integration, both in space and in time, is more accurate and efficient than that gained by employing a general purpose solver. PMID:26034665
An approximate methods approach to probabilistic structural analysis
NASA Technical Reports Server (NTRS)
Mcclung, R. C.; Millwater, H. R.; Wu, Y.-T.; Thacker, B. H.; Burnside, O. H.
1989-01-01
A major research and technology program in Probabilistic Structural Analysis Methods (PSAM) is currently being sponsored by the NASA Lewis Research Center with Southwest Research Institute as the prime contractor. This program is motivated by the need to accurately predict structural response in an environment where the loadings, the material properties, and even the structure may be considered random. The heart of PSAM is a software package which combines advanced structural analysis codes with a fast probability integration (FPI) algorithm for the efficient calculation of stochastic structural response. The basic idea of PAAM is simple: make an approximate calculation of system response, including calculation of the associated probabilities, with minimal computation time and cost, based on a simplified representation of the geometry, loads, and material. The deterministic solution resulting should give a reasonable and realistic description of performance-limiting system responses, although some error will be inevitable. If the simple model has correctly captured the basic mechanics of the system, however, including the proper functional dependence of stress, frequency, etc. on design parameters, then the response sensitivities calculated may be of significantly higher accuracy.
Approximate registration of point clouds with large scale differences
NASA Astrophysics Data System (ADS)
Novak, D.; Schindler, K.
2013-10-01
3D reconstruction of objects is a basic task in many fields, including surveying, engineering, entertainment and cultural heritage. The task is nowadays often accomplished with a laser scanner, which produces dense point clouds, but lacks accurate colour information, and lacks per-point accuracy measures. An obvious solution is to combine laser scanning with photogrammetric recording. In that context, the problem arises to register the two datasets, which feature large scale, translation and rotation differences. The absence of approximate registration parameters (3D translation, 3D rotation and scale) precludes the use of fine-registration methods such as ICP. Here, we present a method to register realistic photogrammetric and laser point clouds in a fully automated fashion. The proposed method decomposes the registration into a sequence of simpler steps: first, two rotation angles are determined by finding dominant surface normal directions, then the remaining parameters are found with RANSAC followed by ICP and scale refinement. These two steps are carried out at low resolution, before computing a precise final registration at higher resolution.
On approximating hereditary dynamics by systems of ordinary differential equations
NASA Technical Reports Server (NTRS)
Cliff, E. M.; Burns, J. A.
1978-01-01
The paper deals with methods of obtaining approximate solutions to linear retarded functional differential equations (hereditary systems). The basic notion is to project the infinite dimensional space of initial functions for the hereditary system onto a finite dimensional subspace. Within this framework, two particular schemes are discussed. The first uses well-known piecewise constant approximations, while the second is a new method based on piecewise linear approximating functions. Numerical results are given.
NASA Astrophysics Data System (ADS)
Kong, Dali; Zhang, Keke; Schubert, Gerald
2015-12-01
In an important paper, Roberts (1963b) studied the hydrostatic equilibrium of an isolated, self-gravitating, rapidly rotating polytropic gaseous body based on a controversial assumption/approximation that all (outer and internal) equidensity surfaces are in the shape of oblate spheroids whose eccentricities are a function of the equatorial radius and whose axes of symmetry are parallel to the rotation axis. We compute the three-dimensional, finite-element, fully self-consistent, continuous solution for a rapidly rotating polytropic gaseous body with Jupiter-like parameters without making any prior assumptions about its outer shape and internal structure. Upon partially relaxing the Roberts' approximation by assuming that only the outer equidensity surface is in the shape of an oblate spheroid, we also compute a finite-element solution with the same parameters without making any prior assumptions about its internal structure. It is found that all equidensity surfaces of the fully self-consistent solution differ only slightly from the oblate spheroidal shape. It is also found that the characteristic difference between the fully self-consistent solution and the outer-spheroidal-shape solution is insignificantly small. Our results suggest that the Roberts' assumption of spheroidal equidensity surfaces represents a reasonably accurate approximation for rotating polytropic gaseous bodies with Jupiter-like parameters. The numerical accuracy of our finite-element solution is checked by an exact analytic solution based on the Green's function using the spheroidal wave function. The three different solutions in non-spherical geometries - the fully self-consistent numerical solution, the numerical solution with the outer spheroidal shape and the exact analytical solution - can also serve as a useful benchmark for other solutions based on different numerical methods.
Accurate thickness measurement of graphene
NASA Astrophysics Data System (ADS)
Shearer, Cameron J.; Slattery, Ashley D.; Stapleton, Andrew J.; Shapter, Joseph G.; Gibson, Christopher T.
2016-03-01
Graphene has emerged as a material with a vast variety of applications. The electronic, optical and mechanical properties of graphene are strongly influenced by the number of layers present in a sample. As a result, the dimensional characterization of graphene films is crucial, especially with the continued development of new synthesis methods and applications. A number of techniques exist to determine the thickness of graphene films including optical contrast, Raman scattering and scanning probe microscopy techniques. Atomic force microscopy (AFM), in particular, is used extensively since it provides three-dimensional images that enable the measurement of the lateral dimensions of graphene films as well as the thickness, and by extension the number of layers present. However, in the literature AFM has proven to be inaccurate with a wide range of measured values for single layer graphene thickness reported (between 0.4 and 1.7 nm). This discrepancy has been attributed to tip-surface interactions, image feedback settings and surface chemistry. In this work, we use standard and carbon nanotube modified AFM probes and a relatively new AFM imaging mode known as PeakForce tapping mode to establish a protocol that will allow users to accurately determine the thickness of graphene films. In particular, the error in measuring the first layer is reduced from 0.1-1.3 nm to 0.1-0.3 nm. Furthermore, in the process we establish that the graphene-substrate adsorbate layer and imaging force, in particular the pressure the tip exerts on the surface, are crucial components in the accurate measurement of graphene using AFM. These findings can be applied to other 2D materials.
Approximate Genealogies Under Genetic Hitchhiking
Pfaffelhuber, P.; Haubold, B.; Wakolbinger, A.
