NASA Astrophysics Data System (ADS)
von Sydow, Lina
2013-10-01
The discontinuous Galerkin method for time integration of the Black-Scholes partial differential equation for option pricing problems is studied and compared with more standard time-integrators. In space an adaptive finite difference discretization is employed. The results show that the dG method are in most cases at least comparable to standard time-integrators and in some cases superior to them. Together with adaptive spatial grids the suggested pricing method shows great qualities.
A time-space domain stereo finite difference method for 3D scalar wave propagation
NASA Astrophysics Data System (ADS)
Chen, Yushu; Yang, Guangwen; Ma, Xiao; He, Conghui; Song, Guojie
2016-11-01
The time-space domain finite difference methods reduce numerical dispersion effectively by minimizing the error in the joint time-space domain. However, their interpolating coefficients are related with the Courant numbers, leading to significantly extra time costs for loading the coefficients consecutively according to velocity in heterogeneous models. In the present study, we develop a time-space domain stereo finite difference (TSSFD) method for 3D scalar wave equation. The method propagates both the displacements and their gradients simultaneously to keep more information of the wavefields, and minimizes the maximum phase velocity error directly using constant interpolation coefficients for different Courant numbers. We obtain the optimal constant coefficients by combining the truncated Taylor series approximation and the time-space domain optimization, and adjust the coefficients to improve the stability condition. Subsequent investigation shows that the TSSFD can suppress numerical dispersion effectively with high computational efficiency. The maximum phase velocity error of the TSSFD is just 3.09% even with only 2 sampling points per minimum wavelength when the Courant number is 0.4. Numerical experiments show that to generate wavefields with no visible numerical dispersion, the computational efficiency of the TSSFD is 576.9%, 193.5%, 699.0%, and 191.6% of those of the 4th-order and 8th-order Lax-Wendroff correction (LWC) method, the 4th-order staggered grid method (SG), and the 8th-order optimal finite difference method (OFD), respectively. Meanwhile, the TSSFD is compatible to the unsplit convolutional perfectly matched layer (CPML) boundary condition for absorbing artificial boundaries. The efficiency and capability to handle complex velocity models make it an attractive tool in imaging methods such as acoustic reverse time migration (RTM).
NASA Astrophysics Data System (ADS)
Jia, Jinhong; Wang, Hong
2015-07-01
Numerical methods for space-fractional diffusion equations often generate dense or even full stiffness matrices. Traditionally, these methods were solved via Gaussian type direct solvers, which requires O (N3) of computational work per time step and O (N2) of memory to store where N is the number of spatial grid points in the discretization. In this paper we develop a preconditioned fast Krylov subspace iterative method for the efficient and faithful solution of finite difference methods (both steady-state and time-dependent) space-fractional diffusion equations with fractional derivative boundary conditions in one space dimension. The method requires O (N) of memory and O (Nlog N) of operations per iteration. Due to the application of effective preconditioners, significantly reduced numbers of iterations were achieved that further reduces the computational cost of the fast method. Numerical results are presented to show the utility of the method.
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 Adaptive Finite Difference Method for Hyperbolic Systems in OneSpace Dimension
Bolstad, John H.
1982-06-01
Many problems of physical interest have solutions which are generally quite smooth in a large portion of the region of interest, but have local phenomena such as shocks, discontinuities or large gradients which require much more accurate approximations or finer grids for reasonable accuracy. Examples are atmospheric fronts, ocean currents, and geological discontinuities. In this thesis we develop and partially analyze an adaptive finite difference mesh refinement algorithm for the initial boundary value problem for hyperbolic systems in one space dimension. The method uses clusters of uniform grids which can ''move'' along with pulses or steep gradients appearing in the calculation, and which are superimposed over a uniform coarse grid. Such refinements are created, destroyed, merged, separated, recursively nested or moved based on estimates of the local truncation error. We use a four-way linked tree and sequentially allocated deques (double-ended queues) to perform these operations efficiently. The local truncation error in the interior of the region is estimated using a three-step Richardson extrapolation procedure, which can also be considered a deferred correction method. At the boundaries we employ differences to estimate the error. Our algorithm was implemented using a portable, extensible Fortran preprocessor, to which we added records and pointers. The method is applied to three model problems: the first order wave equation, the second order wave equation, and the inviscid Burgers equation. For the first two model problems our algorithm is shown to be three to five times more efficient (in computing time) than the use of a uniform coarse mesh, for the same accuracy. Furthermore, to our knowledge, our algorithm is the only one which adaptively treats time-dependent boundary conditions for hyperbolic systems.
NASA Astrophysics Data System (ADS)
Cole, James B.
2014-09-01
The finite difference time domain (FDTD) algorithm is a popular tool for photonics design and simulations, but it also can yield deep insights into the fundamental nature of light and - more speculatively - into the discretization and connectivity and geometry of space-time. The CFL stability limit in FDTD can be interpreted as a limit on the speed of light. It depends not only on the dimensionality of space-time, but also on its connectivity. Thus the speed of light not only tells us something about the dimensionality of space-time but also about its connectivity. The computational molecule in conventional 2-D FDTD is (х +/- h,y)-(x,+/- y h)-(x-y), where h= triangle x = triangle y . It yields the CFL stability limit ctriangle/h<= t/h 1 √2 . Including diagonal nodes (x+/- h, y +/- h) in the computational molecule changes the connectivity of the space and changes the CFL limit. The FDTD model also predicts precursor signals (which physically exist). The Green's function of the FDTD model, which differs from that of the wave equation, may tell us something about underlying periodicities in space-time. It may be possible to experimentally observe effects of space-time discretization and connectivity in optics experiments.
NASA Technical Reports Server (NTRS)
Collier, Richard S.
1997-01-01
This report describes finite difference computer calculations for the Space Shuttle Launch Pad which predict lightning induced electric currents and electric and magnetic fields caused by a lightning strike to the Lightning Protection System caternary wire. Description of possible lightning threats to Shuttle Payload components together with specifications for protection of these components, result from the calculation of lightning induced electric and magnetic fields inside and outside the during a lightning event. These fields also induce currents and voltages on cables and circuits which may be connected to, or a part of, shuttle payload components. These currents and voltages are also calculated. These threat levels are intended as a guide for designers of payload equipment to specify any shielding and/or lightning protection mitigation which may be required for payload components which are in the process of preparation or being transferred into the Shuttle Orbiter.
Numerical computation of transonic flows by finite-element and finite-difference methods
NASA Technical Reports Server (NTRS)
Hafez, M. M.; Wellford, L. C.; Merkle, C. L.; Murman, E. M.
1978-01-01
Studies on applications of the finite element approach to transonic flow calculations are reported. Different discretization techniques of the differential equations and boundary conditions are compared. Finite element analogs of Murman's mixed type finite difference operators for small disturbance formulations were constructed and the time dependent approach (using finite differences in time and finite elements in space) was examined.
Tan, Sirui; Huang, Lianjie
2014-11-01
For modeling scalar-wave propagation in geophysical problems using finite-difference schemes, optimizing the coefficients of the finite-difference operators can reduce numerical dispersion. Most optimized finite-difference schemes for modeling seismic-wave propagation suppress only spatial but not temporal dispersion errors. We develop a novel optimized finite-difference scheme for numerical scalar-wave modeling to control dispersion errors not only in space but also in time. Our optimized scheme is based on a new stencil that contains a few more grid points than the standard stencil. We design an objective function for minimizing relative errors of phase velocities of waves propagating in all directions within a given range of wavenumbers. Dispersion analysis and numerical examples demonstrate that our optimized finite-difference scheme is computationally up to 2.5 times faster than the optimized schemes using the standard stencil to achieve the similar modeling accuracy for a given 2D or 3D problem. Compared with the high-order finite-difference scheme using the same new stencil, our optimized scheme reduces 50 percent of the computational cost to achieve the similar modeling accuracy. This new optimized finite-difference scheme is particularly useful for large-scale 3D scalar-wave modeling and inversion.
NASA Astrophysics Data System (ADS)
Ono, Tomoya; Egami, Yoshiyuki; Hirose, Kikuji
2012-11-01
We demonstrate an efficient nonequilibrium Green's function transport calculation procedure based on the real-space finite-difference method. The direct inversion of matrices for obtaining the self-energy terms of electrodes is computationally demanding in the real-space method because the matrix dimension corresponds to the number of grid points in the unit cell of electrodes, which is much larger than that of sites in the tight-binding approach. The procedure using the ratio matrices of the overbridging boundary-matching technique [Y. Fujimoto and K. Hirose, Phys. Rev. BPRBMDO1098-012110.1103/PhysRevB.67.195315 67, 195315 (2003)], which is related to the wave functions of a couple of grid planes in the matching regions, greatly reduces the computational effort to calculate self-energy terms without losing mathematical strictness. In addition, the present procedure saves computational time to obtain the Green's function of the semi-infinite system required in the Landauer-Büttiker formula. Moreover, the compact expression to relate Green's functions and scattering wave functions, which provide a real-space picture of the scattering process, is introduced. An example of the calculated results is given for the transport property of the BN ring connected to (9,0) carbon nanotubes. The wave-function matching at the interface reveals that the rotational symmetry of wave functions with respect to the tube axis plays an important role in electron transport. Since the states coming from and going to electrodes show threefold rotational symmetry, the states in the vicinity of the Fermi level, the wave function of which exhibits fivefold symmetry, do not contribute to the electron transport through the BN ring.
Mimetic finite difference method
NASA Astrophysics Data System (ADS)
Lipnikov, Konstantin; Manzini, Gianmarco; Shashkov, Mikhail
2014-01-01
The mimetic finite difference (MFD) method mimics fundamental properties of mathematical and physical systems including conservation laws, symmetry and positivity of solutions, duality and self-adjointness of differential operators, and exact mathematical identities of the vector and tensor calculus. This article is the first comprehensive review of the 50-year long history of the mimetic methodology and describes in a systematic way the major mimetic ideas and their relevance to academic and real-life problems. The supporting applications include diffusion, electromagnetics, fluid flow, and Lagrangian hydrodynamics problems. The article provides enough details to build various discrete operators on unstructured polygonal and polyhedral meshes and summarizes the major convergence results for the mimetic approximations. Most of these theoretical results, which are presented here as lemmas, propositions and theorems, are either original or an extension of existing results to a more general formulation using polyhedral meshes. Finally, flexibility and extensibility of the mimetic methodology are shown by deriving higher-order approximations, enforcing discrete maximum principles for diffusion problems, and ensuring the numerical stability for saddle-point systems.
NASA Astrophysics Data System (ADS)
Iwase, Shigeru; Hoshi, Takeo; Ono, Tomoya
2015-06-01
We propose an efficient procedure to obtain Green's functions by combining the shifted conjugate orthogonal conjugate gradient (shifted COCG) method with the nonequilibrium Green's function (NEGF) method based on a real-space finite-difference (RSFD) approach. The bottleneck of the computation in the NEGF scheme is matrix inversion of the Hamiltonian including the self-energy terms of electrodes to obtain the perturbed Green's function in the transition region. This procedure first computes unperturbed Green's functions and calculates perturbed Green's functions from the unperturbed ones using a mathematically strict relation. Since the matrices to be inverted to obtain the unperturbed Green's functions are sparse, complex-symmetric, and shifted for a given set of sampling energy points, we can use the shifted COCG method, in which once the Green's function for a reference energy point has been calculated the Green's functions for the other energy points can be obtained with a moderate computational cost. We calculate the transport properties of a C60@(10,10) carbon nanotube (CNT) peapod suspended by (10,10)CNTs as an example of a large-scale transport calculation. The proposed scheme opens the possibility of performing large-scale RSFD-NEGF transport calculations using massively parallel computers without the loss of accuracy originating from the incompleteness of the localized basis set.
NASA Astrophysics Data System (ADS)
Ren, Zhiming; Liu, Yang; Zhang, Qunshan
2014-05-01
Full waveform inversion (FWI) has the potential to provide preferable subsurface model parameters. The main barrier of its applications to real seismic data is heavy computational amount. Numerical modelling methods are involved in both forward modelling and backpropagation of wavefield residuals, which spend most of computational time in FWI. We develop a time-space domain finite-difference (FD) method and adaptive variable-length spatial operator scheme in numerical simulation of viscoacoustic equation and extend them into the viscoacoustic FWI. Compared with conventional FD methods, different operator lengths are adopted for different velocities and quality factors, which can reduce the amount of computation without reducing accuracy. Inversion algorithms also play a significant role in FWI. In conventional single-scale methods, it is likely to converge to local minimums especially when the initial model is far from the real model. To tackle the problem, we introduce the second generation wavelet transform to implement the multiscale FWI. Compared to other multiscale methods, our method has advantages of ease of implementation and better time-frequency local analysis ability. The L2 norm is widely used in FWI and gives invalid model estimates when the data is contaminated with strong non-uniform noises. We apply the L1-norm and the Huber-norm criteria in the time-domain FWI to improve its antinoise ability. Our strategies have been successfully applied in synthetic experiments to both onshore and offshore reflection seismic data. The results of the viscoacoustic Marmousi example indicate that our new FWI scheme consumes smaller computer resources. In addition, the viscoacoustic Overthrust example shows its better convergence and more reasonable velocity and quality factor structures. All these results demonstrate that our method can improve inversion accuracy and computational efficiency of FWI.
NASA Technical Reports Server (NTRS)
Sohn, Kiho D.; Ip, Shek-Se P.
1988-01-01
Three-dimensional finite element models were generated and transferred into three-dimensional finite difference models to perform transient thermal analyses for the SSME high pressure fuel turbopump's first stage nozzles and rotor blades. STANCOOL was chosen to calculate the heat transfer characteristics (HTCs) around the airfoils, and endwall effects were included at the intersections of the airfoils and platforms for the steady-state boundary conditions. Free and forced convection due to rotation effects were also considered in hollow cores. Transient HTCs were calculated by taking ratios of the steady-state values based on the flow rates and fluid properties calculated at each time slice. Results are presented for both transient plots and three-dimensional color contour isotherm plots; they were also converted into universal files to be used for FEM stress analyses.
Zhang, X B; Yin, Y F; Yao, H M; Han, Y H; Wang, N; Ge, Z L
2016-07-01
To investigate the stress distribution on the maxillary anterior teeth retracted with sliding mechanics and micro-implant anchorage using different retraction hook heights and positions. DICOM image data including maxilla and upper teeth were obtained with cone-beam CT. The three-dimensional finite element model was constructed using Mimics software. Brackets and archwire model were constructed using Creo software. The models were instantiated using Pro/Engineer software. Abaqus software was used to simulate the sliding mechanics by loading 2 N force on 0, 2, 4, 6, 8, 10 mm retraction hooks and three different positions, repectively. Rotation of the occlusal plane, the initial displacement and stress distribution of teeth were analyzed. Lingual rotation of maxillary central incisor(0.021°), gingival movement of the maxillary first molar(0.005 mm), and clockwise rotation of the maxillary occlusal plane(0.012°) were observed when the force application point located at the archwire level (0 mm). In contrast, 0.235° labial rotation of the maxillary central incisor, 0.015 mm occlusal movement of the maxillary first molar, and 0.075° anti-clockwise rotation of the maxillary occlusal plane were observed when the force application point located at the higher level(10 mm retraction hook). The more the force application point was located posteriorly at the archwire level, the less lingual rotation of the maxillary central incisor and the more buccal displacement of maxillary first molar was observed. Maxillary anterior tooth rotation and retraction, vertical displacement of posterior segment, and rotation of the occlusal plane could be controlled by adjusting the height and position of the retraction hook in space closure using miniscrew and sliding mechanics.
NASA Astrophysics Data System (ADS)
Tsukamoto, Shigeru; Ono, Tomoya; Hirose, Kikuji; Blügel, Stefan
2017-03-01
The self-energy term used in transport calculations, which describes the coupling between electrode and transition regions, is able to be evaluated only from a limited number of the propagating and evanescent waves of a bulk electrode. This obviously contributes toward the reduction of the computational expenses in transport calculations. In this paper, we present a mathematical formula for reducing the computational expenses further without using any approximation and without losing accuracy. So far, the self-energy term has been handled as a matrix with the same dimension as the Hamiltonian submatrix representing the interaction between an electrode and a transition region. In this work, through the singular-value decomposition of the submatrix, the self-energy matrix is handled as a smaller matrix, whose dimension is the rank number of the Hamiltonian submatrix. This procedure is practical in the case of using the pseudopotentials in a separable form, and the computational expenses for determining the self-energy matrix are reduced by 90% when employing a code based on the real-space finite-difference formalism and projector-augmented wave method. In addition, this technique is applicable to the transport calculations using atomic or localized basis sets. Adopting the self-energy matrices obtained from this procedure, we present the calculation of the electron transport properties of C20 molecular junctions. The application demonstrates that the electron transmissions are sensitive to the orientation of the molecule with respect to the electrode surface. In addition, channel decomposition of the scattering wave functions reveals that some unoccupied C20 molecular orbitals mainly contribute to the electron conduction through the molecular junction.
The Relation of Finite Element and Finite Difference Methods
NASA Technical Reports Server (NTRS)
Vinokur, M.
1976-01-01
Finite element and finite difference methods are examined in order to bring out their relationship. It is shown that both methods use two types of discrete representations of continuous functions. They differ in that finite difference methods emphasize the discretization of independent variable, while finite element methods emphasize the discretization of dependent variable (referred to as functional approximations). An important point is that finite element methods use global piecewise functional approximations, while finite difference methods normally use local functional approximations. A general conclusion is that finite element methods are best designed to handle complex boundaries, while finite difference methods are superior for complex equations. It is also shown that finite volume difference methods possess many of the advantages attributed to finite element methods.
Finite Topological Spaces as a Pedagogical Tool
ERIC Educational Resources Information Center
Helmstutler, Randall D.; Higginbottom, Ryan S.
2012-01-01
We propose the use of finite topological spaces as examples in a point-set topology class especially suited to help students transition into abstract mathematics. We describe how carefully chosen examples involving finite spaces may be used to reinforce concepts, highlight pathologies, and develop students' non-Euclidean intuition. We end with a…
Finite Topological Spaces as a Pedagogical Tool
ERIC Educational Resources Information Center
Helmstutler, Randall D.; Higginbottom, Ryan S.
2012-01-01
We propose the use of finite topological spaces as examples in a point-set topology class especially suited to help students transition into abstract mathematics. We describe how carefully chosen examples involving finite spaces may be used to reinforce concepts, highlight pathologies, and develop students' non-Euclidean intuition. We end with a…
Upwind Compact Finite Difference Schemes
NASA Astrophysics Data System (ADS)
Christie, I.
1985-07-01
It was shown by Ciment, Leventhal, and Weinberg ( J. Comput. Phys.28 (1978), 135) that the standard compact finite difference scheme may break down in convection dominated problems. An upwinding of the method, which maintains the fourth order accuracy, is suggested and favorable numerical results are found for a number of test problems.
NASA Astrophysics Data System (ADS)
Yan, Hongyong; Liu, Yang; Zhang, Hao
2013-03-01
With advanced computational power, prestack reverse-time migration (RTM) is being used increasingly in seismic imaging. The accuracy and efficiency of RTM strongly depends on the algorithms used for numerical solutions of wave equations. Hence, how to solve the wave equation accurately and rapidly is very important in the process of RTM. In this paper, in order to improve the accuracy of the numerical solution, we use a time-space domain staggered-grid finite-difference (SFD) method to solve the acoustic wave equation, and develop a new acoustic prestack RTM scheme based on this time-space domain high-order SFD. Synthetic and real data tests demonstrate that the RTM scheme improves the imaging quality significantly compared with the conventional SFD RTM. Meanwhile, in the process of wavefield extrapolation, we apply adaptive variable-length spatial operators to compute spatial derivatives to decrease computational costs effectively with little reduction of the accuracy of the numerical solutions.
NASA Technical Reports Server (NTRS)
Mickens, Ronald E.
1996-01-01
A large class of physical phenomena can be modeled by evolution and wave type Partial Differential Equations (PDE). Few of these equations have known explicit exact solutions. Finite-difference techniques are a popular method for constructing discrete representations of these equations for the purpose of numerical integration. However, the solutions to the difference equations often contain so called numerical instabilities; these are solutions to the difference equations that do not correspond to any solution of the PDE's. For explicit schemes, the elimination of this behavior requires functional relations to exist between the time and space steps-sizes. We show that such functional relations can be obtained for certain PDE's by use of a positivity condition. The PDE's studied are the Burgers, Fisher, and linearized Euler equations.
Adaptive finite difference for seismic wavefield modelling in acoustic media
Yao, Gang; Wu, Di; Debens, Henry Alexander
2016-01-01
Efficient numerical seismic wavefield modelling is a key component of modern seismic imaging techniques, such as reverse-time migration and full-waveform inversion. Finite difference methods are perhaps the most widely used numerical approach for forward modelling, and here we introduce a novel scheme for implementing finite difference by introducing a time-to-space wavelet mapping. Finite difference coefficients are then computed by minimising the difference between the spatial derivatives of the mapped wavelet and the finite difference operator over all propagation angles. Since the coefficients vary adaptively with different velocities and source wavelet bandwidths, the method is capable to maximise the accuracy of the finite difference operator. Numerical examples demonstrate that this method is superior to standard finite difference methods, while comparable to Zhang’s optimised finite difference scheme. PMID:27491333
Adaptive finite difference for seismic wavefield modelling in acoustic media.
Yao, Gang; Wu, Di; Debens, Henry Alexander
2016-08-05
Efficient numerical seismic wavefield modelling is a key component of modern seismic imaging techniques, such as reverse-time migration and full-waveform inversion. Finite difference methods are perhaps the most widely used numerical approach for forward modelling, and here we introduce a novel scheme for implementing finite difference by introducing a time-to-space wavelet mapping. Finite difference coefficients are then computed by minimising the difference between the spatial derivatives of the mapped wavelet and the finite difference operator over all propagation angles. Since the coefficients vary adaptively with different velocities and source wavelet bandwidths, the method is capable to maximise the accuracy of the finite difference operator. Numerical examples demonstrate that this method is superior to standard finite difference methods, while comparable to Zhang's optimised finite difference scheme.
NASA Astrophysics Data System (ADS)
Yang, Qingjie; Mao, Weijian
2017-01-01
The poroelastodynamic equations are used to describe the dynamic solid-fluid interaction in the reservoir. To obtain the intrinsic properties of reservoir rocks from geophysical data measured in both laboratory and field, we need an accurate solution of the wave propagation in porous media. At present, the poroelastic wave equations are mostly solved in the time domain, which involves a difficult and complicated time convolution. In order to avoid the issues caused by the time convolution, we propose a frequency-space domain method. The poroelastic wave equations are composed of a linear system in the frequency domain, which easily takes into account the effects of all frequencies on the dispersion and attenuation of seismic wave. A 25-point weighted-averaging finite different scheme is proposed to discretize the equations. For the finite model, the perfectly matched layer technique is applied at the model boundaries. We validated the proposed algorithm by testing three numerical examples of poroelastic models, which are homogenous, two-layered and heterogeneous with different fluids, respectively. The testing results are encouraging in the aspects of both computational accuracy and efficiency.
NASA Astrophysics Data System (ADS)
Yan, Hongyong; Liu, Yang
2013-02-01
With the increment of seismic exploration precision requirement, it is significant to develop the anisotropic migration methods. Pre-stack reverse-time migration (RTM) is performed based on acoustic vertical transversely isotropic (VTI) wave equations, and the accuracy and efficiency of RTM strongly depend on the algorithms used for wave equation numerical solution. Finite-difference (FD) methods have been widely used in numerical solution of wave equations. The conventional FD method derives spatial FD coefficients from the space domain dispersion relation, and it is difficult to satisfy the time-space domain dispersion relation of the wave equation exactly. In this paper, we adopt a time-space domain FD method to solve acoustic VTI wave equations. Dispersion analysis and numerical modelling results demonstrate that the time-space domain FD method has greater accuracy than the conventional FD method under the same discretizations. The time-space domain high-order FD method is also applied in the wavefield extrapolation of acoustic VTI pre-stack RTM. The model tests demonstrate that the acoustic VTI pre-stack RTM based on the time-space domain FD method can obtain better images than that based on the conventional FD method, and the processing results show that the imaging quality of the acoustic VTI RTM is clearer and more correct than that of acoustic isotropic RTM. Meanwhile, in the process of wavefield forward and backward extrapolation, we employ adaptive variable-length spatial operators to compute spatial derivatives to improve the computational efficiency effectively almost without reducing the imaging accuracy.
Finite difference neuroelectric modeling software.
Dang, Hung V; Ng, Kwong T
2011-06-15
This paper describes a finite difference neuroelectric modeling software (FNS), written in C and MATLAB, which can be executed as a standalone program or integrated with other packages for electroencephalography (EEG) analysis. The package from the Oxford Center for Functional MRI of the Brain (FMRIB), FMRIB Software Library (FSL), is used to segment the anatomical magnetic resonance (MR) image for realistic head modeling. The EEG electrode array is fitted to the realistic head model using the Bioelectromagnetism MATLAB toolbox. The finite difference formulation for a general inhomogeneous anisotropic body is used to obtain the system matrix equation, which is then solved using the conjugate gradient algorithm. The reciprocity theorem is utilized to limit the number of required forward solutions to N-1, where N is the number of electrodes. Results show that the forward solver only requires 500 MB of random-access memory (RAM) for a realistic 256×256×256 head model and that the software can be conveniently combined with inverse algorithms such as beamformers and MUSIC. The software is freely available under the GNU Public License.
Finite elements and finite differences for transonic flow calculations
NASA Technical Reports Server (NTRS)
Hafez, M. M.; Murman, E. M.; Wellford, L. C.
1978-01-01
The paper reviews the chief finite difference and finite element techniques used for numerical solution of nonlinear mixed elliptic-hyperbolic equations governing transonic flow. The forms of the governing equations for unsteady two-dimensional transonic flow considered are the Euler equation, the full potential equation in both conservative and nonconservative form, the transonic small-disturbance equation in both conservative and nonconservative form, and the hodograph equations for the small-disturbance case and the full-potential case. Finite difference methods considered include time-dependent methods, relaxation methods, semidirect methods, and hybrid methods. Finite element methods include finite element Lax-Wendroff schemes, implicit Galerkin method, mixed variational principles, dual iterative procedures, optimal control methods and least squares.
Eigenvalues of singular differential operators by finite difference methods. II.
NASA Technical Reports Server (NTRS)
Baxley, J. V.
1972-01-01
Note is made of an earlier paper which defined finite difference operators for the Hilbert space L2(m), and gave the eigenvalues for these operators. The present work examines eigenvalues for higher order singular differential operators by using finite difference methods. The two self-adjoint operators investigated are defined by a particular value in the same Hilbert space, L2(m), and are strictly positive with compact inverses. A class of finite difference operators is considered, with the idea of application to the theory of Toeplitz matrices. The approximating operators consist of a good approximation plus a perturbing operator.
Real-space renormalization yields finite correlations.
Barthel, Thomas; Kliesch, Martin; Eisert, Jens
2010-07-02
Real-space renormalization approaches for quantum lattice systems generate certain hierarchical classes of states that are subsumed by the multiscale entanglement renormalization Ansatz (MERA). It is shown that, with the exception of one spatial dimension, MERA states are actually states with finite correlations, i.e., projected entangled pair states (PEPS) with a bond dimension independent of the system size. Hence, real-space renormalization generates states which can be encoded with local effective degrees of freedom, and MERA states form an efficiently contractible class of PEPS that obey the area law for the entanglement entropy. It is further pointed out that there exist other efficiently contractible schemes violating the area law.
Sasaki, Akira; Kojo, Masashi; Hirose, Kikuji; Goto, Hidekazu
2011-11-02
The path-integral renormalization group and direct energy minimization method of practical first-principles electronic structure calculations for multi-body systems within the framework of the real-space finite-difference scheme are introduced. These two methods can handle higher dimensional systems with consideration of the correlation effect. Furthermore, they can be easily extended to the multicomponent quantum systems which contain more than two kinds of quantum particles. The key to the present methods is employing linear combinations of nonorthogonal Slater determinants (SDs) as multi-body wavefunctions. As one of the noticeable results, the same accuracy as the variational Monte Carlo method is achieved with a few SDs. This enables us to study the entire ground state consisting of electrons and nuclei without the need to use the Born-Oppenheimer approximation. Recent activities on methodological developments aiming towards practical calculations such as the implementation of auxiliary field for Coulombic interaction, the treatment of the kinetic operator in imaginary-time evolutions, the time-saving double-grid technique for bare-Coulomb atomic potentials and the optimization scheme for minimizing the total-energy functional are also introduced. As test examples, the total energy of the hydrogen molecule, the atomic configuration of the methylene and the electronic structures of two-dimensional quantum dots are calculated, and the accuracy, availability and possibility of the present methods are demonstrated.
Optimal neuronal tuning for finite stimulus spaces.
Brown, W. Michael; Backer, Alejandro
2004-09-01
The efficiency of neuronal encoding in sensory and motor systems has been proposed as a first principle governing response properties within the central nervous system. We present a continuation of a theoretical study presented by Zhang and Sejnowski, where the influence of neuronal tuning properties on encoding accuracy is analyzed using information theory. When a finite stimulus space is considered, we show that the encoding accuracy improves with narrow tuning for one- and two-dimensional stimuli. For three dimensions and higher, there is an optimal tuning width.
Instability of flat space at finite temperature
Gross, D.J.; Perry, M.J.; Yaffe, L.G.
1982-01-15
The instabilities of quantum gravity are investigated using the path-integral formulation of Einstein's theory. A brief review is given of the classical gravitational instabilities, as well as the stability of flat space. The Euclidean path-integral representation of the partition function is employed to discuss the instability of flat space at finite temperature. Semiclassical, or saddle-point, approximations are utilized. We show how the Jeans instability arises as a tachyon in the graviton propagator when small perturbations about hot flat space are considered. The effect due to the Schwarzschild instanton is studied. The small fluctuations about this instanton are analyzed and a negative mode is discovered. This produces, in the semiclassical approximation, an imaginary part of the free energy. This is interpreted as being due to the metastability of hot flat space to nucleate black holes. These then evolve by evaporation or by accretion of thermal gravitons, leading to the instability of hot flat space. The nucleation rate of black holes is calculated as a function of temperature.
Finite-difference computations of rotor loads
NASA Technical Reports Server (NTRS)
Caradonna, F. X.; Tung, C.
1985-01-01
The current and future potential of finite difference methods for solving real rotor problems which now rely largely on empiricism are demonstrated. The demonstration consists of a simple means of combining existing finite-difference, integral, and comprehensive loads codes to predict real transonic rotor flows. These computations are performed for hover and high-advanced-ratio flight. Comparisons are made with experimental pressure data.
Finite-difference computations of rotor loads
NASA Technical Reports Server (NTRS)
Caradonna, F. X.; Tung, C.
1985-01-01
This paper demonstrates the current and future potential of finite-difference methods for solving real rotor problems which now rely largely on empiricism. The demonstration consists of a simple means of combining existing finite-difference, integral, and comprehensive loads codes to predict real transonic rotor flows. These computations are performed for hover and high-advance-ratio flight. Comparisons are made with experimental pressure data.
NASA Technical Reports Server (NTRS)
Roberts, G. O.; Fowlis, W. W.; Miller, T. L.
1984-01-01
Numerical methods are used to design a spherical baroclinic flow model experiment of the large scale atmosphere flow for Spacelab. The dielectric simulation of radial gravity is only dominant in a low gravity environment. Computer codes are developed to study the processes at work in crystal growing systems which are also candidates for space flight. Crystalline materials rarely achieve their potential properties because of imperfections and component concentration variations. Thermosolutal convection in the liquid melt can be the cause of these imperfections. Such convection is suppressed in a low gravity environment. Two and three dimensional finite difference codes are being used for this work. Nonuniform meshes and implicit iterative methods are used. The iterative method for steady solutions is based on time stepping but has the options of different time steps for velocity and temperature and of a time step varying smoothly with position according to specified powers of the mesh spacings. This allows for more rapid convergence. The code being developed for the crystal growth studies allows for growth of the crystal as the solid-liquid interface. The moving interface is followed using finite differences; shape variations are permitted. For convenience in applying finite differences in the solid and liquid, a time dependent coordinate transformation is used to make this interface a coordinate surface.
Applications of an exponential finite difference technique
NASA Technical Reports Server (NTRS)
Handschuh, Robert F.; Keith, Theo G., Jr.
1988-01-01
An exponential finite difference scheme first presented by Bhattacharya for one dimensional unsteady heat conduction problems in Cartesian coordinates was extended. The finite difference algorithm developed was used to solve the unsteady diffusion equation in one dimensional cylindrical coordinates and was applied to two and three dimensional conduction problems in Cartesian coordinates. Heat conduction involving variable thermal conductivity was also investigated. The method was used to solve nonlinear partial differential equations in one and two dimensional Cartesian coordinates. Predicted results are compared to exact solutions where available or to results obtained by other numerical methods.
Finite-Difference Algorithms For Computing Sound Waves
NASA Technical Reports Server (NTRS)
Davis, Sanford
1993-01-01
Governing equations considered as matrix system. Method variant of method described in "Scheme for Finite-Difference Computations of Waves" (ARC-12970). Present method begins with matrix-vector formulation of fundamental equations, involving first-order partial derivatives of primitive variables with respect to space and time. Particular matrix formulation places time and spatial coordinates on equal footing, so governing equations considered as matrix system and treated as unit. Spatial and temporal discretizations not treated separately as in other finite-difference methods, instead treated together by linking spatial-grid interval and time step via common scale factor related to speed of sound.
Simulating Space Capsule Water Landing with Explicit Finite Element Method
NASA Technical Reports Server (NTRS)
Wang, John T.; Lyle, Karen H.
2007-01-01
A study of using an explicit nonlinear dynamic finite element code for simulating the water landing of a space capsule was performed. The finite element model contains Lagrangian shell elements for the space capsule and Eulerian solid elements for the water and air. An Arbitrary Lagrangian Eulerian (ALE) solver and a penalty coupling method were used for predicting the fluid and structure interaction forces. The space capsule was first assumed to be rigid, so the numerical results could be correlated with closed form solutions. The water and air meshes were continuously refined until the solution was converged. The converged maximum deceleration predicted is bounded by the classical von Karman and Wagner solutions and is considered to be an adequate solution. The refined water and air meshes were then used in the models for simulating the water landing of a capsule model that has a flexible bottom. For small pitch angle cases, the maximum deceleration from the flexible capsule model was found to be significantly greater than the maximum deceleration obtained from the corresponding rigid model. For large pitch angle cases, the difference between the maximum deceleration of the flexible model and that of its corresponding rigid model is smaller. Test data of Apollo space capsules with a flexible heat shield qualitatively support the findings presented in this paper.
Compact finite difference method for American option pricing
NASA Astrophysics Data System (ADS)
Zhao, Jichao; Davison, Matt; Corless, Robert M.
2007-09-01
A compact finite difference method is designed to obtain quick and accurate solutions to partial differential equation problems. The problem of pricing an American option can be cast as a partial differential equation. Using the compact finite difference method this problem can be recast as an ordinary differential equation initial value problem. The complicating factor for American options is the existence of an optimal exercise boundary which is jointly determined with the value of the option. In this article we develop three ways of combining compact finite difference methods for American option price on a single asset with methods for dealing with this optimal exercise boundary. Compact finite difference method one uses the implicit condition that solutions of the transformed partial differential equation be nonnegative to detect the optimal exercise value. This method is very fast and accurate even when the spatial step size h is large (h[greater-or-equal, slanted]0.1). Compact difference method two must solve an algebraic nonlinear equation obtained by Pantazopoulos (1998) at every time step. This method can obtain second order accuracy for space x and requires a moderate amount of time comparable with that required by the Crank Nicolson projected successive over relaxation method. Compact finite difference method three refines the free boundary value by a method developed by Barone-Adesi and Lugano [The saga of the American put, 2003], and this method can obtain high accuracy for space x. The last two of these three methods are convergent, moreover all the three methods work for both short term and long term options. Through comparison with existing popular methods by numerical experiments, our work shows that compact finite difference methods provide an exciting new tool for American option pricing.
Finite-difference migration to zero offset
Li, Jianchao.
1992-01-01
Migration to zero offset (MZO), also called dip moveout (DMO) or prestack partial migration, transforms prestack offset seismic data into approximate zero-offset data so as to remove reflection point smear and obtain quality stacked results over a range of reflector dips. MZO has become an important step in standard seismic data processing, and a variety of frequency-wavenumber (f-k) and integral MZO algorithms have been used in practice to date. Here, I present a finite-difference MZO algorithm applied to normal-moveout (NMO)-corrected, common-offset sections. This algorithm employs a traditional poststack 15-degree finite-difference migration algorithm and a special velocity function rather than the true migration velocity. This paper shows results of implementation of this MZO algorithm when velocity varies with depth, and discusses the possibility of applying this algorithm to cases where velocity varies with both depth and horizontal distance.
Finite-difference migration to zero offset
Li, Jianchao
1992-07-01
Migration to zero offset (MZO), also called dip moveout (DMO) or prestack partial migration, transforms prestack offset seismic data into approximate zero-offset data so as to remove reflection point smear and obtain quality stacked results over a range of reflector dips. MZO has become an important step in standard seismic data processing, and a variety of frequency-wavenumber (f-k) and integral MZO algorithms have been used in practice to date. Here, I present a finite-difference MZO algorithm applied to normal-moveout (NMO)-corrected, common-offset sections. This algorithm employs a traditional poststack 15-degree finite-difference migration algorithm and a special velocity function rather than the true migration velocity. This paper shows results of implementation of this MZO algorithm when velocity varies with depth, and discusses the possibility of applying this algorithm to cases where velocity varies with both depth and horizontal distance.
Second Order Accurate Finite Difference Methods
1984-08-20
a study of the idealized material has direct applications to some polymer structures (4, 5). Wave propagation studies in hyperelastic materials have...34Acceleration Wave Propagation in Hyperelastic Rods of Variable Cross- section. Wave Motion, V4, pp. 173-180, 1982. 9. M. Hirao and N. Sugimoto...Waves in Hyperelastic Road," Quart. Appl. Math., V37, pp. 377-399, 1979. 11. G. A. Sod. "A Survey of Several Finite Difference Methods for Systems of
Computations in isometry groups of finite metric spaces
Ganyushkin, A.G.; Sushchanskii, V.I.; Tsvirkunov, V.V.
1995-01-01
A meaningful analysis of the structure or properties of a finite metric space requires using its isometry group. This is the group of permutations on the set of points in the space that preserves the binary relations {open_quotes}the points x and y are at a given distance from one another.{close_quotes} The theory of isometry groups of finite metric spaces is therefore a component of the theory of invariant relations of permutation groups. In the framework of their research, Kaluzhnin and his students have obtained a number of deep results. The theory of invariant relations is very useful also because it leads to an efficient programming system for computations in permutation groups defined by their invariant relations and in the associated combinatorial-algebraic objects. This suggests that the approaches and methodology of the general theory may be applied to study isometry groups of finite metric spaces. Direct extension of the results and the algorithms is not always the best strategy, because the specific features of metric spaces can be exploited to analyze the topic in greater detail and to construct more efficient computer algorithms. Section 1 considers the concept of isomorphism of metric spaces and shows that any finite metric space is isomorphic to a space with a metric whose nonzero values completely fill some interval of the natural series. We introduce the chromatic graph associated with a metric space and count the number of pairwise nonisomorphic metric spaces whose chromatic graphs are isomorphic to the given chromatic graph.
Analysis of a finite difference grid
NASA Technical Reports Server (NTRS)
Klopfer, G. H.
1982-01-01
Some means of assessing the suitability of a mesh network for a finite difference calculation are investigated in this study. This has been done by a study of the nonlinear truncation errors of the scheme. It turns out that the mesh can not be properly assessed a priori. The effect of the mesh on the numerical solution depends on several factors including the mesh itself, the numerical algorithm, and the solution. Several recommendations are made with regard to generating the mesh and to assessing its suitability for a particular numerical calculation.
Finite difference methods for approximating Heaviside functions
NASA Astrophysics Data System (ADS)
Towers, John D.
2009-05-01
We present a finite difference method for discretizing a Heaviside function H(u(x→)), where u is a level set function u:Rn ↦ R that is positive on a bounded region Ω⊂Rn. There are two variants of our algorithm, both of which are adapted from finite difference methods that we proposed for discretizing delta functions in [J.D. Towers, Two methods for discretizing a delta function supported on a level set, J. Comput. Phys. 220 (2007) 915-931; J.D. Towers, Discretizing delta functions via finite differences and gradient normalization, Preprint at http://www.miracosta.edu/home/jtowers/; J.D. Towers, A convergence rate theorem for finite difference approximations to delta functions, J. Comput. Phys. 227 (2008) 6591-6597]. We consider our approximate Heaviside functions as they are used to approximate integrals over Ω. We prove that our first approximate Heaviside function leads to second order accurate quadrature algorithms. Numerical experiments verify this second order accuracy. For our second algorithm, numerical experiments indicate at least third order accuracy if the integrand f and ∂Ω are sufficiently smooth. Numerical experiments also indicate that our approximations are effective when used to discretize certain singular source terms in partial differential equations. We mostly focus on smooth f and u. By this we mean that f is smooth in a neighborhood of Ω, u is smooth in a neighborhood of ∂Ω, and the level set u(x)=0 is a manifold of codimension one. However, our algorithms still give reasonable results if either f or u has jumps in its derivatives. Numerical experiments indicate approximately second order accuracy for both algorithms if the regularity of the data is reduced in this way, assuming that the level set u(x)=0 is a manifold. Numerical experiments indicate that dependence on the placement of Ω with respect to the grid is quite small for our algorithms. Specifically, a grid shift results in an O(hp) change in the computed solution
The Complex-Step-Finite-Difference method
NASA Astrophysics Data System (ADS)
Abreu, Rafael; Stich, Daniel; Morales, Jose
2015-07-01
We introduce the Complex-Step-Finite-Difference method (CSFDM) as a generalization of the well-known Finite-Difference method (FDM) for solving the acoustic and elastic wave equations. We have found a direct relationship between modelling the second-order wave equation by the FDM and the first-order wave equation by the CSFDM in 1-D, 2-D and 3-D acoustic media. We present the numerical methodology in order to apply the introduced CSFDM and show an example for wave propagation in simple homogeneous and heterogeneous models. The CSFDM may be implemented as an extension into pre-existing numerical techniques in order to obtain fourth- or sixth-order accurate results with compact three time-level stencils. We compare advantages of imposing various types of initial motion conditions of the CSFDM and demonstrate its higher-order accuracy under the same computational cost and dispersion-dissipation properties. The introduced method can be naturally extended to solve different partial differential equations arising in other fields of science and engineering.
Efficient discretization in finite difference method
NASA Astrophysics Data System (ADS)
Rozos, Evangelos; Koussis, Antonis; Koutsoyiannis, Demetris
2015-04-01
Finite difference method (FDM) is a plausible and simple method for solving partial differential equations. The standard practice is to use an orthogonal discretization to form algebraic approximate formulations of the derivatives of the unknown function and a grid, much like raster maps, to represent the properties of the function domain. For example, for the solution of the groundwater flow equation, a raster map is required for the characterization of the discretization cells (flow cell, no-flow cell, boundary cell, etc.), and two raster maps are required for the hydraulic conductivity and the storage coefficient. Unfortunately, this simple approach to describe the topology comes along with the known disadvantages of the FDM (rough representation of the geometry of the boundaries, wasted computational resources in the unavoidable expansion of the grid refinement in all cells of the same column and row, etc.). To overcome these disadvantages, Hunt has suggested an alternative approach to describe the topology, the use of an array of neighbours. This limits the need for discretization nodes only for the representation of the boundary conditions and the flow domain. Furthermore, the geometry of the boundaries is described more accurately using a vector representation. Most importantly, graded meshes can be employed, which are capable of restricting grid refinement only in the areas of interest (e.g. regions where hydraulic head varies rapidly, locations of pumping wells, etc.). In this study, we test the Hunt approach against MODFLOW, a well established finite difference model, and the Finite Volume Method with Simplified Integration (FVMSI). The results of this comparison are examined and critically discussed.
TUNED FINITE-DIFFERENCE DIFFUSION OPERATORS
Maron, Jason; Low, Mordecai-Mark Mac E-mail: mordecai@amnh.org
2009-05-15
Finite-difference simulations of fluid dynamics and magnetohydrodynamics generally require an explicit diffusion operator, either to maintain stability by attenuating grid-scale structure, or to implement physical diffusivities such as viscosity or resistivity. If the goal is stability only, the diffusion must act at the grid scale, but should affect structure at larger scales as little as possible. For physical diffusivities the diffusion scale depends on the problem, and diffusion may act at larger scales as well. Diffusivity can undesirably limit the computational time step in both cases. We construct tuned finite-difference diffusion operators that minimally limit the time step while acting as desired near the diffusion scale. Such operators reach peak values at the diffusion scale rather than at the grid scale, but behave as standard operators at larger scales. These operators will be useful for simulations with high magnetic diffusivity or kinematic viscosity such as in the simulation of astrophysical dynamos with magnetic Prandtl number far from unity, or for numerical stabilization using hyperdiffusivity.
NASA Technical Reports Server (NTRS)
Fix, G. J.; Rose, M. E.
1983-01-01
A least squares formulation of the system divu = rho, curlu = zeta is surveyed from the viewpoint of both finite element and finite difference methods. Closely related arguments are shown to establish convergence estimates.
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.
Finite element analysis of a deployable space structure
NASA Technical Reports Server (NTRS)
Hutton, D. V.
1982-01-01
To assess the dynamic characteristics of a deployable space truss, a finite element model of the Scientific Applications Space Platform (SASP) truss has been formulated. The model incorporates all additional degrees of freedom associated with the pin-jointed members. Comparison of results with SPAR models of the truss show that the joints of the deployable truss significantly affect the vibrational modes of the structure only if the truss is relatively short.
Finite difference computation of Casimir forces
NASA Astrophysics Data System (ADS)
Pinto, Fabrizio
2016-09-01
In this Invited paper, we begin by a historical introduction to provide a motivation for the classical problems of interatomic force computation and associated challenges. This analysis will lead us from early theoretical and experimental accomplishments to the integration of these fascinating interactions into the operation of realistic, next-generation micro- and nanodevices both for the advanced metrology of fundamental physical processes and in breakthrough industrial applications. Among several powerful strategies enabling vastly enhanced performance and entirely novel technological capabilities, we shall specifically consider Casimir force time-modulation and the adoption of non-trivial geometries. As to the former, the ability to alter the magnitude and sign of the Casimir force will be recognized as a crucial principle to implement thermodynamical nano-engines. As to the latter, we shall first briefly review various reported computational approaches. We shall then discuss the game-changing discovery, in the last decade, that standard methods of numerical classical electromagnetism can be retooled to formulate the problem of Casimir force computation in arbitrary geometries. This remarkable development will be practically illustrated by showing that such an apparently elementary method as standard finite-differencing can be successfully employed to numerically recover results known from the Lifshitz theory of dispersion forces in the case of interacting parallel-plane slabs. Other geometries will be also be explored and consideration given to the potential of non-standard finite-difference methods. Finally, we shall introduce problems at the computational frontier, such as those including membranes deformed by Casimir forces and the effects of anisotropic materials. Conclusions will highlight the dramatic transition from the enduring perception of this field as an exotic application of quantum electrodynamics to the recent demonstration of a human climbing
Geometrical Series and Phase Space in a Finite Oscillatory Motion
ERIC Educational Resources Information Center
Mareco, H. R. Olmedo
2006-01-01
This article discusses some interesting physical properties of oscillatory motion of a particle on two joined inclined planes. The geometrical series demonstrates that the particle will oscillate during a finite time. Another detail is the converging path to the origin of the phase space. Due to its simplicity, this motion may be used as a…
Geometrical Series and Phase Space in a Finite Oscillatory Motion
ERIC Educational Resources Information Center
Mareco, H. R. Olmedo
2006-01-01
This article discusses some interesting physical properties of oscillatory motion of a particle on two joined inclined planes. The geometrical series demonstrates that the particle will oscillate during a finite time. Another detail is the converging path to the origin of the phase space. Due to its simplicity, this motion may be used as a…
A dispersion reducing convective finite difference scheme
NASA Astrophysics Data System (ADS)
Matus, R. J.; Hindman, R. G.