2006-01-01
The rapid fixation of an advantageous allele leads to a reduction in linked neutral variation around the target of selection. The genealogy at a neutral locus in such a selective sweep can be simulated by first generating a random path of the advantageous allele's frequency and then a structured coalescent in this background. Usually the frequency path is approximated by a logistic growth curve. We discuss an alternative method that approximates the genealogy by a random binary splitting tree, a so-called Yule tree that does not require first constructing a frequency path. Compared to the coalescent in a logistic background, this method gives a slightly better approximation for identity by descent during the selective phase and a much better approximation for the number of lineages that stem from the founder of the selective sweep. In applications such as the approximation of the distribution of Tajima's D, the two approximation methods perform equally well. For relevant parameter ranges, the Yule approximation is faster. PMID:17182733
The measurement of psychological literacy: a first approximation
Roberts, Lynne D.; Heritage, Brody; Gasson, Natalie
2015-01-01
Psychological literacy, the ability to apply psychological knowledge to personal, family, occupational, community and societal challenges, is promoted as the primary outcome of an undergraduate education in psychology. As the concept of psychological literacy becomes increasingly adopted as the core business of undergraduate psychology training courses world-wide, there is urgent need for the construct to be accurately measured so that student and institutional level progress can be assessed and monitored. Key to the measurement of psychological literacy is determining the underlying factor-structure of psychological literacy. In this paper we provide a first approximation of the measurement of psychological literacy by identifying and evaluating self-report measures for psychological literacy. Multi-item and single-item self-report measures of each of the proposed nine dimensions of psychological literacy were completed by two samples (N = 218 and N = 381) of undergraduate psychology students at an Australian university. Single and multi-item measures of each dimension were weakly to moderately correlated. Exploratory and confirmatory factor analyses of multi-item measures indicated a higher order three factor solution best represented the construct of psychological literacy. The three factors were reflective processes, generic graduate attributes, and psychology as a helping profession. For the measurement of psychological literacy to progress there is a need to further develop self-report measures and to identify/develop and evaluate objective measures of psychological literacy. Further approximations of the measurement of psychological literacy remain an imperative, given the construct's ties to measuring institutional efficacy in teaching psychology to an undergraduate audience. PMID:25741300
MEMS accelerometers in accurate mount positioning systems
NASA Astrophysics Data System (ADS)
Mészáros, László; Pál, András.; Jaskó, Attila
2014-07-01
In order to attain precise, accurate and stateless positioning of telescope mounts we apply microelectromechanical accelerometer systems (also known as MEMS accelerometers). In common practice, feedback from the mount position is provided by electronic, optical or magneto-mechanical systems or via real-time astrometric solution based on the acquired images. Hence, MEMS-based systems are completely independent from these mechanisms. Our goal is to investigate the advantages and challenges of applying such devices and to reach the sub-arcminute range { that is well smaller than the field-of-view of conventional imaging telescope systems. We present how this sub-arcminute accuracy can be achieved with very cheap MEMS sensors. Basically, these sensors yield raw output within an accuracy of a few degrees. We show what kind of calibration procedures could exploit spherical and cylindrical constraints between accelerometer output channels in order to achieve the previously mentioned accuracy level. We also demonstrate how can our implementation be inserted in a telescope control system. Although this attainable precision is less than both the resolution of telescope mount drive mechanics and the accuracy of astrometric solutions, the independent nature of attitude determination could significantly increase the reliability of autonomous or remotely operated astronomical observations.
Approximating Markov Chains: What and why
Pincus, S.
1996-06-01
Much of the current study of dynamical systems is focused on geometry (e.g., chaos and bifurcations) and ergodic theory. Yet dynamical systems were originally motivated by an attempt to {open_quote}{open_quote}solve,{close_quote}{close_quote} or at least understand, a discrete-time analogue of differential equations. As such, numerical, analytical solution techniques for dynamical systems would seem desirable. We discuss an approach that provides such techniques, the approximation of dynamical systems by suitable finite state Markov Chains. Steady state distributions for these Markov Chains, a straightforward calculation, will converge to the true dynamical system steady state distribution, with appropriate limit theorems indicated. Thus (i) approximation by a computable, linear map holds the promise of vastly faster steady state solutions for nonlinear, multidimensional differential equations; (ii) the solution procedure is unaffected by the presence or absence of a probability density function for the {ital attractor}, entirely skirting singularity, fractal/multifractal, and renormalization considerations. The theoretical machinery underpinning this development also implies that under very general conditions, steady state measures are weakly continuous with control parameter evolution. This means that even though a system may change periodicity, or become chaotic in its limiting behavior, such statistical parameters as the mean, standard deviation, and tail probabilities change continuously, not abruptly with system evolution. {copyright} {ital 1996 American Institute of Physics.}
Bauer, Sebastian; Mathias, Gerald; Tavan, Paul
2014-03-14
We present a reaction field (RF) method which accurately solves the Poisson equation for proteins embedded in dielectric solvent continua at a computational effort comparable to that of an electrostatics calculation with polarizable molecular mechanics (MM) force fields. The method combines an approach originally suggested by Egwolf and Tavan [J. Chem. Phys. 118, 2039 (2003)] with concepts generalizing the Born solution [Z. Phys. 1, 45 (1920)] for a solvated ion. First, we derive an exact representation according to which the sources of the RF potential and energy are inducible atomic anti-polarization densities and atomic shielding charge distributions. Modeling these atomic densities by Gaussians leads to an approximate representation. Here, the strengths of the Gaussian shielding charge distributions are directly given in terms of the static partial charges as defined, e.g., by standard MM force fields for the various atom types, whereas the strengths of the Gaussian anti-polarization densities are calculated by a self-consistency iteration. The atomic volumes are also described by Gaussians. To account for covalently overlapping atoms, their effective volumes are calculated by another self-consistency procedure, which guarantees that the dielectric function ε(r) is close to one everywhere inside the protein. The Gaussian widths σ{sub i} of the atoms i are parameters of the RF approximation. The remarkable accuracy of the method is demonstrated by comparison with Kirkwood's analytical solution for a spherical protein [J. Chem. Phys. 2, 351 (1934)] and with computationally expensive grid-based numerical solutions for simple model systems in dielectric continua including a di-peptide (Ac-Ala-NHMe) as modeled by a standard MM force field. The latter example shows how weakly the RF conformational free energy landscape depends on the parameters σ{sub i}. A summarizing discussion highlights the achievements of the new theory and of its approximate solution