1986-01-01
A one-parameter family of finite difference schemes for systems of convective equations has been developed and applied to the inviscid Burgers' equation and the one-dimensional, unsteady Euler equations. The parameter, alpha, may be chosen in a way to reduce the phase error of the numerical solution compared to other commonly used second order difference schemes, and computational results are included which show the ability of the scheme, called the alpha-scheme in this paper, to calculate solutions which contain discontinuities with very little oscillation. For linear one-dimensional problems, the scheme reduces to Fromm's zero average phase error method, but the present scheme differs from Fromm's in that it is easily applied to nonlinear systems of equations such as the Euler equations describing inviscid fluid flow. A modified MacCormack scheme and Warming and Beam's predictor-corrector upwind scheme are also members of the family of schemes which can be retrieved for particular choices of the parameter, alpha.
Quantum electrodynamic effects in finite space
NASA Astrophysics Data System (ADS)
Dobiasch, P.; Walther, H.
The modifications of various quantum properties due to a discrete structure of the modes of the vacuum electromagnetic field are discussed. In contrast to the usual case of a continuous spectrum of the free space fluctuations, we consider physical systems in a resonator or in a wave guide. It is shown that the relaxation time of the system can be increased ot decreased, by increasing or decreasing the density of modes with respect to the case of unperturbed vacuum. On the other hand, we predict level shifts due to the reduced mass of the electron and deviations from the Lambshift for hydrogen in a wave guide, which can be detected with the presently feasible high resolution spectroscopy. We propose an experimental set-up. Nous discutons les modifications de diverses propriétés quantiques sous l'influence d'une structure de modes discrets du champ électromagnétique dans le vide. En comparaison du cas habituel d'un spectre continu des fluctuations du vide dans l'espace libre, nous considérons ici des systèmes physiques dans un résonateur ou un guide d'ondes. Il est démontré que le temps de relaxation du système peut être prolongé ou raccourci, ceci en augmentant ou diminuant la densité des modes par rapport à sa valeur dans le vide non-perturbé. D'autre part, nous prédisons des déplacements de niveau dus à la masse réduite de l'électron et des déviations du Lamb shift pour des atomes d'hydrogène dans un guide d'ondes, qui peuvent être détectées grâce à la haute résolution accessible actuellement en spectroscopie. Nous présentons un dispositif expérimental.
Finite element models of the space shuttle main engine
NASA Technical Reports Server (NTRS)
Muller, G. R.
1980-01-01
Finite element models were developed as input to dynamic simulations of the high pressure fuel turbopump (HPFTP), the high pressure oxidizer turbopump (HPOTP), and the space shuttle main engine (SSME). Descriptions are provided for the five basic finite element models: HPFTP rotor, HPFTP case, HPOTP rotor, HPOTP case, and SSME (excluding turbopumps). Modal results are presented for the HPFTP rotor, HPFTP case, HPOTP rotor, coupled HPFTP rotor and case, HPOTP case, coupled HPOTP rotor and case, SSME (excluding turbopumps), and SSME (including turbopumps). Results for the SSME (including turbopumps) model are compared to data from a SSME HPOTP modal survey.
Finite Mathematics and Discrete Mathematics: Is There a Difference?
ERIC Educational Resources Information Center
Johnson, Marvin L.
Discrete mathematics and finite mathematics differ in a number of ways. First, finite mathematics has a longer history and is therefore more stable in terms of course content. Finite mathematics courses emphasize certain particular mathematical tools which are useful in solving the problems of business and the social sciences. Discrete mathematics…
Finite Mathematics and Discrete Mathematics: Is There a Difference?
ERIC Educational Resources Information Center
Johnson, Marvin L.
Discrete mathematics and finite mathematics differ in a number of ways. First, finite mathematics has a longer history and is therefore more stable in terms of course content. Finite mathematics courses emphasize certain particular mathematical tools which are useful in solving the problems of business and the social sciences. Discrete mathematics…
High-Order Entropy Stable Finite Difference Schemes for Nonlinear Conservation Laws: Finite Domains
NASA Technical Reports Server (NTRS)
Fisher, Travis C.; Carpenter, Mark H.
2013-01-01
Developing stable and robust high-order finite difference schemes requires mathematical formalism and appropriate methods of analysis. In this work, nonlinear entropy stability is used to derive provably stable high-order finite difference methods with formal boundary closures for conservation laws. Particular emphasis is placed on the entropy stability of the compressible Navier-Stokes equations. A newly derived entropy stable weighted essentially non-oscillatory finite difference method is used to simulate problems with shocks and a conservative, entropy stable, narrow-stencil finite difference approach is used to approximate viscous terms.
Bieniasz, L K
2003-07-01
Accurate calculation of concentration gradients at the boundaries is crucial in electrochemical kinetic simulations, owing to the frequent occurrence of gradient-dependent boundary conditions, and the importance of the gradient-dependent electric current. By using the information about higher spatial derivatives of the concentrations, contained in the time-dependent, kinetic reaction-diffusion partial differential equation(s) in one-dimensional space geometry, under appropriate assumptions it is possible to increase the accuracy orders of the conventional, one-sided n-point finite-difference formulae for the concentration gradients at the boundaries, without increasing n. In this way a new class of high order accurate gradient approximations is derived, and tested in simulations of potential-step chronoamperometric and current-step chronopotentiometric transients for the Reinert-Berg system. The new formulae possess advantages over the conventional gradient approximations. For example, they allow one to obtain a third order accuracy by using two space points only, or fourth order accuracy by using three points, and yet they yield smaller errors than the conventional four-point, or five-point formulae, respectively. Needing fewer points, for approximating the gradients with a given accuracy, simplifies also the solution of the linear algebraic equations arising from the application of implicit time integration schemes.
Finite-element reentry heat-transfer analysis of space shuttle Orbiter
NASA Technical Reports Server (NTRS)
Ko, William L.; Quinn, Robert D.; Gong, Leslie
1986-01-01
A structural performance and resizing (SPAR) finite-element thermal analysis computer program was used in the heat-transfer analysis of the space shuttle orbiter subjected to reentry aerodynamic heating. Three wing cross sections and one midfuselage cross section were selected for the thermal analysis. The predicted thermal protection system temperatures were found to agree well with flight-measured temperatures. The calculated aluminum structural temperatures also agreed reasonably well with the flight data from reentry to touchdown. The effects of internal radiation and of internal convection were found to be significant. The SPAR finite-element solutions agreed reasonably well with those obtained from the conventional finite-difference method.
Huge Casimir effect at finite temperature in electromagnetic Rindler space.
Zhao, Tian-Ming; Miao, Rong-Xin
2011-12-01
We investigate the Casimir effect at a finite temperature in the electromagnetic Rindler space, and we find that the Casimir energy is proportional to T(4)/d(2) in the high-temperature limit, where T ≈ 27 °C is the temperature and d ≈ 100 nm is a small cutoff. We propose to make metamaterials to mimic the Rindler space and measure the predicted Casimir effect. Because the parameters of metamaterials we proposed are quite simple, this experiment would be easily implemented in the laboratory. © 2011 Optical Society of America
Space-time formulation for finite element modeling of superconductors
Ashworth, Stephen P; Grilli, Francesco; Sirois, Frederic; Laforest, Marc
2008-01-01
In this paper we present a new model for computing the current density and field distributions in superconductors by means of a periodic space-time formulation for finite elements (FE). By considering a space dimension as time, we can use a static model to solve a time dependent problem. This allows overcoming one of the major problems of FE modeling of superconductors: the length of simulations, even for relatively simple cases. We present our first results and compare them to those obtained with a 'standard' time-dependent method and with analytical solutions.
Constructing infrared finite propagators in inflating space-time
Rajaraman, Arvind; Kumar, Jason; Leblond, Louis
2010-07-15
The usual (Bunch-Davies) Feynman propagator of a massless field is not well defined in an expanding universe due to the presence of infrared divergences. We propose a new propagator which yields IR finite answers to any correlation function. The key point is that in a de Sitter space-time there is an ambiguity in the zero mode of the propagator. This ambiguity can be used to cancel the apparent divergences which arise in some loop calculations in eternally (or semieternally) inflating space-time. We refer to this process as zero-mode modification. The residual ambiguity is fixed by observational measurement.
Symmetry reduction of ordinary finite difference equations using moving frames
NASA Astrophysics Data System (ADS)
Benson, Joseph; Valiquette, Francis
2017-05-01
The technique of equivariant moving frames is incorporated into the classical symmetry reduction method of ordinary finite difference equations. Using the recurrence relations for the finite difference invariants, computations are performed symbolically without relying on the coordinate expressions of the canonical variables and the difference invariants.
NASA Technical Reports Server (NTRS)
Bauld, N. R., Jr.; Goree, J. G.
1983-01-01
The accuracy of the finite difference method in the solution of linear elasticity problems that involve either a stress discontinuity or a stress singularity is considered. Solutions to three elasticity problems are discussed in detail: a semi-infinite plane subjected to a uniform load over a portion of its boundary; a bimetallic plate under uniform tensile stress; and a long, midplane symmetric, fiber reinforced laminate subjected to uniform axial strain. Finite difference solutions to the three problems are compared with finite element solutions to corresponding problems. For the first problem a comparison with the exact solution is also made. The finite difference formulations for the three problems are based on second order finite difference formulas that provide for variable spacings in two perpendicular directions. Forward and backward difference formulas are used near boundaries where their use eliminates the need for fictitious grid points.
Finite difference time domain analysis of chirped dielectric gratings
NASA Technical Reports Server (NTRS)
Hochmuth, Diane H.; Johnson, Eric G.
1993-01-01
The finite difference time domain (FDTD) method for solving Maxwell's time-dependent curl equations is accurate, computationally efficient, and straight-forward to implement. Since both time and space derivatives are employed, the propagation of an electromagnetic wave can be treated as an initial-value problem. Second-order central-difference approximations are applied to the space and time derivatives of the electric and magnetic fields providing a discretization of the fields in a volume of space, for a period of time. The solution to this system of equations is stepped through time, thus, simulating the propagation of the incident wave. If the simulation is continued until a steady-state is reached, an appropriate far-field transformation can be applied to the time-domain scattered fields to obtain reflected and transmitted powers. From this information diffraction efficiencies can also be determined. In analyzing the chirped structure, a mesh is applied only to the area immediately around the grating. The size of the mesh is then proportional to the electric size of the grating. Doing this, however, imposes an artificial boundary around the area of interest. An absorbing boundary condition must be applied along the artificial boundary so that the outgoing waves are absorbed as if the boundary were absent. Many such boundary conditions have been developed that give near-perfect absorption. In this analysis, the Mur absorbing boundary conditions are employed. Several grating structures were analyzed using the FDTD method.
A Finite Analog of the AGT Relation I: Finite W-Algebras and Quasimaps' Spaces
NASA Astrophysics Data System (ADS)
Braverman, Alexander; Feigin, Boris; Finkelberg, Michael; Rybnikov, Leonid
2011-12-01
Recently Alday, Gaiotto and Tachikawa [2] proposed a conjecture relating 4-dimensional super-symmetric gauge theory for a gauge group G with certain 2-dimensional conformal field theory. This conjecture implies the existence of certain structures on the (equivariant) intersection cohomology of the Uhlenbeck partial compactification of the moduli space of framed G-bundles on {mathbb{P}^2} . More precisely, it predicts the existence of an action of the corresponding W-algebra on the above cohomology, satisfying certain properties. We propose a "finite analog" of the (above corollary of the) AGT conjecture. Namely, we replace the Uhlenbeck space with the space of based quasi-maps from {mathbb{P}^1} to any partial flag variety G/ P of G and conjecture that its equivariant intersection cohomology carries an action of the finite W-algebra {U(mathfrak{g},e)} associated with the principal nilpotent element in the Lie algebra of the Levi subgroup of P; this action is expected to satisfy some list of natural properties. This conjecture generalizes the main result of [5] when P is the Borel subgroup. We prove our conjecture for G = GL( N), using the works of Brundan and Kleshchev interpreting the algebra {U(mathfrak{g},e)} in terms of certain shifted Yangians.
Finite element method for nonlinear Riesz space fractional diffusion equations on irregular domains
NASA Astrophysics Data System (ADS)
Yang, Z.; Yuan, Z.; Nie, Y.; Wang, J.; Zhu, X.; Liu, F.
2017-02-01
In this paper, we consider two-dimensional Riesz space fractional diffusion equations with nonlinear source term on convex domains. Applying Galerkin finite element method in space and backward difference method in time, we present a fully discrete scheme to solve Riesz space fractional diffusion equations. Our breakthrough is developing an algorithm to form stiffness matrix on unstructured triangular meshes, which can help us to deal with space fractional terms on any convex domain. The stability and convergence of the scheme are also discussed. Numerical examples are given to verify accuracy and stability of our scheme.
Application of a new finite difference algorithm for computational aeroacoustics
NASA Technical Reports Server (NTRS)
Goodrich, John W.
1995-01-01
Acoustic problems have become extremely important in recent years because of research efforts such as the High Speed Civil Transport program. Computational aeroacoustics (CAA) requires a faithful representation of wave propagation over long distances, and needs algorithms that are accurate and boundary conditions that are unobtrusive. This paper applies a new finite difference method and boundary algorithm to the Linearized Euler Equations (LEE). The results demonstrate the ability of a new fourth order propagation algorithm to accurately simulate the genuinely multidimensional wave dynamics of acoustic propagation in two space dimensions with the LEE. The results also show the ability of a new outflow boundary condition and fourth order algorithm to pass the evolving solution from the computational domain with no perceptible degradation of the solution remaining within the domain.
Finite difference modeling of Biot's poroelastic equations atseismic frequencies
Masson, Y.J.; Pride, S.R.; Nihei, K.T.
2006-02-24
Across the seismic band of frequencies (loosely defined as<10 kHz), a seismic wave propagating through a porous material willcreate flow in the pore space that is laminar; that is, in thislow-frequency "seismic limit," the development of viscous boundary layersin the pores need not be modeled. An explicit time steppingstaggered-grid finite difference scheme is presented for solving Biot'sequations of poroelasticity in this low-frequency limit. A key part ofthis work is the establishment of rigorous stability conditions. It isdemonstrated that over a wide range of porous material properties typicalof sedimentary rock and despite the presenceof fluid pressure diffusion(Biot slow waves), the usual Courant condition governs the stability asif the problem involved purely elastic waves. The accuracy of the methodis demonstrated by comparing to exact analytical solutions for both fastcompressional waves and slow waves. Additional numerical modelingexamples are also presented.
High order accurate finite difference schemes based on symmetry preservation
NASA Astrophysics Data System (ADS)
Ozbenli, Ersin; Vedula, Prakash
2016-11-01
A new algorithm for development of high order accurate finite difference schemes for numerical solution of partial differential equations using Lie symmetries is presented. Considering applicable symmetry groups (such as those relevant to space/time translations, Galilean transformation, scaling, rotation and projection) of a partial differential equation, invariant numerical schemes are constructed based on the notions of moving frames and modified equations. Several strategies for construction of invariant numerical schemes with a desired order of accuracy are analyzed. Performance of the proposed algorithm is demonstrated using analysis of one-dimensional partial differential equations, such as linear advection diffusion equations inviscid Burgers equation and viscous Burgers equation, as our test cases. Through numerical simulations based on these examples, the expected improvement in accuracy of invariant numerical schemes (up to fourth order) is demonstrated. Advantages due to implementation and enhanced computational efficiency inherent in our proposed algorithm are presented. Extension of the basic framework to multidimensional partial differential equations is also discussed.
Hybrid finite element-finite difference method for thermal analysis of blood vessels.
Blanchard, C H; Gutierrez, G; White, J A; Roemer, R B
2000-01-01
A hybrid finite-difference/finite-element technique for the thermal analysis of blood vessels embedded in perfused tissue has been developed and evaluated. This method provides efficient and accurate solutions to the conjugated heat transfer problem of convection by blood coupled to conduction in the tissue. The technique uses a previously developed 3D automatic meshing method for creating a finite element mesh in the tissue surrounding the vessels, coupled iteratively with a 1-D marching finite difference method for the interior of the vessels. This hybrid technique retains the flexibility and ease of automated finite-element meshing techniques for modelling the complex geometry of blood vessels and irregularly shaped tissues, and speeds the solution time by using a simple finite-difference method to calculate the bulk mean temperatures within all blood vessels. The use of the 1D finite-difference technique in the blood vessels also eliminates the large computer memory requirements needed to accurately solve large vessel network problems when fine FE meshes are used in the interior of vessels. The accuracy of the hybrid technique has been verified against previously verified numerical solutions. In summary, the hybrid technique combines the accuracy and flexibility found in automated finite-element techniques, with the speed and reduction of computational memory requirements associated with the 1D finite-difference technique, something which has not been done before. This method, thus, has the potential to provide accurate, flexible and relatively fast solutions for the thermal analysis of coupled perfusion/blood vessel problems, and large vessel network problems.
Finite difference solutions to shocked acoustic waves
NASA Technical Reports Server (NTRS)
Walkington, N. J.; Eversman, W.
1983-01-01
The MacCormack, Lambda and split flux finite differencing schemes are used to solve a one dimensional acoustics problem. Two duct configurations were considered, a uniform duct and a converging-diverging nozzle. Asymptotic solutions for these two ducts are compared with the numerical solutions. When the acoustic amplitude and frequency are sufficiently high the acoustic signal shocks. This condition leads to a deterioration of the numerical solutions since viscous terms may be required if the shock is to be resolved. A continuous uniform duct solution is considered to demonstrate how the viscous terms modify the solution. These results are then compared with a shocked solution with and without viscous terms. Generally it is found that the most accurate solutions are those obtained using the minimum possible viscosity coefficients. All of the schemes considered give results accurate enough for acoustic power calculations with no one scheme performing significantly better than the others.
A Time Decomposition Method to Space-Time Finite Elements for the Dirac Equation
NASA Astrophysics Data System (ADS)
Lim, Hyun; Kurlej, Arthur; Comeau, Olivia; Stegmeier, Nicholas; Kimn, Jung-Han
2017-01-01
Dirac equation is a relativistic wave equation that describes spin-1/2 massive particles such as electrons and quarks. Furthermore, this system can be extended with different physical aspects such as electromagnetic interaction. However, most of these system cannot be solved analytically. Therefore, numerical simulations are required to understand the nature of these systems. In this work, we examine the behavior of the gauge free, low-mass regime Dirac equation using space-time finite elements with time decomposition method. The purpose of this research is to present a new computational way for stable parallelizable algorithm of the physical system. We discretize space and time together for the entire domain using a finite element space which does not separate time and space basis functions. We also explore the effectiveness of the time decomposition preconditioner, time-additive Schwarz preconditioner with KSP (Krylov Subspace Methods) solvers for this problem.
Coupled finite-difference/finite-element approach for wing-body aeroelasticity
NASA Technical Reports Server (NTRS)
Guruswamy, Guru P.
1992-01-01
Computational methods using finite-difference approaches for fluids and finite-element approaches for structures have individually advanced to solve almost full-aircraft configurations. However, coupled approaches to solve fluid/structural interaction problems are still in their early stages of development, particularly for complex geometries using complete equations such as the Euler/Navier-Stokes equations. Earlier work demonstrated the success of coupling finite-difference and finite-element methods for simple wing configurations using the Euler/Navier-Stokes equations. In this paper, the same approach is extended for general wing-body configurations. The structural properties are represented by beam-type finite elements. The flow is modeled using the Euler/Navier-Stokes equations. A general procedure to fully couple structural finite-element boundary conditions with fluid finite-difference boundary conditions is developed for wing-body configurations. Computations are made using moving grids that adapt to wing-body structural deformations. Results are illustrated for a typical wing-body configuration.
Application of a novel finite difference method to dynamic crack problems
NASA Technical Reports Server (NTRS)
Chen, Y. M.; Wilkins, M. L.
1976-01-01
A versatile finite difference method (HEMP and HEMP 3D computer programs) was developed originally for solving dynamic problems in continuum mechanics. It was extended to analyze the stress field around cracks in a solid with finite geometry subjected to dynamic loads and to simulate numerically the dynamic fracture phenomena with success. This method is an explicit finite difference method applied to the Lagrangian formulation of the equations of continuum mechanics in two and three space dimensions and time. The calculational grid moves with the material and in this way it gives a more detailed description of the physics of the problem than the Eulerian formulation.
A hybrid finite-volume and finite difference scheme for depth-integrated non-hydrostatic model
NASA Astrophysics Data System (ADS)
Yin, Jing; Sun, Jia-wen; Wang, Xing-gang; Yu, Yong-hai; Sun, Zhao-chen
2017-06-01
A depth-integrated, non-hydrostatic model with hybrid finite difference and finite volume numerical algorithm is proposed in this paper. By utilizing a fraction step method, the governing equations are decomposed into hydrostatic and non-hydrostatic parts. The first part is solved by using the finite volume conservative discretization method, whilst the latter is considered by solving discretized Poisson-type equations with the finite difference method. The second-order accuracy, both in time and space, of the finite volume scheme is achieved by using an explicit predictor-correction step and linear construction of variable state in cells. The fluxes across the cell faces are computed in a Godunov-based manner by using MUSTA scheme. Slope and flux limiting technique is used to equip the algorithm with total variation dimensioning property for shock capturing purpose. Wave breaking is treated as a shock by switching off the non-hydrostatic pressure in the steep wave front locally. The model deals with moving wet/dry front in a simple way. Numerical experiments are conducted to verify the proposed model.
Decomposition of Fuzzy Soft Sets with Finite Value Spaces
Jun, Young Bae
2014-01-01
The notion of fuzzy soft sets is a hybrid soft computing model that integrates both gradualness and parameterization methods in harmony to deal with uncertainty. The decomposition of fuzzy soft sets is of great importance in both theory and practical applications with regard to decision making under uncertainty. This study aims to explore decomposition of fuzzy soft sets with finite value spaces. Scalar uni-product and int-product operations of fuzzy soft sets are introduced and some related properties are investigated. Using t-level soft sets, we define level equivalent relations and show that the quotient structure of the unit interval induced by level equivalent relations is isomorphic to the lattice consisting of all t-level soft sets of a given fuzzy soft set. We also introduce the concepts of crucial threshold values and complete threshold sets. Finally, some decomposition theorems for fuzzy soft sets with finite value spaces are established, illustrated by an example concerning the classification and rating of multimedia cell phones. The obtained results extend some classical decomposition theorems of fuzzy sets, since every fuzzy set can be viewed as a fuzzy soft set with a single parameter. PMID:24558342
Geometrically Nonlinear Finite Element Analysis of a Composite Space Reflector
NASA Technical Reports Server (NTRS)
Lee, Kee-Joo; Leet, Sung W.; Clark, Greg; Broduer, Steve (Technical Monitor)
2001-01-01
Lightweight aerospace structures, such as low areal density composite space reflectors, are highly flexible and may undergo large deflection under applied loading, especially during the launch phase. Accordingly, geometrically nonlinear analysis that takes into account the effect of finite rotation may be needed to determine the deformed shape for a clearance check and the stress and strain state to ensure structural integrity. In this study, deformation of the space reflector is determined under static conditions using a geometrically nonlinear solid shell finite element model. For the solid shell element formulation, the kinematics of deformation is described by six variables that are purely vector components. Because rotational angles are not used, this approach is free of the limitations of small angle increments. This also allows easy connections between substructures and large load increments with respect to the conventional shell formulation using rotational parameters. Geometrically nonlinear analyses were carried out for three cases of static point loads applied at selected points. A chart shows results for a case when the load is applied at the center point of the reflector dish. The computed results capture the nonlinear behavior of the composite reflector as the applied load increases. Also, they are in good agreement with the data obtained by experiments.
Geometrically Nonlinear Finite Element Analysis of a Composite Space Reflector
NASA Technical Reports Server (NTRS)
Lee, Kee-Joo; Leet, Sung W.; Clark, Greg; Broduer, Steve (Technical Monitor)
2001-01-01
Lightweight aerospace structures, such as low areal density composite space reflectors, are highly flexible and may undergo large deflection under applied loading, especially during the launch phase. Accordingly, geometrically nonlinear analysis that takes into account the effect of finite rotation may be needed to determine the deformed shape for a clearance check and the stress and strain state to ensure structural integrity. In this study, deformation of the space reflector is determined under static conditions using a geometrically nonlinear solid shell finite element model. For the solid shell element formulation, the kinematics of deformation is described by six variables that are purely vector components. Because rotational angles are not used, this approach is free of the limitations of small angle increments. This also allows easy connections between substructures and large load increments with respect to the conventional shell formulation using rotational parameters. Geometrically nonlinear analyses were carried out for three cases of static point loads applied at selected points. A chart shows results for a case when the load is applied at the center point of the reflector dish. The computed results capture the nonlinear behavior of the composite reflector as the applied load increases. Also, they are in good agreement with the data obtained by experiments.
Decomposition of fuzzy soft sets with finite value spaces.
Feng, Feng; Fujita, Hamido; Jun, Young Bae; Khan, Madad
2014-01-01
The notion of fuzzy soft sets is a hybrid soft computing model that integrates both gradualness and parameterization methods in harmony to deal with uncertainty. The decomposition of fuzzy soft sets is of great importance in both theory and practical applications with regard to decision making under uncertainty. This study aims to explore decomposition of fuzzy soft sets with finite value spaces. Scalar uni-product and int-product operations of fuzzy soft sets are introduced and some related properties are investigated. Using t-level soft sets, we define level equivalent relations and show that the quotient structure of the unit interval induced by level equivalent relations is isomorphic to the lattice consisting of all t-level soft sets of a given fuzzy soft set. We also introduce the concepts of crucial threshold values and complete threshold sets. Finally, some decomposition theorems for fuzzy soft sets with finite value spaces are established, illustrated by an example concerning the classification and rating of multimedia cell phones. The obtained results extend some classical decomposition theorems of fuzzy sets, since every fuzzy set can be viewed as a fuzzy soft set with a single parameter.
A Finite Difference-Augmented Peridynamics Method for Wave Dispersion
2014-10-21
ARL-RP-0531 ● AUG 2015 US Army Research Laboratory A Finite Difference- Augmented Peridynamics Method for Wave Dispersion by...AUG 2015 US Army Research Laboratory A Finite Difference- Augmented Peridynamics Method for Wave Dispersion by Raymond A Wildman and George...Difference- Augmented Peridynamics Method for Wave Dispersion 5a. CONTRACT NUMBER 5b. GRANT NUMBER 5c. PROGRAM ELEMENT NUMBER 6. AUTHOR(S
Numerical techniques in linear duct acoustics. [finite difference and finite element analyses
NASA Technical Reports Server (NTRS)
Baumeister, K. J.
1980-01-01
Both finite difference and finite element analyses of small amplitude (linear) sound propagation in straight and variable area ducts with flow, as might be found in a typical turboject engine duct, muffler, or industrial ventilation system, are reviewed. Both steady state and transient theories are discussed. Emphasis is placed on the advantages and limitations associated with the various numerical techniques. Examples of practical problems are given for which the numerical techniques have been applied.
Vibration analysis of rotating turbomachinery blades by an improved finite difference method
NASA Technical Reports Server (NTRS)
Subrahmanyam, K. B.; Kaza, K. R. V.
1985-01-01
The problem of calculating the natural frequencies and mode shapes of rotating blades is solved by an improved finite difference procedure based on second-order central differences. Lead-lag, flapping and coupled bending-torsional vibration cases of untwisted blades are considered. Results obtained by using the present improved theory have been observed to be close lower bound solutions. The convergence has been found to be rapid in comparison with the classical first-order finite difference method. While the computational space and time required by the present approach is observed to be almost the same as that required by the first-order theory for a given mesh size, accuracies of practical interest can be obtained by using the improved finite difference procedure with a relatively smaller matrix size, in contrast to the classical finite difference procedure which requires either a larger matrix or an extrapolation procedure for improvement in accuracy.
Comparison of different precondtioners for nonsymmtric finite volume element methods
Mishev, I.D.
1996-12-31
We consider a few different preconditioners for the linear systems arising from the discretization of 3-D convection-diffusion problems with the finite volume element method. Their theoretical and computational convergence rates are compared and discussed.
Thermal finite-element analysis of space shuttle main engine turbine blade
NASA Technical Reports Server (NTRS)
Abdul-Aziz, Ali; Tong, Michael T.; Kaufman, Albert
1987-01-01
Finite-element, transient heat transfer analyses were performed for the first-stage blades of the space shuttle main engine (SSME) high-pressure fuel turbopump. The analyses were based on test engine data provided by Rocketdyne. Heat transfer coefficients were predicted by performing a boundary-layer analysis at steady-state conditions with the STAN5 boundary-layer code. Two different peak-temperature overshoots were evaluated for the startup transient. Cutoff transient conditions were also analyzed. A reduced gas temperature profile based on actual thermocouple data was also considered. Transient heat transfer analyses were conducted with the MARC finite-element computer code.
Minimum divergence viscous flow simulation through finite difference and regularization techniques
NASA Astrophysics Data System (ADS)
Victor, Rodolfo A.; Mirabolghasemi, Maryam; Bryant, Steven L.; Prodanović, Maša
2016-09-01
We develop a new algorithm to simulate single- and two-phase viscous flow through a three-dimensional Cartesian representation of the porous space, such as those available through X-ray microtomography. We use the finite difference method to discretize the governing equations and also propose a new method to enforce the incompressible flow constraint under zero Neumann boundary conditions for the velocity components. Finite difference formulation leads to fast parallel implementation through linear solvers for sparse matrices, allowing relatively fast simulations, while regularization techniques used on solving inverse problems lead to the desired incompressible fluid flow. Tests performed using benchmark samples show good agreement with experimental/theoretical values. Additional tests are run on Bentheimer and Buff Berea sandstone samples with available laboratory measurements. We compare the results from our new method, based on finite differences, with an open source finite volume implementation as well as experimental results, specifically to evaluate the benefits and drawbacks of each method. Finally, we calculate relative permeability by using this modified finite difference technique together with a level set based algorithm for multi-phase fluid distribution in the pore space. To our knowledge this is the first time regularization techniques are used in combination with finite difference fluid flow simulations.
Incoherent systems and coverings in finite dimensional Banach spaces
Temlyakov, V N
2014-05-31
We discuss the construction of coverings of the unit ball of a finite dimensional Banach space. There is a well-known technique based on comparing volumes which gives upper and lower bounds on covering numbers. However, this technique does not provide a method for constructing good coverings. Here we study incoherent systems and apply them to construct good coverings. We use the following strategy. First, we build a good covering using balls with a radius close to one. Second, we iterate this construction to obtain a good covering for any radius. We shall concentrate mainly on the first step of this strategy. Bibliography: 14 titles.
All-electron Kohn-Sham density functional theory on hierarchic finite element spaces
NASA Astrophysics Data System (ADS)
Schauer, Volker; Linder, Christian
2013-10-01
In this work, a real space formulation of the Kohn-Sham equations is developed, making use of the hierarchy of finite element spaces from different polynomial order. The focus is laid on all-electron calculations, having the highest requirement onto the basis set, which must be able to represent the orthogonal eigenfunctions as well as the electrostatic potential. A careful numerical analysis is performed, which points out the numerical intricacies originating from the singularity of the nuclei and the necessity for approximations in the numerical setting, with the ambition to enable solutions within a predefined accuracy. In this context the influence of counter-charges in the Poisson equation, the requirement of a finite domain size, numerical quadratures and the mesh refinement are examined as well as the representation of the electrostatic potential in a high order finite element space. The performance and accuracy of the method is demonstrated in computations on noble gases. In addition the finite element basis proves its flexibility in the calculation of the bond-length as well as the dipole moment of the carbon monoxide molecule.
All-electron Kohn–Sham density functional theory on hierarchic finite element spaces
Schauer, Volker; Linder, Christian
2013-10-01
In this work, a real space formulation of the Kohn–Sham equations is developed, making use of the hierarchy of finite element spaces from different polynomial order. The focus is laid on all-electron calculations, having the highest requirement onto the basis set, which must be able to represent the orthogonal eigenfunctions as well as the electrostatic potential. A careful numerical analysis is performed, which points out the numerical intricacies originating from the singularity of the nuclei and the necessity for approximations in the numerical setting, with the ambition to enable solutions within a predefined accuracy. In this context the influence of counter-charges in the Poisson equation, the requirement of a finite domain size, numerical quadratures and the mesh refinement are examined as well as the representation of the electrostatic potential in a high order finite element space. The performance and accuracy of the method is demonstrated in computations on noble gases. In addition the finite element basis proves its flexibility in the calculation of the bond-length as well as the dipole moment of the carbon monoxide molecule.
Ruess, Jakob
2015-12-28
Many stochastic models of biochemical reaction networks contain some chemical species for which the number of molecules that are present in the system can only be finite (for instance due to conservation laws), but also other species that can be present in arbitrarily large amounts. The prime example of such networks are models of gene expression, which typically contain a small and finite number of possible states for the promoter but an infinite number of possible states for the amount of mRNA and protein. One of the main approaches to analyze such models is through the use of equations for the time evolution of moments of the chemical species. Recently, a new approach based on conditional moments of the species with infinite state space given all the different possible states of the finite species has been proposed. It was argued that this approach allows one to capture more details about the full underlying probability distribution with a smaller number of equations. Here, I show that the result that less moments provide more information can only stem from an unnecessarily complicated description of the system in the classical formulation. The foundation of this argument will be the derivation of moment equations that describe the complete probability distribution over the finite state space but only low-order moments over the infinite state space. I will show that the number of equations that is needed is always less than what was previously claimed and always less than the number of conditional moment equations up to the same order. To support these arguments, a symbolic algorithm is provided that can be used to derive minimal systems of unconditional moment equations for models with partially finite state space.
Direct simulations of turbulent flow using finite-difference schemes
NASA Technical Reports Server (NTRS)
Rai, Man Mohan; Moin, Parviz
1989-01-01
A high-order accurate finite-difference approach is presented for calculating incompressible turbulent flow. The methods used include a kinetic energy conserving central difference scheme and an upwind difference scheme. The methods are evaluated in test cases for the evolution of small-amplitude disturbances and fully developed turbulent channel flow. It is suggested that the finite-difference approach can be applied to complex geometries more easilty than highly accurate spectral methods. It is concluded that the upwind scheme is a good candidate for direct simulations of turbulent flows over complex geometries.
Finite-difference schemes for anisotropic diffusion
Es, Bram van; Koren, Barry; Blank, Hugo J. de
2014-09-01
In fusion plasmas diffusion tensors are extremely anisotropic due to the high temperature and large magnetic field strength. This causes diffusion, heat conduction, and viscous momentum loss, to effectively be aligned with the magnetic field lines. This alignment leads to different values for the respective diffusive coefficients in the magnetic field direction and in the perpendicular direction, to the extent that heat diffusion coefficients can be up to 10{sup 12} times larger in the parallel direction than in the perpendicular direction. This anisotropy puts stringent requirements on the numerical methods used to approximate the MHD-equations since any misalignment of the grid may cause the perpendicular diffusion to be polluted by the numerical error in approximating the parallel diffusion. Currently the common approach is to apply magnetic field-aligned coordinates, an approach that automatically takes care of the directionality of the diffusive coefficients. This approach runs into problems at x-points and at points where there is magnetic re-connection, since this causes local non-alignment. It is therefore useful to consider numerical schemes that are tolerant to the misalignment of the grid with the magnetic field lines, both to improve existing methods and to help open the possibility of applying regular non-aligned grids. To investigate this, in this paper several discretization schemes are developed and applied to the anisotropic heat diffusion equation on a non-aligned grid.
Asymptotic analysis of numerical wave propagation in finite difference equations
NASA Technical Reports Server (NTRS)
Giles, M.; Thompkins, W. T., Jr.
1983-01-01
An asymptotic technique is developed for analyzing the propagation and dissipation of wave-like solutions to finite difference equations. It is shown that for each fixed complex frequency there are usually several wave solutions with different wavenumbers and the slowly varying amplitude of each satisfies an asymptotic amplitude equation which includes the effects of smoothly varying coefficients in the finite difference equations. The local group velocity appears in this equation as the velocity of convection of the amplitude. Asymptotic boundary conditions coupling the amplitudes of the different wave solutions are also derived. A wavepacket theory is developed which predicts the motion, and interaction at boundaries, of wavepackets, wave-like disturbances of finite length. Comparison with numerical experiments demonstrates the success and limitations of the theory. Finally an asymptotic global stability analysis is developed.
A comparison of the finite difference and finite element methods for heat transfer calculations
NASA Technical Reports Server (NTRS)
Emery, A. F.; Mortazavi, H. R.
1982-01-01
The finite difference method and finite element method for heat transfer calculations are compared by describing their bases and their application to some common heat transfer problems. In general it is noted that neither method is clearly superior, and in many instances, the choice is quite arbitrary and depends more upon the codes available and upon the personal preference of the analyst than upon any well defined advantages of one method. Classes of problems for which one method or the other is better suited are defined.
Finite difference approximations for a fractional advection diffusion problem
NASA Astrophysics Data System (ADS)
Sousa, Ercília
2009-06-01
The use of the conventional advection diffusion equation in many physical situations has been questioned by many investigators in recent years and alternative diffusion models have been proposed. Fractional space derivatives are used to model anomalous diffusion or dispersion, where a particle plume spreads at a rate inconsistent with the classical Brownian motion model. When a fractional derivative replaces the second derivative in a diffusion or dispersion model, it leads to enhanced diffusion, also called superdiffusion. We consider a one-dimensional advection-diffusion model, where the usual second-order derivative gives place to a fractional derivative of order α, with 1<α⩽2. We derive explicit finite difference schemes which can be seen as generalizations of already existing schemes in the literature for the advection-diffusion equation. We present the order of accuracy of the schemes and in order to show its convergence we prove they are stable under certain conditions. In the end we present a test problem.
Convergence of finite difference transient response computations for thin shells.
NASA Technical Reports Server (NTRS)
Sobel, L. H.; Geers, T. L.
1973-01-01
Numerical studies pertaining to the limits of applicability of the finite difference method in the solution of linear transient shell response problems are performed, and a computational procedure for the use of the method is recommended. It is found that the only inherent limitation of the finite difference method is its inability to reproduce accurately response discontinuities. This is not a serious limitation in view of natural constraints imposed by the extension of Saint Venant's principle to transient response problems. It is also found that the short wavelength limitations of thin shell (Bernoulli-Euler) theory create significant convergence difficulties in computed response to certain types of transverse excitations. These difficulties may be overcome, however, through proper selection of finite difference mesh dimensions and temporal smoothing of the excitation.
Selecting step sizes in sensitivity analysis by finite differences
NASA Technical Reports Server (NTRS)
Iott, J.; Haftka, R. T.; Adelman, H. M.
1985-01-01
This paper deals with methods for obtaining near-optimum step sizes for finite difference approximations to first derivatives with particular application to sensitivity analysis. A technique denoted the finite difference (FD) algorithm, previously described in the literature and applicable to one derivative at a time, is extended to the calculation of several simultaneously. Both the original and extended FD algorithms are applied to sensitivity analysis for a data-fitting problem in which derivatives of the coefficients of an interpolation polynomial are calculated with respect to uncertainties in the data. The methods are also applied to sensitivity analysis of the structural response of a finite-element-modeled swept wing. In a previous study, this sensitivity analysis of the swept wing required a time-consuming trial-and-error effort to obtain a suitable step size, but it proved to be a routine application for the extended FD algorithm herein.
A comparative analysis of finite element and finite difference methods for free surface transport
Chen, S.C.; Vafai, K. . Dept of Mechanical Engineering)
1993-09-01
The present work consists of the comparative evaluation of the finite element method (FEM) and the finite difference method (FDM) for the analysis of free surface transport within a hollow ampule. The phenomenon of motion reversal of the free surfaces obtained earlier by the FDM is also analyzed by the FEM. It is found that the times at which the motion reversal occurs are independent of the applied pressure difference for any fixed dimension of the hollow ampule. Furthermore, it appears that the displacement of the inner and outer free surfaces varies linearly with the magnitude of the applied pressure difference. Finally, detailed comparative discussion is presented on the differences between the results obtained by FDM and FEM.
A language learning model for finite parameter spaces.
Niyogi, P; Berwick, R C
1996-01-01
This paper shows how to formally characterize language learning in a finite parameter space, for instance, in the principles-and-parameters approach to language, as a Markov structure. New language learning results follow directly; we can explicitly calculate how many positive examples on average ("sample complexity") it will take for a learner to correctly identify a target language with high probability. We show how sample complexity varies with input distributions and learning regimes. In particular we find that the average time to converge under reasonable language input distributions for a simple three-parameter system first described by Gibson and Wexler (1994) is psychologically plausible, in the range of 100-150 positive examples. We further find that a simple random step algorithm-that is, simply jumping from one language hypothesis to another rather than changing one parameter at a time-works faster and always converges to the right target language, in contrast to the single-step, local parameter setting method advocated in some recent work.
NASA Astrophysics Data System (ADS)
Sparrow, Victor Ward
1990-01-01
This study has concerned the propagation of finite amplitude, i.e. weakly non-linear, acoustical blast waves from explosions over hard and porous media models of outdoor ground surfaces. The nonlinear acoustic propagation effects require a numerical solution in the time domain. To model a porous ground surface, which in the frequency domain exhibits a finite impedance, the linear phenomenological porous model of Morse and Ingard was used. The phenomenological equations are solved in the time domain for coupling with the time domain propagation solution in the air. The numerical solution is found through the method of finite differences. The second-order in time and fourth -order in space MacCormack method was used in the air, and the second-order in time and space MacCormack method was used in the porous medium modeling the ground. Two kinds of numerical absorbing boundary conditions were developed for the air propagation equations to truncate the physical domain for solution on a computer. Radiation conditions first were used on those sides of the domain where there were outgoing waves. Characteristic boundary conditions secondly are employed near the acoustic source. The numerical model agreed well with the Pestorius algorithm for the propagation of electric spark pulses in the free field, and with a result of Pfriem for normal plane reflection off a hard surface. In addition, curves of pressure amplification versus incident angle for waves obliquely incident on the hard and porous surfaces were produced which are similar to those in the literature. The model predicted that near grazing finite amplitude acoustic blast waves decay with distance over hard surfaces as r to the power -1.2. This result is consistent with the work of Reed. For propagation over the porous ground surface, the model predicted that this surface decreased the decay rate with distance for the larger blasts compared to the rate expected in the linear acoustics limit.
The finite-difference matrix for beam propagation: eigenvalues and eigenvectors
NASA Astrophysics Data System (ADS)
Paxton, Alan H.
2016-03-01
The partial differential equation for the three dimensional propagation of a light beam may be solved numerically by applying finite-difference techniques. We consider the matrix equation for the finite-difference, alternating direction implicit (ADI), numerical solution of the paraxial wave equation for the free-space propagation of light beams. The matrix is tridiagonal. It is also a Toeplitz matrix; Each diagonal descending from left to right is constant. Eigenvalues and eigenvectors are known for such matrices. The equation can be solved by making use of the orthogonality property of the eigenvectors.
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.
Finite-difference and finite-volume methods for nonlinear standing ultrasonic waves in fluid media.
Vanhille, C; Conde, C; Campos-Pozuelo, C
2004-04-01
In the framework of the application of high-power ultrasonics in industrial processing in fluid media, the mathematical prediction of the acoustical parameters inside resonators should improve the development of practical systems. This can be achieved by the use of numerical tools able to treat the nonlinear acoustics involved in these phenomena. In particular, effects like nonlinear distortion and nonlinear attenuation are fundamental in applications. In this paper, three one-dimensional numerical models in the time domain for calculating the nonlinear acoustic field inside a one-dimensional resonant cavity are presented and compared. They are based on the finite-difference and the finite-volume methods. These different algorithms solve the differential equations, from the linear up to the strongly nonlinear case (including weak shock). Some physical results obtained from the modelling of ultrasonic waves and a comparison of the efficiency of the different algorithms are presented.
Finite difference time domain modelling of particle accelerators
Jurgens, T.G.; Harfoush, F.A.
1989-03-01
Finite Difference Time Domain (FDTD) modelling has been successfully applied to a wide variety of electromagnetic scattering and interaction problems for many years. Here the method is extended to incorporate the modelling of wake fields in particle accelerators. Algorithmic comparisons are made to existing wake field codes, such as MAFIA T3. 9 refs., 7 figs.
Finite Difference Solution for Biopotentials of Axially Symmetric Cells
Klee, Maurice; Plonsey, Robert
1972-01-01
The finite difference equations necessary for calculating the three-dimensional, time-varying biopotentials within and surrounding axially symmetric cells are presented. The method of sucessive overrelaxation is employed to solve these equations and is shown to be rapidly convergent and accurate for the exemplary problem of a spheroidal cell under uniform field stimulation. PMID:4655665
Direct Finite-Difference Simulations Of Turbulent Flow
NASA Technical Reports Server (NTRS)
Rai, Man Mohan; Moin, Parviz
1991-01-01
Report discusses use of upwind-biased finite-difference numerical-integration scheme to simulate evolution of small disturbances and fully developed turbulence in three-dimensional flow of viscous, incompressible fluid in channel. Involves use of computational grid sufficiently fine to resolve motion of fluid at all relevant length scales.
NASA Astrophysics Data System (ADS)
Jian, Wang; Xiaohong, Meng; Hong, Liu; Wanqiu, Zheng; Yaning, Liu; Sheng, Gui; Zhiyang, Wang
2017-03-01
Full waveform inversion and reverse time migration are active research areas for seismic exploration. Forward modeling in the time domain determines the precision of the results, and numerical solutions of finite difference have been widely adopted as an important mathematical tool for forward modeling. In this article, the optimum combined of window functions was designed based on the finite difference operator using a truncated approximation of the spatial convolution series in pseudo-spectrum space, to normalize the outcomes of existing window functions for different orders. The proposed combined window functions not only inherit the characteristics of the various window functions, to provide better truncation results, but also control the truncation error of the finite difference operator manually and visually by adjusting the combinations and analyzing the characteristics of the main and side lobes of the amplitude response. Error level and elastic forward modeling under the proposed combined system were compared with outcomes from conventional window functions and modified binomial windows. Numerical dispersion is significantly suppressed, which is compared with modified binomial window function finite-difference and conventional finite-difference. Numerical simulation verifies the reliability of the proposed method.
Solving parabolic and hyperbolic equations by the generalized finite difference method
NASA Astrophysics Data System (ADS)
Benito, J. J.; Urena, F.; Gavete, L.