NASA Astrophysics Data System (ADS)
Bauer, Sebastian; Mathias, Gerald; Tavan, Paul
2014-03-01
We present a reaction field (RF) method which accurately solves the Poisson equation for proteins embedded in dielectric solvent continua at a computational effort comparable to that of an electrostatics calculation with polarizable molecular mechanics (MM) force fields. The method combines an approach originally suggested by Egwolf and Tavan [J. Chem. Phys. 118, 2039 (2003)] with concepts generalizing the Born solution [Z. Phys. 1, 45 (1920)] for a solvated ion. First, we derive an exact representation according to which the sources of the RF potential and energy are inducible atomic anti-polarization densities and atomic shielding charge distributions. Modeling these atomic densities by Gaussians leads to an approximate representation. Here, the strengths of the Gaussian shielding charge distributions are directly given in terms of the static partial charges as defined, e.g., by standard MM force fields for the various atom types, whereas the strengths of the Gaussian anti-polarization densities are calculated by a self-consistency iteration. The atomic volumes are also described by Gaussians. To account for covalently overlapping atoms, their effective volumes are calculated by another self-consistency procedure, which guarantees that the dielectric function ɛ(r) is close to one everywhere inside the protein. The Gaussian widths σi of the atoms i are parameters of the RF approximation. The remarkable accuracy of the method is demonstrated by comparison with Kirkwood's analytical solution for a spherical protein [J. Chem. Phys. 2, 351 (1934)] and with computationally expensive grid-based numerical solutions for simple model systems in dielectric continua including a di-peptide (Ac-Ala-NHMe) as modeled by a standard MM force field. The latter example shows how weakly the RF conformational free energy landscape depends on the parameters σi. A summarizing discussion highlights the achievements of the new theory and of its approximate solution particularly by
Bauer, Sebastian; Mathias, Gerald; Tavan, Paul
2014-03-14
We present a reaction field (RF) method which accurately solves the Poisson equation for proteins embedded in dielectric solvent continua at a computational effort comparable to that of an electrostatics calculation with polarizable molecular mechanics (MM) force fields. The method combines an approach originally suggested by Egwolf and Tavan [J. Chem. Phys. 118, 2039 (2003)] with concepts generalizing the Born solution [Z. Phys. 1, 45 (1920)] for a solvated ion. First, we derive an exact representation according to which the sources of the RF potential and energy are inducible atomic anti-polarization densities and atomic shielding charge distributions. Modeling these atomic densities by Gaussians leads to an approximate representation. Here, the strengths of the Gaussian shielding charge distributions are directly given in terms of the static partial charges as defined, e.g., by standard MM force fields for the various atom types, whereas the strengths of the Gaussian anti-polarization densities are calculated by a self-consistency iteration. The atomic volumes are also described by Gaussians. To account for covalently overlapping atoms, their effective volumes are calculated by another self-consistency procedure, which guarantees that the dielectric function ε(r) is close to one everywhere inside the protein. The Gaussian widths σ(i) of the atoms i are parameters of the RF approximation. The remarkable accuracy of the method is demonstrated by comparison with Kirkwood's analytical solution for a spherical protein [J. Chem. Phys. 2, 351 (1934)] and with computationally expensive grid-based numerical solutions for simple model systems in dielectric continua including a di-peptide (Ac-Ala-NHMe) as modeled by a standard MM force field. The latter example shows how weakly the RF conformational free energy landscape depends on the parameters σ(i). A summarizing discussion highlights the achievements of the new theory and of its approximate solution particularly by
Generalized Quasilinear Approximation: Application to Zonal Jets
NASA Astrophysics Data System (ADS)
Marston, J. B.; Chini, G. P.; Tobias, S. M.
2016-05-01
Quasilinear theory is often utilized to approximate the dynamics of fluids exhibiting significant interactions between mean flows and eddies. We present a generalization of quasilinear theory to include dynamic mode interactions on the large scales. This generalized quasilinear (GQL) approximation is achieved by separating the state variables into large and small zonal scales via a spectral filter rather than by a decomposition into a formal mean and fluctuations. Nonlinear interactions involving only small zonal scales are then removed. The approximation is conservative and allows for scattering of energy between small-scale modes via the large scale (through nonlocal spectral interactions). We evaluate GQL for the paradigmatic problems of the driving of large-scale jets on a spherical surface and on the beta plane and show that it is accurate even for a small number of large-scale modes. As GQL is formally linear in the small zonal scales, it allows for the closure of the system and can be utilized in direct statistical simulation schemes that have proved an attractive alternative to direct numerical simulation for many geophysical and astrophysical problems.
NASA Astrophysics Data System (ADS)
Allphin, Devin
Computational fluid dynamics (CFD) solution approximations for complex fluid flow problems have become a common and powerful engineering analysis technique. These tools, though qualitatively useful, remain limited in practice by their underlying inverse relationship between simulation accuracy and overall computational expense. While a great volume of research has focused on remedying these issues inherent to CFD, one traditionally overlooked area of resource reduction for engineering analysis concerns the basic definition and determination of functional relationships for the studied fluid flow variables. This artificial relationship-building technique, called meta-modeling or surrogate/offline approximation, uses design of experiments (DOE) theory to efficiently approximate non-physical coupling between the variables of interest in a fluid flow analysis problem. By mathematically approximating these variables, DOE methods can effectively reduce the required quantity of CFD simulations, freeing computational resources for other analytical focuses. An idealized interpretation of a fluid flow problem can also be employed to create suitably accurate approximations of fluid flow variables for the purposes of engineering analysis. When used in parallel with a meta-modeling approximation, a closed-form approximation can provide useful feedback concerning proper construction, suitability, or even necessity of an offline approximation tool. It also provides a short-circuit pathway for further reducing the overall computational demands of a fluid flow analysis, again freeing resources for otherwise unsuitable resource expenditures. To validate these inferences, a design optimization problem was presented requiring the inexpensive estimation of aerodynamic forces applied to a valve operating on a simulated piston-cylinder heat engine. The determination of these forces was to be found using parallel surrogate and exact approximation methods, thus evidencing the comparative
Median Approximations for Genomes Modeled as Matrices.
Zanetti, Joao Paulo Pereira; Biller, Priscila; Meidanis, Joao
2016-04-01
The genome median problem is an important problem in phylogenetic reconstruction under rearrangement models. It can be stated as follows: Given three genomes, find a fourth that minimizes the sum of the pairwise rearrangement distances between it and the three input genomes. In this paper, we model genomes as matrices and study the matrix median problem using the rank distance. It is known that, for any metric distance, at least one of the corners is a [Formula: see text]-approximation of the median. Our results allow us to compute up to three additional matrix median candidates, all of them with approximation ratios at least as good as the best corner, when the input matrices come from genomes. We also show a class of instances where our candidates are optimal. From the application point of view, it is usually more interesting to locate medians farther from the corners, and therefore, these new candidates are potentially more useful. In addition to the approximation algorithm, we suggest a heuristic to get a genome from an arbitrary square matrix. This is useful to translate the results of our median approximation algorithm back to genomes, and it has good results in our tests. To assess the relevance of our approach in the biological context, we ran simulated evolution tests and compared our solutions to those of an exact DCJ median solver. The results show that our method is capable of producing very good candidates. PMID:27072561
Parameter Choices for Approximation by Harmonic Splines
NASA Astrophysics Data System (ADS)
Gutting, Martin
2016-04-01
The approximation by harmonic trial functions allows the construction of the solution of boundary value problems in geoscience, e.g., in terms of harmonic splines. Due to their localizing properties regional modeling or the improvement of a global model in a part of the Earth's surface is possible with splines. Fast multipole methods have been developed for some cases of the occurring kernels to obtain a fast matrix-vector multiplication. The main idea of the fast multipole algorithm consists of a hierarchical decomposition of the computational domain into cubes and a kernel approximation for the more distant points. This reduces the numerical effort of the matrix-vector multiplication from quadratic to linear in reference to the number of points for a prescribed accuracy of the kernel approximation. The application of the fast multipole method to spline approximation which also allows the treatment of noisy data requires the choice of a smoothing parameter. We investigate different methods to (ideally automatically) choose this parameter with and without prior knowledge of the noise level. Thereby, the performance of these methods is considered for different types of noise in a large simulation study. Applications to gravitational field modeling are presented as well as the extension to boundary value problems where the boundary is the known surface of the Earth itself.
Alternative approximation concepts for space frame synthesis
NASA Technical Reports Server (NTRS)
Lust, R. V.; Schmit, L. A.
1985-01-01
A structural synthesis methodology for the minimum mass design of 3-dimensionall frame-truss structures under multiple static loading conditions and subject to limits on displacements, rotations, stresses, local buckling, and element cross-sectional dimensions is presented. A variety of approximation concept options are employed to yield near optimum designs after no more than 10 structural analyses. Available options include: (A) formulation of the nonlinear mathematcal programming problem in either reciprocal section property (RSP) or cross-sectional dimension (CSD) space; (B) two alternative approximate problem structures in each design space; and (C) three distinct assumptions about element end-force variations. Fixed element, design element linking, and temporary constraint deletion features are also included. The solution of each approximate problem, in either its primal or dual form, is obtained using CONMIN, a feasible directions program. The frame-truss synthesis methodology is implemented in the COMPASS computer program and is used to solve a variety of problems. These problems were chosen so that, in addition to exercising the various approximation concepts options, the results could be compared with previously published work.
Approximation algorithms for planning and control
NASA Technical Reports Server (NTRS)
Boddy, Mark; Dean, Thomas
1989-01-01
A control system operating in a complex environment will encounter a variety of different situations, with varying amounts of time available to respond to critical events. Ideally, such a control system will do the best possible with the time available. In other words, its responses should approximate those that would result from having unlimited time for computation, where the degree of the approximation depends on the amount of time it actually has. There exist approximation algorithms for a wide variety of problems. Unfortunately, the solution to any reasonably complex control problem will require solving several computationally intensive problems. Algorithms for successive approximation are a subclass of the class of anytime algorithms, algorithms that return answers for any amount of computation time, where the answers improve as more time is allotted. An architecture is described for allocating computation time to a set of anytime algorithms, based on expectations regarding the value of the answers they return. The architecture described is quite general, producing optimal schedules for a set of algorithms under widely varying conditions.
Symmetric approximations of the Navier-Stokes equations
Kobel'kov, G M
2002-08-31
A new method for the symmetric approximation of the non-stationary Navier-Stokes equations by a Cauchy-Kovalevskaya-type system is proposed. Properties of the modified problem are studied. In particular, the convergence as {epsilon}{yields}0 of the solutions of the modified problem to the solutions of the original problem on an infinite interval is established.
Symmetric approximations of the Navier-Stokes equations
NASA Astrophysics Data System (ADS)
Kobel'kov, G. M.
2002-08-01
A new method for the symmetric approximation of the non-stationary Navier-Stokes equations by a Cauchy-Kovalevskaya-type system is proposed. Properties of the modified problem are studied. In particular, the convergence as \\varepsilon\\to0 of the solutions of the modified problem to the solutions of the original problem on an infinite interval is established.
Weizsacker-Williams approximation in quantum chromodynamics
NASA Astrophysics Data System (ADS)
Kovchegov, Yuri V.
The Weizsacker-Williams approximation for a large nucleus in quantum chromodynamics is developed. The non-Abelian Wieizsacker Williams field for a large ultrarelativistic nucleus is constructed. This field is an exact solution of the classical Yang-Mills equations of motion in light cone gauge. The connection is made to the McLerran- Venugopalan model of a large nucleus, and the color charge density for a nucleus in this model is found. The density of states distribution, as a function of color charge density, is proved to be Gaussian. We construct the Feynman diagrams in the light cone gauge which correspond to the classical Weizsacker Williams field. Analyzing these diagrams we obtain a limitation on using the quasi-classical approximation for nuclear collisions.
Flow past a porous approximate spherical shell
NASA Astrophysics Data System (ADS)
Srinivasacharya, D.
2007-07-01
In this paper, the creeping flow of an incompressible viscous liquid past a porous approximate spherical shell is considered. The flow in the free fluid region outside the shell and in the cavity region of the shell is governed by the Navier Stokes equation. The flow within the porous annulus region of the shell is governed by Darcy’s Law. The boundary conditions used at the interface are continuity of the normal velocity, continuity of the pressure and Beavers and Joseph slip condition. An exact solution for the problem is obtained. An expression for the drag on the porous approximate spherical shell is obtained. The drag experienced by the shell is evaluated numerically for several values of the parameters governing the flow.
Accurate Cross Sections for Microanalysis
Rez, Peter
2002-01-01
To calculate the intensity of x-ray emission in electron beam microanalysis requires a knowledge of the energy distribution of the electrons in the solid, the energy variation of the ionization cross section of the relevant subshell, the fraction of ionizations events producing x rays of interest and the absorption coefficient of the x rays on the path to the detector. The theoretical predictions and experimental data available for ionization cross sections are limited mainly to K shells of a few elements. Results of systematic plane wave Born approximation calculations with exchange for K, L, and M shell ionization cross sections over the range of electron energies used in microanalysis are presented. Comparisons are made with experimental measurement for selected K shells and it is shown that the plane wave theory is not appropriate for overvoltages less than 2.5 V. PMID:27446747
Exponential approximations in optimal design
NASA Technical Reports Server (NTRS)
Belegundu, A. D.; Rajan, S. D.; Rajgopal, J.
1990-01-01
One-point and two-point exponential functions have been developed and proved to be very effective approximations of structural response. The exponential has been compared to the linear, reciprocal and quadratic fit methods. Four test problems in structural analysis have been selected. The use of such approximations is attractive in structural optimization to reduce the numbers of exact analyses which involve computationally expensive finite element analysis.