2007-12-01
Classical finite difference schemes are in wide use today for approximately solving partial differential equations of mathematical physics. An evolution of the method of finite differences has been the development of generalized finite difference (GFD) method, that can be applied to irregular grids of points. In this paper the extension of the GFD to the explicit solution of parabolic and hyperbolic equations has been developed for partial differential equations with constant coefficients in the cases of considering one, two or three space dimensions. The convergence of the method has been studied and the truncation errors over irregular grids are given. Different examples have been solved using the explicit finite difference formulae and the criterion of stability. This has been expressed in function of the coefficients of the star equation for irregular clouds of nodes in one, two or three space dimensions. The numerical results show the accuracy obtained over irregular grids. This paper also includes the study of the maximum local error and the global error for different examples of parabolic and hyperbolic time-dependent equations.
Time dependent wave envelope finite difference analysis of sound propagation
NASA Technical Reports Server (NTRS)
Baumeister, K. J.
1984-01-01
A transient finite difference wave envelope formulation is presented for sound propagation, without steady flow. Before the finite difference equations are formulated, the governing wave equation is first transformed to a form whose solution tends not to oscillate along the propagation direction. This transformation reduces the required number of grid points by an order of magnitude. Physically, the transformed pressure represents the amplitude of the conventional sound wave. The derivation for the wave envelope transient wave equation and appropriate boundary conditions are presented as well as the difference equations and stability requirements. To illustrate the method, example solutions are presented for sound propagation in a straight hard wall duct and in a two dimensional straight soft wall duct. The numerical results are in good agreement with exact analytical results.
Experimentally constructing finite difference algorithms in numerical relativity
NASA Astrophysics Data System (ADS)
Anderson, Matthew; Neilsen, David; Matzner, Richard
2002-04-01
Computational studies of gravitational waves require numerical algorithms with long-term stability (necessary for convergence). However, constructing stable finite difference algorithms (FDA) for the ADM formulation of the Einstein equations, especially in multiple dimensions, has proven difficult. Most FDA's are constructed using rules of thumb gained from experience with simple model equations. To search for FDA's with improved stability, we adopt a brute-force approach, where we systematically test thousands of numerical schemes. We sort the spatial derivatives of the Einstein equations into groups, and parameterize each group by finite difference type (centered or upwind) and order. Furthermore, terms proportional to the constraints are added to the evolution equations with additional parameters. A spherically symmetric, excised Schwarzschild black hole (one dimension) and linearized waves in multiple dimensions are used as model systems to evaluate the different numerical schemes.
NASA Technical Reports Server (NTRS)
Steger, J. L.; Caradonna, F. X.
1980-01-01
An implicit finite difference procedure is developed to solve the unsteady full potential equation in conservation law form. Computational efficiency is maintained by use of approximate factorization techniques. The numerical algorithm is first order in time and second order in space. A circulation model and difference equations are developed for lifting airfoils in unsteady flow; however, thin airfoil body boundary conditions have been used with stretching functions to simplify the development of the numerical algorithm.
NASA Technical Reports Server (NTRS)
Mccoy, M. J.
1980-01-01
Various finite difference techniques used to solve Laplace's equation are compared. Curvilinear coordinate systems are used on two dimensional regions with irregular boundaries, specifically, regions around circles and airfoils. Truncation errors are analyzed for three different finite difference methods. The false boundary method and two point and three point extrapolation schemes, used when having the Neumann boundary condition are considered and the effects of spacing and nonorthogonality in the coordinate systems are studied.
The Laguerre finite difference one-way equation solver
NASA Astrophysics Data System (ADS)
Terekhov, Andrew V.
2017-05-01
This paper presents a new finite difference algorithm for solving the 2D one-way wave equation with a preliminary approximation of a pseudo-differential operator by a system of partial differential equations. As opposed to the existing approaches, the integral Laguerre transform instead of Fourier transform is used. After carrying out the approximation of spatial variables it is possible to obtain systems of linear algebraic equations with better computing properties and to reduce computer costs for their solution. High accuracy of calculations is attained at the expense of employing finite difference approximations of higher accuracy order that are based on the dispersion-relationship-preserving method and the Richardson extrapolation in the downward continuation direction. The numerical experiments have verified that as compared to the spectral difference method based on Fourier transform, the new algorithm allows one to calculate wave fields with a higher degree of accuracy and a lower level of numerical noise and artifacts including those for non-smooth velocity models. In the context of solving the geophysical problem the post-stack migration for velocity models of the types Syncline and Sigsbee2A has been carried out. It is shown that the images obtained contain lesser noise and are considerably better focused as compared to those obtained by the known Fourier Finite Difference and Phase-Shift Plus Interpolation methods. There is an opinion that purely finite difference approaches do not allow carrying out the seismic migration procedure with sufficient accuracy, however the results obtained disprove this statement. For the supercomputer implementation it is proposed to use the parallel dichotomy algorithm when solving systems of linear algebraic equations with block-tridiagonal matrices.
Algorithmic vs. finite difference Jacobians for infrared atmospheric radiative transfer
NASA Astrophysics Data System (ADS)
Schreier, Franz; Gimeno García, Sebastián; Vasquez, Mayte; Xu, Jian
2015-10-01
Jacobians, i.e. partial derivatives of the radiance and transmission spectrum with respect to the atmospheric state parameters to be retrieved from remote sensing observations, are important for the iterative solution of the nonlinear inverse problem. Finite difference Jacobians are easy to implement, but computationally expensive and possibly of dubious quality; on the other hand, analytical Jacobians are accurate and efficient, but the implementation can be quite demanding. GARLIC, our "Generic Atmospheric Radiation Line-by-line Infrared Code", utilizes algorithmic differentiation (AD) techniques to implement derivatives w.r.t. atmospheric temperature and molecular concentrations. In this paper, we describe our approach for differentiation of the high resolution infrared and microwave spectra and provide an in-depth assessment of finite difference approximations using "exact" AD Jacobians as a reference. The results indicate that the "standard" two-point finite differences with 1 K and 1% perturbation for temperature and volume mixing ratio, respectively, can exhibit substantial errors, and central differences are significantly better. However, these deviations do not transfer into the truncated singular value decomposition solution of a least squares problem. Nevertheless, AD Jacobians are clearly recommended because of the superior speed and accuracy.
Solving wave equation using finite differences and Taylor series
NASA Astrophysics Data System (ADS)
Nečasová, Gabriela; Kocina, Filip; Veigend, Petr; Chaloupka, Jan; Šátek, Václav; Kunovský, Jiří
2017-07-01
The paper deals with the numerical solution of partial differential equations (PDEs), especially wave equation. Two methods are used to obtain numerical solution of the wave equation. The Finite Difference Method (FDM) is used for transformation of wave equation to the system of ordinary differential equations (ODEs), different types of difference formulas are used. The influence of arithmetic to higher order difference formulas is also presented. The Modern Taylor Series Method (MTSM) allows to solve ODEs numerically with extremely high precision. An important feature of this method is an automatic integration order setting, i.e. using as many Taylor series terms as the defined accuracy requires.
Finite difference seismic modeling of axial magma chambers
Swift, S.A.; Dougherty, M.E.; Stephen, R.A. )
1990-11-01
The authors tested the feasibility of using finite difference methods to model seismic propagation at {approximately}10 Hx through a two-dimensional representation of an axial magma chamber with a thin, liquid lid. This technique produces time series of displacement or pressure at seafloor receivers to mimic a seismic refraction experiment and snapshots of P and S energy propagation. The results indicate that the implementation is stable for models with sharp velocity contrasts and complex geometries. The authors observe a high-energy, downward-traveling shear phase, observable only with borehole receivers, that would be useful in studying the nature and shape of magma chambers. The ability of finite difference methods to model high-order wave phenomena makes this method ideal for testing velocity models of spreading axes and for planning near-axis drilling of the East Pacific Rise in order to optimize the benefits from shear wave imaging of sub-axis structure.
A Random Finite Set Approach to Space Junk Tracking and Identification
2014-09-03
Final 3. DATES COVERED (From - To) 31 Jan 13 – 29 Apr 14 4. TITLE AND SUBTITLE A Random Finite Set Approach to Space Junk Tracking and...01-2013 to 29-04-2014 4. TITLE AND SUBTITLE A Random Finite Set Approach to Space Junk Tracking and Identification 5a. CONTRACT NUMBER FA2386-13...Prescribed by ANSI Std Z39-18 A Random Finite Set Approach to Space Junk Tracking and Indentification Ba-Ngu Vo1, Ba-Tuong Vo1, 1Department of
Finite element thermal-structural analysis of cable-stiffened space structues
NASA Technical Reports Server (NTRS)
Thornton, E. A.; Dechaumphai, P.; Pandey, A. K.
1984-01-01
Finite element thermal-structural analyses of large, cable-stiffened space structures are presented. A computational scheme for the calculation of prestresses in the cable-stiffened structures is also described. The determination of thermal loads on orbiting space structures due to environment heating is discussed briefly. Three finite element structural analysis techniques are presented for the analysis of prestressed structures. Linear, stress stiffening, and large displacement analysis techniques were investigated. These three techniques were employed for analysis of prestressed cable structures at different prestress levels. The analyses produced similar results at small prestress, but at higher prestress, differences between the results became significant. For the cable-stiffened structures studied, the linear analysis technique may not provide acceptable results. The stress stiffening analysis technique may yield results of acceptable accuracy depending upon the level of prestress. The large displacement analysis technique produced accurate results over a wide range of prestress and is recommended as a general analysis technique for thermal-structural analysis of cable-stiffened space structures.
Optimized Finite-Difference Coefficients for Hydroacoustic Modeling
NASA Astrophysics Data System (ADS)
Preston, L. A.
2014-12-01
Responsible utilization of marine renewable energy sources through the use of current energy converter (CEC) and wave energy converter (WEC) devices requires an understanding of the noise generation and propagation from these systems in the marine environment. Acoustic noise produced by rotating turbines, for example, could adversely affect marine animals and human-related marine activities if not properly understood and mitigated. We are utilizing a 3-D finite-difference acoustic simulation code developed at Sandia that can accurately propagate noise in the complex bathymetry in the near-shore to open ocean environment. As part of our efforts to improve computation efficiency in the large, high-resolution domains required in this project, we investigate the effects of using optimized finite-difference coefficients on the accuracy of the simulations. We compare accuracy and runtime of various finite-difference coefficients optimized via criteria such as maximum numerical phase speed error, maximum numerical group speed error, and L-1 and L-2 norms of weighted numerical group and phase speed errors over a given spectral bandwidth. We find that those coefficients optimized for L-1 and L-2 norms are superior in accuracy to those based on maximal error and can produce runtimes of 10% of the baseline case, which uses Taylor Series finite-difference coefficients at the Courant time step limit. We will present comparisons of the results for the various cases evaluated as well as recommendations for utilization of the cases studied. Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-AC04-94AL85000.
Finite difference time domain calculations of antenna mutual coupling
NASA Technical Reports Server (NTRS)
Luebbers, Raymond J.; Kunz, Karl S.
1991-01-01
The Finite Difference Time Domain (FDTD) technique was applied to a wide variety of electromagnetic analysis problems, including shielding and scattering. However, the method has not been exclusively applied to antennas. Here, calculations of self and mutual admittances between wire antennas are made using FDTD and compared with results obtained during the method of moments. The agreement is quite good, indicating the possibilities for FDTD application to antenna impedance and coupling.
Finite difference time domain calculations of antenna mutual coupling
NASA Technical Reports Server (NTRS)
Luebbers, Raymond J.; Kunz, Karl S.
1991-01-01
The Finite Difference Time Domain (FDTD) technique has been applied to a wide variety of electromagnetic analysis problems, including shielding and scattering. However, the method has not been extensively applied to antennas. In this short paper calculations of self and mutual admittances between wire antennas are made using FDTD and compared with results obtained using the Method of Moments. The agreement is quite good, indicating the possibilities for FDTD application to antenna impedance and coupling.
Finite difference time domain grid generation from AMC helicopter models
NASA Technical Reports Server (NTRS)
Cravey, Robin L.
1992-01-01
A simple technique is presented which forms a cubic grid model of a helicopter from an Aircraft Modeling Code (AMC) input file. The AMC input file defines the helicopter fuselage as a series of polygonal cross sections. The cubic grid model is used as an input to a Finite Difference Time Domain (FDTD) code to obtain predictions of antenna performance on a generic helicopter model. The predictions compare reasonably well with measured data.
Finite difference schemes for long-time integration
NASA Technical Reports Server (NTRS)
Haras, Zigo; Taasan, Shlomo
1993-01-01
Finite difference schemes for the evaluation of first and second derivatives are presented. These second order compact schemes were designed for long-time integration of evolution equations by solving a quadratic constrained minimization problem. The quadratic cost function measures the global truncation error while taking into account the initial data. The resulting schemes are applicable for integration times fourfold, or more, longer than similar previously studied schemes. A similar approach was used to obtain improved integration schemes.
An analysis of finite-difference and finite-volume formulations of conservation laws
NASA Technical Reports Server (NTRS)
Vinokur, Marcel
1986-01-01
Finite-difference and finite-volume formulations are analyzed in order to clear up the confusion concerning their application to the numerical solution of conservation laws. A new coordinate-free formulation of systems of conservation laws is developed, which clearly distinguishes the role of physical vectors from that of algebraic vectors which characterize the system. The analysis considers general types of equations--potential, Euler, and Navier-Stokes. Three-dimensional unsteady flows with time-varying grids are described using a single, consistent nomeclature for both formulations. Grid motion due to a non-inertial reference frame as well as flow adaptation is covered. In comparing the two formulations, it is found useful to distinguish between differences in numerical methods and differences in grid definition. The former plays a role for non-Cartesian grids, and results in only cosmetic differences in the manner in which geometric terms are handled. The differences in grid definition for the two formulations is found to be more important, since it affects the manner in which boundary conditions, zonal procedures, and grid singularities are handled at computational boundaries. The proper interpretation of strong and weak conservation-law forms for quasi-one-dimensional and axisymmetric flows is brought out.
An analysis of finite-difference and finite-volume formulations of conservation laws
NASA Technical Reports Server (NTRS)
Vinokur, Marcel
1989-01-01
Finite-difference and finite-volume formulations are analyzed in order to clear up the confusion concerning their application to the numerical solution of conservation laws. A new coordinate-free formulation of systems of conservation laws is developed, which clearly distinguishes the role of physical vectors from that of algebraic vectors which characterize the system. The analysis considers general types of equations: potential, Euler, and Navier-Stokes. Three-dimensional unsteady flows with time-varying grids are described using a single, consistent nomenclature for both formulations. Grid motion due to a non-inertial reference frame as well as flow adaptation is covered. In comparing the two formulations, it is found useful to distinguish between differences in numerical methods and differences in grid definition. The former plays a role for non-Cartesian grids, and results in only cosmetic differences in the manner in which geometric terms are handled. The differences in grid definition for the two formulations is found to be more important, since it affects the manner in which boundary conditions, zonal procedures, and grid singularities are handled at computational boundaries. The proper interpretation of strong and weak conservation-law forms for quasi-one-dimensional and axisymmetric flows is brought out.
Introduction to finite-difference methods for numerical fluid dynamics
Scannapieco, E.; Harlow, F.H.
1995-09-01
This work is intended to be a beginner`s exercise book for the study of basic finite-difference techniques in computational fluid dynamics. It is written for a student level ranging from high-school senior to university senior. Equations are derived from basic principles using algebra. Some discussion of partial-differential equations is included, but knowledge of calculus is not essential. The student is expected, however, to have some familiarity with the FORTRAN computer language, as the syntax of the computer codes themselves is not discussed. Topics examined in this work include: one-dimensional heat flow, one-dimensional compressible fluid flow, two-dimensional compressible fluid flow, and two-dimensional incompressible fluid flow with additions of the equations of heat flow and the {Kappa}-{epsilon} model for turbulence transport. Emphasis is placed on numerical instabilities and methods by which they can be avoided, techniques that can be used to evaluate the accuracy of finite-difference approximations, and the writing of the finite-difference codes themselves. Concepts introduced in this work include: flux and conservation, implicit and explicit methods, Lagrangian and Eulerian methods, shocks and rarefactions, donor-cell and cell-centered advective fluxes, compressible and incompressible fluids, the Boussinesq approximation for heat flow, Cartesian tensor notation, the Boussinesq approximation for the Reynolds stress tensor, and the modeling of transport equations. A glossary is provided which defines these and other terms.
On Love-type waves in a finitely deformed magnetoelastic layered half-space
NASA Astrophysics Data System (ADS)
Saxena, Prashant; Ogden, Ray W.
2012-12-01
In this paper, the propagation of Love-type waves in a homogeneously and finitely deformed layered half-space of an incompressible non-conducting magnetoelastic material in the presence of an initial uniform magnetic field is analyzed. The equations and boundary conditions governing linearized incremental motions superimposed on an underlying deformation and magnetic field for a magnetoelastic material are summarized and then specialized to a form appropriate for the study of Love-type waves in a layered half-space. The wave propagation problem is then analyzed for different directions of the initial magnetic field for two different magnetoelastic energy functions, which are generalizations of the standard neo-Hookean and Mooney-Rivlin elasticity models. The resulting wave speed characteristics in general depend significantly on the initial magnetic field as well as on the initial finite deformation, and the results are illustrated graphically for different combinations of these parameters. In the absence of a layer, shear horizontal surface waves do not exist in a purely elastic material, but the presence of a magnetic field normal to the sagittal plane makes such waves possible, these being analogous to Bleustein-Gulyaev waves in piezoelectric materials. Such waves are discussed briefly at the end of the paper.
Macroscopic traffic modeling with the finite difference method
Mughabghab, S.; Azarm, A.; Stock, D.
1996-03-15
A traffic congestion forecasting model (ATOP), developed in the present investigation, is described briefly. Several macroscopic models, based on the solution of the partial differential equation of conservation of vehicles by the finite difference method, were tested using actual traffic data. The functional form, as well as the parameters, of the equation of state which describes the relation between traffic speed and traffic density, were determined for a section of the Long Island Expressway. The Lax method and the forward difference technique were applied. The results of extensive tests showed that the Lax method, in addition to giving very good agreement with the traffic data, produces stable solutions.
Hassanein, A.M.
1987-01-01
The time dependent heat conduction equation that is solved in different coordinate systems is solved subject to various boundary conditions. Boundary conditions include surface heat flux, energy to vaporization of target materials, radiation from surface to surrounding, and possible phase change of material. This system of equations is subject to two moving boundaries. One moving boundary being the melt-solid interface because the surface heat flux may result in melting the surface of the exposed material. Another moving boundary is the receding surface as a result of evaporation of the wall material due to the continuous heating of the melted surface. Finite difference and the finite element methods are used and compared in such solution to these problems. Physical applications to these problems include high energy deposition from electron or ion beams interaction with materials for space and weapons applications, plasma disruption and energy dump on the walls or components of a fusion reactor, and high energy laser welding and annealing of materials. 23 refs., 3 figs.
Modeling and analysis of the space shuttle nose-gear tire with semianalytic finite elements
NASA Technical Reports Server (NTRS)
Kim, Kyun O.; Noor, Ahmed K.; Tanner, John A.
1990-01-01
A computational procedure is presented for the geometrically nonlinear analysis of aircraft tires. The Space Shuttle Orbiter nose gear tire was modeled by using a two-dimensional laminated anisotropic shell theory with the effects of variation in material and geometric parameters included. The four key elements of the procedure are: (1) semianalytic finite elements in which the shell variables are represented by Fourier series in the circumferential direction and piecewise polynominals in the meridional direction; (2) a mixed formulation with the fundamental unknowns consisting of strain parameters, stress-resultant parameters, and generalized displacements; (3) multilevel operator splitting to effect successive simplifications, and to uncouple the equations associated with different Fourier harmonics; and (4) multilevel iterative procedures and reduction techniques to generate the response of the shell. Numerical results of the Space Shuttle Orbiter nose gear tire model are compared with experimental measurements of the tire subjected to inflation loading.
Phase-space finite elements in a least-squares solution of the transport equation
Drumm, C.; Fan, W.; Pautz, S.
2013-07-01
The linear Boltzmann transport equation is solved using a least-squares finite element approximation in the space, angular and energy phase-space variables. The method is applied to both neutral particle transport and also to charged particle transport in the presence of an electric field, where the angular and energy derivative terms are handled with the energy/angular finite elements approximation, in a manner analogous to the way the spatial streaming term is handled. For multi-dimensional problems, a novel approach is used for the angular finite elements: mapping the surface of a unit sphere to a two-dimensional planar region and using a meshing tool to generate a mesh. In this manner, much of the spatial finite-elements machinery can be easily adapted to handle the angular variable. The energy variable and the angular variable for one-dimensional problems make use of edge/beam elements, also building upon the spatial finite elements capabilities. The methods described here can make use of either continuous or discontinuous finite elements in space, angle and/or energy, with the use of continuous finite elements resulting in a smaller problem size and the use of discontinuous finite elements resulting in more accurate solutions for certain types of problems. The work described in this paper makes use of continuous finite elements, so that the resulting linear system is symmetric positive definite and can be solved with a highly efficient parallel preconditioned conjugate gradients algorithm. The phase-space finite elements capability has been built into the Sceptre code and applied to several test problems, including a simple one-dimensional problem with an analytic solution available, a two-dimensional problem with an isolated source term, showing how the method essentially eliminates ray effects encountered with discrete ordinates, and a simple one-dimensional charged-particle transport problem in the presence of an electric field. (authors)
Seismic imaging using finite-differences and parallel computers
Ober, C.C.
1997-12-31
A key to reducing the risks and costs of associated with oil and gas exploration is the fast, accurate imaging of complex geologies, such as salt domes in the Gulf of Mexico and overthrust regions in US onshore regions. Prestack depth migration generally yields the most accurate images, and one approach to this is to solve the scalar wave equation using finite differences. As part of an ongoing ACTI project funded by the US Department of Energy, a finite difference, 3-D prestack, depth migration code has been developed. The goal of this work is to demonstrate that massively parallel computers can be used efficiently for seismic imaging, and that sufficient computing power exists (or soon will exist) to make finite difference, prestack, depth migration practical for oil and gas exploration. Several problems had to be addressed to get an efficient code for the Intel Paragon. These include efficient I/O, efficient parallel tridiagonal solves, and high single-node performance. Furthermore, to provide portable code the author has been restricted to the use of high-level programming languages (C and Fortran) and interprocessor communications using MPI. He has been using the SUNMOS operating system, which has affected many of his programming decisions. He will present images created from two verification datasets (the Marmousi Model and the SEG/EAEG 3D Salt Model). Also, he will show recent images from real datasets, and point out locations of improved imaging. Finally, he will discuss areas of current research which will hopefully improve the image quality and reduce computational costs.
A multigrid algorithm for the cell-centered finite difference scheme
NASA Technical Reports Server (NTRS)
Ewing, Richard E.; Shen, Jian
1993-01-01
In this article, we discuss a non-variational V-cycle multigrid algorithm based on the cell-centered finite difference scheme for solving a second-order elliptic problem with discontinuous coefficients. Due to the poor approximation property of piecewise constant spaces and the non-variational nature of our scheme, one step of symmetric linear smoothing in our V-cycle multigrid scheme may fail to be a contraction. Again, because of the simple structure of the piecewise constant spaces, prolongation and restriction are trivial; we save significant computation time with very promising computational results.
An Exponential Finite Difference Technique for Solving Partial Differential Equations.
1987-06-01
density , kg/N 3 (lbm/ft 3) 91.*,e separation variables (At dimensionless timelAX) 2 vi -W sNiv W- NiW.4%1 1. INTRODUCTION Partial differential equations...competing numerical analysis were run in double precision on either the IBM-3033 or the Cray X-MP mainframes. The computer codes developed for the...is increased. - R P~p~ 15 Effect of Initial and Boundary Conditions on the Exponential Finite Difference Method In this section the effect of
Finite difference time domain modeling of spiral antennas
NASA Technical Reports Server (NTRS)
Penney, Christopher W.; Beggs, John H.; Luebbers, Raymond J.
1992-01-01
The objectives outlined in the original proposal for this project were to create a well-documented computer analysis model based on the finite-difference, time-domain (FDTD) method that would be capable of computing antenna impedance, far-zone radiation patterns, and radar cross-section (RCS). The ability to model a variety of penetrable materials in addition to conductors is also desired. The spiral antennas under study by this project meet these requirements since they are constructed of slots cut into conducting surfaces which are backed by dielectric materials.
NASA Technical Reports Server (NTRS)
Campbell, W.
1981-01-01
A theoretical evaluation of the stability of an explicit finite difference solution of the transient temperature field in a composite medium is presented. The grid points of the field are assumed uniformly spaced, and media interfaces are either vertical or horizontal and pass through grid points. In addition, perfect contact between different media (infinite interfacial conductance) is assumed. A finite difference form of the conduction equation is not valid at media interfaces; therefore, heat balance forms are derived. These equations were subjected to stability analysis, and a computer graphics code was developed that permitted determination of a maximum time step for a given grid spacing.
OBTAINING POTENTIAL FIELD SOLUTIONS WITH SPHERICAL HARMONICS AND FINITE DIFFERENCES
Toth, Gabor; Van der Holst, Bart; Huang Zhenguang
2011-05-10
Potential magnetic field solutions can be obtained based on the synoptic magnetograms of the Sun. Traditionally, a spherical harmonics decomposition of the magnetogram is used to construct the current- and divergence-free magnetic field solution. This method works reasonably well when the order of spherical harmonics is limited to be small relative to the resolution of the magnetogram, although some artifacts, such as ringing, can arise around sharp features. When the number of spherical harmonics is increased, however, using the raw magnetogram data given on a grid that is uniform in the sine of the latitude coordinate can result in inaccurate and unreliable results, especially in the polar regions close to the Sun. We discuss here two approaches that can mitigate or completely avoid these problems: (1) remeshing the magnetogram onto a grid with uniform resolution in latitude and limiting the highest order of the spherical harmonics to the anti-alias limit; (2) using an iterative finite difference algorithm to solve for the potential field. The naive and the improved numerical solutions are compared for actual magnetograms and the differences are found to be rather dramatic. We made our new Finite Difference Iterative Potential-field Solver (FDIPS) a publicly available code so that other researchers can also use it as an alternative to the spherical harmonics approach.
Pencil: Finite-difference Code for Compressible Hydrodynamic Flows
NASA Astrophysics Data System (ADS)
Brandenburg, Axel; Dobler, Wolfgang
2010-10-01
The Pencil code is a high-order finite-difference code for compressible hydrodynamic flows with magnetic fields. It is highly modular and can easily be adapted to different types of problems. The code runs efficiently under MPI on massively parallel shared- or distributed-memory computers, like e.g. large Beowulf clusters. The Pencil code is primarily designed to deal with weakly compressible turbulent flows. To achieve good parallelization, explicit (as opposed to compact) finite differences are used. Typical scientific targets include driven MHD turbulence in a periodic box, convection in a slab with non-periodic upper and lower boundaries, a convective star embedded in a fully nonperiodic box, accretion disc turbulence in the shearing sheet approximation, self-gravity, non-local radiation transfer, dust particle evolution with feedback on the gas, etc. A range of artificial viscosity and diffusion schemes can be invoked to deal with supersonic flows. For direct simulations regular viscosity and diffusion is being used. The code is written in well-commented Fortran90.
A simple finite-difference scheme for handling topography with the first-order wave equation
NASA Astrophysics Data System (ADS)
Mulder, W. A.; Huiskes, M. J.
2017-07-01
One approach to incorporate topography in seismic finite-difference codes is a local modification of the difference operators near the free surface. An earlier paper described an approach for modelling irregular boundaries in a constant-density acoustic finite-difference code, based on the second-order formulation of the wave equation that only involves the pressure. Here, a similar method is considered for the first-order formulation in terms of pressure and particle velocity, using a staggered finite-difference discretization both in space and in time. In one space dimension, the boundary conditions consist in imposing antisymmetry for the pressure and symmetry for particle velocity components. For the pressure, this means that the solution values as well as all even derivatives up to a certain order are zero on the boundary. For the particle velocity, all odd derivatives are zero. In 2D, the 1-D assumption is used along each coordinate direction, with antisymmetry for the pressure along the coordinate and symmetry for the particle velocity component parallel to that coordinate direction. Since the symmetry or antisymmetry should hold along the direction normal to the boundary rather than along the coordinate directions, this generates an additional numerical error on top of the time stepping errors and the errors due to the interior spatial discretization. Numerical experiments in 2D and 3D nevertheless produce acceptable results.
An implicit-explicit finite-difference lattice Boltzmann subgrid method on nonuniform meshes
NASA Astrophysics Data System (ADS)
Qiu, Ruofan; Chen, Rongqian; You, Yancheng
In this paper, an implicit-explicit finite-difference lattice Boltzmann method with subgrid model on nonuniform meshes is proposed. The implicit-explicit Runge-Kutta scheme, which has good convergence rate, is used for the time discretization and a mixed difference scheme, which combines the upwind scheme with the central scheme, is adopted for the space discretization. Meanwhile, the standard Smagorinsky subgrid model is incorporated into the finite-difference lattice Boltzmann scheme. The effects of implicit-explicit Runge-Kutta scheme and nonuniform meshes of present lattice Boltzmann method are discussed through simulations of a two-dimensional lid-driven cavity flow on nonuniform meshes. Moreover, the comparison simulations of the present method and multiple relaxation time lattice Boltzmann subgrid method are conducted qualitatively and quantitatively.
Parallel 3-D viscoelastic finite difference seismic modelling
NASA Astrophysics Data System (ADS)
Bohlen, Thomas
2002-10-01
Computational power has advanced to a state where we can begin to perform wavefield simulations for realistic (complex) 3-D earth models at frequencies of interest to both seismologists and engineers. On serial platforms, however, 3-D calculations are still limited to small grid sizes and short seismic wave traveltimes. To make use of the efficiency of network computers a parallel 3-D viscoelastic finite difference (FD) code is implemented which allows to distribute the work on several PCs or workstations connected via standard ethernet in an in-house network. By using the portable message passing interface standard (MPI) for the communication between processors, running times can be reduced and grid sizes can be increased significantly. Furthermore, the code shows good performance on massive parallel supercomputers which makes the computation of very large grids feasible. This implementation greatly expands the applicability of the 3-D elastic/viscoelastic finite-difference modelling technique by providing an efficient, portable and practical C-program.
Finite Difference Elastic Wave Field Simulation On GPU
NASA Astrophysics Data System (ADS)
Hu, Y.; Zhang, W.
2011-12-01
Numerical modeling of seismic wave propagation is considered as a basic and important aspect in investigation of the Earth's structure, and earthquake phenomenon. Among various numerical methods, the finite-difference method is considered one of the most efficient tools for the wave field simulation. However, with the increment of computing scale, the power of computing has becoming a bottleneck. With the development of hardware, in recent years, GPU shows powerful computational ability and bright application prospects in scientific computing. Many works using GPU demonstrate that GPU is powerful . Recently, GPU has not be used widely in the simulation of wave field. In this work, we present forward finite difference simulation of acoustic and elastic seismic wave propagation in heterogeneous media on NVIDIA graphics cards with the CUDA programming language. We also implement perfectly matched layers on the graphics cards to efficiently absorb outgoing waves on the fictitious edges of the grid Simulations compared with the results on CPU platform shows reliable accuracy and remarkable efficiency. This work proves that GPU can be an effective platform for wave field simulation, and it can also be used as a practical tool for real-time strong ground motion simulation.
Spatial Parallelism of a 3D Finite Difference, Velocity-Stress Elastic Wave Propagation Code
MINKOFF,SUSAN E.
1999-12-09
Finite difference methods for solving the wave equation more accurately capture the physics of waves propagating through the earth than asymptotic solution methods. Unfortunately. finite difference simulations for 3D elastic wave propagation are expensive. We model waves in a 3D isotropic elastic earth. The wave equation solution consists of three velocity components and six stresses. The partial derivatives are discretized using 2nd-order in time and 4th-order in space staggered finite difference operators. Staggered schemes allow one to obtain additional accuracy (via centered finite differences) without requiring additional storage. The serial code is most unique in its ability to model a number of different types of seismic sources. The parallel implementation uses the MP1 library, thus allowing for portability between platforms. Spatial parallelism provides a highly efficient strategy for parallelizing finite difference simulations. In this implementation, one can decompose the global problem domain into one-, two-, and three-dimensional processor decompositions with 3D decompositions generally producing the best parallel speed up. Because i/o is handled largely outside of the time-step loop (the most expensive part of the simulation) we have opted for straight-forward broadcast and reduce operations to handle i/o. The majority of the communication in the code consists of passing subdomain face information to neighboring processors for use as ''ghost cells''. When this communication is balanced against computation by allocating subdomains of reasonable size, we observe excellent scaled speed up. Allocating subdomains of size 25 x 25 x 25 on each node, we achieve efficiencies of 94% on 128 processors. Numerical examples for both a layered earth model and a homogeneous medium with a high-velocity blocky inclusion illustrate the accuracy of the parallel code.
Viscoelastic Finite Difference Modeling Using Graphics Processing Units
NASA Astrophysics Data System (ADS)
Fabien-Ouellet, G.; Gloaguen, E.; Giroux, B.
2014-12-01
Full waveform seismic modeling requires a huge amount of computing power that still challenges today's technology. This limits the applicability of powerful processing approaches in seismic exploration like full-waveform inversion. This paper explores the use of Graphics Processing Units (GPU) to compute a time based finite-difference solution to the viscoelastic wave equation. The aim is to investigate whether the adoption of the GPU technology is susceptible to reduce significantly the computing time of simulations. The code presented herein is based on the freely accessible software of Bohlen (2002) in 2D provided under a General Public License (GNU) licence. This implementation is based on a second order centred differences scheme to approximate time differences and staggered grid schemes with centred difference of order 2, 4, 6, 8, and 12 for spatial derivatives. The code is fully parallel and is written using the Message Passing Interface (MPI), and it thus supports simulations of vast seismic models on a cluster of CPUs. To port the code from Bohlen (2002) on GPUs, the OpenCl framework was chosen for its ability to work on both CPUs and GPUs and its adoption by most of GPU manufacturers. In our implementation, OpenCL works in conjunction with MPI, which allows computations on a cluster of GPU for large-scale model simulations. We tested our code for model sizes between 1002 and 60002 elements. Comparison shows a decrease in computation time of more than two orders of magnitude between the GPU implementation run on a AMD Radeon HD 7950 and the CPU implementation run on a 2.26 GHz Intel Xeon Quad-Core. The speed-up varies depending on the order of the finite difference approximation and generally increases for higher orders. Increasing speed-ups are also obtained for increasing model size, which can be explained by kernel overheads and delays introduced by memory transfers to and from the GPU through the PCI-E bus. Those tests indicate that the GPU memory size
Domain decomposition methods for nonconforming finite element spaces of Lagrange-type
NASA Technical Reports Server (NTRS)
Cowsar, Lawrence C.
1993-01-01
In this article, we consider the application of three popular domain decomposition methods to Lagrange-type nonconforming finite element discretizations of scalar, self-adjoint, second order elliptic equations. The additive Schwarz method of Dryja and Widlund, the vertex space method of Smith, and the balancing method of Mandel applied to nonconforming elements are shown to converge at a rate no worse than their applications to the standard conforming piecewise linear Galerkin discretization. Essentially, the theory for the nonconforming elements is inherited from the existing theory for the conforming elements with only modest modification by constructing an isomorphism between the nonconforming finite element space and a space of continuous piecewise linear functions.
Elastic finite-difference method for irregular grids
Oprsal, I.; Zahradnik, J.
1999-01-01
Finite-difference (FD) modeling of complicated structures requires simple algorithms. This paper presents a new elastic FD method for spatially irregular grids that is simple and, at the same time, saves considerable memory and computing time. Features like faults, low-velocity layers, cavities, and/or nonplanar surfaces are treated on a fine grid, while the remaining parts of the model are, with equal accuracy, represented on a coarse grid. No interpolation is needed between the fine and coarse parts due to the rectangular grid cells. Relatively abrupt transitions between the small and large grid steps produce no numerical artifacts in the present method. Planar or nonplanar free surfaces, including underground cavities, are treated in a way similar to internal grid points but with consideration of the zero-valued elastic parameters and density outside the free surface (vacuum formalism). A theoretical proof that vacuum formalism fulfills the free-surface conditions is given. Numerical validation is performed through comparison with independent methods, comparing FD with explicitly prescribed boundary conditions and finite elements. Memory and computing time needed in the studied models was only about 10 to 40% of that employing regular square grids of equal accuracy. A practical example of a synthetic seismic section, showing clear signatures of a coal seam and cavity, is presented. The method can be extended to three dimensions.
Li, W. P.; Liu, Y.; Long, Q.; Chen, D. H.; Chen, Y. M.
2008-10-15
The electromagnetic field (both E and B fields) is calculated for a solenoidal inductively coupled plasma (ICP) discharge. The model is based on two-dimensional cylindrical coordinates, and the finite difference method is used for solving Maxwell equations in both the radial and axial directions. Through one-turn coil measurements, assuming that the electrical conductivity has a constant value in each cross section of the discharge tube, the calculated E and B fields rise sharply near the tube wall. The nonuniform radial distributions imply that the skin effect plays a significant role in the energy balance of the stable ICP. Damped distributions in the axial direction show that the magnetic flux gradually dissipates into the surrounding space. A finite difference calculation allows prediction of the electrical conductivity and plasma permeability, and the induction coil voltage and plasma current can be calculated, which are verified for correctness.
Linear finite-difference bond graph model of an ionic polymer actuator
NASA Astrophysics Data System (ADS)
Bentefrit, M.; Grondel, S.; Soyer, C.; Fannir, A.; Cattan, E.; Madden, J. D.; Nguyen, T. M. G.; Plesse, C.; Vidal, F.
2017-09-01
With the recent growing interest for soft actuation, many new types of ionic polymers working in air have been developed. Due to the interrelated mechanical, electrical, and chemical properties which greatly influence the characteristics of such actuators, their behavior is complex and difficult to understand, predict and optimize. In light of this challenge, an original linear multiphysics finite difference bond graph model was derived to characterize this ionic actuation. This finite difference scheme was divided into two coupled subparts, each related to a specific physical, electrochemical or mechanical domain, and then converted into a bond graph model as this language is particularly suited for systems from multiple energy domains. Simulations were then conducted and a good agreement with the experimental results was obtained. Furthermore, an analysis of the power efficiency of such actuators as a function of space and time was proposed and allowed to evaluate their performance.
A mimetic finite difference method for the Stokes problem with elected edge bubbles
Lipnikov, K; Berirao, L
2009-01-01
A new mimetic finite difference method for the Stokes problem is proposed and analyzed. The unstable P{sub 1}-P{sub 0} discretization is stabilized by adding a small number of bubble functions to selected mesh edges. A simple strategy for selecting such edges is proposed and verified with numerical experiments. The discretizations schemes for Stokes and Navier-Stokes equations must satisfy the celebrated inf-sup (or the LBB) stability condition. The stability condition implies a balance between discrete spaces for velocity and pressure. In finite elements, this balance is frequently achieved by adding bubble functions to the velocity space. The goal of this article is to show that the stabilizing edge bubble functions can be added only to a small set of mesh edges. This results in a smaller algebraic system and potentially in a faster calculations. We employ the mimetic finite difference (MFD) discretization technique that works for general polyhedral meshes and can accomodate non-uniform distribution of stabilizing bubbles.
Finite difference time domain implementation of surface impedance boundary conditions
NASA Technical Reports Server (NTRS)
Beggs, John H.; Luebbers, Raymond J.; Yee, Kane S.; Kunz, Karl S.
1991-01-01
Surface impedance boundary conditions are employed to reduce the solution volume during the analysis of scattering from lossy dielectric objects. In the finite difference solution, they also can be utilized to avoid using small cells, made necessary by shorter wavelengths in conducting media throughout the solution volume. The standard approach is to approximate the surface impedance over a very small bandwidth by its value at the center frequency, and then use that result in the boundary condition. Here, two implementations of the surface impedance boundary condition are presented. One implementation is a constant surface impedance boundary condition and the other is a dispersive surface impedance boundary condition that is applicable over a very large frequency bandwidth and over a large range of conductivities. Frequency domain results are presented in one dimension for two conductivity values and are compared with exact results. Scattering width results from an infinite square cylinder are presented as a two dimensional demonstration. Extensions to three dimensions should be straightforward.
Finite difference time domain implementation of surface impedance boundary conditions
NASA Technical Reports Server (NTRS)
Beggs, John H.; Luebbers, Raymond J.; Yee, Kane S.; Kunz, Karl S.
1991-01-01
Surface impedance boundary conditions are employed to reduce the solution volume during the analysis of scattering from lossy dielectric objects. In a finite difference solution, they also can be utilized to avoid using small cells, made necessary by shorter wavelengths in conducting media throughout the solution volume. The standard approach is to approximate the surface impedance over a very small bandwidth by its value at the center frequency, and then use that result in the boundary condition. Two implementations of the surface impedance boundary condition are presented. One implementation is a constant surface impedance boundary condition and the other is a dispersive surface impedance boundary condition that is applicable over a very large frequency bandwidth and over a large range of conductivities. Frequency domain results are presented in one dimension for two conductivity values and are compared with exact results. Scattering width results from an infinite square cylinder are presented as a 2-D demonstration. Extensions to 3-D should be straightforward.
3D finite-difference seismic migration with parallel computers
Ober, C.C.; Gjertsen, R.; Minkoff, S.; Womble, D.E.
1998-11-01
The ability to image complex geologies such as salt domes in the Gulf of Mexico and thrusts in mountainous regions is essential for reducing the risk associated with oil exploration. Imaging these structures, however, is computationally expensive as datasets can be terabytes in size. Traditional ray-tracing migration methods cannot handle complex velocity variations commonly found near such salt structures. Instead the authors use the full 3D acoustic wave equation, discretized via a finite difference algorithm. They reduce the cost of solving the apraxial wave equation by a number of numerical techniques including the method of fractional steps and pipelining the tridiagonal solves. The imaging code, Salvo, uses both frequency parallelism (generally 90% efficient) and spatial parallelism (65% efficient). Salvo has been tested on synthetic and real data and produces clear images of the subsurface even beneath complicated salt structures.
Accurate finite difference methods for time-harmonic wave propagation
NASA Technical Reports Server (NTRS)
Harari, Isaac; Turkel, Eli
1994-01-01
Finite difference methods for solving problems of time-harmonic acoustics are developed and analyzed. Multidimensional inhomogeneous problems with variable, possibly discontinuous, coefficients are considered, accounting for the effects of employing nonuniform grids. A weighted-average representation is less sensitive to transition in wave resolution (due to variable wave numbers or nonuniform grids) than the standard pointwise representation. Further enhancement in method performance is obtained by basing the stencils on generalizations of Pade approximation, or generalized definitions of the derivative, reducing spurious dispersion, anisotropy and reflection, and by improving the representation of source terms. The resulting schemes have fourth-order accurate local truncation error on uniform grids and third order in the nonuniform case. Guidelines for discretization pertaining to grid orientation and resolution are presented.
A parallel finite-difference method for computational aerodynamics
NASA Technical Reports Server (NTRS)
Swisshelm, Julie M.
1989-01-01
A finite-difference scheme for solving complex three-dimensional aerodynamic flow on parallel-processing supercomputers is presented. The method consists of a basic flow solver with multigrid convergence acceleration, embedded grid refinements, and a zonal equation scheme. Multitasking and vectorization have been incorporated into the algorithm. Results obtained include multiprocessed flow simulations from the Cray X-MP and Cray-2. Speedups as high as 3.3 for the two-dimensional case and 3.5 for segments of the three-dimensional case have been achieved on the Cray-2. The entire solver attained a factor of 2.7 improvement over its unitasked version on the Cray-2. The performance of the parallel algorithm on each machine is analyzed.
Parallelization of implicit finite difference schemes in computational fluid dynamics
NASA Technical Reports Server (NTRS)
Decker, Naomi H.; Naik, Vijay K.; Nicoules, Michel
1990-01-01
Implicit finite difference schemes are often the preferred numerical schemes in computational fluid dynamics, requiring less stringent stability bounds than the explicit schemes. Each iteration in an implicit scheme involves global data dependencies in the form of second and higher order recurrences. Efficient parallel implementations of such iterative methods are considerably more difficult and non-intuitive. The parallelization of the implicit schemes that are used for solving the Euler and the thin layer Navier-Stokes equations and that require inversions of large linear systems in the form of block tri-diagonal and/or block penta-diagonal matrices is discussed. Three-dimensional cases are emphasized and schemes that minimize the total execution time are presented. Partitioning and scheduling schemes for alleviating the effects of the global data dependencies are described. An analysis of the communication and the computation aspects of these methods is presented. The effect of the boundary conditions on the parallel schemes is also discussed.
Flexible Automatic Discretization for Finite Differences: Eliminating the Human Factor
NASA Astrophysics Data System (ADS)
Pranger, Casper
2017-04-01
In the geophysical numerical modelling community, finite differences are (in part due to their small footprint) a popular spatial discretization method for PDEs in the regular-shaped continuum that is the earth. However, they rapidly become prone to programming mistakes when physics increase in complexity. To eliminate opportunities for human error, we have designed an automatic discretization algorithm using Wolfram Mathematica, in which the user supplies symbolic PDEs, the number of spatial dimensions, and a choice of symbolic boundary conditions, and the script transforms this information into matrix- and right-hand-side rules ready for use in a C++ code that will accept them. The symbolic PDEs are further used to automatically develop and perform manufactured solution benchmarks, ensuring at all stages physical fidelity while providing pragmatic targets for numerical accuracy. We find that this procedure greatly accelerates code development and provides a great deal of flexibility in ones choice of physics.
Explicit and implicit finite difference schemes for fractional Cattaneo equation
NASA Astrophysics Data System (ADS)
Ghazizadeh, H. R.; Maerefat, M.; Azimi, A.
2010-09-01
In this paper, the numerical solution of fractional (non-integer)-order Cattaneo equation for describing anomalous diffusion has been investigated. Two finite difference schemes namely an explicit predictor-corrector and totally implicit schemes have been developed. In developing each scheme, a separate formulation approach for the governing equations has been considered. The explicit predictor-corrector scheme is the fractional generalization of well-known MacCormack scheme and has been called Generalized MacCormack scheme. This scheme solves two coupled low-order equations and simultaneously computes the flux term with the main variable. Fully implicit scheme however solves a single high-order undecomposed equation. For Generalized MacCormack scheme, stability analysis has been studied through Fourier method. Through a numerical test, the experimental order of convergency of both schemes has been found. Then, the domain of applicability and some numerical properties of each scheme have been discussed.
Finite-difference modeling of commercial aircraft using TSAR
Pennock, S.T.; Poggio, A.J.
1994-11-15
Future aircraft may have systems controlled by fiber optic cables, to reduce susceptibility to electromagnetic interference. However, the digital systems associated with the fiber optic network could still experience upset due to powerful radio stations, radars, and other electromagnetic sources, with potentially serious consequences. We are modeling the electromagnetic behavior of commercial transport aircraft in support of the NASA Fly-by-Light/Power-by-Wire program, using the TSAR finite-difference time-domain code initially developed for the military. By comparing results obtained from TSAR with data taken on a Boeing 757 at the Air Force Phillips Lab., we hope to show that FDTD codes can serve as an important tool in the design and certification of U.S. commercial aircraft, helping American companies to produce safe, reliable air transportation.
Visualization of elastic wavefields computed with a finite difference code
Larsen, S.; Harris, D.
1994-11-15
The authors have developed a finite difference elastic propagation model to simulate seismic wave propagation through geophysically complex regions. To facilitate debugging and to assist seismologists in interpreting the seismograms generated by the code, they have developed an X Windows interface that permits viewing of successive temporal snapshots of the (2D) wavefield as they are calculated. The authors present a brief video displaying the generation of seismic waves by an explosive source on a continent, which propagate to the edge of the continent then convert to two types of acoustic waves. This sample calculation was part of an effort to study the potential of offshore hydroacoustic systems to monitor seismic events occurring onshore.
Stability of finite difference models containing two boundaries or interfaces
NASA Technical Reports Server (NTRS)
Trefethen, L. N.
1984-01-01
The stability of finite difference models of hyperbolic initial boundary value problems is connected with the propagation and reflection of parasitic waves. Wave propagation ideas are applied to models containing two boundaires or interfaces, where repeated reflection of trapped wave packets is a potential new source of instability. Various known instability phenomena are accounted for in a unified way. Results show: (1) dissipativity does not ensure stability when three or more formulas are concatenated at a boundary or internal interface; (2) algebraic GKS instabilities can be converted by a second boundary to exponential instabilities only when an infinite numerical reflection coefficient is present; and (3) GKS-stability and P-stability can be established in certain problems by showing that all numerical reflection coefficients have modulus less than 1.
Finite Element Analysis of a Deployable Space Structure
NASA Technical Reports Server (NTRS)
Weeks, G. E.
1983-01-01
The dynamic characteristics of the Solar Array Flight Experiment (SAFE) structure during deployment and retraction are investigated. The SAFE structure consists of a deployable mast with an attached solar blanket designed with accordion type folds to permit packaging in a small volume. The planar form of the blanket geometry during deployment is maintained by a blanket tension/guidewire system. Structurally, the mast is modeled as an Euler beam column with inplane and out of plane bending and finite torsional stiffness. For out of plane motion, the blanket is modeled as a distributed mass uniformly supported by the three guidewires. For inplane motion the blanket displacements are assumed to vary linearly from the mast base to the mast tip. The mathematical model uses a virtual work formulation, required because the axial loading on the mast is nonconservative, combined with assumed beam modes to derive the differential equations of motion. Consideration of the time dependent boundary conditions results in an infinite set of ODE with time dependent coefficients. Finally, correlation of mast tip accelerations to mast base bending moments for specified modal motions are indicated.
A finite difference solution for the propagation of sound in near sonic flows
NASA Technical Reports Server (NTRS)
Hariharan, S. I.; Lester, H. C.
1984-01-01
An explicit time/space finite difference procedure is used to model the propagation of sound in a quasi one-dimensional duct containing high Mach number subsonic flow. Nonlinear acoustic equations are derived by perturbing the time-dependent Euler equations about a steady, compressible mean flow. The governing difference relations are based on a fourth-order, two-step (predictor-corrector) MacCormack scheme. The solution algorithm functions by switching on a time harmonic source and allowing the difference equations to iterate to a steady state. The principal effect of the non-linearities was to shift aocustical energy to higher harmonics. With increased source strength, wave steepening was observed. This phenomenon suggests that the acoustical response may approach a shock behavior at higher sound pressure level as the throat Mach number approaches unity. On a peak level basis, good agreement between the nonlinear finite difference and linear finite element solutions was observed, even through a peak sound pressure level of about 150 dB occurred in the throat region. Nonlinear steady state waveform solutions are shown to be in excellent agreement with a nonlinear asymptotic theory. Previously announced in STAR as N83-30167
A finite difference solution for the propagation of sound in near sonic flows
NASA Technical Reports Server (NTRS)
Hariharan, S. I.; Lester, H. C.
1983-01-01
An explicit time/space finite difference procedure is used to model the propagation of sound in a quasi one-dimensional duct containing high Mach number subsonic flow. Nonlinear acoustic equations are derived by perturbing the time-dependent Euler equations about a steady, compressible mean flow. The governing difference relations are based on a fourth-order, two-step (predictor-corrector) MacCormack scheme. The solution algorithm functions by switching on a time harmonic source and allowing the difference equations to iterate to a steady state. The principal effect of the non-linearities was to shift acoustical energy to higher harmonics. With increased source strengths, wave steepening was observed. This phenomenon suggests that the acoustical response may approach a shock behavior at at higher sound pressure level as the throat Mach number aproaches unity. On a peak level basis, good agreement between the nonlinear finite difference and linear finite element solutions was observed, even through a peak sound pressure level of about 150 dB occurred in the throat region. Nonlinear steady state waveform solutions are shown to be in excellent agreement with a nonlinear asymptotic theory.
Higher-order finite-difference formulation of periodic Orbital-free Density Functional Theory
Ghosh, Swarnava; Suryanarayana, Phanish
2016-02-15
We present a real-space formulation and higher-order finite-difference implementation of periodic Orbital-free Density Functional Theory (OF-DFT). Specifically, utilizing a local reformulation of the electrostatic and kernel terms, we develop a generalized framework for performing OF-DFT simulations with different variants of the electronic kinetic energy. In particular, we propose a self-consistent field (SCF) type fixed-point method for calculations involving linear-response kinetic energy functionals. In this framework, evaluation of both the electronic ground-state and forces on the nuclei are amenable to computations that scale linearly with the number of atoms. We develop a parallel implementation of this formulation using the finite-difference discretization. We demonstrate that higher-order finite-differences can achieve relatively large convergence rates with respect to mesh-size in both the energies and forces. Additionally, we establish that the fixed-point iteration converges rapidly, and that it can be further accelerated using extrapolation techniques like Anderson's mixing. We validate the accuracy of the results by comparing the energies and forces with plane-wave methods for selected examples, including the vacancy formation energy in Aluminum. Overall, the suitability of the proposed formulation for scalable high performance computing makes it an attractive choice for large-scale OF-DFT calculations consisting of thousands of atoms.
Subresolution Displacements in Finite Difference Simulations of Ultrasound Propagation and Imaging.
Pinton, Gianmarco F
2017-03-01
Time domain finite difference simulations are used extensively to simulate wave propagation. They approximate the wave field on a discrete domain with a grid spacing that is typically on the order of a tenth of a wavelength. The smallest displacements that can be modeled by this type of simulation are thus limited to discrete values that are integer multiples of the grid spacing. This paper presents a method to represent continuous and subresolution displacements by varying the impedance of individual elements in a multielement scatterer. It is demonstrated that this method removes the limitations imposed by the discrete grid spacing by generating a continuum of displacements as measured by the backscattered signal. The method is first validated on an ideal perfect correlation case with a single scatterer. It is subsequently applied to a more complex case with a field of scatterers that model an acoustic radiation force-induced displacement used in ultrasound elasticity imaging. A custom finite difference simulation tool is used to simulate propagation from ultrasound imaging pulses in the scatterer field. These simulated transmit-receive events are then beamformed into images, which are tracked with a correlation-based algorithm to determine the displacement. A linear predictive model is developed to analytically describe the relationship between element impedance and backscattered phase shift. The error between model and simulation is λ/ 1364 , where λ is the acoustical wavelength. An iterative method is also presented that reduces the simulation error to λ/ 5556 over one iteration. The proposed technique therefore offers a computationally efficient method to model continuous subresolution displacements of a scattering medium in ultrasound imaging. This method has applications that include ultrasound elastography, blood flow, and motion tracking. This method also extends generally to finite difference simulations of wave propagation, such as electromagnetic or
Performance analysis and optimization of finite-difference schemes for wave propagation problems
NASA Astrophysics Data System (ADS)
Pirozzoli, Sergio
2007-03-01
In the present paper, we gauge the performance of finite-difference schemes with Runge-Kutta time integration for wave propagation problems by rigorously defining appropriate cost and error metrics in a simple setting represented by the linear advection equation. Optimal values of the grid spacing and of the time step are obtained as a result of a cost minimization (for given error level) procedure. The theory suggests superior performance of high-order schemes when highly accurate solutions are sought for, and in several space dimensions even more. The analysis of the global discretization error shows the occurrence of two (approximately independent) sources of error, associated with the space and time discretizations. The improvement of the performance of finite-difference schemes can then be achieved by trying to separately minimize the two contributions. General guidelines for the design of problem-tailored, optimized schemes are provided, suggesting that significant reductions of the computational cost are in principle possible. The application of the analysis to wave propagation problems in a two-dimensional environment demonstrates that the analysis carried out for the scalar case directly applies to the propagation of monochromatic sound waves. For problems of sound propagation involving disparate length-scales the analysis still provides useful insight for the optimal exploitation of computational resources; however, the actual advantage provided by optimized schemes is not as evident as in the single-scale, scalar case.
Neutral aggregation in finite-length genotype space
NASA Astrophysics Data System (ADS)
Houchmandzadeh, Bahram
2017-01-01
The advent of modern genome sequencing techniques allows for a more stringent test of the neutrality hypothesis of Darwinian evolution, where all individuals have the same fitness. Using the individual-based model of Wright and Fisher, we compute the amplitude of neutral aggregation in the genome space, i.e., the probability of finding two individuals at genetic (Hamming) distance k as a function of the genome size L , population size N , and mutation probability per base ν . In well-mixed populations, we show that for N ν <1 /L , neutral aggregation is the dominant force and most individuals are found at short genetic distances from each other. For N ν >1 , on the contrary, individuals are randomly dispersed in genome space. The results are extended to a geographically dispersed population, where the controlling parameter is shown to be a combination of mutation and migration probability. The theory we develop can be used to test the neutrality hypothesis in various ecological and evolutionary systems.
HEMP 3D -- a finite difference program for calculating elastic-plastic flow
Wilkins, M.L.
1993-05-26
The HEMP 3D program can be used to solve problems in solid mechanics involving dynamic plasticity and time dependent material behavior and problems in gas dynamics. The equations of motion, the conservation equations, and the constitutive relations are solved by finite difference methods following the format of the HEMP computer simulation program formulated in two space dimensions and time. Presented here is an update of the 1975 report on the HEMP 3D numerical technique. The present report includes the sliding surface routines programmed by Robert Gulliford.
The limitations of staggered grid finite differences in plasticity problems
NASA Astrophysics Data System (ADS)
Pranger, Casper; Herrendörfer, Robert; Le Pourhiet, Laetitia
2017-04-01
Most crustal-scale applications operate at grid sizes much larger than those at which plasticity occurs in nature. As a consequence, plastic shear bands often localize to the scale of one grid cell, and numerical ploys — like introducing an artificial length scale — are needed to counter this. If for whatever reasons (good or bad) this is not done, we find that problems may arise due to the fact that in the staggered grid finite difference discretization, unknowns like components of the stress tensor and velocity vector are located in physically different positions. This incurs frequent interpolation, reducing the accuracy of the discretization. For purely stress-dependent plasticity problems the adverse effects might be contained because the magnitude of the stress discontinuity across a plastic shear band is limited. However, we find that when rate-dependence of friction is added in the mix, things become ugly really fast and the already hard-to-solve and highly nonlinear problem of plasticity incurs an extra penalty.
A finite difference model for free surface gravity drainage
Couri, F.R.; Ramey, H.J. Jr.
1993-09-01
The unconfined gravity flow of liquid with a free surface into a well is a classical well test problem which has not been well understood by either hydrologists or petroleum engineers. Paradigms have led many authors to treat an incompressible flow as compressible flow to justify the delayed yield behavior of a time-drawdown test. A finite-difference model has been developed to simulate the free surface gravity flow of an unconfined single phase, infinitely large reservoir into a well. The model was verified with experimental results in sandbox models in the literature and with classical methods applied to observation wells in the Groundwater literature. The simulator response was also compared with analytical Theis (1935) and Ramey et al. (1989) approaches for wellbore pressure at late producing times. The seepage face in the sandface and the delayed yield behavior were reproduced by the model considering a small liquid compressibility and incompressible porous medium. The potential buildup (recovery) simulated by the model evidenced a different- phenomenon from the drawdown, contrary to statements found in the Groundwater literature. Graphs of buildup potential vs time, buildup seepage face length vs time, and free surface head and sand bottom head radial profiles evidenced that the liquid refills the desaturating cone as a flat moving surface. The late time pseudo radial behavior was only approached after exaggerated long times.
Nonlinear wave propagation using three different finite difference schemes (category 2 application)
NASA Technical Reports Server (NTRS)
Pope, D. Stuart; Hardin, J. C.
1995-01-01
Three common finite difference schemes are used to examine the computation of one-dimensional nonlinear wave propagation. The schemes are studied for their responses to numerical parameters such as time step selection, boundary condition implementation, and discretization of governing equations. The performance of the schemes is compared and various numerical phenomena peculiar to each is discussed.
Passivity control with practically finite-time convergence for large space structures
NASA Astrophysics Data System (ADS)
Hu, Quan; Li, Jinyue; Zhang, Jingrui
2017-02-01
A nonlinear output feedback control law based on passivity is proposed to reduce the vibration of large space structures. The considered system is assumed to be equipped with collocated actuators and sensors. The concept of practically finite-time stability is first developed to describe the finite-time convergence of a passive system. Then, an output feedback is introduced to drive the trajectories of a passive system into a small set around the origin in finite time. Finally, the proposed control strategy is applied to the vibration suppression of large space structures with distributed thrusters and velocity sensors or torque outputting devices and angular rate sensors. Numerical simulations are conducted to validate the effectiveness of the proposed controller.
On Generalized Difference Hahn Sequence Spaces
Raj, Kuldip; Kiliçman, Adem
2014-01-01
We construct some generalized difference Hahn sequence spaces by mean of sequence of modulus functions. The topological properties and some inclusion relations of spaces h p(ℱ, u, Δr) are investigated. Also we compute the dual of these spaces, and some matrix transformations are characterized. PMID:25025085
A fast finite volume method for conservative space-fractional diffusion equations in convex domains
NASA Astrophysics Data System (ADS)
Jia, Jinhong; Wang, Hong
2016-04-01
We develop a fast finite volume method for variable-coefficient, conservative space-fractional diffusion equations in convex domains via a volume-penalization approach. The method has an optimal storage and an almost linear computational complexity. The method retains second-order accuracy without requiring a Richardson extrapolation. Numerical results are presented to show the utility of the method.
Finite-difference numerical simulations of underground explosion cavity decoupling
NASA Astrophysics Data System (ADS)
Aldridge, D. F.; Preston, L. A.; Jensen, R. P.
2012-12-01
Earth models containing a significant portion of ideal fluid (e.g., air and/or water) are of increasing interest in seismic wave propagation simulations. Examples include a marine model with a thick water layer, and a land model with air overlying a rugged topographic surface. The atmospheric infrasound community is currently interested in coupled seismic-acoustic propagation of low-frequency signals over long ranges (~tens to ~hundreds of kilometers). Also, accurate and efficient numerical treatment of models containing underground air-filled voids (caves, caverns, tunnels, subterranean man-made facilities) is essential. In support of the Source Physics Experiment (SPE) conducted at the Nevada National Security Site (NNSS), we are developing a numerical algorithm for simulating coupled seismic and acoustic wave propagation in mixed solid/fluid media. Solution methodology involves explicit, time-domain, finite-differencing of the elastodynamic velocity-stress partial differential system on a three-dimensional staggered spatial grid. Conditional logic is used to avoid shear stress updating within the fluid zones; this approach leads to computational efficiency gains for models containing a significant proportion of ideal fluid. Numerical stability and accuracy are maintained at air/rock interfaces (where the contrast in mass density is on the order of 1 to 2000) via a finite-difference operator "order switching" formalism. The fourth-order spatial FD operator used throughout the bulk of the earth model is reduced to second-order in the immediate vicinity of a high-contrast interface. Current modeling efforts are oriented toward quantifying the amount of atmospheric infrasound energy generated by various underground seismic sources (explosions and earthquakes). Source depth and orientation, and surface topography play obvious roles. The cavity decoupling problem, where an explosion is detonated within an air-filled void, is of special interest. A point explosion
A hybrid finite-difference and analytic element groundwater model.
Haitjema, H M; Feinstein, D T; Hunt, R J; Gusyev, M A
2010-01-01
Regional finite-difference models tend to have large cell sizes, often on the order of 1-2 km on a side. Although the regional flow patterns in deeper formations may be adequately represented by such a model, the intricate surface water and groundwater interactions in the shallower layers are not. Several stream reaches and nearby wells may occur in a single cell, precluding any meaningful modeling of the surface water and groundwater interactions between the individual features. We propose to replace the upper MODFLOW layer or layers, in which the surface water and groundwater interactions occur, by an analytic element model (GFLOW) that does not employ a model grid; instead, it represents wells and surface waters directly by the use of point-sinks and line-sinks. For many practical cases it suffices to provide GFLOW with the vertical leakage rates calculated in the original coarse MODFLOW model in order to obtain a good representation of surface water and groundwater interactions. However, when the combined transmissivities in the deeper (MODFLOW) layers dominate, the accuracy of the GFLOW solution diminishes. For those cases, an iterative coupling procedure, whereby the leakages between the GFLOW and MODFLOW model are updated, appreciably improves the overall solution, albeit at considerable computational cost. The coupled GFLOW-MODFLOW model is applicable to relatively large areas, in many cases to the entire model domain, thus forming an attractive alternative to local grid refinement or inset models.
QED multi-dimensional vacuum polarization finite-difference solver
NASA Astrophysics Data System (ADS)
Carneiro, Pedro; Grismayer, Thomas; Silva, Luís; Fonseca, Ricardo
2015-11-01
The Extreme Light Infrastructure (ELI) is expected to deliver peak intensities of 1023 - 1024 W/cm2 allowing to probe nonlinear Quantum Electrodynamics (QED) phenomena in an unprecedented regime. Within the framework of QED, the second order process of photon-photon scattering leads to a set of extended Maxwell's equations [W. Heisenberg and H. Euler, Z. Physik 98, 714] effectively creating nonlinear polarization and magnetization terms that account for the nonlinear response of the vacuum. To model this in a self-consistent way, we present a multi dimensional generalized Maxwell equation finite difference solver with significantly enhanced dispersive properties, which was implemented in the OSIRIS particle-in-cell code [R.A. Fonseca et al. LNCS 2331, pp. 342-351, 2002]. We present a detailed numerical analysis of this electromagnetic solver. As an illustration of the properties of the solver, we explore several examples in extreme conditions. We confirm the theoretical prediction of vacuum birefringence of a pulse propagating in the presence of an intense static background field [arXiv:1301.4918 [quant-ph
Assessment of Linear Finite-Difference Poisson-Boltzmann Solvers
Wang, Jun; Luo, Ray
2009-01-01
CPU time and memory usage are two vital issues that any numerical solvers for the Poisson-Boltzmann equation have to face in biomolecular applications. In this study we systematically analyzed the CPU time and memory usage of five commonly used finite-difference solvers with a large and diversified set of biomolecular structures. Our comparative analysis shows that modified incomplete Cholesky conjugate gradient and geometric multigrid are the most efficient in the diversified test set. For the two efficient solvers, our test shows that their CPU times increase approximately linearly with the numbers of grids. Their CPU times also increase almost linearly with the negative logarithm of the convergence criterion at very similar rate. Our comparison further shows that geometric multigrid performs better in the large set of tested biomolecules. However, modified incomplete Cholesky conjugate gradient is superior to geometric multigrid in molecular dynamics simulations of tested molecules. We also investigated other significant components in numerical solutions of the Poisson-Boltzmann equation. It turns out that the time-limiting step is the free boundary condition setup for the linear systems for the selected proteins if the electrostatic focusing is not used. Thus, development of future numerical solvers for the Poisson-Boltzmann equation should balance all aspects of the numerical procedures in realistic biomolecular applications. PMID:20063271
The geometry of finite difference discretizations of semilinear elliptic operators
NASA Astrophysics Data System (ADS)
Teles, Eduardo; Tomei, Carlos
2012-04-01
Discretizations by finite differences of some semilinear elliptic equations lead to maps F(u) = Au - f(u), u \\in {{R}}^n , given by nonlinear convex diagonal perturbations of symmetric matrices A. For natural nonlinearity classes, we consider the equation F(u) = y - tp, where t is a large positive number and p is a vector with negative coordinates. As the range of the derivative f'i of the coordinates of f encloses more eigenvalues of A, the number of solutions increases geometrically, eventually reaching 2n. This phenomenon, somewhat in contrast with behaviour associated with the Lazer-McKenna conjecture, has a very simple geometric explanation: a perturbation of a multiple fold gives rise to a function which sends connected components of its critical set to hypersurfaces with large rotation numbers with respect to vectors with very negative coordinates. Strictly speaking, the results have nothing to do with elliptic equations: they are properties of the interaction of a (self-adjoint) linear map with increasingly stronger nonlinear convex diagonal interactions.
Finite-difference time-domain simulations of metamaterials
NASA Astrophysics Data System (ADS)
Hao, Zhengwei
Metamaterials are periodic structures created by many identical scattering objects which are stationary and small compared to the wavelength of electromagnetic wave applied to it so that when combined with different elements, these materials have the potential to be coupled to the applied electromagnetic wave without modifying the structure. Due to their unusual properties that are not readily available in nature, metamaterials have been drawing significant attentions in many research areas, including theoretical, experimental as well as numerical investigations. As one of the major computational electromagnetic modeling methods, finite-difference time-domain (FDTD) technique tackles problems by providing a full wave solution. FDTD, which is able to show transient evolution of interactions between electromagnetic wave and physical objects, not only has the advantage in dispersive and nonlinear material simulations, but also has the ability to model circuit elements including semiconductor devices. All these features make FDTD a competitive candidate in numerical methods of metamaterial simulations. This dissertation presents the implementation of FDTD technique to deal with three dimensional (3D) problems characterized with metamaterial structures. We endeavor to make the FDTD engine multi-functional and fast, as depicted in the following three efforts: (1) We incorporated FDTD engine with the stable and highly efficient model for materials with dispersion, nonlinearity and gain properties. (2) We coupled FDTD engine with SPICE, the general-purpose and powerful analog electronic circuit simulator. This makes FDTD ready to simulate complex semiconductor devices and provides a variety of possibilities for novel metamaterials. (3) We investigated the cutting-edge area of Graphics Processing Units (GPU) computing module to speed up the FDTD engine, and implemented subgridding system to target more efficient modeling for metamaterial applications with embedded fine
Wang, Wei; Shu, Chi-Wang; Yee, H.C.; Sjögreen, Björn
2012-01-01
A new high order finite-difference method utilizing the idea of Harten ENO subcell resolution method is proposed for chemical reactive flows and combustion. In reaction problems, when the reaction time scale is very small, e.g., orders of magnitude smaller than the fluid dynamics time scales, the governing equations will become very stiff. Wrong propagation speed of discontinuity may occur due to the underresolved numerical solution in both space and time. The present proposed method is a modified fractional step method which solves the convection step and reaction step separately. In the convection step, any high order shock-capturing method can be used. In the reaction step, an ODE solver is applied but with the computed flow variables in the shock region modified by the Harten subcell resolution idea. For numerical experiments, a fifth-order finite-difference WENO scheme and its anti-diffusion WENO variant are considered. A wide range of 1D and 2D scalar and Euler system test cases are investigated. Studies indicate that for the considered test cases, the new method maintains high order accuracy in space for smooth flows, and for stiff source terms with discontinuities, it can capture the correct propagation speed of discontinuities in very coarse meshes with reasonable CFL numbers.
A semi-Lagrangian finite difference WENO scheme for scalar nonlinear conservation laws
NASA Astrophysics Data System (ADS)
Huang, Chieh-Sen; Arbogast, Todd; Hung, Chen-Hui
2016-10-01
For a nonlinear scalar conservation law in one-space dimension, we develop a locally conservative semi-Lagrangian finite difference scheme based on weighted essentially non-oscillatory reconstructions (SL-WENO). This scheme has the advantages of both WENO and semi-Lagrangian schemes. It is a locally mass conservative finite difference scheme, it is formally high-order accurate in space, it has small time truncation error, and it is essentially non-oscillatory. The scheme is nearly free of a CFL time step stability restriction for linear problems, and it has a relaxed CFL condition for nonlinear problems. The scheme can be considered as an extension of the SL-WENO scheme of Qiu and Shu (2011) [2] developed for linear problems. The new scheme is based on a standard sliding average formulation with the flux function defined using WENO reconstructions of (semi-Lagrangian) characteristic tracings of grid points. To handle nonlinear problems, we use an approximate, locally frozen trace velocity and a flux correction step. A special two-stage WENO reconstruction procedure is developed that is biased to the upstream direction. A Strang splitting algorithm is used for higher-dimensional problems. Numerical results are provided to illustrate the performance of the scheme and verify its formal accuracy. Included are applications to the Vlasov-Poisson and guiding-center models of plasma flow.
Evaluation of a thin-slot formalism for finite-difference time-domain electromagnetics codes
Turner, C.D.; Bacon, L.D.
1987-03-01
A thin-slot formalism for use with finite-difference time-domain (FDTD) electromagnetics codes has been evaluated in both two and three dimensions. This formalism allows narrow slots to be modeled in the wall of a scatterer without reducing the space grid size to the gap width. In two dimensions, the evaluation involves the calculation of the total fields near two infinitesimally thin coplanar strips separated by a gap. A method-of-moments (MoM) solution of the same problem is used as a benchmark for comparison. Results in two dimensions show that up to 10% error can be expected in total electric and magnetic fields both near (lambda/40) and far (1 lambda) from the slot. In three dimensions, the evaluation is similar. The finite-length slot is placed in a finite plate and an MoM surface patch solution is used for the benchmark. These results, although less extensive than those in two dimensions, show that slightly larger errors can be expected. Considering the approximations made near the slot in incorporating the formalism, the results are very promising. Possibilities also exist for applying this formalism to walls of arbitrary thickness and to other types of slots, such as overlapping joints. 11 refs., 25 figs., 6 tabs.
NASA Technical Reports Server (NTRS)
Siegel, R.; Molls, F. B.
1992-01-01
Transient solutions were obtained for a square region of heat conducting semitransparent material cooling by thermal radiation. The region is in a vacuum environment, so energy is dissipated only by radiation from within the medium leaving through its boundaries. The effect of heat conduction during the transient is to partially equalize the internal temperature distribution. As the optical thickness of the region is increased, the temperature gradients increase near the boundaries and corners, unless heat conduction is large. The solution procedure must provide accurate temperature distributions in these regions to prevent error in the calculated radiation losses. Two-dimensional numerical Gaussian integration is used to obtain the local radiative source term. A finite difference procedure with variable space and time increments is used to solve the transient energy equation. Variable spacing was used to concentrate grid points in regions with large temperature gradients.
A formula for the weight of a minimal filling of a finite metric space
Eremin, A Yu
2013-09-30
We consider the problem of finding a minimal filling for a finite metric space, that is, a weighted graph of minimal weight joining a given finite metric space. We obtain a minimax formula for the weight of the minimal filling, which we use to prove various properties of minimal fillings. Bibliography: 10 titles.
Dynamic and thermal response finite element models of multi-body space structural configurations
NASA Technical Reports Server (NTRS)
Edighoffer, Harold H.
1987-01-01
Presented is structural dynamics modeling of two multibody space structural configurations. The first configuration is a generic space station model of a cylindrical habitation module, two solar array panels, radiator panel, and central connecting tube. The second is a 15-m hoop-column antenna. Discussed is the special joint elimination sequence used for these large finite element models, so that eigenvalues could be extracted. The generic space station model aided test configuration design and analysis/test data correlation. The model consisted of six finite element models, one of each substructure and one of all substructures as a system. Static analysis and tests at the substructure level fine-tuned the finite element models. The 15-m hoop-column antenna is a truss column and structural ring interconnected with tension stabilizing cables. To the cables, pretensioned mesh membrane elements were attached to form four parabolic shaped antennae, one per quadrant. Imposing thermal preloads in the cables and mesh elements produced pretension in the finite element model. Thermal preload variation in the 96 control cables was adjusted to maintain antenna shape within the required tolerance and to give pointing accuracy.
Examples of Bosonic de Finetti States over Finite Dimensional Hilbert Spaces
NASA Astrophysics Data System (ADS)
Gottlieb, Alex D.
2005-11-01
According to the Quantum de Finetti Theorem, locally normal infinite particle states with Bose-Einstein symmetry can be represented as mixtures of infinite tensor powers of vector states. This note presents examples of infinite-particle states with Bose-Einstein symmetry that arise as limits of Gibbs ensembles on finite dimensional spaces, and displays their de Finetti representations. We consider Gibbs ensembles for systems of bosons in a finite dimensional setting and discover limits as the number of particles tends to infinity, provided the temperature is scaled in proportion to particle number
3D Finite Difference Modelling of Basaltic Region
NASA Astrophysics Data System (ADS)
Engell-Sørensen, L.
2003-04-01
The main purpose of the work was to generate realistic data to be applied for testing of processing and migration tools for basaltic regions. The project is based on the three - dimensional finite difference code (FD), TIGER, made by Sintef. The FD code was optimized (parallelized) by the author, to run on parallel computers. The parallel code enables us to model large-scale realistic geological models and to apply traditional seismic and micro seismic sources. The parallel code uses multiple processors in order to manipulate subsets of large amounts of data simultaneously. The general anisotropic code uses 21 elastic coefficients. Eight independent coefficients are needed as input parameters for the general TI medium. In the FD code, the elastic wave field computation is implemented by a higher order FD solution to the elastic wave equation and the wave fields are computed on a staggered grid, shifted half a node in one or two directions. The geological model is a gridded basalt model, which covers from 24 km to 37 km of a real shot line in horizontal direction and from the water surface to the depth of 3.5 km. The 2frac {1}{2}D model has been constructed using the compound modeling software from Norsk Hydro. The vertical parameter distribution is obtained from observations in two wells. At The depth of between 1100 m to 1500 m, a basalt horizon covers the whole sub surface layers. We have shown that it is possible to simulate a line survey in realistic (3D) geological models in reasonable time by using high performance computers. The author would like to thank Norsk Hydro, Statoil, GEUS, and SINTEF for very helpful discussions and Parallab for being helpful with the new IBM, p690 Regatta system.
Discretizing delta functions via finite differences and gradient normalization
NASA Astrophysics Data System (ADS)
Towers, John D.
2009-06-01
In [J.D. Towers, Two methods for discretizing a delta function supported on a level set, J. Comput. Phys. 220 (2007) 915-931] the author presented two closely related finite difference methods (referred to here as FDM1 and FDM2) for discretizing a delta function supported on a manifold of codimension one defined by the zero level set of a smooth mapping u :Rn ↦ R . These methods were shown to be consistent (meaning that they converge to the true solution as the mesh size h → 0) in the codimension one setting. In this paper, we concentrate on n ⩽ 3 , but generalize our methods to codimensions other than one - now the level set function is generally a vector valued mapping u → :Rn ↦Rm, 1 ⩽ m ⩽ n ⩽ 3 . Seemingly reasonable algorithms based on simple products of approximate delta functions are not generally consistent when applied to these problems. Motivated by this, we instead use the wedge product formalism to generalize our FDM algorithms, and this approach results in accurate, often consistent approximations. With the goal of ensuring consistency in general, we propose a new gradient normalization process that is applied before our FDM algorithms. These combined algorithms seem to be consistent in all reasonable situations, with numerical experiments indicating O (h2) convergence for our new gradient-normalized FDM2 algorithm. In the full codimension setting (m = n) , our gradient normalization processing also improves accuracy when using more standard approximate delta functions. This combination also yields approximations that appear to be consistent.
Space Object Detection and Tracking Within a Finite Set Statistics Framework
2017-04-13
AFRL-AFOSR-CL-TR-2017-0005 Space Object Detection & Tracking Within a Finite Set Statistics Framework Martin Adams Department of Electrical...MM-YYYY) 21-04-2017 2. REPORT TYPE Final 3. DATES COVERED (From - To) 01 Feb 2015 to 31 Jan 2017 4. TITLE AND SUBTITLE Space Object Detection...Grant No. FA9550-15-1-0069, devoted to the investigation and improvement of the detection and tracking methods of inactive Resident Space Objects (RSOs
NASA Technical Reports Server (NTRS)
Gladden, Herbert J.; Ko, Ching L.; Boddy, Douglas E.
1995-01-01
A higher-order finite-difference technique is developed to calculate the developing-flow field of steady incompressible laminar flows in the entrance regions of circular pipes. Navier-Stokes equations governing the motion of such a flow field are solved by using this new finite-difference scheme. This new technique can increase the accuracy of the finite-difference approximation, while also providing the option of using unevenly spaced clustered nodes for computation such that relatively fine grids can be adopted for regions with large velocity gradients. The velocity profile at the entrance of the pipe is assumed to be uniform for the computation. The velocity distribution and the surface pressure drop of the developing flow then are calculated and compared to existing experimental measurements reported in the literature. Computational results obtained are found to be in good agreement with existing experimental correlations and therefore, the reliability of the new technique has been successfully tested.
NASA Technical Reports Server (NTRS)
Ransom, Jonathan B.
2002-01-01
A multifunctional interface method with capabilities for variable-fidelity modeling and multiple method analysis is presented. The methodology provides an effective capability by which domains with diverse idealizations can be modeled independently to exploit the advantages of one approach over another. The multifunctional method is used to couple independently discretized subdomains, and it is used to couple the finite element and the finite difference methods. The method is based on a weighted residual variational method and is presented for two-dimensional scalar-field problems. A verification test problem and a benchmark application are presented, and the computational implications are discussed.
Finite element analysis of the Space Shuttle 2.5-inch frangible nut
NASA Technical Reports Server (NTRS)
McKinnis, Darin N.
1994-01-01
Finite element analysis of the Space Shuttle 2.5-inch frangible nut was conducted to improve understanding of the current design and proposed design changes to this explosively-actuated nut. The 2.5-inch frangible nut is used in two places to attach the aft end of the Space Shuttle Orbiter to the External Tank. Both 2.5-inch frangible nuts must function to complete safe separation. The 2.5-inch frangible nut contains two explosive boosters containing RDX explosive each capable of splitting the nut in half, on command from the Orbiter computers. To ensure separation, the boosters are designed to be redundant. The detonation of one booster is sufficient to split the nut in half. However, beginning in 1987 some production lots of 2.5-inch frangible nuts have demonstrated an inability to separate using only a single booster. The cause of the failure has been attributed to differences in the material properties and response of the Inconel 718 from which the 2.5-inch frangible nut is manufactured. Subsequent tests have resulted in design modifications of the boosters and frangible nut. Model development and initial analysis was conducted by Sandia National Laboratories (SNL) under funding from NASA Lyndon B. Johnson Space Center (NASA-JSC) starting in 1992. Modeling codes previously developed by SNL were transferred to NASA-JSC for further analysis on this and other devices. An explosive bolt with NASA Standard Detonator (NSD) charge, a 3/4-inch frangible nut, and the Super*Zip linear separation system are being modeled by NASA-JSC.
Finite element analysis of the Space Shuttle 2.5-inch frangible nut
NASA Technical Reports Server (NTRS)
McKinnis, Darin N.
1994-01-01
Finite element analysis of the Space Shuttle 2.5-inch frangible nut was conducted to improve understanding of the current design and proposed design changes to this explosively-actuated nut. The 2.5-inch frangible nut is used in two places to attach the aft end of the Space Shuttle Orbiter to the External Tank. Both 2.5-inch frangible nuts must function to complete safe separation. The 2.5-inch frangible nut contains two explosive boosters containing RDX explosive each capable of splitting the nut in half, on command from the Orbiter computers. To ensure separation, the boosters are designed to be redundant. The detonation of one booster is sufficient to split the nut in half. However, beginning in 1987 some production lots of 2.5-inch frangible nuts have demonstrated an inability to separate using only a single booster. The cause of the failure has been attributed to differences in the material properties and response of the Inconel 718 from which the 2.5-inch frangible nut is manufactured. Subsequent tests have resulted in design modifications of the boosters and frangible nut. Model development and initial analysis was conducted by Sandia National Laboratories (SNL) under funding from NASA Lyndon B. Johnson Space Center (NASA-JSC) starting in 1992. Modeling codes previously developed by SNL were transferred to NASA-JSC for further analysis on this and other devices. An explosive bolt with NASA Standard Detonator (NSD) charge, a 3/4-inch frangible nut, and the Super*Zip linear separation system are being modeled by NASA-JSC.
1988-06-01
passes through zero degrees. FDM-A, FDM-B, FDM-C and FEM-C represent the same physical solution, which is called the consensus solution. These sol - utions...Fig. 18). All the models except FDM-C depict the same shape as the phase consensus and FEM-C is again closest to the consensus sol - ution. Note that...models are closer than the finite element models to the consensus sol - ution for grids A and C. FDM-B and FEM-B are nearly identical. FDM-C is closest
NASA Astrophysics Data System (ADS)
Li, Gangqiang; Zhu, Zheng H.; Ruel, Stephane; Meguid, S. A.
2017-08-01
This paper developed a new multiphysics finite element method for the elastodynamic analysis of space debris deorbit by a bare flexible electrodynamic tether. Orbital motion limited theory and dynamics of flexible electrodynamic tethers are discretized by the finite element method, where the motional electric field is variant along the tether and coupled with tether deflection and motion. Accordingly, the electrical current and potential bias profiles of tether are solved together with the tether dynamics by the nodal position finite element method. The newly proposed multiphysics finite element method is applied to analyze the deorbit dynamics of space debris by electrodynamic tethers with a two-stage energy control strategy to ensure an efficient and stable deorbit process. Numerical simulations are conducted to study the coupled effect between the motional electric field and the tether dynamics. The results reveal that the coupling effect has a significant influence on the tether stability and the deorbit performance. It cannot be ignored when the libration and deflection of the tether are significant.
Finite-difference time-domain simulation of GPR data
NASA Astrophysics Data System (ADS)
Chen, How-Wei; Huang, Tai-Min
1998-10-01
Simulation of digital ground penetrating radar (GPR) wave propagation in two-dimensional (2-D) media is developed, tested, implemented, and applied using a time-domain staggered-grid finite-difference (FD) numerical method. Three types of numerical algorithms for constructing synthetic common-shot, constant-offset radar profiles based on an actual transmitter-to-receiver configuration and based on the exploding reflector concept are demonstrated to mimic different types of radar survey geometries. Frequency-dependent attenuation is also incorporated to account for amplitude decay and time shift in the recorded responses. The algorithms are based on an explicit FD solution to Maxwell's curl equations. In addition, the first-order TE mode responses of wave propagation phenomena are considered due to the operating frequency of current GPR instruments. The staggered-grid technique is used to sample the fields and approximate the spatial derivatives with fourth-order FDs. The temporal derivatives are approximated by an explicit second-order difference time-marching scheme. By combining paraxial approximation of the one-way wave equation ( A2) and the damping mechanisms (sponge filter), we propose a new composite absorbing boundary conditions (ABC) algorithm that effectively absorb both incoming and outgoing waves. To overcome the angle- and frequency-dependent characteristic of the absorbing behaviors, each ABC has two types of absorption mechanism. The first ABC uses a modified Clayton and Enquist's A2 condition. Moreover, a fixed and a floating A2 ABC that operates at one grid point is proposed. The second ABC uses a damping mechanism. By superimposing artificial damping and by alternating the physical attenuation properties and impedance contrast of the media within the absorbing region, those waves impinging on the boundary can be effectively attenuated and can prevent waves from reflecting back into the grid. The frequency-dependent characteristic of the damping
A finite different field solver for dipole modes
Nelson, E.M.
1992-08-01
A finite element field solver for dipole modes in axisymmetric structures has been written. The second-order elements used in this formulation yield accurate mode frequencies with no spurious modes. Quasi-periodic boundaries are included to allow travelling waves in periodic structures. The solver is useful in applications requiring precise frequency calculations such as detuned accelerator structures for linear colliders. Comparisons are made with measurements and with the popular but less accurate field solver URMEL.
Application of Finite Element Method to the structure design of the Space Solar Telescope
NASA Astrophysics Data System (ADS)
Zhang, Rui; Chen, Zhi-Yuan; Chen, Zhi-Ping; Yang, Shi-Mo
2005-12-01
Finite Element Method (FEM), the primary numerical means to process structure analysis in the modern engineering field, is adopted widely in the design of astronomical instruments at present. It can help designers to find out various characteristics of the object, to discover the weakness in stiffness and strength, and to improve and optimize the design as well. It is also used widely in many processes during the designing of the Space Solar Telescope (SST), such as in the main truss and the primary cell. From the beginning of the geometry modeling and the finite element creating, many aspects such as linear static, modal analysis, transient response and thermal analysis are demonstrated in SST. The error existing in the FEM, why it exists, and how to reduce it are discussed. Finally, the development trend of FEM in the astronomical instruments especially the space astronomical instruments is presented.
Chelikowsky, J.R.; Oeguet, S.; Jing, X.; Wu, K.; Stathopoulos, A.; Saad, Y.
1996-12-31
Determining the electronic and structural properties of semiconductor clusters is one of the outstanding problems in materials science. The existence of numerous structures with nearly identical energies makes it very difficult to determine a realistic ground state structure. Moreover, even if an effective procedure can be devised to predict the ground state structure, questions can arise about the relevancy of the structure at finite temperatures. Kinetic effects and non-equilibrium structures may dominate the structural configurations present in clusters created under laboratory conditions. The authors illustrate theoretical techniques for predicting the structure and electronic properties of small germanium clusters. Specifically, they illustrate that the detailed agreement between theoretical and experimental features can be exploited to identify the relevant isomers present under experimental conditions.
Improved finite-difference computation of the van der Waals force: One-dimensional case
Pinto, Fabrizio
2009-10-15
We present an improved demonstration of the calculation of Casimir forces in one-dimensional systems based on the recently proposed numerical imaginary frequency Green's function computation approach. The dispersion force on two thick lossy dielectric slabs separated by an empty gap and placed within a perfectly conducting cavity is obtained from the Green's function of the modified Helmholtz equation by means of an ordinary finite-difference method. In order to demonstrate the possibility to develop algorithms to explore complex geometries in two and three dimensions to higher order in the mesh spacing, we generalize existing classical electromagnetism algebraic methods to generate the difference equations for dielectric boundaries not coinciding with any grid points. Diagnostic tests are presented to monitor the accuracy of our implementation of the method and follow-up applications in higher dimensions are introduced.
Improved finite-difference computation of the van der Waals force: One-dimensional case
NASA Astrophysics Data System (ADS)
Pinto, Fabrizio
2009-10-01
We present an improved demonstration of the calculation of Casimir forces in one-dimensional systems based on the recently proposed numerical imaginary frequency Green’s function computation approach. The dispersion force on two thick lossy dielectric slabs separated by an empty gap and placed within a perfectly conducting cavity is obtained from the Green’s function of the modified Helmholtz equation by means of an ordinary finite-difference method. In order to demonstrate the possibility to develop algorithms to explore complex geometries in two and three dimensions to higher order in the mesh spacing, we generalize existing classical electromagnetism algebraic methods to generate the difference equations for dielectric boundaries not coinciding with any grid points. Diagnostic tests are presented to monitor the accuracy of our implementation of the method and follow-up applications in higher dimensions are introduced.
NASA Technical Reports Server (NTRS)
Joshi, S. M.; Groom, N. J.
1980-01-01
A finite element structural model of a 30.48 m x 30.48 m x 2.54 mm completely free aluminum plate is described and modal frequencies and mode shape data for the first 44 modes are presented. An explanation of the procedure for using the data is also presented. The model should prove useful for the investigation of controller design approaches for large flexible space structures.
Higher-Order, Space-Time Adaptive Finite Volume Methods: Algorithms, Analysis and Applications
Minion, Michael
2014-04-29
The four main goals outlined in the proposal for this project were: 1. Investigate the use of higher-order (in space and time) finite-volume methods for fluid flow problems. 2. Explore the embedding of iterative temporal methods within traditional block-structured AMR algorithms. 3. Develop parallel in time methods for ODEs and PDEs. 4. Work collaboratively with the Center for Computational Sciences and Engineering (CCSE) at Lawrence Berkeley National Lab towards incorporating new algorithms within existing DOE application codes.
Equivalent Continuum Finite Element Modelling of Plate-Like Space Lattice Structures.
1985-08-01
regulation cost of the structure as a function of the structural design parameters. A micropolar plate continuum model of large plate-like repetitive space...lattice structures with rigid joints is derived. A plate finite element is derived based on this continuum model with micropolar rotations and transverse...by rigid joints which makes use of the higher order micropolar beam continuum formulation. 8 Detailed Models For this research the baseline against
Boiling Fluids Behave Quite Differently in Space
The boiling process is really different in space, since the vapor phase of a boiling liquid does not rise via buoyancy. Spacecraft and Earth-based systems use boiling to efficiently remove large am...
Finite element analysis of space debris removal by high-power lasers
NASA Astrophysics Data System (ADS)
Xue, Li; Jiang, Guanlei; Yu, Shuang; Li, Ming
2015-08-01
With the development of space station technologies, irradiation of space debris by space-based high-power lasers, can locally generate high-temperature plasmas and micro momentum, which may achieve the removal of debris through tracking down. Considered typical square-shaped space debris of material Ti with 5cm×5cm size, whose thermal conductivity, density, specific heat capacity and emissivity are 7.62W/(m·°C), 4500kg/m3, 0.52J/(kg·°C) and 0.3,respectively, based on the finite element analysis of ANSYS, each irradiation of space debris by high-power lasers with power density 106W/m2 and weapons-grade lasers with power density 3000W/m2 are simulated under space environment, and the temperature curves due to laser thermal irradiation are obtained and compared. Results show only 2s is needed for high-power lasers to make the debris temperature reach to about 10000K, which is the threshold temperature for plasmas-state conversion. While for weapons-grade lasers, it is 13min needed. Using two line elements (TLE), and combined with the coordinate transformation from celestial coordinate system to site coordinate system, the visible period of space debris is calculated as 5-10min. That is, in order to remove space debris by laser plasmas, the laser power density should be further improved. The article provides an intuitive and visual feasibility analysis method of space debris removal, and the debris material and shape, laser power density and spot characteristics are adjustable. This finite element analysis method is low-cost, repeatable and adaptable, which has an engineering-prospective applications.
High-stability Finite-Length Silicon Nanowires: A Real Space Theoretical Study
NASA Astrophysics Data System (ADS)
Koukaras, E. N.; Zdetsis, A. D.; Garoufalis, C. S.
2007-12-01
We demonstrate by real-space density functional calculations that unreconstructed low-stability finite size hydrogenated silicon nanowires could bend through relaxation under the influence of internal strains, contrary to high-stability "magic" nanowires. The strains and the resulting bending depend on the distribution and orientation of silicon dihydrides on the nanowire's surface. This and other related effects cannot be accounted for by the usual k-space supercell techniques. We also demonstrate that reconstructed (2×1) nanowires, although bend they are practically as stable as the "magic" unreconstructed nanowires. Our calculations are in full agreement with the experimental work of Ma et al. [Science 299, 1874, (2003)].
The Benard problem: A comparison of finite difference and spectral collocation eigen value solutions
NASA Technical Reports Server (NTRS)
Skarda, J. Raymond Lee; Mccaughan, Frances E.; Fitzmaurice, Nessan
1995-01-01
The application of spectral methods, using a Chebyshev collocation scheme, to solve hydrodynamic stability problems is demonstrated on the Benard problem. Implementation of the Chebyshev collocation formulation is described. The performance of the spectral scheme is compared with that of a 2nd order finite difference scheme. An exact solution to the Marangoni-Benard problem is used to evaluate the performance of both schemes. The error of the spectral scheme is at least seven orders of magnitude smaller than finite difference error for a grid resolution of N = 15 (number of points used). The performance of the spectral formulation far exceeded the performance of the finite difference formulation for this problem. The spectral scheme required only slightly more effort to set up than the 2nd order finite difference scheme. This suggests that the spectral scheme may actually be faster to implement than higher order finite difference schemes.