Analytic approximations to the modon dispersion relation. [in oceanography
NASA Technical Reports Server (NTRS)
Boyd, J. P.
1981-01-01
Three explicit analytic approximations are given to the modon dispersion relation developed by Flierl et al. (1980) to describe Gulf Stream rings and related phenomena in the oceans and atmosphere. The solutions are in the form of k(q), and are developed in the form of a power series in q for small q, an inverse power series in 1/q for large q, and a two-point Pade approximant. The low order Pade approximant is shown to yield a solution for the dispersion relation with a maximum relative error for the lowest branch of the function equal to one in 700 in the q interval zero to infinity.
Accurate Estimation of the Entropy of Rotation-Translation Probability Distributions.
Fogolari, Federico; Dongmo Foumthuim, Cedrix Jurgal; Fortuna, Sara; Soler, Miguel Angel; Corazza, Alessandra; Esposito, Gennaro
2016-01-12
The estimation of rotational and translational entropies in the context of ligand binding has been the subject of long-time investigations. The high dimensionality (six) of the problem and the limited amount of sampling often prevent the required resolution to provide accurate estimates by the histogram method. Recently, the nearest-neighbor distance method has been applied to the problem, but the solutions provided either address rotation and translation separately, therefore lacking correlations, or use a heuristic approach. Here we address rotational-translational entropy estimation in the context of nearest-neighbor-based entropy estimation, solve the problem numerically, and provide an exact and an approximate method to estimate the full rotational-translational entropy. PMID:26605696
Accurate ab Initio Spin Densities
2012-01-01
We present an approach for the calculation of spin density distributions for molecules that require very large active spaces for a qualitatively correct description of their electronic structure. Our approach is based on the density-matrix renormalization group (DMRG) algorithm to calculate the spin density matrix elements as a basic quantity for the spatially resolved spin density distribution. The spin density matrix elements are directly determined from the second-quantized elementary operators optimized by the DMRG algorithm. As an analytic convergence criterion for the spin density distribution, we employ our recently developed sampling-reconstruction scheme [J. Chem. Phys.2011, 134, 224101] to build an accurate complete-active-space configuration-interaction (CASCI) wave function from the optimized matrix product states. The spin density matrix elements can then also be determined as an expectation value employing the reconstructed wave function expansion. Furthermore, the explicit reconstruction of a CASCI-type wave function provides insight into chemically interesting features of the molecule under study such as the distribution of α and β electrons in terms of Slater determinants, CI coefficients, and natural orbitals. The methodology is applied to an iron nitrosyl complex which we have identified as a challenging system for standard approaches [J. Chem. Theory Comput.2011, 7, 2740]. PMID:22707921
On the loop approximation in nucleon QCD sum rules
Drukarev, E. G. Ryskin, M. G.; Sadovnikova, V. A.
2015-10-15
There was a general belief that the nucleon QCD sum rules which include only the quark loops and thus contain only the condensates of dimension d = 3 and d = 4 have only a trivial solution. We demonstrate that there is also a nontrivial solution. We show that it can be treated as the lowest order approximation to the solution which includes the higher terms of the Operator Product Expansion. Inclusion of the radiative corrections improves the convergence of the series.
Solving Math Problems Approximately: A Developmental Perspective
Ganor-Stern, Dana
2016-01-01
Although solving arithmetic problems approximately is an important skill in everyday life, little is known about the development of this skill. Past research has shown that when children are asked to solve multi-digit multiplication problems approximately, they provide estimates that are often very far from the exact answer. This is unfortunate as computation estimation is needed in many circumstances in daily life. The present study examined 4th graders, 6th graders and adults’ ability to estimate the results of arithmetic problems relative to a reference number. A developmental pattern was observed in accuracy, speed and strategy use. With age there was a general increase in speed, and an increase in accuracy mainly for trials in which the reference number was close to the exact answer. The children tended to use the sense of magnitude strategy, which does not involve any calculation but relies mainly on an intuitive coarse sense of magnitude, while the adults used the approximated calculation strategy which involves rounding and multiplication procedures, and relies to a greater extent on calculation skills and working memory resources. Importantly, the children were less accurate than the adults, but were well above chance level. In all age groups performance was enhanced when the reference number was smaller (vs. larger) than the exact answer and when it was far (vs. close) from it, suggesting the involvement of an approximate number system. The results suggest the existence of an intuitive sense of magnitude for the results of arithmetic problems that might help children and even adults with difficulties in math. The present findings are discussed in the context of past research reporting poor estimation skills among children, and the conditions that might allow using children estimation skills in an effective manner. PMID:27171224
NASA Technical Reports Server (NTRS)
Kory, Carol L.
1999-01-01
The phenomenal growth of commercial communications has created a great demand for traveling-wave tube (TWT) amplifiers. Although the helix slow-wave circuit remains the mainstay of the TWT industry because of its exceptionally wide bandwidth, until recently it has been impossible to accurately analyze a helical TWT using its exact dimensions because of the complexity of its geometrical structure. For the first time, an accurate three-dimensional helical model was developed that allows accurate prediction of TWT cold-test characteristics including operating frequency, interaction impedance, and attenuation. This computational model, which was developed at the NASA Lewis Research Center, allows TWT designers to obtain a more accurate value of interaction impedance than is possible using experimental methods. Obtaining helical slow-wave circuit interaction impedance is an important part of the design process for a TWT because it is related to the gain and efficiency of the tube. This impedance cannot be measured directly; thus, conventional methods involve perturbing a helical circuit with a cylindrical dielectric rod placed on the central axis of the circuit and obtaining the difference in resonant frequency between the perturbed and unperturbed circuits. A mathematical relationship has been derived between this frequency difference and the interaction impedance (ref. 1). However, because of the complex configuration of the helical circuit, deriving this relationship involves several approximations. In addition, this experimental procedure is time-consuming and expensive, but until recently it was widely accepted as the most accurate means of determining interaction impedance. The advent of an accurate three-dimensional helical circuit model (ref. 2) made it possible for Lewis researchers to fully investigate standard approximations made in deriving the relationship between measured perturbation data and interaction impedance. The most prominent approximations made
Hydration thermodynamics beyond the linear response approximation.