Two-dimensional time-domain finite-difference modeling for viscoelastic seismic wave propagation
NASA Astrophysics Data System (ADS)
Fan, Na; Zhao, Lian-Feng; Xie, Xiao-Bi; Ge, Zengxi; Yao, Zhen-Xing
2016-09-01
Real Earth media are not perfectly elastic. Instead, they attenuate propagating mechanical waves. This anelastic phenomenon in wave propagation can be modeled by a viscoelastic mechanical model consisting of several standard linear solids. Using this viscoelastic model, we approximate a constant Q over a frequency band of interest. We use a four-element viscoelastic model with a trade-off between accuracy and computational costs to incorporate Q into 2-D time-domain first-order velocity-stress wave equations. To improve the computational efficiency, we limit the Q in the model to a list of discrete values between 2 and 1000. The related stress and strain relaxation times that characterize the viscoelastic model are pre-calculated and stored in a database for use by the finite-difference calculation. A viscoelastic finite-difference scheme that is second order in time and fourth order in space is developed based on the MacCormack algorithm. The new method is validated by comparing the numerical result with analytical solutions that are calculated using the generalized reflection/transmission coefficient method. The synthetic seismograms exhibit greater than 95 per cent consistency in a two-layer viscoelastic model. The dispersion generated from the simulation is consistent with the Kolsky-Futterman dispersion relationship.
NASA Technical Reports Server (NTRS)
Baumeister, Kenneth J.; Kreider, Kevin L.
1996-01-01
An explicit finite difference iteration scheme is developed to study harmonic sound propagation in aircraft engine nacelles. To reduce storage requirements for large 3D problems, the time dependent potential form of the acoustic wave equation is used. To insure that the finite difference scheme is both explicit and stable, time is introduced into the Fourier transformed (steady-state) acoustic potential field as a parameter. Under a suitable transformation, the time dependent governing equation in frequency space is simplified to yield a parabolic partial differential equation, which is then marched through time to attain the steady-state solution. The input to the system is the amplitude of an incident harmonic sound source entering a quiescent duct at the input boundary, with standard impedance boundary conditions on the duct walls and duct exit. The introduction of the time parameter eliminates the large matrix storage requirements normally associated with frequency domain solutions, and time marching attains the steady-state quickly enough to make the method favorable when compared to frequency domain methods. For validation, this transient-frequency domain method is applied to sound propagation in a 2D hard wall duct with plug flow.
NASA Astrophysics Data System (ADS)
Zehner, Björn; Hellwig, Olaf; Linke, Maik; Görz, Ines; Buske, Stefan
2016-01-01
3D geological underground models are often presented by vector data, such as triangulated networks representing boundaries of geological bodies and geological structures. Since models are to be used for numerical simulations based on the finite difference method, they have to be converted into a representation discretizing the full volume of the model into hexahedral cells. Often the simulations require a high grid resolution and are done using parallel computing. The storage of such a high-resolution raster model would require a large amount of storage space and it is difficult to create such a model using the standard geomodelling packages. Since the raster representation is only required for the calculation, but not for the geometry description, we present an algorithm and concept for rasterizing geological models on the fly for the use in finite difference codes that are parallelized by domain decomposition. As a proof of concept we implemented a rasterizer library and integrated it into seismic simulation software that is run as parallel code on a UNIX cluster using the Message Passing Interface. We can thus run the simulation with realistic and complicated surface-based geological models that are created using 3D geomodelling software, instead of using a simplified representation of the geological subsurface using mathematical functions or geometric primitives. We tested this set-up using an example model that we provide along with the implemented library.
NASA Technical Reports Server (NTRS)
Baumeister, K. J.; Kreider, K. L.
1996-01-01
An explicit finite difference iteration scheme is developed to study harmonic sound propagation in ducts. To reduce storage requirements for large 3D problems, the time dependent potential form of the acoustic wave equation is used. To insure that the finite difference scheme is both explicit and stable, time is introduced into the Fourier transformed (steady-state) acoustic potential field as a parameter. Under a suitable transformation, the time dependent governing equation in frequency space is simplified to yield a parabolic partial differential equation, which is then marched through time to attain the steady-state solution. The input to the system is the amplitude of an incident harmonic sound source entering a quiescent duct at the input boundary, with standard impedance boundary conditions on the duct walls and duct exit. The introduction of the time parameter eliminates the large matrix storage requirements normally associated with frequency domain solutions, and time marching attains the steady-state quickly enough to make the method favorable when compared to frequency domain methods. For validation, this transient-frequency domain method is applied to sound propagation in a 2D hard wall duct with plug flow.
An angularly refineable phase space finite element method with approximate sweeping procedure
Kophazi, J.; Lathouwers, D.
2013-07-01
An angularly refineable phase space finite element method is proposed to solve the neutron transport equation. The method combines the advantages of two recently published schemes. The angular domain is discretized into small patches and patch-wise discontinuous angular basis functions are restricted to these patches, i.e. there is no overlap between basis functions corresponding to different patches. This approach yields block diagonal Jacobians with small block size and retains the possibility for S{sub n}-like approximate sweeping of the spatially discontinuous elements in order to provide efficient preconditioners for the solution procedure. On the other hand, the preservation of the full FEM framework (as opposed to collocation into a high-order S{sub n} scheme) retains the possibility of the Galerkin interpolated connection between phase space elements at arbitrary levels of discretization. Since the basis vectors are not orthonormal, a generalization of the Riemann procedure is introduced to separate the incoming and outgoing contributions in case of unstructured meshes. However, due to the properties of the angular discretization, the Riemann procedure can be avoided at a large fraction of the faces and this fraction rapidly increases as the level of refinement increases, contributing to the computational efficiency. In this paper the properties of the discretization scheme are studied with uniform refinement using an iterative solver based on the S{sub 2} sweep order of the spatial elements. The fourth order convergence of the scalar flux is shown as anticipated from earlier schemes and the rapidly decreasing fraction of required Riemann faces is illustrated. (authors)
NASA Technical Reports Server (NTRS)
Panczak, Tim; Ring, Steve; Welch, Mark
1999-01-01
Thermal engineering has long been left out of the concurrent engineering environment dominated by CAD (computer aided design) and FEM (finite element method) software. Current tools attempt to force the thermal design process into an environment primarily created to support structural analysis, which results in inappropriate thermal models. As a result, many thermal engineers either build models "by hand" or use geometric user interfaces that are separate from and have little useful connection, if any, to CAD and FEM systems. This paper describes the development of a new thermal design environment called the Thermal Desktop. This system, while fully integrated into a neutral, low cost CAD system, and which utilizes both FEM and FD methods, does not compromise the needs of the thermal engineer. Rather, the features needed for concurrent thermal analysis are specifically addressed by combining traditional parametric surface based radiation and FD based conduction modeling with CAD and FEM methods. The use of flexible and familiar temperature solvers such as SINDA/FLUINT (Systems Improved Numerical Differencing Analyzer/Fluid Integrator) is retained.
NASA Technical Reports Server (NTRS)
Panczak, Tim; Ring, Steve; Welch, Mark
1999-01-01
Thermal engineering has long been left out of the concurrent engineering environment dominated by CAD (computer aided design) and FEM (finite element method) software. Current tools attempt to force the thermal design process into an environment primarily created to support structural analysis, which results in inappropriate thermal models. As a result, many thermal engineers either build models "by hand" or use geometric user interfaces that are separate from and have little useful connection, if any, to CAD and FEM systems. This paper describes the development of a new thermal design environment called the Thermal Desktop. This system, while fully integrated into a neutral, low cost CAD system, and which utilizes both FEM and FD methods, does not compromise the needs of the thermal engineer. Rather, the features needed for concurrent thermal analysis are specifically addressed by combining traditional parametric surface based radiation and FD based conduction modeling with CAD and FEM methods. The use of flexible and familiar temperature solvers such as SINDA/FLUINT (Systems Improved Numerical Differencing Analyzer/Fluid Integrator) is retained.
Study of two-dimensional transient cavity fields using the finite-difference time-domain technique
Crisp, J.L.
1988-06-01
This work is intended to be a study into the application of the finite-difference time-domain, or FD-TD technique, to some of the problems faced by designers of equipment used in modern accelerators. In particular it discusses using the FD-TD algorithm to study the field distribution of a simple two-dimensional cavity in both space and time. 18 refs.
NASA Technical Reports Server (NTRS)
Shu, Chi-Wang
1998-01-01
This project is about the development of high order, non-oscillatory type schemes for computational fluid dynamics. Algorithm analysis, implementation, and applications are performed. Collaborations with NASA scientists have been carried out to ensure that the research is relevant to NASA objectives. The combination of ENO finite difference method with spectral method in two space dimension is considered, jointly with Cai [3]. The resulting scheme behaves nicely for the two dimensional test problems with or without shocks. Jointly with Cai and Gottlieb, we have also considered one-sided filters for spectral approximations to discontinuous functions [2]. We proved theoretically the existence of filters to recover spectral accuracy up to the discontinuity. We also constructed such filters for practical calculations.
NASA Astrophysics Data System (ADS)
Azmir, O. Shahrul; Azwadi, C. S. Nor
2010-06-01
This paper presents numerical study of flow behavior from a heated concentric annulus cylinder at various Rayleigh number Ra, Prandtl number Pr while the aspect ratio is fixed to 5.0 of the outer and inner cylinders. The Finite Different Lattice Boltzmann Method (FDLBM) numerical scheme is proposed to improve the computational efficiency and numerical stability of the conventional method. The proposed FELBM applied UTOPIA approach (third order accuracy in space) to study the temperature distribution and the vortex formation in the annulus cylinder. The comparison of the flow pattern and temperature distribution for every case via streamline, vortices and temperature distribution contour with published paper in literature were carried out for the validation purposes. Current investigation concluded that the UTOPIA FDLBM is an efficient approach for the current problem in hand and good agreement with the benchmark solution.
NASA Technical Reports Server (NTRS)
Kaul, Upender K. (Inventor)
2009-01-01
Modeling and simulation of free and forced structural vibrations is essential to an overall structural health monitoring capability. In the various embodiments, a first principles finite-difference approach is adopted in modeling a structural subsystem such as a mechanical gear by solving elastodynamic equations in generalized curvilinear coordinates. Such a capability to generate a dynamic structural response is widely applicable in a variety of structural health monitoring systems. This capability (1) will lead to an understanding of the dynamic behavior of a structural system and hence its improved design, (2) will generate a sufficiently large space of normal and damage solutions that can be used by machine learning algorithms to detect anomalous system behavior and achieve a system design optimization and (3) will lead to an optimal sensor placement strategy, based on the identification of local stress maxima all over the domain.
Black-Scholes finite difference modeling in forecasting of call warrant prices in Bursa Malaysia
NASA Astrophysics Data System (ADS)
Mansor, Nur Jariah; Jaffar, Maheran Mohd
2014-07-01
Call warrant is a type of structured warrant in Bursa Malaysia. It gives the holder the right to buy the underlying share at a specified price within a limited period of time. The issuer of the structured warrants usually uses European style to exercise the call warrant on the maturity date. Warrant is very similar to an option. Usually, practitioners of the financial field use Black-Scholes model to value the option. The Black-Scholes equation is hard to solve analytically. Therefore the finite difference approach is applied to approximate the value of the call warrant prices. The central in time and central in space scheme is produced to approximate the value of the call warrant prices. It allows the warrant holder to forecast the value of the call warrant prices before the expiry date.
Convergence Rates of Finite Difference Stochastic Approximation Algorithms
2016-06-01
was due to Chung (1954) and was formulated in the present form by Fabian (1971). Lemma 1. Let s, t, B,An, bn be real numbers, 0 < s ≤ 1, t ≥ 0, B > 0...Define b+ = 0 if s < 1 and b+ = t if s = 1 and assume that c = limn→∞An − b+ exists and is finite. If for 5 n ≥ n0, bn +1 ≤ bn (1− An ns ) + B ns+t and...Denote An = 1 + σ n −K3an − σ n K3an +O( 1 n2 ), Bn = (1 + 1 n )σnσ(banδ β n(1 + E[εn])− anE[τnθn]). Then zn+1 = Anzn − Bn .(16) Note that an = an −α, 0
NASA Astrophysics Data System (ADS)
Dumbser, Michael; Hidalgo, Arturo; Zanotti, Olindo
2014-01-01
We present a class of high order finite volume schemes for the solution of non-conservative hyperbolic systems that combines the one-step ADER-WENO finite volume approach with space-time adaptive mesh refinement (AMR). The resulting algorithm, which is particularly well suited for the treatment of material interfaces in compressible multi-phase flows, is based on: (i) high order of accuracy in space obtained through WENO reconstruction, (ii) a high order one-step time discretization via a local space-time discontinuous Galerkin predictor method, and (iii) the use of a path conservative scheme for handling the non-conservative terms of the equations. The AMR property with time accurate local time stepping, which has been treated according to a 'cell-by-cell' strategy, strongly relies on the high order one-step time discretization, which naturally allows a high order accurate and consistent computation of the jump terms at interfaces between elements using different time steps. The new scheme has been successfully validated on some test problems for the Baer-Nunziato model of compressible multiphase flows.
NASA Technical Reports Server (NTRS)
Byun, Chansup; Guruswamy, Guru P.
1993-01-01
This paper presents a procedure for computing the aeroelasticity of wing-body configurations on multiple-instruction, multiple-data (MIMD) parallel computers. In this procedure, fluids are modeled using Euler equations discretized by a finite difference method, and structures are modeled using finite element equations. The procedure is designed in such a way that each discipline can be developed and maintained independently by using a domain decomposition approach. A parallel integration scheme is used to compute aeroelastic responses by solving the coupled fluid and structural equations concurrently while keeping modularity of each discipline. The present procedure is validated by computing the aeroelastic response of a wing and comparing with experiment. Aeroelastic computations are illustrated for a High Speed Civil Transport type wing-body configuration.
Wang, Shumin; Duyn, Jeff H
2008-05-21
A hybrid method that combines the finite-difference time-domain (FDTD) method and the finite-element time-domain (FETD) method is presented for simulating radio-frequency (RF) coils in magnetic resonance imaging. This method applies a high-fidelity FETD method to RF coils, while the human body is modeled with a low-cost FDTD method. Since the FDTD and the FETD methods are applied simultaneously, the dynamic interaction between RF coils and the human body is fully accounted for. In order to simplify the treatment of the highly irregular FDTD/FETD interface, composite elements are proposed. Two examples are provided to demonstrate the validity and effectiveness of the hybrid method in high-field receive-and-transmit coil design. This approach is also applicable to general bio-electromagnetic simulations.
Finite-difference time-domain simulation of spacetime cloak.
Cornelius, Jason; Liu, Jinjie; Brio, Moysey
2014-05-19
In this work, we present a numerical method that remedies the instabilities of the conventional FDTD approach for solving Maxwell's equations in a space-time dependent magneto-electric medium with direct application to the simulation of the recently proposed spacetime cloak. We utilize a dual grid FDTD method overlapped in the time domain to provide a stable approach for the simulation of a magneto-electric medium with time and space varying permittivity, permeability and coupling coefficient. The developed method can be applied to explore other new physical possibilities offered by spacetime cloaking, metamaterials, and transformation optics.
Various finite-difference schemes for transient three-dimensional heat conduction
Yalamanchili, R.; Yalamanchili, S.R.
1992-03-01
The motivation for this task comes from the needs of future hypervelocity projectile surrounded by asymmetric flow due to angle of attack and/or fins in case of kinetic energy projectile. In either case, unsteady and three-dimensional effects, large and nonuniform heat fluxes, tedious and repetitive number crunching capabilities of supercomputers dictate optimum numerical techniques and predictive critical time steps for successful and practical solutions. Finite element modeling is ideal whenever there is geometrical complexity, coatings, composite and multi materials. However, classical finite element technique yields a particular equation. There may be some finite difference schemes superior to classical finite element technique. Therefore, various finite difference schemes are derived and their characteristics are discussed applicable to transient three dimensional heat conduction problems.
Simulation of transonic separated airfoil flow by finite difference viscous-inviscid interaction
NASA Technical Reports Server (NTRS)
Vandalsem, W. R.; Steger, J. L.
1984-01-01
A finite difference viscous inviscid interaction program has been developed for simulating the separated transonic flow about lifting airfoils, including the wake. In contrast to most interaction programs, this code combines a finite difference boundary layer algorithm with the inviscid program. The recently developed finite difference boundary layer code efficiently simulates attached and reversed compressible boundary layer and wake flows. New viscous inviscid interaction algorithms were also developed to couple the boundary layer code with the inviscid transonic full potential program. Transonic cases with shock induced and trailing edge separation are computed and compared with experimental and Navier-Stokes results.
Similarity and generalized finite-difference solutions of parabolic partial differential equations.
NASA Technical Reports Server (NTRS)
Clausing, A. M.
1971-01-01
Techniques are presented for obtaining generalized finite-difference solutions to partial differential equations of the parabolic type. It is shown that the advantages of similarity in the solution of similar problems are generally not lost if the solution to the original partial differential equations is effected in the physical plane by finite-difference methods. The analysis results in a considerable saving in computational effort in the solution of both similar and nonsimilar problems. Several examples, including both the heat-conduction equation and the boundary-layer equations, are given. The analysis also provides a practical means of estimating the accuracy of finite-difference solutions to parabolic equations.
Mattila, Keijo Kalervo; Hegele Júnior, Luiz Adolfo; Philippi, Paulo Cesar
2014-01-01
We propose isotropic finite differences for high-accuracy approximation of high-rank derivatives. These finite differences are based on direct application of lattice-Boltzmann stencils. The presented finite-difference expressions are valid in any dimension, particularly in two and three dimensions, and any lattice-Boltzmann stencil isotropic enough can be utilized. A theoretical basis for the proposed utilization of lattice-Boltzmann stencils in the approximation of high-rank derivatives is established. In particular, the isotropy and accuracy properties of the proposed approximations are derived directly from this basis. Furthermore, in this formal development, we extend the theory of Hermite polynomial tensors in the case of discrete spaces and present expressions for the discrete inner products between monomials and Hermite polynomial tensors. In addition, we prove an equivalency between two approaches for constructing lattice-Boltzmann stencils. For the numerical verification of the presented finite differences, we introduce 5th-, 6th-, and 8th-order two-dimensional lattice-Boltzmann stencils. PMID:24688360
Mattila, Keijo Kalervo; Hegele Júnior, Luiz Adolfo; Philippi, Paulo Cesar
2014-01-01
We propose isotropic finite differences for high-accuracy approximation of high-rank derivatives. These finite differences are based on direct application of lattice-Boltzmann stencils. The presented finite-difference expressions are valid in any dimension, particularly in two and three dimensions, and any lattice-Boltzmann stencil isotropic enough can be utilized. A theoretical basis for the proposed utilization of lattice-Boltzmann stencils in the approximation of high-rank derivatives is established. In particular, the isotropy and accuracy properties of the proposed approximations are derived directly from this basis. Furthermore, in this formal development, we extend the theory of Hermite polynomial tensors in the case of discrete spaces and present expressions for the discrete inner products between monomials and Hermite polynomial tensors. In addition, we prove an equivalency between two approaches for constructing lattice-Boltzmann stencils. For the numerical verification of the presented finite differences, we introduce 5th-, 6th-, and 8th-order two-dimensional lattice-Boltzmann stencils.
High-order cyclo-difference techniques: An alternative to finite differences
NASA Technical Reports Server (NTRS)
Carpenter, Mark H.; Otto, John C.
1993-01-01
The summation-by-parts energy norm is used to establish a new class of high-order finite-difference techniques referred to here as 'cyclo-difference' techniques. These techniques are constructed cyclically from stable subelements, and require no numerical boundary conditions; when coupled with the simultaneous approximation term (SAT) boundary treatment, they are time asymptotically stable for an arbitrary hyperbolic system. These techniques are similar to spectral element techniques and are ideally suited for parallel implementation, but do not require special collocation points or orthogonal basis functions. The principal focus is on methods of sixth-order formal accuracy or less; however, these methods could be extended in principle to any arbitrary order of accuracy.
The finite difference method in electronic structure calculations
Fattebert, Jean -Luc
2015-11-21
Since the development of quantum mechanics, we know the equations describing the behavior of atoms and electrons at the microscopic level. The Schroedinger equation is however too difficult to solve for more than a few particles because of the high dimensional space of the solution - 3N for N particles. So various simplified models have been developed. Furthermore, the first simplification usually introduced is the Born-Oppenhaimer approximation in which atomic nuclei are treated as classical particles surrounded by quantum electrons.
ADER-WENO finite volume schemes with space-time adaptive mesh refinement
NASA Astrophysics Data System (ADS)
Dumbser, Michael; Zanotti, Olindo; Hidalgo, Arturo; Balsara, Dinshaw S.
2013-09-01
We present the first high order one-step ADER-WENO finite volume scheme with adaptive mesh refinement (AMR) in multiple space dimensions. High order spatial accuracy is obtained through a WENO reconstruction, while a high order one-step time discretization is achieved using a local space-time discontinuous Galerkin predictor method. Due to the one-step nature of the underlying scheme, the resulting algorithm is particularly well suited for an AMR strategy on space-time adaptive meshes, i.e. with time-accurate local time stepping. The AMR property has been implemented 'cell-by-cell', with a standard tree-type algorithm, while the scheme has been parallelized via the message passing interface (MPI) paradigm. The new scheme has been tested over a wide range of examples for nonlinear systems of hyperbolic conservation laws, including the classical Euler equations of compressible gas dynamics and the equations of magnetohydrodynamics (MHD). High order in space and time have been confirmed via a numerical convergence study and a detailed analysis of the computational speed-up with respect to highly refined uniform meshes is also presented. We also show test problems where the presented high order AMR scheme behaves clearly better than traditional second order AMR methods. The proposed scheme that combines for the first time high order ADER methods with space-time adaptive grids in two and three space dimensions is likely to become a useful tool in several fields of computational physics, applied mathematics and mechanics.
NASA Astrophysics Data System (ADS)
Wu, Shun-Der; Glytsis, Elias N.
2002-10-01
The effects of finite number of periods (FNP) and finite incident beams on the diffraction efficiencies of holographic gratings are investigated by the finite-difference frequency-domain (FDFD) method. Gratings comprising 20, 15, 10, 5, and 3 periods illuminated by TE and TM incident light with various beam sizes are analyzed with the FDFD method and compared with the rigorous coupled-wave analysis (RCWA). Both unslanted and slanted gratings are treated in transmission as well as in reflection configurations. In general, the effect of the FNP is a decrease in the diffraction efficiency with a decrease in the number of periods of the grating. Similarly, a decrease in incident-beam width causes a decrease in the diffraction efficiency. Exceptions appear in off-Bragg incidence in which a smaller beam width could result in higher diffraction efficiency. For beam widths greater than 10 grating periods and for gratings with more than 20 periods in width, the diffraction efficiencies slowly converge to the values predicted by the RCWA (infinite incident beam and infinite-number-of-periods grating) for both TE and TM polarizations. Furthermore, the effects of FNP holographic gratings on their diffraction performance are found to be comparable to their counterparts of FNP surface-relief gratings. 2002 Optical Society of America
Castellani, Marco; Giuli, Massimiliano
2016-02-15
We study pseudomonotone and quasimonotone quasivariational inequalities in a finite dimensional space. In particular we focus our attention on the closedness of some solution maps associated to a parametric quasivariational inequality. From this study we derive two results on the existence of solutions of the quasivariational inequality. On the one hand, assuming the pseudomonotonicity of the operator, we get the nonemptiness of the set of the classical solutions. On the other hand, we show that the quasimonoticity of the operator implies the nonemptiness of the set of nonzero solutions. An application to traffic network is also considered.
Finite-key security analyses on passive decoy-state QKD protocols with different unstable sources.
Song, Ting-Ting; Qin, Su-Juan; Wen, Qiao-Yan; Wang, Yu-Kun; Jia, Heng-Yue
2015-10-16
In quantum communication, passive decoy-state QKD protocols can eliminate many side channels, but the protocols without any finite-key analyses are not suitable for in practice. The finite-key securities of passive decoy-state (PDS) QKD protocols with two different unstable sources, type-II parametric down-convention (PDC) and phase randomized weak coherent pulses (WCPs), are analyzed in our paper. According to the PDS QKD protocols, we establish an optimizing programming respectively and obtain the lower bounds of finite-key rates. Under some reasonable values of quantum setup parameters, the lower bounds of finite-key rates are simulated. The simulation results show that at different transmission distances, the affections of different fluctuations on key rates are different. Moreover, the PDS QKD protocol with an unstable PDC source can resist more intensity fluctuations and more statistical fluctuation.
Finite-key security analyses on passive decoy-state QKD protocols with different unstable sources
NASA Astrophysics Data System (ADS)
Song, Ting-Ting; Qin, Su-Juan; Wen, Qiao-Yan; Wang, Yu-Kun; Jia, Heng-Yue
2015-10-01
In quantum communication, passive decoy-state QKD protocols can eliminate many side channels, but the protocols without any finite-key analyses are not suitable for in practice. The finite-key securities of passive decoy-state (PDS) QKD protocols with two different unstable sources, type-II parametric down-convention (PDC) and phase randomized weak coherent pulses (WCPs), are analyzed in our paper. According to the PDS QKD protocols, we establish an optimizing programming respectively and obtain the lower bounds of finite-key rates. Under some reasonable values of quantum setup parameters, the lower bounds of finite-key rates are simulated. The simulation results show that at different transmission distances, the affections of different fluctuations on key rates are different. Moreover, the PDS QKD protocol with an unstable PDC source can resist more intensity fluctuations and more statistical fluctuation.
Finite-difference scheme for the numerical solution of the Schroedinger equation
NASA Technical Reports Server (NTRS)
Mickens, Ronald E.; Ramadhani, Issa
1992-01-01
A finite-difference scheme for numerical integration of the Schroedinger equation is constructed. Asymptotically (r goes to infinity), the method gives the exact solution correct to terms of order r exp -2.
A non-linear constrained optimization technique for the mimetic finite difference method
Manzini, Gianmarco; Svyatskiy, Daniil; Bertolazzi, Enrico; Frego, Marco
2014-09-30
This is a strategy for the construction of monotone schemes in the framework of the mimetic finite difference method for the approximation of diffusion problems on unstructured polygonal and polyhedral meshes.
Lisitsa, Vadim; Tcheverda, Vladimir; Botter, Charlotte
2016-04-15
We present an algorithm for the numerical simulation of seismic wave propagation in models with a complex near surface part and free surface topography. The approach is based on the combination of finite differences with the discontinuous Galerkin method. The discontinuous Galerkin method can be used on polyhedral meshes; thus, it is easy to handle the complex surfaces in the models. However, this approach is computationally intense in comparison with finite differences. Finite differences are computationally efficient, but in general, they require rectangular grids, leading to the stair-step approximation of the interfaces, which causes strong diffraction of the wavefield. In this research we present a hybrid algorithm where the discontinuous Galerkin method is used in a relatively small upper part of the model and finite differences are applied to the main part of the model.
Finite-key security analyses on passive decoy-state QKD protocols with different unstable sources
Song, Ting-Ting; Qin, Su-Juan; Wen, Qiao-Yan; Wang, Yu-Kun; Jia, Heng-Yue
2015-01-01
In quantum communication, passive decoy-state QKD protocols can eliminate many side channels, but the protocols without any finite-key analyses are not suitable for in practice. The finite-key securities of passive decoy-state (PDS) QKD protocols with two different unstable sources, type-II parametric down-convention (PDC) and phase randomized weak coherent pulses (WCPs), are analyzed in our paper. According to the PDS QKD protocols, we establish an optimizing programming respectively and obtain the lower bounds of finite-key rates. Under some reasonable values of quantum setup parameters, the lower bounds of finite-key rates are simulated. The simulation results show that at different transmission distances, the affections of different fluctuations on key rates are different. Moreover, the PDS QKD protocol with an unstable PDC source can resist more intensity fluctuations and more statistical fluctuation. PMID:26471947
Exact finite difference schemes for the non-linear unidirectional wave equation
NASA Technical Reports Server (NTRS)
Mickens, R. E.
1985-01-01
Attention is given to the construction of exact finite difference schemes for the nonlinear unidirectional wave equation that describes the nonlinear propagation of a wave motion in the positive x-direction. The schemes constructed for these equations are compared with those obtained by using the usual procedures of numerical analysis. It is noted that the order of the exact finite difference models is equal to the order of the differential equation.
NASA Technical Reports Server (NTRS)
Beggs, John H.; Luebbers, Raymond J.; Kunz, Karl S.; Yee, Kane S.
1991-01-01
Surface impedance boundary conditions are employed to reduce the solution volume during the analysis of scattering from lossy dielectric objects. In a finite difference solution, they also can be utilized to avoid using small cells, made necessary by shorter wavelengths in conducting media, throughout the solution volume. A 1-D implementation for a surface impedance boundary condition for good conductors in the Finite Difference Time Domain (FDTD) technique.
Finite-difference solution to the radiative-transfer equation for in-water radiance
NASA Astrophysics Data System (ADS)
Helliwell, W. S.
1985-08-01
The radiative-transfer equation is solved using a finite-difference method. For the first time to the author's knowledge, finite-difference solutions are obtained for in-water radiance. Comparison with data is good for radiance distributions that are due to the sun at depths down to 26 attenuation lengths in a lake. Comparisons are also made with off-axis radiance measurements from an ocean laser experiment together with the solution from a small-angle approximation model.
Optimal search strategies of space-time coupled random walkers with finite lifetimes.
Campos, D; Abad, E; Méndez, V; Yuste, S B; Lindenberg, K
2015-05-01
We present a simple paradigm for detection of an immobile target by a space-time coupled random walker with a finite lifetime. The motion of the walker is characterized by linear displacements at a fixed speed and exponentially distributed duration, interrupted by random changes in the direction of motion and resumption of motion in the new direction with the same speed. We call these walkers "mortal creepers." A mortal creeper may die at any time during its motion according to an exponential decay law characterized by a finite mean death rate ω(m). While still alive, the creeper has a finite mean frequency ω of change of the direction of motion. In particular, we consider the efficiency of the target search process, characterized by the probability that the creeper will eventually detect the target. Analytic results confirmed by numerical results show that there is an ω(m)-dependent optimal frequency ω=ω(opt) that maximizes the probability of eventual target detection. We work primarily in one-dimensional (d=1) domains and examine the role of initial conditions and of finite domain sizes. Numerical results in d=2 domains confirm the existence of an optimal frequency of change of direction, thereby suggesting that the observed effects are robust to changes in dimensionality. In the d=1 case, explicit expressions for the probability of target detection in the long time limit are given. In the case of an infinite domain, we compute the detection probability for arbitrary times and study its early- and late-time behavior. We further consider the survival probability of the target in the presence of many independent creepers beginning their motion at the same location and at the same time. We also consider a version of the standard "target problem" in which many creepers start at random locations at the same time.
Finite energy monopoles in non-Abelian gauge theories on odd-dimensional spaces
Kihara, Hironobu
2009-02-15
In higher-dimensional gauge theory, we need energies with higher power terms of field strength in order to realize pointwise monopoles. We consider new models with higher power terms of field strength and extraordinary kinetic terms of the scalar field. Monopole charges are computed as integrals over spheres and they are related to mapping class degree. Hedgehog solutions are investigated in these models. Every differential equation for these solutions is Abel's differential equation. A condition for the existence of a finite energy solution is shown. The spaces of 1-jets of these equations are defined as sets of zeros of polynomials. Those spaces can be interpreted as singular quartic surfaces in three-dimensional complex projective spa0008.
Calculation of electrical potentials on the surface of a realistic head model by finite differences.
Lemieux, L; McBride, A; Hand, J W
1996-07-01
We present a method for the calculation of electrical potentials at the surface of realistic head models from a point dipole generator based on a 3D finite-difference algorithm. The model was validated by comparing calculated values with those obtained algebraically for a three-shell spherical model. For a 1.25 mm cubic grid size, the mean error was 4.9% for a superficial dipole (3.75 mm from the inner surface of the skull) pointing in the radial direction. The effect of generator discretization and node spacing on the accuracy of the model was studied. Three values of the node spacing were considered: 1, 1.25 and 1.5 mm. The mean relative errors were 4.2, 6.3 and 9.3% respectively. The quality of the approximation of a point dipole by an array of nodes in a spherical neighbourhood did not depend significantly on the number of nodes used. The application of the method to a conduction model derived from MRI data is demonstrated.
NASA Astrophysics Data System (ADS)
Nikkar, Samira; Nordström, Jan
2015-06-01
A time-dependent coordinate transformation of a constant coefficient hyperbolic system of equations which results in a variable coefficient system of equations is considered. By applying the energy method, well-posed boundary conditions for the continuous problem are derived. Summation-by-Parts (SBP) operators for the space and time discretization, together with a weak imposition of boundary and initial conditions using Simultaneously Approximation Terms (SATs) lead to a provable fully-discrete energy-stable conservative finite difference scheme. We show how to construct a time-dependent SAT formulation that automatically imposes boundary conditions, when and where they are required. We also prove that a uniform flow field is preserved, i.e. the Numerical Geometric Conservation Law (NGCL) holds automatically by using SBP-SAT in time and space. The developed technique is illustrated by considering an application using the linearized Euler equations: the sound generated by moving boundaries. Numerical calculations corroborate the stability and accuracy of the new fully discrete approximations.
NASA Astrophysics Data System (ADS)
Dunn, David B., Jr.
1994-12-01
This paper extends past analysis of an optimal source distribution around a homogeneous sphere of muscle tissue by using a 3-D finite difference time domain (FDTD) scenario in which an anatomically correct human head model is irradiated. It first duplicates the analytical solution within an FDTD space using an FDTD computer code developed at Penn State University. This duplication uses a 9.45 cm radius sphere represented in an FDTD space of 2.35 mm cubic cells. FDTD simulations are then performed on four, three, and two layer laminated spheres, designed to provide simple models of a head. Finally, four simulations were performed in FDTD on the human head model developed at Penn State from an MRI scan of an actual human head. The comparison of analytic simulations to the FDTD simulations on a homogeneous sphere showed a pixel by pixel average of 5.34% error between the two with a standard deviation of 7.84%. The layered sphere models showed considerable spiking at the two poles along with a small amount of spiking due to the stair-step approximation of the spheres. None of these spikes increased the power beyond that at the surface and hence were not critical. The simulations on a true human head showed improvement in depth due to the low-loss of the bone tissue. This study demonstrates that microwave hyperthermia with good resolution is possible in an anatomically correct head model.
NASA Astrophysics Data System (ADS)
Zhang, Wei; Chen, Xiaofei
2006-10-01
In this study, we propose a new numerical method, named as Traction Image method, to accurately and efficiently implement the traction-free boundary conditions in finite difference simulation in the presence of surface topography. In this algorithm, the computational domain is discretized by boundary-conforming grids, in which the irregular surface is transformed into a `flat' surface in computational space. Thus, the artefact of staircase approximation to arbitrarily irregular surface can be avoided. Such boundary-conforming gridding is equivalent to a curvilinear coordinate system, in which the first-order partial differential velocity-stress equations are numerically updated by an optimized high-order non-staggered finite difference scheme, that is, DRP/opt MacCormack scheme. To satisfy the free surface boundary conditions, we extend the Stress Image method for planar surface to Traction Image method for arbitrarily irregular surface by antisymmetrically setting the values of normal traction on the grid points above the free surface. This Traction Image method can be efficiently implemented. To validate this new method, we perform numerical tests to several complex models by comparing our results with those computed by other independent accurate methods. Although some of the testing examples have extremely sloped topography, all tested results show an excellent agreement between our results and those from the reference solutions, confirming the validity of our method for modelling seismic waves in the heterogeneous media with arbitrary shape topography. Numerical tests also demonstrate the efficiency of this method. We find about 10 grid points per shortest wavelength is enough to maintain the global accuracy of the simulation. Although the current study is for 2-D P-SV problem, it can be easily extended to 3-D problem.
NASA Technical Reports Server (NTRS)
Baumeister, K. J.; Eversman, W.; Astley, R. J.; White, J. W.
1981-01-01
Sound propagation without flow in a rectangular duct with a converging-diverging area variation was studied experimentally and theoretically. The area variation was of sufficient magnitude to produce large reflections and induce modal scattering. The rms (root-mean-squared) pressure and phase angle on both the flat and curved surface were measured and tabulated. The steady state finite element theory and the transient finite difference theory are in good agreement with the data. It is concluded that numerical finite difference and finite element theories appear ideally suited for handling duct propagation problems which encounter large area variations.
A finite element approach for large motion dynamic analysis of multibody structures in space
NASA Technical Reports Server (NTRS)
Chang, Che-Wei
1989-01-01
A three-dimensional finite element formulation for modeling the transient dynamics of constrained multibody space sructures with truss-like configurations is presented. Convected coordinate systems are used to define rigid-body motion of individual elements in the system. These systems are located at one end of each element and are oriented such that one axis passes through the other end of the element. Deformation of each element, relative to its convected coordinate system, is defined by cubic flexural shape functions as used in finite element methods of structural analysis. The formulation is oriented toward joint dominated structures and places the generalized coordinates at the joint. A transformation matrix is derived to integrate joint degree-of-freedom into the equations of motion of the element. Based on the derivation, a general-purpose code LATDYN (Large Angle Transient DYNamics) was developed. Two examples are presented to illustrate the application of the code. For the spin-up of a flexible beam, results are compared with existing solutions available in the literature. For the deployment of one bay of a deployable space truss (the Minimast), results are verified by the geometric knowledge of the system and converged solution of a successively refined model.
On the monotonicity of multidimensional finite difference schemes
NASA Astrophysics Data System (ADS)
Kovyrkina, O.; Ostapenko, V.
2016-10-01
The classical concept of monotonicity, introduced by Godunov for linear one-dimensional difference schemes, is extended to multidimensional case. Necessary and sufficient conditions of monotonicity are obtained for linear multidimensional difference schemes of first order. The constraints on the numerical viscosity are given that ensure the monotonicity of a difference scheme in the multidimensional case. It is proposed a modification of the second order multidimensional CABARET scheme that preserves the monotonicity of one-dimensional discrete solutions and, as a result, ensures higher smoothness in the computation of multidimensional discontinuous solutions. The results of two-dimensional test computations illustrating the advantages of the modified CABARET scheme are presented.
Sun, Yongzheng; Li, Wang; Zhao, Donghua
2012-06-01
In this paper, the finite-time stochastic outer synchronization between two different complex dynamical networks with noise perturbation is investigated. By using suitable controllers, sufficient conditions for finite-time stochastic outer synchronization are derived based on the finite-time stability theory of stochastic differential equations. It is noticed that the coupling configuration matrix is not necessary to be symmetric or irreducible, and the inner coupling matrix need not be symmetric. Finally, numerical examples are examined to illustrate the effectiveness of the analytical results. The effect of control parameters on the settling time is also numerically demonstrated.
Finite-Difference Seismic Modeling of Discrete Fractures in a San Juan Basin Gas Reservoir
NASA Astrophysics Data System (ADS)
Daley, T. M.; Nihei, K. T.; Myer, L. R.; Majer, E. L.; Queen, J. H.; Fortuna, M. A.; Murphy, J. O.; Coates, R. T.
2001-12-01
As part of a Dept. of Energy sponsored program in fractured gas production, we are conducting numerical modeling of seismic wave propagation in fractured media. The current modeling algorithm is a 2-D, anisotropic, elastic, finite-difference implementation. Fractures are discrete (one grid point wide), vertical, and are described by two parameters, the normal and tangential fracture stiffness, which are converted to anisotropic, elastic constants and placed in an isotropic background. A five-layer, 2250 m2 model with 3 m grid point spacing is used study the effects of fracturing on two scales: long, compliant fractures (i.e. joints) at wide spacing (650 m) and short, stiff fractures at narrower spacing (21 m). The fracture spacing is approximately equal to bed thickness. The fracture stiffness value for the stiff, short fractures was derived from a conceptual model of regularly spaced, infinitely thin openings which are 30 % of the fracture length. The joints were arbitrarily assigned a stiffness 10 times lower (more compliant). The normal and tangential stiffness were assumed equal (for a model of gas-filled fractures). The layer properties (P- and S-velocity and density) and the model's scale are based on well information from the San Juan basin, focusing on the Mesa Verde unit and its Cliffhouse sandstone member. Surface seismic (including CMP gathers) and VSP geometries, as modeled, were based on field data acquired in the basin. The model results (including wavefield time snapshots, and two-component seismograms) show discrete P- and S-wave scattered events from the compliant joints which have large amplitude P-to-S converted phases. These converted waves can be observed in surface seismic acquisition geometry when they are reflected by the horizontal velocity interfaces. In VSP geometry the downgoing fracture-scattered phases can be directly observed. The closely spaced, stiffer fractures generate multiple scattering which is observed as lower amplitude
NASA Astrophysics Data System (ADS)
Zhang, Rui; Wen, Lihua; Naboulsi, Sam; Eason, Thomas; Vasudevan, Vijay K.; Qian, Dong
2016-08-01
A multiscale space-time finite element method based on time-discontinuous Galerkin and enrichment approach is presented in this work with a focus on improving the computational efficiencies for high cycle fatigue simulations. While the robustness of the TDG-based space-time method has been extensively demonstrated, a critical barrier for the extensive application is the large computational cost due to the additional temporal dimension and enrichment that are introduced. The present implementation focuses on two aspects: firstly, a preconditioned iterative solver is developed along with techniques for optimizing the matrix storage and operations. Secondly, parallel algorithms based on multi-core graphics processing unit are established to accelerate the progressive damage model implementation. It is shown that the computing time and memory from the accelerated space-time implementation scale with the number of degree of freedom N through ˜ O(N^{1.6}) and ˜ O(N), respectively. Finally, we demonstrate the accelerated space-time FEM simulation through benchmark problems.
Finite time attitude takeover control for combination via tethered space robot
NASA Astrophysics Data System (ADS)
Lu, Yingbo; Huang, Panfeng; Meng, Zhongjie; Hu, Yongxin; Zhang, Fan; Zhang, Yizhai
2017-07-01
Up to April 6, 2016, there are 17,385 large debris in orbit around the Earth, which poses a serious hazard to near-Earth space activities. As a promising on-orbit debris capture strategy, tethered space robots (TSRs) have wide applications in future on-orbit service owing to its flexibility and great workspace. However, lots of problems may arise in the Tethered Space Robots (TSRs) system from the approaching, capturing, postcapturing and towing phases. The postcapture combination attitude takeover control by the TSR is studied in this paper. Taking control constraints, tether oscillations and external disturbances into consideration, a fast terminal sliding mode control (FTSMC) methodology with dual closed loops for the flexible combination attitude takeover control is designed. The unknown upper bounds of the uncertainties, external disturbances are estimated through adaptive techniques. Stability of the dual closed loop control system and finite time convergence of system states are proved via Lyapunov stability theory. Besides, null space intersection control allocation was adopted to distribute the required control moment over TSR's redundant thrusters. Simulation studies have been conducted to demonstrate the effectiveness of the proposed controller with the conventional sliding mode control(SMC).
Viscosity approximation methods for a finite family of nonexpansive mappings in Banach spaces
NASA Astrophysics Data System (ADS)
Chang, Shih-Sen
2006-11-01
By using viscosity approximation methods for a finite family of nonexpansive mappings in Banach spaces, some sufficient and necessary conditions for the iterative sequence to converging to a common fixed point are obtained. The results presented in the paper extend and improve some recent results in [H.K. Xu, Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl. 298 (2004) 279-291; H.K. Xu, Remark on an iterative method for nonexpansive mappings, Comm. Appl. Nonlinear Anal. 10 (2003) 67-75; H.H. Bauschke, The approximation of fixed points of compositions of nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 202 (1996) 150-159; B. Halpern, Fixed points of nonexpansive maps, Bull. Amer. Math. Soc. 73 (1967) 957-961; J.S. Jung, Iterative approaches to common fixed points of nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 302 (2005) 509-520; P.L. Lions, Approximation de points fixes de contractions', C. R. Acad. Sci. Paris Ser. A 284 (1977) 1357-1359; A. Moudafi, Viscosity approximation methods for fixed point problems, J. Math. Anal. Appl. 241 (2000) 46-55; S. Reich, Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. Math. Anal. Appl. 75 (1980) 128-292; R. Wittmann, Approximation of fixed points of nonexpansive mappings, Arch. Math. 58 (1992) 486-491].
NASA Astrophysics Data System (ADS)
Liebendörfer, Matthias; Messer, O. E. Bronson; Mezzacappa, Anthony; Bruenn, Stephen W.; Cardall, Christian Y.; Thielemann, F.-K.
2004-01-01
We present an implicit finite difference representation for general relativistic radiation hydrodynamics in spherical symmetry. Our code, AGILE-BOLTZTRAN, solves the Boltzmann transport equation for the angular and spectral neutrino distribution functions in self-consistent simulations of stellar core collapse and postbounce evolution. It implements a dynamically adaptive grid in comoving coordinates. A comoving frame in the momentum phase space facilitates the evaluation and tabulation of neutrino-matter interaction cross sections but produces a multitude of observer corrections in the transport equation. Most macroscopically interesting physical quantities are defined by expectation values of the distribution function. We optimize the finite differencing of the microscopic transport equation for a consistent evolution of important expectation values. We test our code in simulations launched from progenitor stars with 13 solar masses and 40 solar masses. Half a second after core collapse and bounce, the protoneutron star in the latter case reaches its maximum mass and collapses further to form a black hole. When the hydrostatic gravitational contraction sets in, we find a transient increase in electron flavor neutrino luminosities due to a change in the accretion rate. The μ- and τ-neutrino luminosities and rms energies, however, continue to rise because previously shock-heated material with a nondegenerate electron gas starts to replace the cool degenerate material at their production site. We demonstrate this by supplementing the concept of neutrinospheres with a more detailed statistical description of the origin of escaping neutrinos. Adhering to our tradition, we compare the evolution of the 13 Msolar progenitor star to corresponding simulations with the multigroup flux-limited diffusion approximation, based on a recently developed flux limiter. We find similar results in the postbounce phase and validate this MGFLD approach for the spherically symmetric
Comparison of Finite Differences and WKB approximation Methods for PT symmetric complex potentials
NASA Astrophysics Data System (ADS)
Naceri, Leila; Chekkal, Meziane; Hammou, Amine B.
2016-10-01
We consider the one dimensional schrödinger eigenvalue problem on a finite domain (Strum-Liouville problem) for several PT-symmetric complex potentials, studied by Bender and Jones using the WKB approximation method. We make a comparison between the solutions of theses PT-symmetric complex potentials using both the finite difference method (FDM) and the WKB approximation method and show quantitative and qualitative agreement between the two methods.
Trew, Mark L; Smaill, Bruce H; Bullivant, David P; Hunter, Peter J; Pullan, Andrew J
2005-12-01
A generalized finite difference (GFD) method is presented that can be used to solve the bi-domain equations modeling cardiac electrical activity. Classical finite difference methods have been applied by many researchers to the bi-domain equations. However, these methods suffer from the limitation of requiring computational meshes that are structured and orthogonal. Finite element or finite volume methods enable the bi-domain equations to be solved on unstructured meshes, although implementations of such methods do not always cater for meshes with varying element topology. The GFD method solves the bi-domain equations on arbitrary and irregular computational meshes without any need to specify element basis functions. The method is useful as it can be easily applied to activation problems using existing meshes that have originally been created for use by finite element or finite difference methods. In addition, the GFD method employs an innovative approach to enforcing nodal and non-nodal boundary conditions. The GFD method performs effectively for a range of two and three-dimensional test problems and when computing bi-domain electrical activation moving through a fully anisotropic three-dimensional model of canine ventricles.