Raineri, Fernando O
2016-10-19
The solvation energetics associated with the transformation of a solute molecule at infinite dilution in water from an initial state A to a final state B is reconsidered. The two solute states have different potentials energies of interaction, [Formula: see text] and [Formula: see text], with the solvent environment. Throughout the A [Formula: see text] B transformation of the solute, the solvation system is described by a Hamiltonian [Formula: see text] that changes linearly with the coupling parameter ξ. By focusing on the characterization of the probability density [Formula: see text] that the dimensionless perturbational solute-solvent interaction energy [Formula: see text] has numerical value y when the coupling parameter is ξ, we derive a hierarchy of differential equation relations between the ξ-dependent cumulant functions of various orders in the expansion of the appropriate cumulant generating function. On the basis of this theoretical framework we then introduce an inherently nonlinear solvation model for which we are able to find analytical results for both [Formula: see text] and for the solvation thermodynamic functions. The solvation model is based on the premise that there is an upper or a lower bound (depending on the nature of the interactions considered) to the amplitude of the fluctuations of Y in the solution system at equilibrium. The results reveal essential differences in behavior for the model when compared with the linear response approximation to solvation, particularly with regards to the probability density [Formula: see text]. The analytical expressions for the solvation properties show, however, that the linear response behavior is recovered from the new model when the room for the thermal fluctuations in Y is not restricted by the existence of a nearby bound. We compare the predictions of the model with the results from molecular dynamics computer simulations for aqueous solvation, in which either (1) the solute
Finite difference approximation of hedging quantities in the Heston model
NASA Astrophysics Data System (ADS)
in't Hout, Karel
2012-09-01
This note concerns the hedging quantities Delta and Gamma in the Heston model for European-style financial options. A modification of the discretization technique from In 't Hout & Foulon (2010) is proposed, which enables a fast and accurate approximation of these important quantities. Numerical experiments are given that illustrate the performance.
Discrete approximation methods for parameter identification in delay systems
NASA Technical Reports Server (NTRS)
Rosen, I. G.
1984-01-01
Approximation schemes for parameter identification problems in which the governing state equation is a linear functional differential equation of retarded type are constructed. The basis of the schemes is the replacement of the parameter identification problem having an infinite dimensional state equation by a sequence of approximating parameter identification problems in which the states are given by finite dimensional discrete difference equations. The difference equations are constructed using linear semigroup theory and rational function approximations to the exponential. Sufficient conditions are given for the convergence of solutions to the approximating problems, which can be obtained using conventional methods, to solutions to the original parameter identification problem. Finite difference and spline based schemes using Paderational function approximations to the exponential are constructed, and shown to satisfy the sufficient conditions for convergence. A discussion and analysis of numerical results obtained through the application of the schemes to several examples is included.
Wavelet Sparse Approximate Inverse Preconditioners
NASA Technical Reports Server (NTRS)
Chan, Tony F.; Tang, W.-P.; Wan, W. L.
1996-01-01
There is an increasing interest in using sparse approximate inverses as preconditioners for Krylov subspace iterative methods. Recent studies of Grote and Huckle and Chow and Saad also show that sparse approximate inverse preconditioner can be effective for a variety of matrices, e.g. Harwell-Boeing collections. Nonetheless a drawback is that it requires rapid decay of the inverse entries so that sparse approximate inverse is possible. However, for the class of matrices that, come from elliptic PDE problems, this assumption may not necessarily hold. Our main idea is to look for a basis, other than the standard one, such that a sparse representation of the inverse is feasible. A crucial observation is that the kind of matrices we are interested in typically have a piecewise smooth inverse. We exploit this fact, by applying wavelet techniques to construct a better sparse approximate inverse in the wavelet basis. We shall justify theoretically and numerically that our approach is effective for matrices with smooth inverse. We emphasize that in this paper we have only presented the idea of wavelet approximate inverses and demonstrated its potential but have not yet developed a highly refined and efficient algorithm.
Approximate entropy of network parameters.
West, James; Lacasa, Lucas; Severini, Simone; Teschendorff, Andrew
2012-04-01
We study the notion of approximate entropy within the framework of network theory. Approximate entropy is an uncertainty measure originally proposed in the context of dynamical systems and time series. We first define a purely structural entropy obtained by computing the approximate entropy of the so-called slide sequence. This is a surrogate of the degree sequence and it is suggested by the frequency partition of a graph. We examine this quantity for standard scale-free and Erdös-Rényi networks. By using classical results of Pincus, we show that our entropy measure often converges with network size to a certain binary Shannon entropy. As a second step, with specific attention to networks generated by dynamical processes, we investigate approximate entropy of horizontal visibility graphs. Visibility graphs allow us to naturally associate with a network the notion of temporal correlations, therefore providing the measure a dynamical garment. We show that approximate entropy distinguishes visibility graphs generated by processes with different complexity. The result probes to a greater extent these networks for the study of dynamical systems. Applications to certain biological data arising in cancer genomics are finally considered in the light of both approaches. PMID:22680542
Approximate entropy of network parameters
NASA Astrophysics Data System (ADS)
West, James; Lacasa, Lucas; Severini, Simone; Teschendorff, Andrew
2012-04-01
We study the notion of approximate entropy within the framework of network theory. Approximate entropy is an uncertainty measure originally proposed in the context of dynamical systems and time series. We first define a purely structural entropy obtained by computing the approximate entropy of the so-called slide sequence. This is a surrogate of the degree sequence and it is suggested by the frequency partition of a graph. We examine this quantity for standard scale-free and Erdös-Rényi networks. By using classical results of Pincus, we show that our entropy measure often converges with network size to a certain binary Shannon entropy. As a second step, with specific attention to networks generated by dynamical processes, we investigate approximate entropy of horizontal visibility graphs. Visibility graphs allow us to naturally associate with a network the notion of temporal correlations, therefore providing the measure a dynamical garment. We show that approximate entropy distinguishes visibility graphs generated by processes with different complexity. The result probes to a greater extent these networks for the study of dynamical systems. Applications to certain biological data arising in cancer genomics are finally considered in the light of both approaches.
Kilcrease, D. P.; Brookes, S.
2013-08-19
The modeling of NLTE plasmas requires the solution of population rate equations to determine the populations of the various atomic levels relevant to a particular problem. The equations require many cross sections for excitation, de-excitation, ionization and recombination. Additionally, a simple and computational fast way to calculate electron collisional excitation cross-sections for ions is by using the plane-wave Born approximation. This is essentially a high-energy approximation and the cross section suffers from the unphysical problem of going to zero near threshold. Various remedies for this problem have been employed with varying degrees of success. We present a correction procedure formore » the Born cross-sections that employs the Elwert–Sommerfeld factor to correct for the use of plane waves instead of Coulomb waves in an attempt to produce a cross-section similar to that from using the more time consuming Coulomb Born approximation. We compare this new approximation with other, often employed correction procedures. Furthermore, we also look at some further modifications to our Born Elwert procedure and its combination with Y.K. Kim's correction of the Coulomb Born approximation for singly charged ions that more accurately approximate convergent close coupling calculations.« less
Kilcrease, D. P.; Brookes, S.