NASA Technical Reports Server (NTRS)
Byun, Chansup; Guruswamy, Guru P.; Kutler, Paul (Technical Monitor)
1994-01-01
In recent years significant advances have been made for parallel computers in both hardware and software. Now parallel computers have become viable tools in computational mechanics. Many application codes developed on conventional computers have been modified to benefit from parallel computers. Significant speedups in some areas have been achieved by parallel computations. For single-discipline use of both fluid dynamics and structural dynamics, computations have been made on wing-body configurations using parallel computers. However, only a limited amount of work has been completed in combining these two disciplines for multidisciplinary applications. The prime reason is the increased level of complication associated with a multidisciplinary approach. In this work, procedures to compute aeroelasticity on parallel computers using direct coupling of fluid and structural equations will be investigated for wing-body configurations. The parallel computer selected for computations is an Intel iPSC/860 computer which is a distributed-memory, multiple-instruction, multiple data (MIMD) computer with 128 processors. In this study, the computational efficiency issues of parallel integration of both fluid and structural equations will be investigated in detail. The fluid and structural domains will be modeled using finite-difference and finite-element approaches, respectively. Results from the parallel computer will be compared with those from the conventional computers using a single processor. This study will provide an efficient computational tool for the aeroelastic analysis of wing-body structures on MIMD type parallel computers.
Effect of Finite Computational Domain on Turbulence Scaling Law in Both Physical and Spectral Spaces
NASA Technical Reports Server (NTRS)
Hou, Thomas Y.; Wu, Xiao-Hui; Chen, Shiyi; Zhou, Ye
1998-01-01
The well-known translation between the power law of energy spectrum and that of the correlation function or the second order structure function has been widely used in analyzing random data. Here, we show that the translation is valid only in proper scaling regimes. The regimes of valid translation are different for the correlation function and the structure function. Indeed, they do not overlap. Furthermore, in practice, the power laws exist only for a finite range of scales. We show that this finite range makes the translation inexact even in the proper scaling regime. The error depends on the scaling exponent. The current findings are applicable to data analysis in fluid turbulence and other stochastic systems.
Effect of finite computational domain on turbulence scaling law in both physical and spectral spaces
NASA Astrophysics Data System (ADS)
Hou, Thomas Y.; Wu, Xiao-Hui; Chen, Shiyi; Zhou, Ye
1998-11-01
The well-known translation between the power law of the energy spectrum and that of the correlation function or the second order structure function has been widely used in analyzing random data. Here, we show that the translation is valid only in proper scaling regimes. The regimes of valid translation are different for the correlation function and the structure function. Indeed, they do not overlap. Furthermore, in practice, the power laws exist only for a finite range of scales. We show that this finite range makes the translation inexact even in the proper scaling regime. The error depends on the scaling exponent. The current findings are applicable to data analysis in fluid turbulence and other stochastic systems.
Garvie, Marcus R
2007-04-01
We present two finite-difference algorithms for studying the dynamics of spatially extended predator-prey interactions with the Holling type II functional response and logistic growth of the prey. The algorithms are stable and convergent provided the time step is below a (non-restrictive) critical value. This is advantageous as it is well-known that the dynamics of approximations of differential equations (DEs) can differ significantly from that of the underlying DEs themselves. This is particularly important for the spatially extended systems that are studied in this paper as they display a wide spectrum of ecologically relevant behavior, including chaos. Furthermore, there are implementational advantages of the methods. For example, due to the structure of the resulting linear systems, standard direct, and iterative solvers are guaranteed to converge. We also present the results of numerical experiments in one and two space dimensions and illustrate the simplicity of the numerical methods with short programs MATLAB: . Users can download, edit, and run the codes from http://www.uoguelph.ca/~mgarvie/, to investigate the key dynamical properties of spatially extended predator-prey interactions.
Simulation of axi-symmetric flow towards wells: A finite-difference approach
NASA Astrophysics Data System (ADS)
Louwyck, Andy; Vandenbohede, Alexander; Bakker, Mark; Lebbe, Luc
2012-07-01
A detailed finite-difference approach is presented for the simulation of transient radial flow in multi-layer systems. The proposed discretization scheme simulates drawdown within the well more accurately than commonly applied schemes. The solution is compared to existing (semi) analytical models for the simulation of slug tests and pumping tests with constant discharge in single- and multi-layer systems. For all cases, it is concluded that the finite-difference model approximates drawdown to acceptable accuracy. The main advantage of finite-difference approaches is the ability to account for the varying saturated thickness in unconfined top layers. Additionally, it is straightforward to include radial variation of hydraulic parameters, which is useful to simulate the effect of a finite-thickness well skin. Aquifer tests with variable pumping rate and/or multiple wells may be simulated by superposition. The finite-difference solution is implemented in MAxSym, a MATLAB tool which is designed specifically to simulate axi-symmetric flow. Alternatively, the presented equations can be solved using a standard finite-difference model. A procedure is outlined to apply the same approach with MODFLOW. The required modifications to the input parameters are much larger for MODFLOW than for MAxSym, but the results are virtually identical. The presented finite-difference solution may be used, for example, as a forward model in parameter estimation algorithms. Since it is applicable to multi-layer systems, its use is not limited to the simulation of traditional pumping and slug tests, but also includes advanced aquifer tests, such as multiple pumping tests or multi-level slug tests.
Huang, Qihua; Wang, Hao
2016-08-01
The question of the effects of environmental toxins on ecological communities is of great interest from both environmental and conservational points of view. Mathematical models have been applied increasingly to predict the effects of toxins on a variety of ecological processes. Motivated by the fact that individuals with different sizes may have different sensitivities to toxins, we develop a toxin-mediated size-structured model which is given by a system of first order fully nonlinear partial differential equations (PDEs). It is very possible that this work represents the first derivation of a PDE model in the area of ecotoxicology. To solve the model, an explicit finite difference approximation to this PDE system is developed. Existence-uniqueness of the weak solution to the model is established and convergence of the finite difference approximation to this unique solution is proved. Numerical examples are provided by numerically solving the PDE model using the finite difference scheme.
NASA Technical Reports Server (NTRS)
Melis, Matthew E.
2003-01-01
NASA Glenn Research Center s Structural Mechanics Branch has years of expertise in using explicit finite element methods to predict the outcome of ballistic impact events. Shuttle engineers from the NASA Marshall Space Flight Center and NASA Kennedy Space Flight Center required assistance in assessing the structural loads that a newly proposed thrust vector control system for the space shuttle solid rocket booster (SRB) aft skirt would expect to see during its recovery splashdown.
Use of the finite-difference time-domain method in electromagnetic dosimetry
Sullivan, D.M.
1987-01-01
Although there are acceptable methods for calculating whole body electromagnetic absorption, no completely acceptable method for calculating the local specific absorption rate (SAR) at points within the body has been developed. Frequency domain methods, such as the method of moments (MoM) have achieved some success; however, the MoM requires computer storage on the order of (3N)/sup 2/, and computation time on the order of (3N)/sup 3/ where N is the number of cells. The finite-difference time-domain (FDTD) method has been employed extensively in calculating the scattering from metallic objects, and recently is seeing some use in calculating the interaction of EM fields with complex, lossy dielectric bodies. Since the FDTD method has storage and time requirements proportional to N, it presents an attractive alternative to calculating SAR distribution in large bodies. This dissertation describes the FDTD method and evaluates it by comparing its results with analytic solutions in 2 and 3 dimensions. The results obtained demonstrate that the FDTD method is capable of calculating internal SAR distribution with acceptable accuracy. The construction of a data base to provide detailed, inhomogeneous man models for use with the FDTD method is described. Using this construction method, a model of 40,000 1.31 cm. cells is developed for use at 350 MHz, and another model consisting of 5000 2.62 cm. cells is developed for use at 100 MHz. To add more realism to the problem, a ground plane is added to the FDTD software. The needed changes to the software are described, along with a test which confirms its accuracy. Using the CRAY II supercomputer, SAR distributions in human models are calculated using incident frequencies of 100 MHz and 350 MHz for three different cases: (1) A homogeneous man model in free space, (2) an inhomogeneous man model in free space, and (3) an inhomogeneous man model standing on a ground plane.
SOME NEW FINITE DIFFERENCE METHODS FOR HELMHOLTZ EQUATIONS ON IRREGULAR DOMAINS OR WITH INTERFACES.
Wan, Xiaohai; Li, Zhilin
2012-06-01
Solving a Helmholtz equation Δu + λu = f efficiently is a challenge for many applications. For example, the core part of many efficient solvers for the incompressible Navier-Stokes equations is to solve one or several Helmholtz equations. In this paper, two new finite difference methods are proposed for solving Helmholtz equations on irregular domains, or with interfaces. For Helmholtz equations on irregular domains, the accuracy of the numerical solution obtained using the existing augmented immersed interface method (AIIM) may deteriorate when the magnitude of λ is large. In our new method, we use a level set function to extend the source term and the PDE to a larger domain before we apply the AIIM. For Helmholtz equations with interfaces, a new maximum principle preserving finite difference method is developed. The new method still uses the standard five-point stencil with modifications of the finite difference scheme at irregular grid points. The resulting coefficient matrix of the linear system of finite difference equations satisfies the sign property of the discrete maximum principle and can be solved efficiently using a multigrid solver. The finite difference method is also extended to handle temporal discretized equations where the solution coefficient λ is inversely proportional to the mesh size.
Improving sub-grid scale accuracy of boundary features in regional finite-difference models
Panday, Sorab; Langevin, Christian D.
2012-01-01
As an alternative to grid refinement, the concept of a ghost node, which was developed for nested grid applications, has been extended towards improving sub-grid scale accuracy of flow to conduits, wells, rivers or other boundary features that interact with a finite-difference groundwater flow model. The formulation is presented for correcting the regular finite-difference groundwater flow equations for confined and unconfined cases, with or without Newton Raphson linearization of the nonlinearities, to include the Ghost Node Correction (GNC) for location displacement. The correction may be applied on the right-hand side vector for a symmetric finite-difference Picard implementation, or on the left-hand side matrix for an implicit but asymmetric implementation. The finite-difference matrix connectivity structure may be maintained for an implicit implementation by only selecting contributing nodes that are a part of the finite-difference connectivity. Proof of concept example problems are provided to demonstrate the improved accuracy that may be achieved through sub-grid scale corrections using the GNC schemes.
Improving sub-grid scale accuracy of boundary features in regional finite-difference models
NASA Astrophysics Data System (ADS)
Panday, Sorab; Langevin, Christian D.
2012-06-01
As an alternative to grid refinement, the concept of a ghost node, which was developed for nested grid applications, has been extended towards improving sub-grid scale accuracy of flow to conduits, wells, rivers or other boundary features that interact with a finite-difference groundwater flow model. The formulation is presented for correcting the regular finite-difference groundwater flow equations for confined and unconfined cases, with or without Newton Raphson linearization of the nonlinearities, to include the Ghost Node Correction (GNC) for location displacement. The correction may be applied on the right-hand side vector for a symmetric finite-difference Picard implementation, or on the left-hand side matrix for an implicit but asymmetric implementation. The finite-difference matrix connectivity structure may be maintained for an implicit implementation by only selecting contributing nodes that are a part of the finite-difference connectivity. Proof of concept example problems are provided to demonstrate the improved accuracy that may be achieved through sub-grid scale corrections using the GNC schemes.
SOME NEW FINITE DIFFERENCE METHODS FOR HELMHOLTZ EQUATIONS ON IRREGULAR DOMAINS OR WITH INTERFACES
Wan, Xiaohai; Li, Zhilin
2012-01-01
Solving a Helmholtz equation Δu + λu = f efficiently is a challenge for many applications. For example, the core part of many efficient solvers for the incompressible Navier-Stokes equations is to solve one or several Helmholtz equations. In this paper, two new finite difference methods are proposed for solving Helmholtz equations on irregular domains, or with interfaces. For Helmholtz equations on irregular domains, the accuracy of the numerical solution obtained using the existing augmented immersed interface method (AIIM) may deteriorate when the magnitude of λ is large. In our new method, we use a level set function to extend the source term and the PDE to a larger domain before we apply the AIIM. For Helmholtz equations with interfaces, a new maximum principle preserving finite difference method is developed. The new method still uses the standard five-point stencil with modifications of the finite difference scheme at irregular grid points. The resulting coefficient matrix of the linear system of finite difference equations satisfies the sign property of the discrete maximum principle and can be solved efficiently using a multigrid solver. The finite difference method is also extended to handle temporal discretized equations where the solution coefficient λ is inversely proportional to the mesh size. PMID:22701346
Performance prediction of finite-difference solvers for different computer architectures
NASA Astrophysics Data System (ADS)
Louboutin, Mathias; Lange, Michael; Herrmann, Felix J.; Kukreja, Navjot; Gorman, Gerard
2017-08-01
The life-cycle of a partial differential equation (PDE) solver is often characterized by three development phases: the development of a stable numerical discretization; development of a correct (verified) implementation; and the optimization of the implementation for different computer architectures. Often it is only after significant time and effort has been invested that the performance bottlenecks of a PDE solver are fully understood, and the precise details varies between different computer architectures. One way to mitigate this issue is to establish a reliable performance model that allows a numerical analyst to make reliable predictions of how well a numerical method would perform on a given computer architecture, before embarking upon potentially long and expensive implementation and optimization phases. The availability of a reliable performance model also saves developer effort as it both informs the developer on what kind of optimisations are beneficial, and when the maximum expected performance has been reached and optimisation work should stop. We show how discretization of a wave-equation can be theoretically studied to understand the performance limitations of the method on modern computer architectures. We focus on the roofline model, now broadly used in the high-performance computing community, which considers the achievable performance in terms of the peak memory bandwidth and peak floating point performance of a computer with respect to algorithmic choices. A first principles analysis of operational intensity for key time-stepping finite-difference algorithms is presented. With this information available at the time of algorithm design, the expected performance on target computer systems can be used as a driver for algorithm design.
Space-Frequency Sampling Criteria for Electromagnetic Scattering of a Finite Object.
1985-08-01
3 *W2**2 -0 PUNCTMN FUNC2 (R, W2) I r C2ud1./ 3 .)*(sIntRIW2))**2. ED 161 C C FUNCrIOt4: FUNC3 C THIS FUNCION WILL PROCC THE LINEAR ITERKLATION )RITDN...8217N’) GO M 50 IF ( OMEGA . NE. ’Y’) GO) MI 3 0 CALL TRAN2D (DATA, MSIZE,-1) CALL rLUT(DATA,MSIZ EARAY1 .ARRAY2 .ARRAYP) GO MD25 C C INVERSE FOURE...AO-A162 553 SPACE-FREQUENCY SAMPLING CRITERIA FOR ELECTROINNINETIC 1/ 3 SCATTERING OF A FINITE UBJECT(U) OHI10 STATE UNIV COLUMBUS ELECTROSCIENCE LAB
Harmonic finite-element thermoelastic analysis of space frames and trusses
Givoli, D.; Rand, O. )
1993-09-01
A numerical procedure is devised for the thermoelastic analysis of three-dimensional frame- or truss-type space structures exposed to solar radiation. Thin-walled frame or truss members with cross sections of arbitrary shape are considered. Tension-compression, bending, shear, and torsional effects due to the temperature distribution induced by the solar radiation are all taken into account. The procedure proposed involves finite element discretization in the axial direction and a harmonic analysts in the circumferential direction of each member. This procedure is an extension of the one employed previously to obtain the temperature field in trusses. A multibay frame structure serves as a model to demonstrate the performance of the proposed method. The temperature, displacement, and stress fields in the frame are found in various cases. 23 refs.
NASA Astrophysics Data System (ADS)
Darrall, Bradley T.
For the first time true variational principles are formulated for the analysis of the continuum problems of heat diffusion, dynamic thermoelasticity, poroelasticity, and time-dependent quantum mechanics. This is accomplished by considering the stationarity of a mixed convolved action, which can be seen as a modern counterpart to the original actions posed in Hamilton's principle and its many extensions. By including fractional derivatives, convolution integrals, and mixed variables into the definition of the action these new variational principles overcome the shortcomings of the many other variational methods based on Hamilton's principle, namely the inability to include dissipation in a consistent manner and the unjustified need to constrain variations on the primary unknowns of a system at the end of the time interval. These new variational principles then provide ideal weak forms from which novel time-space finite element methods having certain attractive properties are formulated.
Ezzinbi, Khalil; Ndambomve, Patrice
2016-01-01
In this work, we consider the control system governed by some partial functional integrodifferential equations with finite delay in Banach spaces. We assume that the undelayed part admits a resolvent operator in the sense of Grimmer. Firstly, some suitable conditions are established to guarantee the existence and uniqueness of mild solutions for a broad class of partial functional integrodifferential infinite dimensional control systems. Secondly, it is proved that, under generally mild conditions of cost functional, the associated Lagrange problem has an optimal solution, and that for each optimal solution there is a minimizing sequence of the problem that converges to the optimal solution with respect to the trajectory, the control, and the functional in appropriate topologies. Our results extend and complement many other important results in the literature. Finally, a concrete example of application is given to illustrate the effectiveness of our main results.
NASA Technical Reports Server (NTRS)
Lee, L. C.
1976-01-01
The cross correlation of the intensity fluctuations between different frequencies and finite bandwidth effects on the intensity correlations based on the Markov approximation were calculated. Results may be applied to quite general turbulence spectra for an extended turbulent medium. Calculations of the cross-correlation function and of finite bandwidth effects are explicitly carried out for both Gaussian and Kolmogorov turbulence spectra. The increases of the correlation scale of intensity fluctuations are different for these two spectra and the difference can be used to determine whether the interstellar turbulent medium has a Gaussian or a Kolmogorov spectrum.
NASA Astrophysics Data System (ADS)
Cole, James B.; Okada, Naoki
2013-09-01
The use of Green's functions to solve inhomogeneous differential equations, such as the Maxwell's equations with source currents is well known. Unfortunately, it is usually difficult - if not impossible - to find a Green's function which satisfies the boundary conditions. The finite difference time domain (FDTD) algorithm is derived from a finite difference equation (FDE) of Maxwell's equations. FDTD, which automatically takes boundary conditions into account, is often used to solve the FDE, but its computational cost increases much faster than the accuracy as the grid spacing (h) decreases; moreover small h must be used to capture fine features of structures such as subwavelength gratings. A discrete Green's function (DGF), computed using FDTD, can be used to overcome some of the shortcomings of FDTD alone. A DGF computed using FDTD automatically includes the boundary conditions of the problem. In this paper we use a DGF based on what is called a nonstandard finite difference model of Maxwell's equations to compute scattering off subwavelength structures. We verify the accuracy of our method by comparing our calculations with analytic solutions given by Mie theory.
Effects of finite volume on the KL – KS mass difference
Christ, N. H.; Feng, X.; Martinelli, G.; ...
2015-06-24
Phenomena that involve two or more on-shell particles are particularly sensitive to the effects of finite volume and require special treatment when computed using lattice QCD. In this paper we generalize the results of Lüscher and Lellouch and Lüscher, which determine the leading-order effects of finite volume on the two-particle spectrum and two-particle decay amplitudes to determine the finite-volume effects in the second-order mixing of the K⁰ and K⁰⁻ states. We extend the methods of Kim, Sachrajda, and Sharpe to provide a direct, uniform treatment of these three, related, finite-volume corrections. In particular, the leading, finite-volume corrections to the KLmore » – KS mass difference ΔMK and the CP-violating parameter εK are determined, including the potentially large effects which can arise from the near degeneracy of the kaon mass and the energy of a finite-volume, two-pion state.« less
NASA Technical Reports Server (NTRS)
Tam, Christopher K. W.; Webb, Jay C.
1994-01-01
In this paper finite-difference solutions of the Helmholtz equation in an open domain are considered. By using a second-order central difference scheme and the Bayliss-Turkel radiation boundary condition, reasonably accurate solutions can be obtained when the number of grid points per acoustic wavelength used is large. However, when a smaller number of grid points per wavelength is used excessive reflections occur which tend to overwhelm the computed solutions. Excessive reflections are due to the incompability between the governing finite difference equation and the Bayliss-Turkel radiation boundary condition. The Bayliss-Turkel radiation boundary condition was developed from the asymptotic solution of the partial differential equation. To obtain compatibility, the radiation boundary condition should be constructed from the asymptotic solution of the finite difference equation instead. Examples are provided using the improved radiation boundary condition based on the asymptotic solution of the governing finite difference equation. The computed results are free of reflections even when only five grid points per wavelength are used. The improved radiation boundary condition has also been tested for problems with complex acoustic sources and sources embedded in a uniform mean flow. The present method of developing a radiation boundary condition is also applicable to higher order finite difference schemes. In all these cases no reflected waves could be detected. The use of finite difference approximation inevita bly introduces anisotropy into the governing field equation. The effect of anisotropy is to distort the directional distribution of the amplitude and phase of the computed solution. It can be quite large when the number of grid points per wavelength used in the computation is small. A way to correct this effect is proposed. The correction factor developed from the asymptotic solutions is source independent and, hence, can be determined once and for all. The
NASA Technical Reports Server (NTRS)
Tam, Christopher K. W.; Webb, Jay C.
1994-01-01
In this paper finite-difference solutions of the Helmholtz equation in an open domain are considered. By using a second-order central difference scheme and the Bayliss-Turkel radiation boundary condition, reasonably accurate solutions can be obtained when the number of grid points per acoustic wavelength used is large. However, when a smaller number of grid points per wavelength is used excessive reflections occur which tend to overwhelm the computed solutions. Excessive reflections are due to the incompability between the governing finite difference equation and the Bayliss-Turkel radiation boundary condition. The Bayliss-Turkel radiation boundary condition was developed from the asymptotic solution of the partial differential equation. To obtain compatibility, the radiation boundary condition should be constructed from the asymptotic solution of the finite difference equation instead. Examples are provided using the improved radiation boundary condition based on the asymptotic solution of the governing finite difference equation. The computed results are free of reflections even when only five grid points per wavelength are used. The improved radiation boundary condition has also been tested for problems with complex acoustic sources and sources embedded in a uniform mean flow. The present method of developing a radiation boundary condition is also applicable to higher order finite difference schemes. In all these cases no reflected waves could be detected. The use of finite difference approximation inevita bly introduces anisotropy into the governing field equation. The effect of anisotropy is to distort the directional distribution of the amplitude and phase of the computed solution. It can be quite large when the number of grid points per wavelength used in the computation is small. A way to correct this effect is proposed. The correction factor developed from the asymptotic solutions is source independent and, hence, can be determined once and for all. The
GEOTHERM: A finite difference code for testing metamorphic P-T-t paths and tectonic models
NASA Astrophysics Data System (ADS)
Casini, Leonardo; Puccini, Antonio; Cuccuru, Stefano; Maino, Matteo; Oggiano, Giacomo
2013-09-01
Here, time-dependent solutions for the heat conduction equation are numerically evaluated in 1D space using a fully implicit algorithm based on the finite difference method, assuming temperature-dependence of thermal conductivity. The method is implemented using the package 'GEOTHERM', comprising 13 MATLAB-derived scripts and 3 Excel spreadsheets. In the package, the initial state of the modeled crust, including its thickness, average density, and average heat production rate, can be configured by the user. The exhumation/burial history and metamorphic evolution of the crust are simulated by changing these initial values to fit the vertical displacement rates of the crust imposed by the user. Once the inputs have been made, the variations with depth of temperature, proportion of melt, and shear stress, as well as average values of heat flow at the surface and across the Moho, are calculated and displayed in five separate plots. The code is demonstrated with respect to the Carboniferous evolution of the South Variscan Belt. The best fit to independent petrologic constraints derived from thermobarometry is obtained with an early Carboniferous (342 Ma) slab break-off and a shear strain rate of 10-13 s-1 between 318 and 305 Ma.
Kemp, Jonathan A; Bilbao, Stefan; McMaster, James; Smith, Richard A
2013-08-01
Wave separation within a trumpet is presented using three high pressure microphones to measure pressure waves within the curved, constant cross-section tuning slide of the instrument while the instrument was being played by a virtuoso trumpet player. A closer inter-microphone spacing was possible in comparison to previous work through the use of time domain windowing on non-causal transfer functions and performing wave separation in the frequency domain. Time domain plots of the experimental wave separation were then compared to simulations using a physical model based on a time domain finite difference simulation of the trumpet bore coupled to a one mass, two degree of freedom lip model. The time domain and frequency spectra of the measured and synthesized sounds showed a similar profile, with the sound produced by the player showing broader spectral peaks in experimental data. Using a quality factor of 5 for the lip model was found to give greater agreement between the simulated and experimental starting transients in comparison to the values in the range 1-3 often assumed. Deviations in the spectral content and wave shape provide insights into the areas where future research may be directed in improving the accuracy of physical modeling synthesis.
NASA Astrophysics Data System (ADS)
Guan, Zhen; Heinonen, Vili; Lowengrub, John; Wang, Cheng; Wise, Steven M.
2016-09-01
In this paper we construct an energy stable finite difference scheme for the amplitude expansion equations for the two-dimensional phase field crystal (PFC) model. The equations are formulated in a periodic hexagonal domain with respect to the reciprocal lattice vectors to achieve a provably unconditionally energy stable and solvable scheme. To our knowledge, this is the first such energy stable scheme for the PFC amplitude equations. The convexity of each part in the amplitude equations is analyzed, in both the semi-discrete and fully-discrete cases. Energy stability is based on a careful convexity analysis for the energy (in both the spatially continuous and discrete cases). As a result, unique solvability and unconditional energy stability are available for the resulting scheme. Moreover, we show that the scheme is point-wise stable for any time and space step sizes. An efficient multigrid solver is devised to solve the scheme, and a few numerical experiments are presented, including grain rotation and shrinkage and grain growth studies, as examples of the strength and robustness of the proposed scheme and solver.
M2Di: MATLAB 2D Stokes solvers using the Finite Difference method
NASA Astrophysics Data System (ADS)
Räss, Ludovic; Duretz, Thibault; Schmalholz, Stefan; Podladchikov, Yury
2017-04-01
The study of coupled processes in Earth Sciences leads to the development of multiphysics modelling tools. Mechanical solvers represent the essential ingredient of any of these tools such that their performance and robustness is generally dictated by that of the mechanical solver. Here, we present M2Di, a collection of MATLAB routines designed for studying 2D linear and power law incompressible viscous flow using Finite Difference discretisation. The scripts are written in a concise vectorised MATLAB fashion and rely on fast and robust linear and non-linear solvers (Picard and Newton iterations). As a result, time to solution of 22 seconds for linear viscous flow with 104 viscosity jump on 10002 grid points can be achieved on a standard personal computer. We will present a numerous example of applications that span from high resolution crystal-melt dynamics, deformation of heterogeneous power law viscous fluids, instantaneous mantle flow patterns in cylindrical coordinates, and calculation of pressure gradients around inclusions using variable grid spacing. We use analytical solution for linear viscous flow with highly variable viscosity to validate the linear flow solver. Validation of the non-linear solver is achieved by comparing numerical solution to analytic and benchmark solutions of power law viscous folding and necking. The M2Di codes are open source and can hence be used for research or educational purposes.
NASA Astrophysics Data System (ADS)
Do, Seongju; Li, Haojun; Kang, Myungjoo
2017-06-01
In this paper, we present an accurate and efficient wavelet-based adaptive weighted essentially non-oscillatory (WENO) scheme for hydrodynamics and ideal magnetohydrodynamics (MHD) equations arising from the hyperbolic conservation systems. The proposed method works with the finite difference weighted essentially non-oscillatory (FD-WENO) method in space and the third order total variation diminishing (TVD) Runge-Kutta (RK) method in time. The philosophy of this work is to use the lifted interpolating wavelets as not only detector for singularities but also interpolator. Especially, flexible interpolations can be performed by an inverse wavelet transformation. When the divergence cleaning method introducing auxiliary scalar field ψ is applied to the base numerical schemes for imposing divergence-free condition to the magnetic field in a MHD equation, the approximations to derivatives of ψ require the neighboring points. Moreover, the fifth order WENO interpolation requires large stencil to reconstruct high order polynomial. In such cases, an efficient interpolation method is necessary. The adaptive spatial differentiation method is considered as well as the adaptation of grid resolutions. In order to avoid the heavy computation of FD-WENO, in the smooth regions fixed stencil approximation without computing the non-linear WENO weights is used, and the characteristic decomposition method is replaced by a component-wise approach. Numerical results demonstrate that with the adaptive method we are able to resolve the solutions that agree well with the solution of the corresponding fine grid.
Three-dimensional finite difference viscoelastic wave modelling including surface topography
NASA Astrophysics Data System (ADS)
Hestholm, Stig
1999-12-01
I have undertaken 3-D finite difference (FD) modelling of seismic scattering fromfree-surface topography. Exact free-surface boundary conditions for arbitrary 3-D topographies have been derived for the particle velocities. The boundary conditions are combined with a velocity-stress formulation of the full viscoelastic wave equations. A curved grid represents the physical medium and its upper boundary represents the free-surface topography. The wave equations are numerically discretized by an eighth-order FD method on a staggered grid in space, and a leap-frog technique and the Crank-Nicholson method in time. I simulate scattering from teleseismic P waves by using plane incident wave fronts and real topography from a 60 x 60 km area that includes the NORESS array of seismic receiver stations in southeastern Norway. Synthetic snapshots and seismograms of the wavefield show clear conversion from P to Rg (short-period fundamental-mode Rayleigh) waves in areas of rough topography, which is consistent with numerous observations. By parallelization on fast supercomputers, it is possible to model higher frequencies and/or larger areas than before.
A composite Chebyshev finite difference method for nonlinear optimal control problems
NASA Astrophysics Data System (ADS)
Marzban, H. R.; Hoseini, S. M.
2013-06-01
In this paper, a composite Chebyshev finite difference method is introduced and is successfully employed for solving nonlinear optimal control problems. The proposed method is an extension of the Chebyshev finite difference scheme. This method can be regarded as a non-uniform finite difference scheme and is based on a hybrid of block-pulse functions and Chebyshev polynomials using the well-known Chebyshev-Gauss-Lobatto points. The convergence of the method is established. The nice properties of hybrid functions are then used to convert the nonlinear optimal control problem into a nonlinear mathematical programming one that can be solved efficiently by a globally convergent algorithm. The validity and applicability of the proposed method are demonstrated through some numerical examples. The method is simple, easy to implement and yields very accurate results.
Explicit finite-difference time domain for nonlinear analysis of waveguide modes
NASA Astrophysics Data System (ADS)
Barakat, N. M.; Shabat, M. M.; El-Azab, S.; Jaeger, Dieter
2003-07-01
The Finite Difference Time Domain Technique is at present the most widely used tool employed in the study of light propagation in various photonic waveguide structure. In this paper we derived an explicit finite-difference time-domain (FDTD) method for solving the wave equation in a four optical waveguiding rectangular structure. We derive the stability condition to achieve the stability in nonlinear media region, we also check that the wave equation used is consistence and convergent with the approximate finite difference equation. Our method is tested against some previous problems and we find a high degree of accuracy, moreover it is easy for programming. Numerical results are illustrated for a rectangular waveguide with four layers, where one of these layers is a nonlinear medium.
Relative and Absolute Error Control in a Finite-Difference Method Solution of Poisson's Equation
ERIC Educational Resources Information Center
Prentice, J. S. C.
2012-01-01
An algorithm for error control (absolute and relative) in the five-point finite-difference method applied to Poisson's equation is described. The algorithm is based on discretization of the domain of the problem by means of three rectilinear grids, each of different resolution. We discuss some hardware limitations associated with the algorithm,…
Propagation of 3-D Beams Using a Finite-Difference Algorithm: Practical Considerations
2011-05-22
difference optical propagation, including non-paraxial methods, was reviewed and augmented by Bekker .2 2. FINITE DIFFERENCE APPROXIMATION TO THE...unstable resonator calculations with laser medium,” Applied Optics 13(11), 2546–2561 (1974). [2] Bekker , E. V., et al., “Wide-angle alternating-direction
Relative and Absolute Error Control in a Finite-Difference Method Solution of Poisson's Equation
ERIC Educational Resources Information Center
Prentice, J. S. C.
2012-01-01
An algorithm for error control (absolute and relative) in the five-point finite-difference method applied to Poisson's equation is described. The algorithm is based on discretization of the domain of the problem by means of three rectilinear grids, each of different resolution. We discuss some hardware limitations associated with the algorithm,…
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.
Bubble Dynamics Calculations Using the DYSMAS/E Finite Difference Code
1988-07-01
NSWC TR 88-226 AD-A241 549 BUBBLE DYNAMICS CALCULATIONS USING THE DYSMAS/E FINITE DIFFERENCE CODE BY STEPHEN A. WILKERSON (NSWC) DR. HANS SCHITrKE...62314N RJ I4W27 1t. TITLE (include Securfry CJalssticdtti) Bubble Dynamics Calculations Using the l)YSMAS/E Finite D~ifference Code 12, PERSONAL AUTHOR...FIELD GROUP SUB. GR. bubble divnamics DN’SNAS/E, code 19 09 bubble collapse detonation 19. ABSTRACT (Continue on rovotse if noceisary and idenrty by block
Locally conformal finite-difference time-domain techniques for particle-in-cell plasma simulation
NASA Astrophysics Data System (ADS)
Clark, R. E.; Welch, D. R.; Zimmerman, W. R.; Miller, C. L.; Genoni, T. C.; Rose, D. V.; Price, D. W.; Martin, P. N.; Short, D. J.; Jones, A. W. P.; Threadgold, J. R.
2011-02-01
The Dey-Mittra [S. Dey, R. Mitra, A locally conformal finite-difference time-domain (FDTD) algorithm for modeling three-dimensional perfectly conducting objects, IEEE Microwave Guided Wave Lett. 7 (273) 1997] finite-difference time-domain partial cell method enables the modeling of irregularly shaped conducting surfaces while retaining second-order accuracy. We present an algorithm to extend this method to include charged particle emission and absorption in particle-in-cell codes. Several examples are presented that illustrate the possible improvements that can be realized using the new algorithm for problems relevant to plasma simulation.
NASA Technical Reports Server (NTRS)
Beggs, John H.; Luebbers, Raymond J.; Kunz, Karl S.; Yee, Kane S.
1991-01-01
Surface impedance boundary conditions are used to reduce the solution volume during the analysis of scattering from lossy dielectric objects. In a finite difference solution, they also can be used to avoid using small cells, made necessary by shorter wavelengths in conducting media, throughout the solution volume. A one dimensional implementation is presented for a surface impedance boundary condition for good conductors in the Finite Difference Time Domain (FDTD) technique. In order to illustrate the FDTD surface impedance boundary condition, a planar air-lossy dielectric interface is considered.
Poroelastic Wave Propagation With a 3D Velocity-Stress-Pressure Finite-Difference Algorithm
NASA Astrophysics Data System (ADS)
Aldridge, D. F.; Symons, N. P.; Bartel, L. C.
2004-12-01
Seismic wave propagation within a three-dimensional, heterogeneous, isotropic poroelastic medium is numerically simulated with an explicit, time-domain, finite-difference algorithm. A system of thirteen, coupled, first-order, partial differential equations is solved for the particle velocity vector components, the stress tensor components, and the pressure associated with solid and fluid constituents of the two-phase continuum. These thirteen dependent variables are stored on staggered temporal and spatial grids, analogous to the scheme utilized for solution of the conventional velocity-stress system of isotropic elastodynamics. Centered finite-difference operators possess 2nd-order accuracy in time and 4th-order accuracy in space. Seismological utility is enhanced by an optional stress-free boundary condition applied on a horizontal plane representing the earth's surface. Absorbing boundary conditions are imposed on the flanks of the 3D spatial grid via a simple wavefield amplitude taper approach. A massively parallel computational implementation, utilizing the spatial domain decomposition strategy, allows investigation of large-scale earth models and/or broadband wave propagation within reasonable execution times. Initial algorithm testing indicates that a point force density and/or moment density source activated within a poroelastic medium generates diverging fast and slow P waves (and possibly an S-wave)in accord with Biot theory. Solid and fluid particle velocities are in-phase for the fast P-wave, whereas they are out-of-phase for the slow P-wave. Conversions between all wave types occur during reflection and transmission at interfaces. Thus, although the slow P-wave is regarded as difficult to detect experimentally, its presence is strongly manifest within the complex of waves generated at a lithologic or fluid boundary. Very fine spatial and temporal gridding are required for high-fidelity representation of the slow P-wave, without inducing excessive
NASA Astrophysics Data System (ADS)
Huang, Binke; Zhao, Chongfeng
2014-01-01
The 2-D finite-difference frequency-domain method (FDFD) combined with the surface impedance boundary condition (SIBC) was employed to analyze the propagation characteristics of hollow rectangular waveguides at Terahertz (THz) frequencies. The electromagnetic field components, in the interior of the waveguide, were discretized using central finite-difference schemes. Considering the hollow rectangular waveguide surrounded by a medium of finite conductivity, the electric and magnetic tangential field components on the metal surface were related by the SIBC. The surface impedance was calculated by the Drude dispersion model at THz frequencies, which was used to characterize the conductivity of the metal. By solving the Eigen equations, the propagation constants, including the attenuation constant and the phase constant, were obtained for a given frequency. The proposed method shows good applicability for full-wave analysis of THz waveguides with complex boundaries.
NASA Astrophysics Data System (ADS)
Etemadsaeed, Leila; Moczo, Peter; Kristek, Jozef; Ansari, Anooshiravan; Kristekova, Miriam
2016-10-01
We investigate the problem of finite-difference approximations of the velocity-stress formulation of the equation of motion and constitutive law on the staggered grid (SG) and collocated grid (CG). For approximating the first spatial and temporal derivatives, we use three approaches: Taylor expansion (TE), dispersion-relation preserving (DRP), and combined TE-DRP. The TE and DRP approaches represent two fundamental extremes. We derive useful formulae for DRP and TE-DRP approximations. We compare accuracy of the numerical wavenumbers and numerical frequencies of the basic TE, DRP and TE-DRP approximations. Based on the developed approximations, we construct and numerically investigate 14 basic TE, DRP and TE-DRP finite-difference schemes on SG and CG. We find that (1) the TE second-order in time, TE fourth-order in space, 2-point in time, 4-point in space SG scheme (that is the standard (2,4) VS SG scheme, say TE-2-4-2-4-SG) is the best scheme (of the 14 investigated) for large fractions of the maximum possible time step, or, in other words, in a homogeneous medium; (2) the TE second-order in time, combined TE-DRP second-order in space, 2-point in time, 4-point in space SG scheme (say TE-DRP-2-2-2-4-SG) is the best scheme for small fractions of the maximum possible time step, or, in other words, in models with large velocity contrasts if uniform spatial grid spacing and time step are used. The practical conclusion is that in computer codes based on standard TE-2-4-2-4-SG, it is enough to redefine the values of the approximation coefficients by those of TE-DRP-2-2-2-4-SG for increasing accuracy of modelling in models with large velocity contrast between rock and sediments.
NASA Astrophysics Data System (ADS)
Moretti, Valter; Pastorello, Davide
2016-12-01
This work concerns some issues about the interplay of standard and geometric (Hamiltonian) approaches to finite-dimensional quantum mechanics, formulated in the projective space. Our analysis relies upon the notion and the properties of so-called frame functions, introduced by Gleason to prove his celebrated theorem. In particular, the problem of associating quantum states with positive Liouville densities is tackled from an axiomatic point of view, proving a theorem classifying all possible correspondences. A similar result is established for classical-like observables (i.e. real scalar functions on the projective space) representing quantum ones. These correspondences turn out to be encoded in a one-parameter class and, in both cases, the classical-like objects representing quantum ones result to be frame functions. The requirements of U(n) covariance and (convex) linearity play a central role in the proof of those theorems. A new characterization of classical-like observables describing quantum observables is presented, together with a geometric description of the C∗-algebra structure of the set of quantum observables in terms of classical-like ones.
Estimation of geosynchronous space objects using finite set statistics filtering methods
NASA Astrophysics Data System (ADS)
Gehly, Steve
The use of near Earth space has increased dramatically in the past few decades, and operational satellites are an integral part of modern society. The increased presence in space has led to an increase in the amount of orbital debris, which poses a growing threat to current and future space missions. Characterization of the debris environment is crucial to our continued use of high value orbit regimes such as the geosynchronous (GEO) belt. Objects in GEO pose unique challenges, by virtue of being densely spaced and tracked by a limited number of sensors in short observation windows. This research examines the use of a new class of multitarget filters to approach the problem of orbit determination for the large number of objects present. The filters make use of a recently developed mathematical toolbox derived from point process theory known as Finite Set Statistics (FISST). Details of implementing FISST-derived filters are discussed, and a qualitative and quantitative comparison between FISST and traditional multitarget estimators demonstrates the suitability of the new methods for space object estimation. Specific challenges in the areas of sensor allocation and initial orbit determination are addressed in the framework. The sensor allocation scheme makes use of information gain functionals as formulated for FISST to efficiently collect measurements on the full multitarget system. Results from a simulated network of three ground stations tracking a large catalog of geosynchronous objects demonstrate improved performance as compared to simpler, non-information theoretic tasking schemes. Further studies incorporate an initial orbit determination technique to initiate new tracks in the multitarget filter. Together with a sensor allocation scheme designed to search for new targets and maintain knowledge of the existing catalog, the method comprises a solution to the search-detect-track problem. Simulation results for a single sensor case show that the problem can be
A guide to differences between stochastic point-source and stochastic finite-fault simulations
Atkinson, G.M.; Assatourians, K.; Boore, D.M.; Campbell, K.; Motazedian, D.
2009-01-01
Why do stochastic point-source and finite-fault simulation models not agree on the predicted ground motions for moderate earthquakes at large distances? This question was posed by Ken Campbell, who attempted to reproduce the Atkinson and Boore (2006) ground-motion prediction equations for eastern North America using the stochastic point-source program SMSIM (Boore, 2005) in place of the finite-source stochastic program EXSIM (Motazedian and Atkinson, 2005) that was used by Atkinson and Boore (2006) in their model. His comparisons suggested that a higher stress drop is needed in the context of SMSIM to produce an average match, at larger distances, with the model predictions of Atkinson and Boore (2006) based on EXSIM; this is so even for moderate magnitudes, which should be well-represented by a point-source model. Why? The answer to this question is rooted in significant differences between point-source and finite-source stochastic simulation methodologies, specifically as implemented in SMSIM (Boore, 2005) and EXSIM (Motazedian and Atkinson, 2005) to date. Point-source and finite-fault methodologies differ in general in several important ways: (1) the geometry of the source; (2) the definition and application of duration; and (3) the normalization of finite-source subsource summations. Furthermore, the specific implementation of the methods may differ in their details. The purpose of this article is to provide a brief overview of these differences, their origins, and implications. This sets the stage for a more detailed companion article, "Comparing Stochastic Point-Source and Finite-Source Ground-Motion Simulations: SMSIM and EXSIM," in which Boore (2009) provides modifications and improvements in the implementations of both programs that narrow the gap and result in closer agreement. These issues are important because both SMSIM and EXSIM have been widely used in the development of ground-motion prediction equations and in modeling the parameters that control
Marchiolli, Marcelo A.; Ruzzi, Maurizio; Galetti, Diogenes
2005-10-15
By means of a mod(N)-invariant operator basis, s-parametrized phase-space functions associated with bounded operators in a finite-dimensional Hilbert space are introduced in the context of the extended Cahill-Glauber formalism, and their properties are discussed in details. The discrete Glauber-Sudarshan, Wigner, and Husimi functions emerge from this formalism as specific cases of s-parametrized phase-space functions where, in particular, a hierarchical process among them is promptly established. In addition, a phase-space description of quantum tomography and quantum teleportation is presented and new results are obtained.
Finite difference methods for transient signal propagation in stratified dispersive media
NASA Technical Reports Server (NTRS)
Lam, D. H.
1975-01-01
Explicit difference equations are presented for the solution of a signal of arbitrary waveform propagating in an ohmic dielectric, a cold plasma, a Debye model dielectric, and a Lorentz model dielectric. These difference equations are derived from the governing time-dependent integro-differential equations for the electric fields by a finite difference method. A special difference equation is derived for the grid point at the boundary of two different media. Employing this difference equation, transient signal propagation in an inhomogeneous media can be solved provided that the medium is approximated in a step-wise fashion. The solutions are generated simply by marching on in time. It is concluded that while the classical transform methods will remain useful in certain cases, with the development of the finite difference methods described, an extensive class of problems of transient signal propagating in stratified dispersive media can be effectively solved by numerical methods.
Phase space matching and finite lifetime effects for top-pair production close to threshold
Hoang, Andre H.; Reisser, Christoph J.; Ruiz-Femenia, Pedro
2010-07-01
The top-pair tt production cross section close to threshold in e{sup +}e{sup -} collisions is strongly affected by the small lifetime of the top quark. Since the cross section is defined through final states containing the top decay products, a consistent definition of the cross section depends on prescriptions of how these final states are accounted for the cross section. Experimentally, these prescriptions are implemented, for example, through cuts on kinematic quantities such as the reconstructed top quark invariant masses. As long as these cuts do not reject final states that can arise from the decay of a top and an antitop quark with a small off-shellness compatible with the nonrelativistic power counting, they can be implemented through imaginary phase space matching conditions in nonrelativistic QCD. The prescription-dependent cross section can then be determined from the optical theorem using the e{sup +}e{sup -} forward scattering amplitude. We compute the phase space matching conditions associated to cuts on the top and antitop invariant masses at next-to-next-to-leading logarithmic order and partially at next-to-next-to-next-to-leading logarithmic order in the nonrelativistic expansion accounting also for higher order QCD effects. Together with finite lifetime and electroweak effects known from previous work, we analyze their numerical impact on the tt cross section. We show that the phase space matching contributions are essential to make reliable nonrelativistic QCD predictions, particularly for energies below the peak region, where the cross section is small. We find that irreducible background contributions associated to final states that do not come from top decays are strongly suppressed and can be neglected for the theoretical predictions.
ERIC Educational Resources Information Center
Song, Hairong; Ferrer, Emilio
2009-01-01
This article presents a state-space modeling (SSM) technique for fitting process factor analysis models directly to raw data. The Kalman smoother via the expectation-maximization algorithm to obtain maximum likelihood parameter estimates is used. To examine the finite sample properties of the estimates in SSM when common factors are involved, a…
ERIC Educational Resources Information Center
Song, Hairong; Ferrer, Emilio
2009-01-01
This article presents a state-space modeling (SSM) technique for fitting process factor analysis models directly to raw data. The Kalman smoother via the expectation-maximization algorithm to obtain maximum likelihood parameter estimates is used. To examine the finite sample properties of the estimates in SSM when common factors are involved, a…
NASA Astrophysics Data System (ADS)
Lin, M. C.; Nieter, C.; Stoltz, P. H.; Smithe, D. N.
2009-05-01
This work introduces a conformal finite difference time domain (CFDTD) method to accurately determine the dispersion relation of an A6 relativistic magnetron. The accuracy is measured by comparing with accurate SUPERFISH calculations based on finite element method. The results show that an accuracy of 99.4% can be achieved by using only 10,000 mesh points with Dey-Mittra algorithm. By comparison, a mesh number of 360,000 is needed to preserve 99% accuracy using conventional FDTD method. This suggests one can efficiently and accurately study the hot tests of microwave tubes using CFDTD particle-in-cell method instead of conventional FDTD one.
NASA Technical Reports Server (NTRS)
Baumeister, K. J.
1981-01-01
The cutoff mode instability problem associated with a transient finite difference solution to the wave equation is explained. The steady-state impedance boundary condition is found to produce acoustic reflections during the initial transient, which cause finite instabilities in the cutoff modes. The stability problem is resolved by extending the duct length to prevent transient reflections. Numerical calculations are presented at forcing frequencies above, below, and nearly at the cutoff frequency, and exit impedance models are presented for use in the practical design of turbofan inlets.
NASA Astrophysics Data System (ADS)
Kharytonov, Oleksii M.; Kiforenko, Boris M.
2011-08-01
The nuclear thermal rocket (NTR) propulsion is one of the leading promising technologies for primary space propulsion for manned exploration of the solar system due to its high specific impulse capability and sufficiently high thrust-to-weight ratio. Another benefit of NTR is its possible bimodal design, when nuclear reactor is used for generation of a jet thrust in a high-thrust mode and (with an appropriate power conversion system) as a source of electric power to supply the payload and the electric engines in a low-thrust mode. The model of the NTR thrust control was developed considering high-thrust NTR as a propulsion system of limited power and exhaust velocity. For the proposed model the control of the thrust value is accomplished by the regulation of reactor thermal power and propellant mass flow rate. The problem of joint optimization of the combination of high- and low-thrust arcs and the parameters of bimodal NTR (BNTR) propulsion system is considered for the interplanetary transfers. The interplanetary trajectory of the space vehicle is formed by the high-thrust NTR burns, which define planet-centric maneuvers and by the low-thrust heliocentric arcs where the nuclear electric propulsion (NEP) is used. The high-thrust arcs are analyzed using finite-thrust approach. The motion of the corresponding dynamical system is realized in three phase spaces concerning the departure planet-centric maneuver by means of high-thrust NTR propulsion, the low-thrust NEP heliocentric maneuver and the approach high-thrust NTR planet-centric maneuver. The phase coordinates are related at the time instants of the change of the phase spaces due to the relations between the space vehicle masses. The optimal control analysis is performed using Pontryagin's maximum principle. The numerical results are analyzed for Earth-Mars "sprint" transfer. The optimal values of the parameters that define the masses of NTR and NEP subsystems have been evaluated. It is shown that the low
Finite-Difference Algorithm for 3D Orthorhombic Elastic Wave Propagation
NASA Astrophysics Data System (ADS)
Jensen, R.; Preston, L. A.; Aldridge, D. F.
2016-12-01
Many geophysicists concur that an orthorhombic elastic medium, characterized by three mutually orthogonal symmetry planes, constitutes a realistic representation of seismic anisotropy in shallow crustal rocks. This symmetry condition typically arises via a dense system of vertically-aligned microfractures superimposed on a finely-layered horizontal geology. Mathematically, the elastic stress-strain constitutive relations for an orthorhombic body contain nine independent moduli. In turn, these moduli can be determined by observing (or prescribing) nine independent P-wave and S-wave phase speeds along different propagation directions. We are developing an explicit time-domain finite-difference (FD) algorithm for simulating 3D elastic wave propagation in a heterogeneous orthorhombic medium. The components of the particle velocity vector and the stress tensor are governed by a set of nine, coupled, first-order, linear, partial differential equations (PDEs) called the velocity-stress system. All time and space derivatives are discretized with centered and staggered FD operators possessing second- and fourth-order numerical accuracy, respectively. Simplified FD updating formulae (with significantly reduced operation counts) for stress components are obtained by restricting the principle axes of the modulus tensor to be parallel to the global rectangular coordinate axes. Moreover, restriction to a piecewise homogeneous earth model reduces computational memory demand for storing the ten (including mass density) model parameters. These restrictions will be relaxed in the future. Novel perfectly matched layer (PML) absorbing boundary conditions, specifically designed for orthorhombic media, effectively suppress grid boundary reflections. Initial modeling results reveal the well-established anisotropic seismic phenomena of complex wavefront shapes, split (fast and slow) S-waves, and shear waves generated by a spherically-symmetric explosion in a homogeneous body.
Rupture Dynamics Simulation for Non-Planar fault by a Curved Grid Finite Difference Method
NASA Astrophysics Data System (ADS)
Zhang, Z.; Zhu, G.; Chen, X.
2011-12-01
We first implement the non-staggered finite difference method to solve the dynamic rupture problem, with split-node, for non-planar fault. Split-node method for dynamic simulation has been used widely, because of that it's more precise to represent the fault plane than other methods, for example, thick fault, stress glut and so on. The finite difference method is also a popular numeric method to solve kinematic and dynamic problem in seismology. However, previous works focus most of theirs eyes on the staggered-grid method, because of its simplicity and computational efficiency. However this method has its own disadvantage comparing to non-staggered finite difference method at some fact for example describing the boundary condition, especially the irregular boundary, or non-planar fault. Zhang and Chen (2006) proposed the MacCormack high order non-staggered finite difference method based on curved grids to precisely solve irregular boundary problem. Based upon on this non-staggered grid method, we make success of simulating the spontaneous rupture problem. The fault plane is a kind of boundary condition, which could be irregular of course. So it's convinced that we could simulate rupture process in the case of any kind of bending fault plane. We will prove this method is valid in the case of Cartesian coordinate first. In the case of bending fault, the curvilinear grids will be used.
NASA Technical Reports Server (NTRS)
Mickens, R. E.
1984-01-01
Work on the construction of finite difference models of differential equations having zero truncation errors is summarized. Both linear and nonlinear unidirectional wave equations are discussed. Results regarding the construction of zero truncation error schemes for the full wave equation and Burger's equation are also briefly reported.
Implementing measured source signatures in a coarse-grid, finite-difference modeling scheme
Landroe, M.; Mittet, R.; Sollie, R. . Sintef Group)
1993-12-01
In a marine seismic air-gun array, each gun location does not necessarily coincide with a node in a finite-difference grid. Especially for coarse-grid, finite-difference modeling, this problem must be handled with care since there might be up to three or four air guns between adjacent grid points. The real sources are represented by fictitious monopole- and dipole source functions located at grid nodes. The effective sources are estimated from the extrapolated pressure field at a horizontal surface located below the sources. The authors find that an array consisting of eight guns separated by a distance of 3 m and located at 7.5 m depth can be approximated by six monopole- and dipole source functions distributed on a finite-difference grid with 10 m spatial sampling. The residual error energy norm between the actual wavefield and the corresponding finite-difference wavefield observed on a fictitious streamer placed at 95 m depth is less than 0.5 percent.
High Order Finite Difference Methods, Multidimensional Linear Problems and Curvilinear Coordinates
NASA Technical Reports Server (NTRS)
Nordstrom, Jan; Carpenter, Mark H.
1999-01-01
Boundary and interface conditions are derived for high order finite difference methods applied to multidimensional linear problems in curvilinear coordinates. The boundary and interface conditions lead to conservative schemes and strict and strong stability provided that certain metric conditions are met.
Positivity-preserving High Order Finite Difference WENO Schemes for Compressible Euler Equations
2011-07-15
schemes are preferred, for example, cosmological simulation [5], finite difference WENO scheme [10] is more favored than DG schemes [2, 3] and the...densities, Journal of Computational Physics, 92 (1991), 273-295. [5] L.-L. Feng, C.-W. Shu and M. Zhang, A hybrid cosmological hydrodynamic/N-body code
Parallel electromagnetic simulator based on the Finite-Difference Time Domain method
NASA Astrophysics Data System (ADS)
Walendziuk, Wojciech
2006-03-01
In the following paper the parallel tool for electromagnetic field distribution analysis is presented. The main simulation programme is based on the parallel algorithm of the Finite-Difference Time-Domain method and use Message Passing Interface as a communication library. In the paper also ways of communications among computation nodes in a parallel environment and efficiency of the parallel algorithm are presented.
Streamline-coordinate finite-difference method for hot metal deformations
Chung, S.G. ); Kuwahara, K. ); Richmond, O. )
1993-09-01
The hot metal deformation in the rolling process is a typical example of near-steady, quasi two-dimensional non-Newtonian flows. An isotropic work-hardening model characterized by a dislocation energy-density is presented and analyzed the streamline-coordinate finite-difference method. 21 refs., 4 figs., 3 tabs.
Optimized compact-difference-based finite-volume schemes for linear wave phenomena
Gaitonde, D.; Shang, J.S.
1997-12-01
This paper discusses a numerical method to analyze linear wave propagation phenomena with emphasis on electromagnetic in the time-domain. The numerical methods is based on a compact-difference-based finite-volume method at higher-orders. This scheme is evaluated using a classical fourth-order Runge-Kutta technique.
A FINITE-DIFFERENCE, DISCRETE-WAVENUMBER METHOD FOR CALCULATING RADAR TRACES
A hybrid of the finite-difference method and the discrete-wavenumber method is developed to calculate radar traces. The method is based on a three-dimensional model defined in the Cartesian coordinate system; the electromagnetic properties of the model are symmetric with respect ...
The role of finite-difference methods in design and analysis for supersonic cruise
NASA Technical Reports Server (NTRS)
Townsend, J. C.
1976-01-01
Finite-difference methods for analysis of steady, inviscid supersonic flows are described, and their present state of development is assessed with particular attention to their applicability to vehicles designed for efficient cruise flight. Current work is described which will allow greater geometric latitude, improve treatment of embedded shock waves, and relax the requirement that the axial velocity must be supersonic.
A FINITE-DIFFERENCE, DISCRETE-WAVENUMBER METHOD FOR CALCULATING RADAR TRACES
A hybrid of the finite-difference method and the discrete-wavenumber method is developed to calculate radar traces. The method is based on a three-dimensional model defined in the Cartesian coordinate system; the electromagnetic properties of the model are symmetric with respect ...
Finite difference micromagnetic simulation with self-consistent currents and smooth surfaces
Cerjan, C; Gibbons, M R; Hewett, D W; Parker, G
1999-05-27
A micromagnetic algorithm has been developed using the finite difference method (FDM). Elliptic field equations are solved on the mesh using the efficient Dynamic Alternating Direction Implicit method. Smooth surfaces have been included in the FDM formulation so structures of irregular shape can be modeled. The current distribution and temperature of devices are also calculated. Keywords: Micromagnetic simulation, Magnetic dots, Read heads, Thermal Effects
A FINITE-DIFFERENCE, DISCRETE-WAVENUMBER METHOD FOR CALCULATING RADAR TRACES
A hybrid of the finite-difference method and the discrete-wavenumber method is developed to calculate radar traces. The method is based on a three-dimensional model defined in the Cartesian coordinate system; the electromag-netic properties of the model are symmetric with respect...
A FINITE-DIFFERENCE, DISCRETE-WAVENUMBER METHOD FOR CALCULATING RADAR TRACES
A hybrid of the finite-difference method and the discrete-wavenumber method is developed to calculate radar traces. The method is based on a three-dimensional model defined in the Cartesian coordinate system; the electromag-netic properties of the model are symmetric with respect...
Optimal convergence rate of the explicit finite difference scheme for American option valuation
NASA Astrophysics Data System (ADS)
Hu, Bei; Liang, Jin; Jiang, Lishang
2009-08-01
An optimal convergence rate O([Delta]x) for an explicit finite difference scheme for a variational inequality problem is obtained under the stability condition using completely PDE methods. As a corollary, a binomial tree scheme of an American put option (where ) is convergent unconditionally with the rate O(([Delta]t)1/2).
Finite-difference, spectral and Galerkin methods for time-dependent problems
NASA Technical Reports Server (NTRS)
Tadmor, E.
1983-01-01
Finite difference, spectral and Galerkin methods for the approximate solution of time dependent problems are surveyed. A unified discussion on their accuracy, stability and convergence is given. In particular, the dilemma of high accuracy versus stability is studied in some detail.
An Eigenvalue Analysis of finite-difference approximations for hyperbolic IBVPs
NASA Technical Reports Server (NTRS)
Warming, Robert F.; Beam, Richard M.
1989-01-01
The eigenvalue spectrum associated with a linear finite-difference approximation plays a crucial role in the stability analysis and in the actual computational performance of the discrete approximation. The eigenvalue spectrum associated with the Lax-Wendroff scheme applied to a model hyperbolic equation was investigated. For an initial-boundary-value problem (IBVP) on a finite domain, the eigenvalue or normal mode analysis is analytically intractable. A study of auxiliary problems (Dirichlet and quarter-plane) leads to asymptotic estimates of the eigenvalue spectrum and to an identification of individual modes as either benign or unstable. The asymptotic analysis establishes an intuitive as well as quantitative connection between the algebraic tests in the theory of Gustafsson, Kreiss, and Sundstrom and Lax-Richtmyer L(sub 2) stability on a finite domain.
NASA Astrophysics Data System (ADS)
Ahmad, Azhar; Azmi, Amirah; Majid, Ahmad Abd.; Hamid, Nur Nadiah Abd
2017-08-01
In this paper, Nonlinear Schrödinger (NLS) equation with Neumann boundary conditions is solved using finite difference method (FDM) and cubic B-spline interpolation method (CuBSIM). First, the approach is based on the FDM applied on the time and space discretization with the help of theta-weighted method. However, our main interest is the second approach, whereby FDM is applied on the time discretization and cubic B-spline is utilized as an interpolation function in the space dimension with the same help of theta-weighted method. The CuBSIM is shown to be stable by using von Neumann stability analysis. The proposed method is tested on a test problem with single soliton motion of the NLS equation. The accuracy of the numerical results is measured by the Euclidean-norm and infinity-norm. CuBSIM is found to produce more accurate results than the FDM.
Dynamic Buckling of Elastic Bar under Axial Impact Based on Finite Difference Method
NASA Astrophysics Data System (ADS)
Ma, Hao; Yang, Qiang; Han, Zhi-Jun; Lu, Guo-Yun
2016-05-01
Considering first order shear deformation theory, the dynamic buckling governing equations of elastic bar with initial imperfections, transverse inertia and axial inertia are derived by Hamilton principle. The equations are converted into the form of non-dimension. Based on the finite difference method, the equations are solved approximately. The buckling mode of elastic bar under different axial impact velocities has been obtained. The influence of different axial impact velocity on the dynamic buckling of elastic bar is discussed.
A nine-point finite difference scheme for one-dimensional wave equation
NASA Astrophysics Data System (ADS)
Szyszka, Barbara
2017-07-01
The paper is devoted to an implicit finite difference method (FDM) for solving initial-boundary value problems (IBVP) for one-dimensional wave equation. The second-order derivatives in the wave equation have been approximated at the four intermediate points, as a consequence, an implicit nine-point difference scheme has been obtained. Von Neumann stability analysis has been conducted and we have demonstrated, that the presented difference scheme is unconditionally stable.
Choi, A P C; Zheng, Y P
2005-03-01
Young's modulus and Poisson's ratio of a tissue can be simultaneously obtained using two indentation tests with two different sized indentors in two indentations. Owing to the assumption of infinitesimal deformation of the indentation, the finite deformation effect of indentation on the calculated material parameters was not fully understood in the double indentation approach. However, indentation tests with infinitesimal deformation are not practical for the measurement of real tissues. Accordingly, finite element models were developed to simulate the indentation with different indentor diameters and different deformation ratios to investigate the finite deformation effect of indentation. The results indicated that Young's modulus E increased with the increase in the indentation deformation w, if the finite deformation effect of indentation was not considered. This phenomenon became obvious when Poisson's ratio v approached 0.5 and/or the ratio of indentor radius and tissue thickness a/h increased. The calculated Young's modulus could be different by 23% at 10% deformation in comparison with its real value. The results also demonstrated that the finite deformation effect to indentation on the calculation of Poisson's ratio v was much smaller. After the finite deformation effect of indentation was considered, the error of the calculated Young's modulus could be controlled within 5% (a/h = 1) and 2% (a/h = 2) for deformation up to 10%.
NASA Technical Reports Server (NTRS)
Baumeister, K. J.; Eversman, W.; Astley, R. J.; White, J. W.
1981-01-01
Experimental data are presented for sound propagation in a simulated infinite hard wall duct with a large change in duct cross sectional area. The data are conveniently tabulated for further use. The 'steady' state finite element theory of Astley and Eversman (1981) and the transient finite difference theory of White (1981) are in good agreement with the data for both the axial and transverse pressure profiles and the axial phase angle. Therefore, numerical finite difference and finite element theories appear to be ideally suited for handling duct propagation problems which encounter large axial gradients in acoustic parameters. The measured energy reflection coefficient agrees with the values from the Astley-Eversman modal coupling model.
An improved finite-difference analysis of uncoupled vibrations of tapered cantilever beams
NASA Technical Reports Server (NTRS)
Subrahmanyam, K. B.; Kaza, K. R. V.
1983-01-01
An improved finite difference procedure for determining the natural frequencies and mode shapes of tapered cantilever beams undergoing uncoupled vibrations is presented. Boundary conditions are derived in the form of simple recursive relations involving the second order central differences. Results obtained by using the conventional first order central differences and the present second order central differences are compared, and it is observed that the present second order scheme is more efficient than the conventional approach. An important advantage offered by the present approach is that the results converge to exact values rapidly, and thus the extrapolation of the results is not necessary. Consequently, the basic handicap with the classical finite difference method of solution that requires the Richardson's extrapolation procedure is eliminated. Furthermore, for the cases considered herein, the present approach produces consistent lower bound solutions.
Modeling anisotropic flow and heat transport by using mimetic finite differences
NASA Astrophysics Data System (ADS)
Chen, Tao; Clauser, Christoph; Marquart, Gabriele; Willbrand, Karen; Büsing, Henrik
2016-08-01
Modeling anisotropic flow in porous or fractured rock often assumes that the permeability tensor is diagonal, which means that its principle directions are always aligned with the coordinate axes. However, the permeability of a heterogeneous anisotropic medium usually is a full tensor. For overcoming this shortcoming, we use the mimetic finite difference method (mFD) for discretizing the flow equation in a hydrothermal reservoir simulation code, SHEMAT-Suite, which couples this equation with the heat transport equation. We verify SHEMAT-Suite-mFD against analytical solutions of pumping tests, using both diagonal and full permeability tensors. We compare results from three benchmarks for testing the capability of SHEMAT-Suite-mFD to handle anisotropic flow in porous and fractured media. The benchmarks include coupled flow and heat transport problems, three-dimensional problems and flow through a fractured porous medium with full equivalent permeability tensor. It shows firstly that the mimetic finite difference method can model anisotropic flow both in porous and in fractured media accurately and its results are better than those obtained by the multi-point flux approximation method in highly anisotropic models, secondly that the asymmetric permeability tensor can be included and leads to improved results compared the symmetric permeability tensor in the equivalent fracture models, and thirdly that the method can be easily implemented in existing finite volume or finite difference codes, which has been demonstrated successfully for SHEMAT-Suite.
NASA Astrophysics Data System (ADS)
Chen, M.; Wei, S.
2016-12-01
The serious damage of Mexico City caused by the 1985 Michoacan earthquake 400 km away indicates that urban areas may be affected by remote earthquakes. To asses earthquake risk of urban areas imposed by distant earthquakes, we developed a hybrid Frequency Wavenumber (FK) and Finite Difference (FD) code implemented with MPI, since the computation of seismic wave propagation from a distant earthquake using a single numerical method (e.g. Finite Difference, Finite Element or Spectral Element) is very expensive. In our approach, we compute the incident wave field (ud) at the boundaries of the excitation box, which surrounding the local structure, using a paralleled FK method (Zhu and Rivera, 2002), and compute the total wave field (u) within the excitation box using a parallelled 2D FD method. We apply perfectly matched layer (PML) absorbing condition to the diffracted wave field (u-ud). Compared to previous Generalized Ray Theory and Finite Difference (Wen and Helmberger, 1998), Frequency Wavenumber and Spectral Element (Tong et al., 2014), and Direct Solution Method and Spectral Element hybrid method (Monteiller et al., 2013), our absorbing boundary condition dramatically suppress the numerical noise. The MPI implementation of our method can greatly speed up the calculation. Besides, our hybrid method also has a potential use in high resolution array imaging similar to Tong et al. (2014).
NASA Astrophysics Data System (ADS)
Pettit, J. R.; Walker, A.; Lowe, M. J. S.
2015-01-01
A common goal when using Finite Element (FE) modelling in time domain wave scattering problems is to minimise model size by only considering a region immediately surrounding a scatterer or feature of interest. The model boundaries must simulate infinite space by minimising the reflection of incident waves. This is a significant and long-standing challenge that has only achieved partial success. Industrial companies wishing to perform such modelling are keen to use established commercial FE packages that offer a thorough history of validation and testing. Unfortunately, this limits the flexibility available to modellers preventing the use of popular research tools such as Perfectly Matched Layers (PML). Unlike PML, Absorbing Layers by Increasing Damping (ALID) have proven successful offering practical implementation into any solver that has representation of material damping. Despite good performance further improvements are desirable. Here, a Stiffness Reduction Method (SRM) has been developed and optimised to operate within a significantly reduced spatial domain. The technique is applied by altering damping and stiffness matrices, inducing decay of incident waves. Variables are expressed as a function of known model constants, easing implementation for generic problems. Analytical and numerical solutions have shown that SRM out performs ALID, with results approaching those of PML.
Finite-time Properties of the Navier-Stokes Equations Under Lebesque Space Disturbances
NASA Astrophysics Data System (ADS)
Bobba, Kumar
2006-11-01
A complete understanding of the stability characteristics of the Navier-Stokes equations involve understanding both the transient response and the steady state response. The steady state (or infinite-time) response of the Navier-Stokes equations is characterized by the point spectrum and has been well studied. In this work, we study the transient (or finite-time) response of the unsteady Navier-Stokes equations linearized about plane Couette base flow under spatial and temporal varying disturbance forcing. The forcing and response are assumed to belong to infinite-dimensional Lebesque function spaces, L2 and L∞. An analytical characterization is given for the induced norms that characterize the response. It is shown that the L2 induced norm is tightly bounded by the H∞ norm of the transfer function operator and the L∞ induced norm is upper bounded by the L1 norm of the impulse response operator. The structure of the worst case disturbances and their amplification rates are computed using spectral methods---with Fourier modes in homogeneous direction and Chebyshev collocation in non-homogeneous direction. The relevance of the present results to the channel flow laminar-turbulent transition experiments will be discussed.
M2Di: Concise and efficient MATLAB 2-D Stokes solvers using the Finite Difference Method
NASA Astrophysics Data System (ADS)
Räss, Ludovic; Duretz, Thibault; Podladchikov, Yury Y.; Schmalholz, Stefan M.
2017-02-01
Recent development of many multiphysics modeling tools reflects the currently growing interest for studying coupled processes in Earth Sciences. The core of such tools should rely on fast and robust mechanical solvers. Here we provide M2Di, a set of routines for 2-D linear and power law incompressible viscous flow based on Finite Difference discretizations. The 2-D codes are written in a concise vectorized MATLAB fashion and can achieve a time to solution of 22 s for linear viscous flow on 10002 grid points using a standard personal computer. We provide application examples spanning from finely resolved crystal-melt dynamics, deformation of heterogeneous power law viscous fluids to instantaneous models of mantle flow in cylindrical coordinates. The routines are validated against analytical solution for linear viscous flow with highly variable viscosity and compared against analytical and numerical solutions of power law viscous folding and necking. In the power law case, both Picard and Newton iterations schemes are implemented. For linear Stokes flow and Picard linearization, the discretization results in symmetric positive-definite matrix operators on Cartesian grids with either regular or variable grid spacing allowing for an optimized solving procedure. For Newton linearization, the matrix operator is no longer symmetric and an adequate solving procedure is provided. The reported performance of linear and power law Stokes flow is finally analyzed in terms of wall time. All MATLAB codes are provided and can readily be used for educational as well as research purposes. The M2Di routines are available from Bitbucket and the University of Lausanne Scientific Computing Group website, and are also supplementary material to this article.
NASA Technical Reports Server (NTRS)
Chen, G.; Zheng, Q.; Coleman, M.; Weerakoon, S.
1983-01-01
This paper briefly reviews convergent finite difference schemes for hyperbolic initial boundary value problems and their applications to boundary control systems of hyperbolic type which arise in the modelling of vibrations. These difference schemes are combined with the primal and the dual approaches to compute the optimal control in the unconstrained case, as well as the case when the control is subject to inequality constraints. Some of the preliminary numerical results are also presented.
An outgoing energy flux boundary condition for finite difference ICRP antenna models
Batchelor, D.B.; Carter, M.D.
1992-11-01
For antennas at the ion cyclotron range of frequencies (ICRF) modeling in vacuum can now be carried out to a high level of detail such that shaping of the current straps, isolating septa, and discrete Faraday shield structures can be included. An efficient approach would be to solve for the fields in the vacuum region near the antenna in three dimensions by finite methods and to match this solution at the plasma-vacuum interface to a solution obtained in the plasma region in one dimension by Fourier methods. This approach has been difficult to carry out because boundary conditions must be imposed at the edge of the finite difference grid on a point-by-point basis, whereas the condition for outgoing energy flux into the plasma is known only in terms of the Fourier transform of the plasma fields. A technique is presented by which a boundary condition can be imposed on the computational grid of a three-dimensional finite difference, or finite element, code by constraining the discrete Fourier transform of the fields at the boundary points to satisfy an outgoing energy flux condition appropriate for the plasma. The boundary condition at a specific grid point appears as a coupling to other grid points on the boundary, with weighting determined by a kemel calctdated from the plasma surface impedance matrix for the various plasma Fourier modes. This boundary condition has been implemented in a finite difference solution of a simple problem in two dimensions, which can also be solved directly by Fourier transformation. Results are presented, and it is shown that the proposed boundary condition does enforce outgoing energy flux and yields the same solution as is obtained by Fourier methods.
NASA Astrophysics Data System (ADS)
Sharapudinov, I. I.
2014-02-01
The paper deals with the space L^{p(x)} consisting of classes of real measurable functions f(x) on \\lbrack 0,1 \\rbrack with finite integral \\displaystyle\\int_0^1\\vert f(x)\\vert^{p(x)}\\,dx. If 1\\le p(x)\\le \\overline p\\lt\\infty, then the space L^{p(x)} can be made into a Banach space with the norm \\displaystyle\\Vert f\\Vert _{p(\\cdot)}=\\inf\\biggl\\{\\alpha\\,{\\gt}\\,0: \\int_0^1 \\vert{f(x)/\\alpha}\\vert^{p(x)}\\,dx\\le\
NASA Technical Reports Server (NTRS)
Miner, E. W.; Lewis, C. H.
1972-01-01
An implicit finite difference method has been applied to tangential slot injection into supersonic turbulent boundary layer flows. In addition, the effects induced by the interaction between the boundary layer displacement thickness and the external pressure field are considered. In the present method, three different eddy viscosity models have been used to specify the turbulent momentum exchange. One model depends on the species concentration profile and the species conservation equation has been included in the system of governing partial differential equations. Results are compared with experimental data at stream Mach numbers of 2.4 and 6.0 and with results of another finite difference method. Good agreement was generally obtained for the reduction of wall skin friction with slot injection and with experimental Mach number and pitot pressure profiles. Calculations with the effects of pressure interaction included showed these effects to be smaller than effects of changing eddy viscosity models.
The modified equation approach to the stability and accuracy analysis of finite-difference methods
NASA Technical Reports Server (NTRS)
Warming, R. F.; Hyett, B. J.
1974-01-01
The stability and accuracy of finite-difference approximations to simple linear partial differential equations are analyzed by studying the modified partial differential equation. Aside from round-off error, the modified equation represents the actual partial differential equation solved when a numerical solution is computed using a finite-difference equation. The modified equation is derived by first expanding each term of a difference scheme in a Taylor series and then eliminating time derivatives higher than first order by certain algebraic manipulations. The connection between 'heuristic' stability theory based on the modified equation approach and the von Neumann (Fourier) method is established. In addition to the determination of necessary and sufficient conditions for computational stability, a truncated version of the modified equation can be used to gain insight into the nature of both dissipative and dispersive errors.
NASA Astrophysics Data System (ADS)
Subrahmanyam, K. B.; Kaza, K. R. V.
1985-03-01
Theoretical natural frequencies of the first three modes of torsional vibration of pre-twisted, rotating cantilever beams are determined for various thickness and aspect ratios. Conclusions concerning individual and collective effects of warping, pretwist, tension-torsion coupling and tennis racket effect (twist-rotational coupling) terms on the natural frequencies are drawn from numerical results obtained by using a finite difference procedure with first order central differences. The relative importance of structural warping, inertial warping, pretwist, tension-torsion and twist-rotational coupling terms is discussed for various rotational speeds. The accuracy of results obtained by using the finite difference approach is verified by a comparison with the exact solution for specialized simple cases of the equation of motion used in this paper.
Energy stable and high-order-accurate finite difference methods on staggered grids
NASA Astrophysics Data System (ADS)
O'Reilly, Ossian; Lundquist, Tomas; Dunham, Eric M.; Nordström, Jan
2017-10-01
For wave propagation over distances of many wavelengths, high-order finite difference methods on staggered grids are widely used due to their excellent dispersion properties. However, the enforcement of boundary conditions in a stable manner and treatment of interface problems with discontinuous coefficients usually pose many challenges. In this work, we construct a provably stable and high-order-accurate finite difference method on staggered grids that can be applied to a broad class of boundary and interface problems. The staggered grid difference operators are in summation-by-parts form and when combined with a weak enforcement of the boundary conditions, lead to an energy stable method on multiblock grids. The general applicability of the method is demonstrated by simulating an explosive acoustic source, generating waves reflecting against a free surface and material discontinuity.
NASA Technical Reports Server (NTRS)
Subrahmanyam, K. B.; Kaza, K. R. V.
1985-01-01
Theoretical natural frequencies of the first three modes of torsional vibration of pretwisted, rotating cantilever beams are determined for various thickness and aspect ratios. Conclusions concerning individual and collective effects of warping, pretwist, tension-torsion coupling and tennis racket effect (twist-rotational coupling) terms on the natural frequencies are drawn from numerical results obtained by using a finite difference procedure with first order central differences. The relative importance of structural warping, inertial warping, pretwist, tension-torsion and twist-rotational coupling terms is discussed for various rotational speeds. The accuracy of results obtained by using the finite difference approach is verified by a comparison with the exact solution for specialized simple cases of the equation of motion used in this paper.
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.
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.
A Multi-CPU/GPU implementation of RBF-generated finite differences for PDEs on a Sphere
NASA Astrophysics Data System (ADS)
Bollig, E. F.; Flyer, N.; Erlebacher, G.
2011-12-01
Numerical methods leveraging Radial Basis Functions (RBFs) are on the rise in computational science. With natural extensions into higher dimensions, functionality in the face of unstructured grids, stability for large time-steps, competitive accuracy and convergence when compared to other state-of-the-art methods, it is hard to ignore these simple-to-code alternatives. RBF-generated finite differences (RBF-FD) hold a promising future in that they have the advantages of global RBFs but have the ability to be highly parallelizable on multi-core machines. They differ from classical finite differences in that the test functions used to calculate the differentiation weights are n-dimensional RBFs rather than one-dimensional polynomials. This allows for generalization to n-dimensional space on completely scattered node layouts. We present an ongoing effort to develop fast and efficient implementations of RBF-FD for the geosciences. Specifically, we introduce a multi-CPU/GPU implementation for the solution of parabolic and hyperbolic PDEs. This work targets the NSF funded Keeneland GPU cluster, which---like many of the latest HPC systems around the world---offers significantly more GPU accelerators than CPU counterparts. We will discuss parallelization strategies, algorithms and data-structures used to span computation across the heterogeneous architecture.
On One-Dimensional Stretching Functions for Finite-Difference Calculations
NASA Technical Reports Server (NTRS)
Vinokur, M.
1980-01-01
The class of one dimensional stretching function used in finite difference calculations is studied. For solutions containing a highly localized region of rapid variation, simple criteria for a stretching function are derived using a truncation error analysis. These criteria are used to investigate two types of stretching functions. One is an interior stretching function, for which the location and slope of an interior clustering region are specified. The simplest such function satisfying the criteria is found to be one based on the inverse hyperbolic sine. The other type of function is a two sided stretching function, for which the arbitrary slopes at the two ends of the one dimensional interval are specified. The simplest such general function is found to be one based on the inverse tangent. The general two sided function has many applications in the construction of finite difference grids.
NASA Astrophysics Data System (ADS)
Dai, Liyi
2016-05-01
Stochastic optimization is a fundamental problem that finds applications in many areas including biological and cognitive sciences. The classical stochastic approximation algorithm for iterative stochastic optimization requires gradient information of the sample object function that is typically difficult to obtain in practice. Recently there has been renewed interests in derivative free approaches to stochastic optimization. In this paper, we examine the rates of convergence for the Kiefer-Wolfowitz algorithm and the mirror descent algorithm, by approximating gradient using finite differences generated through common random numbers. It is shown that the convergence of these algorithms can be accelerated by controlling the implementation of the finite differences. Particularly, it is shown that the rate can be increased to n-2/5 in general and to n-1/2, the best possible rate of stochastic approximation, in Monte Carlo optimization for a broad class of problems, in the iteration number n.
Convergence rates of finite difference stochastic approximation algorithms part I: general sampling
NASA Astrophysics Data System (ADS)
Dai, Liyi
2016-05-01
Stochastic optimization is a fundamental problem that finds applications in many areas including biological and cognitive sciences. The classical stochastic approximation algorithm for iterative stochastic optimization requires gradient information of the sample object function that is typically difficult to obtain in practice. Recently there has been renewed interests in derivative free approaches to stochastic optimization. In this paper, we examine the rates of convergence for the Kiefer-Wolfowitz algorithm and the mirror descent algorithm, under various updating schemes using finite differences as gradient approximations. The analysis is carried out under a general framework covering a wide range of updating scenarios. It is shown that the convergence of these algorithms can be accelerated by controlling the implementation of the finite differences.
A semi-implicit finite difference model for three-dimensional tidal circulation,
Casulli, V.; Cheng, R.T.
1992-01-01
A semi-implicit finite difference formulation for the numerical solution of three-dimensional tidal circulation is presented. The governing equations are the three-dimensional Reynolds equations in which the pressure is assumed to be hydrostatic. A minimal degree of implicitness has been introduced in the finite difference formula so that in the absence of horizontal viscosity the resulting algorithm is unconditionally stable at a minimal computational cost. When only one vertical layer is specified this method reduces, as a particular case, to a semi-implicit scheme for the solutions of the corresponding two-dimensional shallow water equations. The resulting two- and three-dimensional algorithm is fast, accurate and mass conservative. This formulation includes the simulation of flooding and drying of tidal flats, and is fully vectorizable for an efficient implementation on modern vector computers.
A mapped finite difference study of noise propagation in nonuniform ducts with mean flow
NASA Astrophysics Data System (ADS)
Raad, Peter E.; White, James W.
1987-10-01
The primary objective of this work is to study noise propagation in acoustically lined variable area ducts with mean fluid flow. The method of study is numerical in nature and involves a body-fitted grid mapping procedure in conjunction with a factored-implicit finite difference technique. The mean fluid flow model used is two-dimensional, inviscid, irrotational, incompressible, and nonheat conducting. Fully-coupled solutions of the linearized gasdynamic equations are obtained for both positive and negative Mach numbers as well as for hard and soft wall conditions. The factored-implicit finite difference technique used did give rise to short wavelength perturbations, but these were dampened by the introduction of higher order artificial dissipation terms into the scheme. Results compared favorably with available numerical and experimental data.
Thermal Analysis of AC Contactor Using Thermal Network Finite Difference Analysis Method
NASA Astrophysics Data System (ADS)
Niu, Chunping; Chen, Degui; Li, Xingwen; Geng, Yingsan
To predict the thermal behavior of switchgear quickly, the Thermal Network Finite Difference Analysis method (TNFDA) is adopted in thermal analysis of AC contactor in the paper. The thermal network model is built with nodes, thermal resistors and heat generators, and it is solved using finite difference method (FDM). The main circuit and the control system are connected by thermal resistors network, which solves the problem of multi-sources interaction in the application of TNFDA. The temperature of conducting wires is calculated according to the heat transfer process and the fundamental equations of thermal conduction. It provides a method to solve the problem of boundary conditions in applying the TNFDA. The comparison between the results of TNFDA and measurements shows the feasibility and practicability of the method.
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.
Solving moving interface problems using a higher order accurate finite difference scheme
NASA Astrophysics Data System (ADS)
Mittal, H. V. R.; Ray, Rajendra K.
2017-07-01
A new finite difference scheme is applied to solve partial differential equations in domains with discontinuities due to the presence of time dependent moving or deforming interfaces. This scheme is an extension of the finite difference idea developed for solving incompressible, steady stokes equations in discontinuous domains with fixed interfaces [1]. This new idea is applied at the irregular points at each time step in conjunction with the Crank-Nicolson (CN) implicit scheme and a recently developed Higher Order Compact (HOC) scheme at regular points. For validation, Stefan's problem is considered with a moving interface in one dimension. In two dimensions, heat equation is considered on a square domain with a circular interface whose radius is continuously changing with time. HOC scheme is found to produce better results and the order of accuracy is slightly better than that of the CN scheme. However, both the schemes show around second order accuracy and good agreement with the analytical solution.
Accurate finite-difference time-domain simulation of anisotropic media by subpixel smoothing.
Oskooi, Ardavan F; Kottke, Chris; Johnson, Steven G
2009-09-15
Finite-difference time-domain methods suffer from reduced accuracy when discretizing discontinuous materials. We previously showed that accuracy can be significantly improved by using subpixel smoothing of the isotropic dielectric function, but only if the smoothing scheme is properly designed. Using recent developments in perturbation theory that were applied to spectral methods, we extend this idea to anisotropic media and demonstrate that the generalized smoothing consistently reduces the errors and even attains second-order convergence with resolution.
Finite-difference models of ordinary differential equations - Influence of denominator functions
NASA Technical Reports Server (NTRS)
Mickens, Ronald E.; Smith, Arthur
1990-01-01
This paper discusses the influence on the solutions of finite-difference schemes of using a variety of denominator functions in the discrete modeling of the derivative for any ordinary differential equation. The results obtained are a consequence of using a generalized definition of the first derivative. A particular example of the linear decay equation is used to illustrate in detail the various solution possibilities that can occur.
Grid cell distortion and MODFLOW's integrated finite-difference numerical solution.
Romero, Dave M; Silver, Steven E
2006-01-01
The ground water flow model MODFLOW inherently implements a nongeneralized integrated finite-difference (IFD) numerical scheme. The IFD numerical scheme allows for construction of finite-difference model grids with curvilinear (piecewise linear) rows. The resulting grid comprises model cells in the shape of trapezoids and is distorted in comparison to a traditional MODFLOW finite-difference grid. A version of MODFLOW-88 (herein referred to as MODFLOW IFD) with the code adapted to make the one-dimensional DELR and DELC arrays two dimensional, so that equivalent conductance between distorted grid cells can be calculated, is described. MODFLOW IFD is used to inspect the sensitivity of the numerical head and velocity solutions to the level of distortion in trapezoidal grid cells within a converging radial flow domain. A test problem designed for the analysis implements a grid oriented such that flow is parallel to columns with converging widths. The sensitivity analysis demonstrates MODFLOW IFD's capacity to numerically derive a head solution and resulting intercell volumetric flow when the internal calculation of equivalent conductance accounts for the distortion of the grid cells. The sensitivity of the velocity solution to grid cell distortion indicates criteria for distorted grid design. In the radial flow test problem described, the numerical head solution is not sensitive to grid cell distortion. The accuracy of the velocity solution is sensitive to cell distortion with error <1% if the angle between the nonparallel sides of trapezoidal cells is <12.5 degrees. The error of the velocity solution is related to the degree to which the spatial discretization of a curve is approximated with piecewise linear segments. Curvilinear finite-difference grid construction adds versatility to spatial discretization of the flow domain. MODFLOW-88's inherent IFD numerical scheme and the test problem results imply that more recent versions of MODFLOW 2000, with minor
NASA Technical Reports Server (NTRS)
Sorenson, R. L.
1986-01-01
An elliptic grid-generation method for finite-difference computations about complex aerodynamic configurations is developed. A zonal approach is used, which involves first making a coarse global grid filling the entire physical domain and then subdividing regions of that grid to make the individual zone grids. The details of the grid-generation method are presented along with results of the present application, a wing-body configuration based on the F-16 fighter aircraft.
NASA Astrophysics Data System (ADS)
Korpusik, Adam
2017-02-01
We present a nonstandard finite difference scheme for a basic model of cellular immune response to viral infection. The main advantage of this approach is that it preserves the essential qualitative features of the original continuous model (non-negativity and boundedness of the solution, equilibria and their stability conditions), while being easy to implement. All of the qualitative features are preserved independently of the chosen step-size. Numerical simulations of our approach and comparison with other conventional simulation methods are presented.
ADI Finite Difference Discretization of the Heston-Hull-White PDE
NASA Astrophysics Data System (ADS)
Haentjens, Tinne; Hout, Karel in't.
2010-09-01
This paper concerns the efficient numerical solution of the time-dependent, three-dimensional Heston-Hull-White PDE for the fair prices of European call options. The numerical solution method described in this paper consists of a finite difference discretization on non-uniform spatial grids followed by an Alternating Direction Implicit scheme for the time discretization and extends the method recently proved effective by In't Hout & Foulon (2010) for the simpler, two-dimensional Heston PDE.
Finite Difference Methods for Time-Dependent, Linear Differential Algebraic Equations
1993-10-27
Time-Dependent, Linear Differential Algebraic Equations ’ BY PATRICK J. RABIER AND WERNER C. RHEINBOLDT 2 T r e n - sa le; its tot puba"- c. 2 ed...1993 Finite Difference Methods for Time-Dependent, I Linear Differential Algebraic Equations ’ BY PATRICK J. RABIER AND WERNER C. RHEINBOLDT2...LINEAR DIFFERENTIAL ALGEBRAIC EQUATIONS 1 BY PATRICK J. RABIER AND WERNER C. RHEINBOLDT 2 ABSTRACT. Recently the authors developed a global reduction
Double absorbing boundaries for finite-difference time-domain electromagnetics
NASA Astrophysics Data System (ADS)
LaGrone, John; Hagstrom, Thomas
2016-12-01
We describe the implementation of optimal local radiation boundary condition sequences for second order finite difference approximations to Maxwell's equations and the scalar wave equation using the double absorbing boundary formulation. Numerical experiments are presented which demonstrate that the design accuracy of the boundary conditions is achieved and, for comparable effort, exceeds that of a convolution perfectly matched layer with reasonably chosen parameters. An advantage of the proposed approach is that parameters can be chosen using an accurate a priori error bound.
Numerical solution of multiparameter spectral problems by high order finite different schemes
NASA Astrophysics Data System (ADS)
Amodio, Pierluigi; Settanni, Giuseppina
2016-10-01
We report on the progress achieved in the numerical simulation of self-adjoint multiparameter spectral problems for ordinary differential equations. We describe how to obtain a discrete problem by means of High Order Finite Difference Schemes and discuss its numerical solution. Based on this approach, we also define a recursive algorithm to compute approximations of the parameters by means of the solution of a set of problems converging to the original one.
Transport and dispersion of pollutants in surface impoundments: a finite difference model
Yeh, G.T.
1980-07-01
A surface impoundment model by finite-difference (SIMFD) has been developed. SIMFD computes the flow rate, velocity field, and the concentration distribution of pollutants in surface impoundments with any number of islands located within the region of interest. Theoretical derivations and numerical algorithm are described in detail. Instructions for the application of SIMFD and listings of the FORTRAN IV source program are provided. Two sample problems are given to illustrate the application and validity of the model.
Gabran, S R I; Saad, J H; Salama, M M A; Mansour, R R
2009-01-01
This paper demonstrates the electromagnetic modeling and simulation of an implanted Medtronic deep brain stimulation (DBS) electrode using finite difference time domain (FDTD). The model is developed using Empire XCcel and represents the electrode surrounded with brain tissue assuming homogenous and isotropic medium. The model is created to study the parameters influencing the electric field distribution within the tissue in order to provide reference and benchmarking data for DBS and intra-cortical electrode development.
Spiegel, R.J.; Fatmi, M.B.; Ward, T.R.
1987-01-01
The rate of the electromagnetic energy deposition and the resultant thermoregulatory response of a block model of a squirrel monkey exposed to plane-wave fields at 350 MHz were calculated using a finite-difference procedure. Noninvasive temperature measurements in live squirrel monkeys under similar exposure conditions were obtained using Vitek probes. Calculations exhibit reasonable correlation with the measured data, especially for the rise in colonic temperature.
NASA Technical Reports Server (NTRS)
Dey, C.; Dey, S. K.
1983-01-01
An explicit finite difference scheme consisting of a predictor and a corrector has been developed and applied to solve some hyperbolic partial differential equations (PDEs). The corrector is a convex-type function which is applied at each time level and at each mesh point. It consists of a parameter which may be estimated such that for larger time steps the algorithm should remain stable and generate a fast speed of convergence to the steady-state solution. Some examples have been given.
Properties of finite difference models of non-linear conservative oscillators
NASA Technical Reports Server (NTRS)
Mickens, R. E.
1988-01-01
Finite-difference (FD) approaches to the numerical solution of the differential equations describing the motion of a nonlinear conservative oscillator are investigated analytically. A generalized formulation of the Duffing and modified Duffing equations is derived and analyzed using several FD techniques, and it is concluded that, although it is always possible to contstruct FD models of conservative oscillators which are themselves conservative, caution is required to avoid numerical solutions which do not accurately reflect the properties of the original equation.
Finite-difference model for 3-D flow in bays and estuaries
Smith, Peter E.; Larock, Bruce E.; ,
1993-01-01
This paper describes a semi-implicit finite-difference model for the numerical solution of three-dimensional flow in bays and estuaries. The model treats the gravity wave and vertical diffusion terms in the governing equations implicitly, and other terms explicitly. The model achieves essentially second-order accurate and stable solutions in strongly nonlinear problems by using a three-time-level leapfrog-trapezoidal scheme for the time integration.
NASA Technical Reports Server (NTRS)
Abramopoulos, Frank
1988-01-01
The conditions under which finite difference schemes for the shallow water equations can conserve both total energy and potential enstrophy are considered. A method of deriving such schemes using operator formalism is developed. Several such schemes are derived for the A-, B- and C-grids. The derived schemes include second-order schemes and pseudo-fourth-order schemes. The simplest B-grid pseudo-fourth-order schemes are presented.
Simulation of realistic rotor blade-vortex interactions using a finite-difference technique
NASA Technical Reports Server (NTRS)
Hassan, Ahmed A.; Charles, Bruce D.
1989-01-01
A numerical finite-difference code has been used to predict helicopter blade loads during realistic self-generated three-dimensional blade-vortex interactions. The velocity field is determined via a nonlinear superposition of the rotor flowfield. Data obtained from a lifting-line helicopter/rotor trim code are used to determine the instantaneous position of the interaction vortex elements with respect to the blade. Data obtained for three rotor advance ratios show a reasonable correlation with wind tunnel data.
Gradient Approximation on Uniform Meshes by Finite Differences and Cubic Spline Interpolation
NASA Astrophysics Data System (ADS)
Sablonnière, P.