2013-08-19
The modeling of NLTE plasmas requires the solution of population rate equations to determine the populations of the various atomic levels relevant to a particular problem. The equations require many cross sections for excitation, de-excitation, ionization and recombination. Additionally, a simple and computational fast way to calculate electron collisional excitation cross-sections for ions is by using the plane-wave Born approximation. This is essentially a high-energy approximation and the cross section suffers from the unphysical problem of going to zero near threshold. Various remedies for this problem have been employed with varying degrees of success. We present a correction procedure for the Born cross-sections that employs the Elwert–Sommerfeld factor to correct for the use of plane waves instead of Coulomb waves in an attempt to produce a cross-section similar to that from using the more time consuming Coulomb Born approximation. We compare this new approximation with other, often employed correction procedures. Furthermore, we also look at some further modifications to our Born Elwert procedure and its combination with Y.K. Kim's correction of the Coulomb Born approximation for singly charged ions that more accurately approximate convergent close coupling calculations.
Relativistic regular approximations revisited: An infinite-order relativistic approximation
Dyall, K.G.; van Lenthe, E.
1999-07-01
The concept of the regular approximation is presented as the neglect of the energy dependence of the exact Foldy{endash}Wouthuysen transformation of the Dirac Hamiltonian. Expansion of the normalization terms leads immediately to the zeroth-order regular approximation (ZORA) and first-order regular approximation (FORA) Hamiltonians as the zeroth- and first-order terms of the expansion. The expansion may be taken to infinite order by using an un-normalized Foldy{endash}Wouthuysen transformation, which results in the ZORA Hamiltonian and a nonunit metric. This infinite-order regular approximation, IORA, has eigenvalues which differ from the Dirac eigenvalues by order E{sup 3}/c{sup 4} for a hydrogen-like system, which is a considerable improvement over the ZORA eigenvalues, and similar to the nonvariational FORA energies. A further perturbation analysis yields a third-order correction to the IORA energies, TIORA. Results are presented for several systems including the neutral U atom. The IORA eigenvalues for all but the 1s spinor of the neutral system are superior even to the scaled ZORA energies, which are exact for the hydrogenic system. The third-order correction reduces the IORA error for the inner orbitals to a very small fraction of the Dirac eigenvalue. {copyright} {ital 1999 American Institute of Physics.}
Convergence of the Approximation Scheme to American Option Pricing via the Discrete Morse Semiflow
Ishii, Katsuyuki; Omata, Seiro
2011-12-15
We consider the approximation scheme to the American call option via the discrete Morse semiflow, which is a minimizing scheme of a time semi-discretized variational functional. In this paper we obtain a rate of convergence of approximate solutions and the convergence of approximate free boundaries. We mainly apply the theory of variational inequalities and that of viscosity solutions to prove our results.
Fast Approximate Quadratic Programming for Graph Matching
Vogelstein, Joshua T.; Conroy, John M.; Lyzinski, Vince; Podrazik, Louis J.; Kratzer, Steven G.; Harley, Eric T.; Fishkind, Donniell E.; Vogelstein, R. Jacob; Priebe, Carey E.
2015-01-01
Quadratic assignment problems arise in a wide variety of domains, spanning operations research, graph theory, computer vision, and neuroscience, to name a few. The graph matching problem is a special case of the quadratic assignment problem, and graph matching is increasingly important as graph-valued data is becoming more prominent. With the aim of efficiently and accurately matching the large graphs common in big data, we present our graph matching algorithm, the Fast Approximate Quadratic assignment algorithm. We empirically demonstrate that our algorithm is faster and achieves a lower objective value on over 80% of the QAPLIB benchmark library, compared with the previous state-of-the-art. Applying our algorithm to our motivating example, matching C. elegans connectomes (brain-graphs), we find that it efficiently achieves performance. PMID:25886624
Fast approximate quadratic programming for graph matching.
Vogelstein, Joshua T; Conroy, John M; Lyzinski, Vince; Podrazik, Louis J; Kratzer, Steven G; Harley, Eric T; Fishkind, Donniell E; Vogelstein, R Jacob; Priebe, Carey E
2015-01-01
Quadratic assignment problems arise in a wide variety of domains, spanning operations research, graph theory, computer vision, and neuroscience, to name a few. The graph matching problem is a special case of the quadratic assignment problem, and graph matching is increasingly important as graph-valued data is becoming more prominent. With the aim of efficiently and accurately matching the large graphs common in big data, we present our graph matching algorithm, the Fast Approximate Quadratic assignment algorithm. We empirically demonstrate that our algorithm is faster and achieves a lower objective value on over 80% of the QAPLIB benchmark library, compared with the previous state-of-the-art. Applying our algorithm to our motivating example, matching C. elegans connectomes (brain-graphs), we find that it efficiently achieves performance. PMID:25886624
Comparing numerical and analytic approximate gravitational waveforms
NASA Astrophysics Data System (ADS)
Afshari, Nousha; Lovelace, Geoffrey; SXS Collaboration
2016-03-01
A direct observation of gravitational waves will test Einstein's theory of general relativity under the most extreme conditions. The Laser Interferometer Gravitational-Wave Observatory, or LIGO, began searching for gravitational waves in September 2015 with three times the sensitivity of initial LIGO. To help Advanced LIGO detect as many gravitational waves as possible, a major research effort is underway to accurately predict the expected waves. In this poster, I will explore how the gravitational waveform produced by a long binary-black-hole inspiral, merger, and ringdown is affected by how fast the larger black hole spins. In particular, I will present results from simulations of merging black holes, completed using the Spectral Einstein Code (black-holes.org/SpEC.html), including some new, long simulations designed to mimic black hole-neutron star mergers. I will present comparisons of the numerical waveforms with analytic approximations.
Inverse solutions for a Risley prism scanner with iterative refinement by a forward solution.
Li, Anhu; Gao, Xinjian; Sun, Wansong; Yi, Wanli; Bian, Yongming; Liu, Hongzhan; Liu, Liren
2015-11-20
Risley prism scanners are increasingly used for laser beam steering due to their wide angular scanning range and high resolution. However, the inverse problem, which focuses on obtaining the required prisms' orientations for a given target position, has not been perfectly solved so far. The existing inverse solutions are not accurate or efficient enough for high-accuracy and real-time tracking. An iterative method that combines an approximate inverse solution with an iterative refinement by the forward solution is set forth in this paper. Two case studies indicate that the rotation motions of Risley prism pairs controlled by iterative solutions can slew the beam to create the desired tracking pattern quickly and accurately. Based on this method, a Risley prism scanner developed as a standard trajectory generator is implemented for the error measurement of a robotic manipulator in our experiments. The simulation and experimental results show that the inverse solution for one target point can be obtained within nine iterations for a prescribed tracking error threshold. PMID:26836567
How to Solve Schroedinger Problems by Approximating the Potential Function
Ledoux, Veerle; Van Daele, Marnix
2010-09-30
We give a survey over the efforts in the direction of solving the Schroedinger equation by using piecewise approximations of the potential function. Two types of approximating potentials have been considered in the literature, that is piecewise constant and piecewise linear functions. For polynomials of higher degree the approximating problem is not so easy to integrate analytically. This obstacle can be circumvented by using a perturbative approach to construct the solution of the approximating problem, leading to the so-called piecewise perturbation methods (PPM). We discuss the construction of a PPM in its most convenient form for applications and show that different PPM versions (CPM,LPM) are in fact equivalent.
Solution of Dirac equation in Reissner-Nordström de Sitter space
NASA Astrophysics Data System (ADS)
Lyu, Yan; Cui, Song
2009-02-01
The radial parts of the Dirac equation between the outer black hole horizon and the cosmological horizon are solved in Reissner-Nordström de Sitter (RNdS) space numerically. An accurate approximation, the polynomial approximation, is used to approximate the modified tortoise coordinate \\hat r_* , which leads to the inverse function r = r(\\hat r_* ) and the potential V(\\hat r_* ). The potential V(\\hat r_* ) is replaced by a collection of step functions in sequence. Then the solution of the wave equation as well as the reflection and transmission coefficients is computed by a quantum mechanical method.
Approximate techniques for predicting size effects on cleavage fracture toughness (J{sub c})
Kirk, M.T.; Dodds, R.H. Jr.
1993-07-01
This investigation examines the ability of an elastic T-stress analysis coupled with modified boundary layer (MBL) solution to predict stresses ahead of a crack tip in a variety of planar geometries. The approximate stresses are used as input to estimate the effective driving force for cleavage fracture (J{sub 0}) using the micromechanically based approach introduced by Dodds and Anderson. Finite element analyses for a wide variety of planar cracked geometries are conducted which have elastic biaxiality parameters ({beta}) ranging from {minus}0.99 (very low constraint) to +2.96 (very high constraint). The magnitude and sign of {beta} indicate the rate at which crack-tip constraint changes with increasing applied load. All results pertain to a moderately strain hardening material (strain hardening exponent ({eta}) of 10). These analyses suggest that {beta} is an effective indicator of both the accuracy of T-MBL estimates of J{sub 0} and of applicability limits on evolving fracture analysis methodologies (i.e. T-MBL, J-Q, and J/J{sub 0}). Specifically, when 1{beta}1>0.4 these analyses show that the T-MBL approximation of J{sub 0} is accurate to within 20% of a detailed finite-element analysis. As ``structural type`` configurations, i.e. shallow cracks in tension, generally have 1{beta}1>0.4, it appears that only an elastic analysis may be needed to determine reasonably accurate J{sub 0} values for structural conditions.
Frequency domain measurements on turbid media with strong absorption using the PN approximation.
Baltes, Christof; Faris, Gregory W
2009-06-01
We have applied the frequency-domain technique to measurement of the optical properties of turbid media with strong absorption in the infinite medium limit. Absorption coefficients up to 2.3 cm(-1) for a modified scattering coefficient of 4.3 cm(-1) are studied, which corresponds to a reduced scattering albedo of 0.65. Low phase noise and good phase stability are required for these low albedo conditions. As the degree of absorption increases, the phase changes are reduced while amplitude changes increase. For this reason, correction of amplitude-phase cross talk is essential to achieve accurate measurements with strong absorption. Careful control of stray reflections is required to properly measure amplitude-phase cross talk. Because the diffusion approximation becomes less accurate, measurements are compared to calculations performed in the PN approximation, which is essentially an exact solution for the infinite medium limit. Agreement between theory and experiment is only obtained when correction for amplitude-phase cross talk is performed. These measurements can provide a good method for testing amplitude-phase cross talk. PMID:19488110
Gadgets, approximation, and linear programming
Trevisan, L.; Sudan, M.; Sorkin, G.B.; Williamson, D.P.
1996-12-31
We present a linear-programming based method for finding {open_quotes}gadgets{close_quotes}, i.e., combinatorial structures reducing constraints of one optimization problems to constraints of another. A key step in this method is a simple observation which limits the search space to a finite one. Using this new method we present a number of new, computer-constructed gadgets for several different reductions. This method also answers a question posed by on how to prove the optimality of gadgets-we show how LP duality gives such proofs. The new gadgets improve hardness results for MAX CUT and MAX DICUT, showing that approximating these problems to within factors of 60/61 and 44/45 respectively is N P-hard. We also use the gadgets to obtain an improved approximation algorithm for MAX 3SAT which guarantees an approximation ratio of .801. This improves upon the previous best bound of .7704.
Heat pipe transient response approximation
NASA Astrophysics Data System (ADS)
Reid, Robert S.
2002-01-01
A simple and concise routine that approximates the response of an alkali metal heat pipe to changes in evaporator heat transfer rate is described. This analytically based routine is compared with data from a cylindrical heat pipe with a crescent-annular wick that undergoes gradual (quasi-steady) transitions through the viscous and condenser boundary heat transfer limits. The sonic heat transfer limit can also be incorporated into this routine for heat pipes with more closely coupled condensers. The advantages and obvious limitations of this approach are discussed. For reference, a source code listing for the approximation appears at the end of this paper. .
Kinetic energy density dependent approximations to the exchange energy
NASA Astrophysics Data System (ADS)
Ernzerhof, Matthias; Scuseria, Gustavo E.
1999-07-01
Two nonempirical kinetic energy density dependent approximations are introduced. First, the local τ approximation (LTA) is proposed in which the exchange energy Ex depends only on a kinetic energy density τ. This LTA scheme appears to be complementary to the local spin density (LSD) approximation in the sense that its exchange contribution to the atomization energy ΔEx=Exatoms-Exmolecule is fairly accurate for systems where LSD fails. On the other hand, in cases where LSD works well LTA results for ΔEx are worse. Secondly, the τPBE approximation to Ex is developed which combines some of the advantages of LTA and of the Perdew-Burke-Ernzerhof (PBE) exchange functional. Like the PBE exchange functional, τPBE is free of empirical parameters. Furthermore, it yields improved atomization energies compared to the PBE approximation.