For the approximation of gradients from data values at vertices of a uniform grid, we compare two methods based on cubic spline interpolation with a classical method based on finite differences. For univariate cubic splines, we use the so-called de Boor’s Not a Knot property and a new method giving pretty good slopes. Then these methods are used on parallels to the axes for estimating gradients on bivariate grids. They are illustrated by several numerical examples.
Double absorbing boundaries for finite-difference time-domain electromagnetics
LaGrone, John Hagstrom, Thomas
2016-12-01
We describe the implementation of optimal local radiation boundary condition sequences for second order finite difference approximations to Maxwell's equations and the scalar wave equation using the double absorbing boundary formulation. Numerical experiments are presented which demonstrate that the design accuracy of the boundary conditions is achieved and, for comparable effort, exceeds that of a convolution perfectly matched layer with reasonably chosen parameters. An advantage of the proposed approach is that parameters can be chosen using an accurate a priori error bound.
NASA Technical Reports Server (NTRS)
Abramopoulos, Frank
1988-01-01
The conditions under which finite difference schemes for the shallow water equations can conserve both total energy and potential enstrophy are considered. A method of deriving such schemes using operator formalism is developed. Several such schemes are derived for the A-, B- and C-grids. The derived schemes include second-order schemes and pseudo-fourth-order schemes. The simplest B-grid pseudo-fourth-order schemes are presented.
NASA Technical Reports Server (NTRS)
Lansing, Faiza S.; Rascoe, Daniel L.
1993-01-01
This paper presents a modified Finite-Difference Time-Domain (FDTD) technique using a generalized conformed orthogonal grid. The use of the Conformed Orthogonal Grid, Finite Difference Time Domain (GFDTD) enables the designer to match all the circuit dimensions, hence eliminating a major source o error in the analysis.
Physicists make a difference in space cryogenics
NASA Technical Reports Server (NTRS)
Petrac, D.
1990-01-01
An evaluation is made of space cryogenics technology development challenges in whose treatment the unique abilities of physicists have been notably valuable. In addition to such broad concerns as the quantum properties of superfluid He, the basic laws of thermodynamics and entropy, and quantization of such fields as those of phonons and magnons, productive efforts have been made by physicists in the basic principles of SQUIDs, magnetic shielding due to the Meissner effect in superconductors, adiabatic demagnetization, and dilution cooling of He-3 and He-4. These contributions will be of fundamental importance to the IR Telescope for Space and Advanced X-ray Astronomical Facility, which are currently under development.
Finite-Difference Lattice Boltzmann Scheme for High-Speed Compressible Flow: Two-Dimensional Case
NASA Astrophysics Data System (ADS)
Gan, Yan-Biao; Xu, Ai-Guo; Zhang, Guang-Cai; Zhang, Ping; Zhang, Lei; Li, Ying-Jun
2008-07-01
Lattice Boltzmann (LB) modeling of high-speed compressible flows has long been attempted by various authors. One common weakness of most of previous models is the instability problem when the Mach number of the flow is large. In this paper we present a finite-difference LB model, which works for flows with flexible ratios of specific heats and a wide range of Mach number, from 0 to 30 or higher. Besides the discrete-velocity-model by Watari [Physica A 382 (2007) 502], a modified Lax Wendroff finite difference scheme and an artificial viscosity are introduced. The combination of the finite-difference scheme and the adding of artificial viscosity must find a balance of numerical stability versus accuracy. The proposed model is validated by recovering results of some well-known benchmark tests: shock tubes and shock reflections. The new model may be used to track shock waves and/or to study the non-equilibrium procedure in the transition between the regular and Mach reflections of shock waves, etc.
NASA Astrophysics Data System (ADS)
Kaus, Boris; Popov, Anton; Püsök, Adina
2014-05-01
In order to solve high-resolution 3D problems in computational geodynamics it is crucial to use multigrid solvers in combination with parallel computers. A number of approaches are currently in use in the community, which can broadly be divided into coupled and decoupled approaches. In the decoupled approach, an algebraic or geometric multigrid method is used as a preconditioner for the velocity equations only while an iterative approach such as Schur complement reduction used to solve the outer velocity-pressure equations. In the coupled approach, on the other hand, a multigrid approach is applied to both the velocity and pressure equations. The coupled multigrid approaches are typically employed in combination with staggered finite difference discretizations, whereas the decoupled approach is the method of choice in many of the existing finite element codes. Yet, it is unclear whether there are differences in speed between the two approaches, and if so, how this depends on the initial guess. Here, we implemented both approaches in combination with a staggered finite difference discretization and test the robustness of the two approaches with respect to large jumps in viscosity contrast, as well as their computational efficiency as a function of the initial guess. Acknowledgements. Funding was provided by the European Research Council under the European Community's Seventh Framework Program (FP7/2007-2013) / ERC Grant agreement #258830. Numerical computations have been performed on JUQUEEN of the Jülich high-performance computing center.
Gallego, Rafael; Castro, Mario; López, Juan M
2007-11-01
We present a comparison between finite differences schemes and a pseudospectral method applied to the numerical integration of stochastic partial differential equations that model surface growth. We have studied, in 1+1 dimensions, the Kardar, Parisi, and Zhang model (KPZ) and the Lai, Das Sarma, and Villain model (LDV). The pseudospectral method appears to be the most stable for a given time step for both models. This means that the time up to which we can follow the temporal evolution of a given system is larger for the pseudospectral method. Moreover, for the KPZ model, a pseudospectral scheme gives results closer to the predictions of the continuum model than those obtained through finite difference methods. On the other hand, some numerical instabilities appearing with finite difference methods for the LDV model are absent when a pseudospectral integration is performed. These numerical instabilities give rise to an approximate multiscaling observed in earlier numerical simulations. With the pseudospectral approach no multiscaling is seen in agreement with the continuum model.
Rayleigh Wave Numerical Dispersion in a 3D Finite-Difference Algorithm
NASA Astrophysics Data System (ADS)
Preston, L. A.; Aldridge, D. F.
2010-12-01
A Rayleigh wave propagates laterally without dispersion in the vicinity of the plane stress-free surface of a homogeneous and isotropic elastic halfspace. The phase speed is independent of frequency and depends only on the Poisson ratio of the medium. However, after temporal and spatial discretization, a Rayleigh wave simulated by a 3D staggered-grid finite-difference (FD) seismic wave propagation algorithm suffers from frequency- and direction-dependent numerical dispersion. The magnitude of this dispersion depends critically on FD algorithm implementation details. Nevertheless, proper gridding can control numerical dispersion to within an acceptable level, leading to accurate Rayleigh wave simulations. Many investigators have derived dispersion relations appropriate for body wave propagation by various FD algorithms. However, the situation for surface waves is less well-studied. We have devised a numerical search procedure to estimate Rayleigh phase speed and group speed curves for 3D O(2,2) and O(2,4) staggered-grid FD algorithms. In contrast with the continuous time-space situation (where phase speed is obtained by extracting the appropriate root of the Rayleigh cubic), we cannot develop a closed-form mathematical formula governing the phase speed. Rather, we numerically seek the particular phase speed that leads to a solution of the discrete wave propagation equations, while holding medium properties, frequency, horizontal propagation direction, and gridding intervals fixed. Group speed is then obtained by numerically differentiating the phase speed with respect to frequency. The problem is formulated for an explicit stress-free surface positioned at two different levels within the staggered spatial grid. Additionally, an interesting variant involving zero-valued medium properties above the surface is addressed. We refer to the latter as an implicit free surface. Our preliminary conclusion is that an explicit free surface, implemented with O(4) spatial FD
3D Staggered-Grid Finite-Difference Simulation of Acoustic Waves in Turbulent Moving Media
NASA Astrophysics Data System (ADS)
Symons, N. P.; Aldridge, D. F.; Marlin, D.; Wilson, D. K.; Sullivan, P.; Ostashev, V.
2003-12-01
Acoustic wave propagation in a three-dimensional heterogeneous moving atmosphere is accurately simulated with a numerical algorithm recently developed under the DOD Common High Performance Computing Software Support Initiative (CHSSI). Sound waves within such a dynamic environment are mathematically described by a set of four, coupled, first-order partial differential equations governing small-amplitude fluctuations in pressure and particle velocity. The system is rigorously derived from fundamental principles of continuum mechanics, ideal-fluid constitutive relations, and reasonable assumptions that the ambient atmospheric motion is adiabatic and divergence-free. An explicit, time-domain, finite-difference (FD) numerical scheme is used to solve the system for both pressure and particle velocity wavefields. The atmosphere is characterized by 3D gridded models of sound speed, mass density, and the three components of the wind velocity vector. Dependent variables are stored on staggered spatial and temporal grids, and centered FD operators possess 2nd-order and 4th-order space/time accuracy. Accurate sound wave simulation is achieved provided grid intervals are chosen appropriately. The gridding must be fine enough to reduce numerical dispersion artifacts to an acceptable level and maintain stability. The algorithm is designed to execute on parallel computational platforms by utilizing a spatial domain-decomposition strategy. Currently, the algorithm has been validated on four different computational platforms, and parallel scalability of approximately 85% has been demonstrated. Comparisons with analytic solutions for uniform and vertically stratified wind models indicate that the FD algorithm generates accurate results with either a vanishing pressure or vanishing vertical-particle velocity boundary condition. Simulations are performed using a kinematic turbulence wind profile developed with the quasi-wavelet method. In addition, preliminary results are presented
NASA Astrophysics Data System (ADS)
Ghosh, Swarnava; Suryanarayana, Phanish
2017-03-01
As the first component of SPARC (Simulation Package for Ab-initio Real-space Calculations), we present an accurate and efficient finite-difference formulation and parallel implementation of Density Functional Theory (DFT) for isolated clusters. Specifically, utilizing a local reformulation of the electrostatics, the Chebyshev polynomial filtered self-consistent field iteration, and a reformulation of the non-local component of the force, we develop a framework using the finite-difference representation that enables the efficient evaluation of energies and atomic forces to within the desired accuracies in DFT. Through selected examples consisting of a variety of elements, we demonstrate that SPARC obtains exponential convergence in energy and forces with domain size; systematic convergence in the energy and forces with mesh-size to reference plane-wave result at comparably high rates; forces that are consistent with the energy, both free from any noticeable 'egg-box' effect; and accurate ground-state properties including equilibrium geometries and vibrational spectra. In addition, for systems consisting up to thousands of electrons, SPARC displays weak and strong parallel scaling behavior that is similar to well-established and optimized plane-wave implementations, but with a significantly reduced prefactor. Overall, SPARC represents an attractive alternative to plane-wave codes for practical DFT simulations of isolated clusters.
NASA Astrophysics Data System (ADS)
Ghosh, Swarnava; Suryanarayana, Phanish
2017-07-01
As the second component of SPARC (Simulation Package for Ab-initio Real-space Calculations), we present an accurate and efficient finite-difference formulation and parallel implementation of Density Functional Theory (DFT) for extended systems. Specifically, employing a local formulation of the electrostatics, the Chebyshev polynomial filtered self-consistent field iteration, and a reformulation of the non-local force component, we develop a finite-difference framework wherein both the energy and atomic forces can be efficiently calculated to within desired accuracies in DFT. We demonstrate using a wide variety of materials systems that SPARC achieves high convergence rates in energy and forces with respect to spatial discretization to reference plane-wave result; exponential convergence in energies and forces with respect to vacuum size for slabs and wires; energies and forces that are consistent and display negligible 'egg-box' effect; accurate properties of crystals, slabs, and wires; and negligible drift in molecular dynamics simulations. We also demonstrate that the weak and strong scaling behavior of SPARC is similar to well-established and optimized plane-wave implementations for systems consisting up to thousands of electrons, but with a significantly reduced prefactor. Overall, SPARC represents an attractive alternative to plane-wave codes for performing DFT simulations of extended systems.
Finite difference time domain analysis of microwave ferrite devices and mobile antenna systems
NASA Astrophysics Data System (ADS)
Yildirim, Bahadir Suleyman
This dissertation presents analysis and design of shielded mobile antenna systems and microwave ferrite devices using a finite-difference time-domain method. Novel shielded antenna structures suitable for cellular communications have been analyzed and designed with emphasize on reducing excessive radiated energy absorbed in user's head and hand, while keeping the antenna performance at its peak in the presence of user. These novel antennas include a magnetically shielded antenna, a dual-resonance shielded antenna and, a shorted and truncated microstrip antenna. The effect of magnetic coating on the performance of a shielded monopole antenna is studied extensively. A parametric study is performed to analyze the dual-resonance phenomenon observed in the dual-resonance shielded antenna, optimize the antenna design within the cellular communications band, and improve the antenna performance. Input impedance, near and far fields of the dual-resonance shielded antenna are calculated using the finite-difference time-domain method. Experimental validation is also presented. In addition, performance of a shorted and truncated microstrip antenna has been investigated over a wide range of substrate parameters and dimensions. Objectives of the research work also include development of a finite-difference time-domain technique to accurately model magnetically anisotropic media, including the effect of non-uniform magnetization within the finite-size ferrite material due to demagnetizing fields. A slow wave thin film isolator and a stripline disc junction circulator are analyzed. An extensive parametric study calculates wide-band frequency-dependent parameters of these devices for various device dimensions and material parameters. Finally, a ferrite-filled stripline configuration is analyzed to study the non- linear behaviour of ferrite by introducing a modified damping factor.
Same but Different: Space, Time and Narrative
ERIC Educational Resources Information Center
Bansel, Peter
2013-01-01
In this paper, I give an account of the ways in which narratives and identities change over space and time. I give an account of a mobile and changing human subject, one who does not simply express or represent her- or himself through narrative, but is constructed and reconstructed through narrative. I draw on Paul Ricoeur's concepts of "narrative…
Same but Different: Space, Time and Narrative
ERIC Educational Resources Information Center
Bansel, Peter
2013-01-01
In this paper, I give an account of the ways in which narratives and identities change over space and time. I give an account of a mobile and changing human subject, one who does not simply express or represent her- or himself through narrative, but is constructed and reconstructed through narrative. I draw on Paul Ricoeur's concepts of "narrative…
Modelling migration in multilayer systems by a finite difference method: the spherical symmetry case
NASA Astrophysics Data System (ADS)
Hojbotǎ, C. I.; Toşa, V.; Mercea, P. V.
2013-08-01
We present a numerical model based on finite differences to solve the problem of chemical impurity migration within a multilayer spherical system. Migration here means diffusion of chemical species in conditions of concentration partitioning at layer interfaces due to different solubilities of the migrant in different layers. We detail here the numerical model and discuss the results of its implementation. To validate the method we compare it with cases where an analytic solution exists. We also present an application of our model to a practical problem in which we compute the migration of caprolactam from the packaging multilayer foil into the food.
NASA Astrophysics Data System (ADS)
Wang, Cheng; Dong, XinZhuang; Shu, Chi-Wang
2015-10-01
For numerical simulation of detonation, computational cost using uniform meshes is large due to the vast separation in both time and space scales. Adaptive mesh refinement (AMR) is advantageous for problems with vastly different scales. This paper aims to propose an AMR method with high order accuracy for numerical investigation of multi-dimensional detonation. A well-designed AMR method based on finite difference weighted essentially non-oscillatory (WENO) scheme, named as AMR&WENO is proposed. A new cell-based data structure is used to organize the adaptive meshes. The new data structure makes it possible for cells to communicate with each other quickly and easily. In order to develop an AMR method with high order accuracy, high order prolongations in both space and time are utilized in the data prolongation procedure. Based on the message passing interface (MPI) platform, we have developed a workload balancing parallel AMR&WENO code using the Hilbert space-filling curve algorithm. Our numerical experiments with detonation simulations indicate that the AMR&WENO is accurate and has a high resolution. Moreover, we evaluate and compare the performance of the uniform mesh WENO scheme and the parallel AMR&WENO method. The comparison results provide us further insight into the high performance of the parallel AMR&WENO method.
Mimetic finite difference method for the stokes problem on polygonal meshes
Lipnikov, K; Beirao Da Veiga, L; Gyrya, V; Manzini, G
2009-01-01
Various approaches to extend the finite element methods to non-traditional elements (pyramids, polyhedra, etc.) have been developed over the last decade. Building of basis functions for such elements is a challenging task and may require extensive geometry analysis. The mimetic finite difference (MFD) method has many similarities with low-order finite element methods. Both methods try to preserve fundamental properties of physical and mathematical models. The essential difference is that the MFD method uses only the surface representation of discrete unknowns to build stiffness and mass matrices. Since no extension inside the mesh element is required, practical implementation of the MFD method is simple for polygonal meshes that may include degenerate and non-convex elements. In this article, we develop a MFD method for the Stokes problem on arbitrary polygonal meshes. The method is constructed for tensor coefficients, which will allow to apply it to the linear elasticity problem. The numerical experiments show the second-order convergence for the velocity variable and the first-order for the pressure.
Mimetic finite difference method for the Stokes problem on polygonal meshes
NASA Astrophysics Data System (ADS)
Beirão da Veiga, L.; Gyrya, V.; Lipnikov, K.; Manzini, G.
2009-10-01
Various approaches to extend finite element methods to non-traditional elements (general polygons, pyramids, polyhedra, etc.) have been developed over the last decade. The construction of basis functions for such elements is a challenging task and may require extensive geometrical analysis. The mimetic finite difference (MFD) method works on general polygonal meshes and has many similarities with low-order finite element methods. Both schemes try to preserve the fundamental properties of the underlying physical and mathematical models. The essential difference between the two schemes is that the MFD method uses only the surface representation of discrete unknowns to build the stiffness and mass matrices. Since no extension of basis functions inside the mesh elements is required, practical implementation of the MFD method is simple for polygonal meshes that may include degenerate and non-convex elements. In this article, we present a new MFD method for the Stokes problem on arbitrary polygonal meshes and analyze its stability. The method is developed for the general case of tensor coefficients, which allows us to apply it to a linear elasticity problem, as well. Numerical experiments show, for the velocity variable, second-order convergence in a discrete L2 norm and first-order convergence in a discrete H1 norm. For the pressure variable, first-order convergence is shown in the L2 norm.
Experiments with explicit filtering for LES using a finite-difference method
NASA Technical Reports Server (NTRS)
Lund, T. S.; Kaltenbach, H. J.
1995-01-01
The equations for large-eddy simulation (LES) are derived formally by applying a spatial filter to the Navier-Stokes equations. The filter width as well as the details of the filter shape are free parameters in LES, and these can be used both to control the effective resolution of the simulation and to establish the relative importance of different portions of the resolved spectrum. An analogous, but less well justified, approach to filtering is more or less universally used in conjunction with LES using finite-difference methods. In this approach, the finite support provided by the computational mesh as well as the wavenumber-dependent truncation errors associated with the finite-difference operators are assumed to define the filter operation. This approach has the advantage that it is also 'automatic' in the sense that no explicit filtering: operations need to be performed. While it is certainly convenient to avoid the explicit filtering operation, there are some practical considerations associated with finite-difference methods that favor the use of an explicit filter. Foremost among these considerations is the issue of truncation error. All finite-difference approximations have an associated truncation error that increases with increasing wavenumber. These errors can be quite severe for the smallest resolved scales, and these errors will interfere with the dynamics of the small eddies if no corrective action is taken. Years of experience at CTR with a second-order finite-difference scheme for high Reynolds number LES has repeatedly indicated that truncation errors must be minimized in order to obtain acceptable simulation results. While the potential advantages of explicit filtering are rather clear, there is a significant cost associated with its implementation. In particular, explicit filtering reduces the effective resolution of the simulation compared with that afforded by the mesh. The resolution requirements for LES are usually set by the need to capture
Optimizing for minimum weight when two different finite element models and analyses are required
NASA Technical Reports Server (NTRS)
Hall, Jeffrey C.
1989-01-01
The Finite Element Structural Optimization Program's (FESOP) ability to perform minimum weight optimization using two different finite element analyses and models is discussed. FESOP uses the ADS optimizer developed by Dr. Garret Vanderplaats to solve the nonlinear constrained optimization problem. The design optimization problem requires a response spectrum analysis and model to evaluate the stress and displacement constraints. However, the problem needs a frequency analysis and model to calculate the natural frequencies used to evaluate the frequency range constraints. The results of both the successful and unsuccessful approaches used to solve this difficult weight minimization problem are summarized. The results show that no one ADS optimization algorithm worked in all cases. However, the Sequential Convex Programming and Modified Method of Feasible Directions algorithms were the most successful.
3D finite-difference modeling algorithm and anomaly features of ZTEM
NASA Astrophysics Data System (ADS)
Wang, Tao; Tan, Han-Dong; Li, Zhi-Qiang; Wang, Kun-Peng; Hu, Zhi-Ming; Zhang, Xing-Dong
2016-09-01
The Z-Axis tipper electromagnetic (ZTEM) technique is based on a frequency-domain airborne electromagnetic system that measures the natural magnetic field. A survey area was divided into several blocks by using the Maxwell's equations, and the magnetic components at the center of each edge of the grid cell are evaluated by applying the staggered-grid finite-difference method. The tipper and its divergence are derived to complete the 3D ZTEM forward modeling algorithm. A synthetic model is then used to compare the responses with those of 2D finite-element forward modeling to verify the accuracy of the algorithm. ZTEM offers high horizontal resolution to both simple and complex distributions of conductivity. This work is the theoretical foundation for the interpretation of ZTEM data and the study of 3D ZTEM inversion.
NASA Technical Reports Server (NTRS)
Kishoni, Doron; Taasan, Shlomo
1994-01-01
Solution of the wave equation using techniques such as finite difference or finite element methods can model elastic wave propagation in solids. This requires mapping the physical geometry into a computational domain whose size is governed by the size of the physical domain of interest and by the required resolution. This computational domain, in turn, dictates the computer memory requirements as well as the calculation time. Quite often, the physical region of interest is only a part of the whole physical body, and does not necessarily include all the physical boundaries. Reduction of the calculation domain requires positioning an artificial boundary or region where a physical boundary does not exist. It is important however that such a boundary, or region, will not affect the internal domain, i.e., it should not cause reflections that propagate back into the material. This paper concentrates on the issue of constructing such a boundary region.
NASA Astrophysics Data System (ADS)
Wang, Ying; Zhou, Hui; Yuan, Sanyi; Ye, Yameng
2017-01-01
The fourth order accuracy finite difference scheme is known advantageous in reducing memory and improving efficiency. Summation-by-parts finite difference operator is a natural way for wavefield simulation in complicated domains containing surface topography and irregular interfaces. The application of summation-by-parts method guarantees the stability of numerical approximation for heterogeneous media on curvilinear grids. This paper extends the second order summation-by-parts finite difference method to the fourth order case for the discretization of acoustic wave equation and perfect matched layer in boundary-conforming grids. In particular, the implementation of the fourth order method for wavefield simulation and reverse time migration in complicated domains can significantly improve the efficiency and decrease the storage. The elliptic method is applied for boundary-conforming grid generation in complicated domains. Under such grids, the two-dimensional acoustic wave equation in second order displacement formulation is compactly reformulated for forward modeling and reverse time migration, and the symmetric and compact form of perfectly matched layers expressed in a curvilinear coordinate system are applied to suppress artificial reflections. The discretizations of the acoustic wave equation and perfectly matched layer formula are fourth and second order accuracy in space and time respectively, where the spatial discretization satisfies the principle of summation-by-parts and is stable. Numerical experiments are presented to compare the accuracy of the second with fourth order summation-by-parts finite difference methods and to evaluate the efficiency of reverse time migration by using these two methods. As well, comparisons are performed between the fourth order accuracy summation-by-parts finite difference method and central finite difference method to illustrate the stability superiority of summation-by-parts operators.
Vanrumste, B; Van Hoey, G; Van de Walle, R; D'Havé, M R; Lemahieu, I A; Boon, P A
2001-01-01
The performance of the finite difference reciprocity method (FDRM) to solve the inverse problem in EEG dipole source analysis is investigated in the analytically solvable three-shell spherical head model for a large set of test dipoles. The location error for a grid with 2 mm and 3 mm node spacing is in general, not larger than twice the internode distance, hence 4 mm and 6 mm, respectively. Increasing the number of scalp electrodes from 27 to 44 only marginally improves the location error. The orientation error is always smaller than 4 degrees for all the test dipoles considered. We have also compared the sensitivity to noise using FDRM in EEG dipole source analysis with the sensitivity to noise using the analytical expression for the forward problem. FDRM is not more sensitive to noise than the method using the analytical expression.
Son, Sang-Kil
2011-03-01
We introduce a new numerical grid-based method on unstructured grids in the three-dimensional real-space to investigate the electronic structure of polyatomic molecules. The Voronoi-cell finite difference (VFD) method realizes a discrete Laplacian operator based on Voronoi cells and their natural neighbors, featuring high adaptivity and simplicity. To resolve multicenter Coulomb singularity in all-electron calculations of polyatomic molecules, this method utilizes highly adaptive molecular grids which consist of spherical atomic grids. It provides accurate and efficient solutions for the Schroedinger equation and the Poisson equation with the all-electron Coulomb potentials regardless of the coordinate system and the molecular symmetry. For numerical examples, we assess accuracy of the VFD method for electronic structures of one-electron polyatomic systems, and apply the method to the density-functional theory for many-electron polyatomic molecules.
NASA Technical Reports Server (NTRS)
Harten, A.; Tal-Ezer, H.
1981-01-01
An implicit finite difference method of fourth order accuracy in space and time is introduced for the numerical solution of one-dimensional systems of hyperbolic conservation laws. The basic form of the method is a two-level scheme which is unconditionally stable and nondissipative. The scheme uses only three mesh points at level t and three mesh points at level t + delta t. The dissipative version of the basic method given is conditionally stable under the CFL (Courant-Friedrichs-Lewy) condition. This version is particularly useful for the numerical solution of problems with strong but nonstiff dynamic features, where the CFL restriction is reasonable on accuracy grounds. Numerical results are provided to illustrate properties of the proposed method.
NASA Astrophysics Data System (ADS)
Liu, Q.; Liu, F.; Turner, I.; Anh, V.
2007-03-01
In this paper we present a random walk model for approximating a Lévy-Feller advection-dispersion process, governed by the Lévy-Feller advection-dispersion differential equation (LFADE). We show that the random walk model converges to LFADE by use of a properly scaled transition to vanishing space and time steps. We propose an explicit finite difference approximation (EFDA) for LFADE, resulting from the Grünwald-Letnikov discretization of fractional derivatives. As a result of the interpretation of the random walk model, the stability and convergence of EFDA for LFADE in a bounded domain are discussed. Finally, some numerical examples are presented to show the application of the present technique.
NASA Technical Reports Server (NTRS)
Noor, A. K.; Stephens, W. B.
1973-01-01
Several finite difference schemes are applied to the stress and free vibration analysis of homogeneous isotropic and layered orthotropic shells of revolution. The study is based on a form of the Sanders-Budiansky first-approximation linear shell theory modified such that the effects of shear deformation and rotary inertia are included. A Fourier approach is used in which all the shell stress resultants and displacements are expanded in a Fourier series in the circumferential direction, and the governing equations reduce to ordinary differential equations in the meridional direction. While primary attention is given to finite difference schemes used in conjunction with first order differential equation formulation, comparison is made with finite difference schemes used with other formulations. These finite difference discretization models are compared with respect to simplicity of application, convergence characteristics, and computational efficiency. Numerical studies are presented for the effects of variations in shell geometry and lamination parameters on the accuracy and convergence of the solutions obtained by the different finite difference schemes. On the basis of the present study it is shown that the mixed finite difference scheme based on the first order differential equation formulation and two interlacing grids for the different fundamental unknowns combines a number of advantages over other finite difference schemes previously reported in the literature.
Dispersion and stability analysis for a finite difference beam propagation method.
de-Oliva-Rubio, J; Molina-Fernández, I; Godoy-Rubio, R
2008-06-09
Applying continuous and discrete transformation techniques, new analytical expressions to calculate dispersion and stability of a Runge- Kutta Finite Difference Beam Propagation Method (RK-FDBPM) are obtained. These expressions give immediate insight about the discretization errors introduced by the numerical method in the plane-wave spectrum domain. From these expressions a novel strategy to adequately set the mesh steps sizes of the RK-FDBPM is presented. Assessment of the method is performed by studying the propagation in several linear and nonlinear photonic devices for different spatial discretizations.
Vincenti, H.; Vay, J. -L.
2015-11-22
Due to discretization effects and truncation to finite domains, many electromagnetic simulations present non-physical modifications of Maxwell's equations in space that may generate spurious signals affecting the overall accuracy of the result. Such modifications for instance occur when Perfectly Matched Layers (PMLs) are used at simulation domain boundaries to simulate open media. Another example is the use of arbitrary order Maxwell solver with domain decomposition technique that may under some condition involve stencil truncations at subdomain boundaries, resulting in small spurious errors that do eventually build up. In each case, a careful evaluation of the characteristics and magnitude of themore » errors resulting from these approximations, and their impact at any frequency and angle, requires detailed analytical and numerical studies. To this end, we present a general analytical approach that enables the evaluation of numerical discretization errors of fully three-dimensional arbitrary order finite-difference Maxwell solver, with arbitrary modification of the local stencil in the simulation domain. The analytical model is validated against simulations of domain decomposition technique and PMLs, when these are used with very high-order Maxwell solver, as well as in the infinite order limit of pseudo-spectral solvers. Results confirm that the new analytical approach enables exact predictions in each case. It also confirms that the domain decomposition technique can be used with very high-order Maxwell solver and a reasonably low number of guard cells with negligible effects on the whole accuracy of the simulation.« less
Vincenti, H.; Vay, J. -L.
2015-11-22
Due to discretization effects and truncation to finite domains, many electromagnetic simulations present non-physical modifications of Maxwell's equations in space that may generate spurious signals affecting the overall accuracy of the result. Such modifications for instance occur when Perfectly Matched Layers (PMLs) are used at simulation domain boundaries to simulate open media. Another example is the use of arbitrary order Maxwell solver with domain decomposition technique that may under some condition involve stencil truncations at subdomain boundaries, resulting in small spurious errors that do eventually build up. In each case, a careful evaluation of the characteristics and magnitude of the errors resulting from these approximations, and their impact at any frequency and angle, requires detailed analytical and numerical studies. To this end, we present a general analytical approach that enables the evaluation of numerical discretization errors of fully three-dimensional arbitrary order finite-difference Maxwell solver, with arbitrary modification of the local stencil in the simulation domain. The analytical model is validated against simulations of domain decomposition technique and PMLs, when these are used with very high-order Maxwell solver, as well as in the infinite order limit of pseudo-spectral solvers. Results confirm that the new analytical approach enables exact predictions in each case. It also confirms that the domain decomposition technique can be used with very high-order Maxwell solver and a reasonably low number of guard cells with negligible effects on the whole accuracy of the simulation.
An overlapped grid method for multigrid, finite volume/difference flow solvers: MaGGiE
NASA Technical Reports Server (NTRS)
Baysal, Oktay; Lessard, Victor R.
1990-01-01
The objective is to develop a domain decomposition method via overlapping/embedding the component grids, which is to be used by upwind, multi-grid, finite volume solution algorithms. A computer code, given the name MaGGiE (Multi-Geometry Grid Embedder) is developed to meet this objective. MaGGiE takes independently generated component grids as input, and automatically constructs the composite mesh and interpolation data, which can be used by the finite volume solution methods with or without multigrid convergence acceleration. Six demonstrative examples showing various aspects of the overlap technique are presented and discussed. These cases are used for developing the procedure for overlapping grids of different topologies, and to evaluate the grid connection and interpolation data for finite volume calculations on a composite mesh. Time fluxes are transferred between mesh interfaces using a trilinear interpolation procedure. Conservation losses are minimal at the interfaces using this method. The multi-grid solution algorithm, using the coaser grid connections, improves the convergence time history as compared to the solution on composite mesh without multi-gridding.
High-order finite difference methods for earthquake rupture dynamics in complex geometries
NASA Astrophysics Data System (ADS)
O'Reilly, O.; Kozdon, J. E.; Dunham, E. M.; Nordström, J.
2010-12-01
In this work we continue our development of high-order summation-by-parts (SBP) finite difference methods for earthquake rupture dynamics. SBP methods use centered spatial differences in the interior and one-sided differences near the boundary. The transition to one-sided differences is done in a particular manner that permits one to provably maintain stability and accuracy. In many methods the boundary conditions are strongly enforced by modifying the difference operator at the boundary so that the solution there exactly satisfies the boundary condition. Though conceptually straightforward, this approach can introduce instabilities. In contrast, when boundary conditions are enforced weakly by adding a penalty term to the spatial discretization, it is possible to prove that the method is strictly stable, dissipating energy slightly faster than the continuous problem (with the additional dissipation vanishing under grid refinement). Another benefit of SBP operators is their built-in inner product which, if correctly constructed, can be interpreted as a quadrature operator. Thus, important integrated quantities such as the total mechanical energy in the system, the energy dissipation rate along faults, and the radiated energy flux through exterior boundaries can be rigorously calculated. These numerically integrated quantities converge to their true values with the same order of accuracy as the difference approximation. Though standard SBP methods are based on uniform Cartesian grids, it is possible to use the methods for problems with nonplanar faults, free surface topography, and branching faults through the use of coordinate transforms. Recently, it has also been shown how second-order SBP methods can be extended to unstructured grids. Due to the SBP character of both the finite difference and node-centered finite volume method they can be used together in a stable and accurate way. Inclusion of these techniques will be important for problems that have regions
NASA Astrophysics Data System (ADS)
Lo, F. S.; Lu, P. S.; Ragan-Kelley, B.; Minnich, A. J.; Lee, T. H.; Lin, M. C.; Verboncoeur, J. P.
2014-02-01
A thermionic energy converter (TEC) is a static device that converts heat directly into electricity by boiling electrons off a hot emitter surface across a small inter-electrode gap to a cooler collector surface. The main challenge in TECs is overcoming the space charge limit, which limits the current transmitted across a gap of a given voltage and width. We have verified the feasibility of studying and developing a TEC using a bounded finite-difference time-domain particle-in-cell plasma simulation code, OOPD1, developed by Plasma Theory and Simulation Group, formerly at UC Berkeley and now at Michigan State University. In this preliminary work, a TEC has been modeled kinetically using OOPD1, and the accuracy has been verified by comparing with an analytically solvable case, giving good agreement. With further improvement of the code, one will be able to quickly and cheaply analyze space charge effects, and seek designs that mitigate the space charge effect, allowing TECs to become more efficient and cost-effective.
Lo, F. S.; Lee, T. H.; Lu, P. S.; Ragan-Kelley, B.; Minnich, A.; Lin, M. C.; Verboncoeur, J. P.
2014-02-15
A thermionic energy converter (TEC) is a static device that converts heat directly into electricity by boiling electrons off a hot emitter surface across a small inter-electrode gap to a cooler collector surface. The main challenge in TECs is overcoming the space charge limit, which limits the current transmitted across a gap of a given voltage and width. We have verified the feasibility of studying and developing a TEC using a bounded finite-difference time-domain particle-in-cell plasma simulation code, OOPD1, developed by Plasma Theory and Simulation Group, formerly at UC Berkeley and now at Michigan State University. In this preliminary work, a TEC has been modeled kinetically using OOPD1, and the accuracy has been verified by comparing with an analytically solvable case, giving good agreement. With further improvement of the code, one will be able to quickly and cheaply analyze space charge effects, and seek designs that mitigate the space charge effect, allowing TECs to become more efficient and cost-effective.
Space-time finite-element objects: Efficiently modeling physically complex flows
Dilts, G.A.
1996-03-28
Accurate modeling of high-explosive systems requires detailed consideration of many different physical properties and processes: These diverse processes generally occur in localized regions of the problem. Thus the very partial differential equations used to mathematically model the problem change from one region of space and time to another. The numerical algorithms generally used to solve these equations are frequently conceived in terms of data values for physical field variables u{sup i} defined at a number of spatial points indexed by multi-integer subscripts x{sub J}, resulting in a number of discrete state variables u{sup i}{sub J}. Instead of using as the fundamental object a physical field, which naturally maps to an array, the authors imagine a small piece of space modeled for a small amount of time, a space-time ``element``. Within it, various physical processes occur at various times. Self-contained, it gives account of what happens within its borders. It cooperates with a set of neighbors that organize into meshes, which organize into problems. The authors achieve in the software model a decoupling between the where and the how and the what, lack of which historically has been the source of a great deal of the software overhead of modelling continuum systems, and which is a necessary consequence of writing down u{sup i}{sub J}. An efficient implementation of this idea requires a reformulation of the discretization and solution of systems of conservation laws, and careful class design. A working prototype for systems in one space dimension using Mathematica and C++ is provided.
a Mapped Finite Difference Study of Noise Transmission in Nonuniform Ducts
NASA Astrophysics Data System (ADS)
Raad, Peter Emile
The primary objective of this work was to study a class of problems involving noise propagation in acoustically lined variable area ducts with or without mean fluid flow. The method of study was numerical in nature and included body -fitted grid mapping procedures in conjunction with implicit finite difference techniques. The work resulted in several general FORTRAN programs that were tested for cases with or without mean fluid flow, including soft wall or hard wall acoustic liner conditions, and plane wave or far field exit conditions. The results were compared to available theoretical and experimental data. The automated, body-fitted grid mapping procedure was found to be robust, simple to use, and capable of mapping very complicated geometries simply by defining the grid distribution on the boundaries. In general, the solution of the wave equation was found to be successful when using a plane wave exit condition, whereas a problem was encountered with reflections from the particular far field exit condition being applied. The problem was determined to be the result of the proximity of the far field boundary to the noise source as well as its applicability to exactly cylindrical wave expansions only. The fully-coupled solution of the linearized gas dynamic equations was successful for both positive and negative Mach numbers as well as for hard and soft wall conditions. The mean fluid flow considered was two-dimensional, inviscid, irrotational, incompressible, and nonheat conducting. The factored-implicit finite difference technique used did give rise to short wavelength perturbations, but these were dampened by the introduction of higher order artificial dissipation terms into the scheme. In the different problems that this study considered, the finite difference theory was found to be well-suited for the simulation of noise transmission in nonuniform ducts.
Appelo, D; Petersson, N A
2007-12-17
The isotropic elastic wave equation governs the propagation of seismic waves caused by earthquakes and other seismic events. It also governs the propagation of waves in solid material structures and devices, such as gas pipes, wave guides, railroad rails and disc brakes. In the vast majority of wave propagation problems arising in seismology and solid mechanics there are free surfaces. These free surfaces have, in general, complicated shapes and are rarely flat. Another feature, characterizing problems arising in these areas, is the strong heterogeneity of the media, in which the problems are posed. For example, on the characteristic length scales of seismological problems, the geological structures of the earth can be considered piecewise constant, leading to models where the values of the elastic properties are also piecewise constant. Large spatial contrasts are also found in solid mechanics devices composed of different materials welded together. The presence of curved free surfaces, together with the typical strong material heterogeneity, makes the design of stable, efficient and accurate numerical methods for the elastic wave equation challenging. Today, many different classes of numerical methods are used for the simulation of elastic waves. Early on, most of the methods were based on finite difference approximations of space and time derivatives of the equations in second order differential form (displacement formulation), see for example [1, 2]. The main problem with these early discretizations were their inability to approximate free surface boundary conditions in a stable and fully explicit manner, see e.g. [10, 11, 18, 20]. The instabilities of these early methods were especially bad for problems with materials with high ratios between the P-wave (C{sub p}) and S-wave (C{sub s}) velocities. For rectangular domains, a stable and explicit discretization of the free surface boundary conditions is presented in the paper [17] by Nilsson et al. In summary
Determination of cutoff frequencies of simple waveguides using finite difference method
NASA Astrophysics Data System (ADS)
Kolagani, Sridhar
Waveguides are used to transfer electromagnetic energy from one location to another. Within many electronic circles, waveguides are commonly used for microwave RF signals; the same principle can be used for many forms of waves from sound to light. They have been used in many technologies like acoustic waveguide speaker technology, high-performance passive waveguide technologies for remote sensing and communication, optical computing, robotic-vision, biochemical sensing and many more. Modern waveguide technology employs a variety of waveguides with different cross sections and perturbations, the cutoff frequencies and mode shapes of many of these waveguides are ill-suited for determination by an analytical method. In this thesis, we solve this type of waveguides by employing the numerical procedure of finite difference method. By adopting finite difference approach with an application of eigenvalue method, we discuss about few different types of these waveguides in determining the cutoff frequencies of supported modes, and extracting the possible degenerate modes and their field distributions. To validate the method and its accuracy, it is applied to the two well known rectangular waveguides, viz. PEC Rectangular Waveguide and Artificial Rectangular Waveguide (consists of PEC and PMC walls) and compared with the analytical solutions.
A modular three-dimensional finite-difference ground-water flow model
McDonald, Michael G.; Harbaugh, Arlen W.
1988-01-01
This report presents a finite-difference model and its associated modular computer program. The model simulates flow in three dimensions. The report includes detailed explanations of physical and mathematical concepts on which the model is based and an explanation of how those concepts are incorporated in the modular structure of the computer program. The modular structure consists of a Main Program and a series of highly independent subroutines called 'modules.' The modules are grouped into 'packages.' Each package deals with a specific feature of the hydrologic system which is to be simulated, such as flow from rivers or flow into drains, or with a specific method of solving linear equations which describe the flow system, such as the Strongly Implicit Procedure or Slice-Successive Overrelaxation. The division of the program into modules permits the user to examine specific hydrologic features of the model independently. This also facilita development of additional capabilities because new packages can be added to the program without modifying the existing packages. The input and output systems of the computer program are also designed to permit maximum flexibility. Ground-water flow within the aquifer is simulated using a block-centered finite-difference approach. Layers can be simulated as confined, unconfined, or a combination of confined and unconfined. Flow associated with external stresses, such as wells, areal recharge, evapotranspiration, drains, and streams, can also be simulated. The finite-difference equations can be solved using either the Strongly Implicit Procedure or Slice-Successive Overrelaxation. The program is written in FORTRAN 77 and will run without modification on most computers that have a FORTRAN 77 compiler. For each program ,module, this report includes a narrative description, a flow chart, a list of variables, and a module listing.
A modular three-dimensional finite-difference ground-water flow model
McDonald, M.G.; Harbaugh, A.W.
1984-01-01
This report presents a finite-difference model and its associated modular computer program. The model simulates flow in three dimensions. The report includes detailed explanations of physical and mathematical concepts on which the model is based and an explanation of how those concepts were incorporated in the modular structure of the computer program. The modular structure consists of a Main Program and a series of highly independent subroutines called 'modules.' The modules are grouped into 'packages.' Each package deals with a specific feature of the hydrologic system which is to be simulated, such as flow from rivers or flow into drains, or with a specific method of solving linear equations which describe the flow system, such as the Strongly Implicit Procedure or Slice-Successive Overrelaxation. The division of the program into modules permits the user to examine specific hydrologic features of the model independently. This also facilitates development of additional capabilities because new modules or packages can be added to the program without modifying the existing modules or packages. The input and output systems of the computer program are also designed to permit maximum flexibility. Ground-water flow within the aquifer is simulated using a block-centered finite-difference approach. Layers can be simulated as confined, unconfined, or a combination of confined and unconfined. Flow from external stresses, such as flow to wells, areal recharge, evapotranspiration, flow to drains, and flow through riverbeds, can also be simulated. The finite-difference equations can be solved using either the Strongly Implicit Procedure or Slice-Successive Overrelaxation. The program is written in FORTRAN '66 and will run without modification on most computers which have a FORTRAN '66 compiler. It will also run, without modification, with most extended FORTRAN '77 compilers and with minor modifications on standard FORTRAN '77 compilers. Documentation presented in this report
A Modular Three-Dimensional Finite-Difference Ground-Water Flow Model
McDonald, Michael G.; Harbaugh, Arlen W.; Guo, Weixing; Lu, Guoping
1988-01-01
This report presents a finite-difference model and its associated modular computer program. The model simulates flow in three dimensions. The report includes detailed explanations of physical and mathematical concepts on which the model is based and an explanation of how those concepts are incorporated in the modular structure of the computer program. The modular structure consists of a Main Program and a series of highly independent subroutines called 'modules.' The modules are grouped into 'packages.' Each package deals with a specific feature of the hydrologic system which is to be simulated, such as flow from rivers or flow into drains, or with a specific method of solving linear equations which describe the flow system, such as the Strongly Implicit Procedure or Slice-Successive Overrelaxation. The division of the program into modules permits the user to examine specific hydrologic features of the model independently. This also facilita development of additional capabilities because new packages can be added to the program without modifying the existing packages. The input and output systems of the computer program are also designed to permit maximum flexibility. Ground-water flow within the aquifer is simulated using a block-centered finite-difference approach. Layers can be simulated as confined, unconfined, or a combination of confined and unconfined. Flow associated with external stresses, such as wells, areal recharge, evapotranspiration, drains, and streams, can also be simulated. The finite-difference equations can be solved using either the Strongly Implicit Procedure or Slice-Successive Overrelaxation. The program is written in FORTRAN 77 and will run without modification on most computers that have a FORTRAN 77 compiler. For each program ,module, this report includes a narrative description, a flow chart, a list of variables, and a module listing.
Ahmed, S.
1992-01-01
The physical processes involving leachate flow in a solid waste landfill are described by the unsaturated flow through the refuse to the saturated leachate mound at the bottom of a landfill. The moisture-flow in the unsaturated zone helps build up the saturated leachate mound at the bottom of a landfill. The moisture content in the unsaturated zone is obtained by solving the two-dimensional unsaturated moisture-flow equation using numerical techniques. A two-dimensional unsteady sate Flow Investigation for Landfill Leachate (FILL) model, based on the implicit finite-difference technique, has been developed to describe the leachate flow process in a landfill. To obtain accuracy and efficiency in numerical molding, it is important to investigate the numerical solution techniques suitable to solve the governing equations. Accuracy and efficiency of the boundary integral method over the finite-difference methods has been investigated. Two approaches, direct Green's function and perturbation Green's function formulations have been developed to solve the unsaturated flow problem. Direct Green's function and perturbation Green's function boundary integral solutions are found to be more accurate than both the Gauss-Seidel iteration and Gauss-Jordon elimination method of finite-difference solution. The efficiency of the boundary integral formulation for the computation of the moisture-flux is an advantage that is useful to estimate leachate of the moisture-flux is an advantage that is useful to estimate leachate accretion in a landfill. A close agreement of the internal fluxes with the exact solution shows the ability of the boundary integral methods to compute accurate recharge from the unsaturated zone to the saturated leachate mound.
A study of unstable rock failures using finite difference and discrete element methods
NASA Astrophysics Data System (ADS)
Garvey, Ryan J.
Case histories in mining have long described pillars or faces of rock failing violently with an accompanying rapid ejection of debris and broken material into the working areas of the mine. These unstable failures have resulted in large losses of life and collapses of entire mine panels. Modern mining operations take significant steps to reduce the likelihood of unstable failure, however eliminating their occurrence is difficult in practice. Researchers over several decades have supplemented studies of unstable failures through the application of various numerical methods. The direction of the current research is to extend these methods and to develop improved numerical tools with which to study unstable failures in underground mining layouts. An extensive study is first conducted on the expression of unstable failure in discrete element and finite difference methods. Simulated uniaxial compressive strength tests are run on brittle rock specimens. Stable or unstable loading conditions are applied onto the brittle specimens by a pair of elastic platens with ranging stiffnesses. Determinations of instability are established through stress and strain histories taken for the specimen and the system. Additional numerical tools are then developed for the finite difference method to analyze unstable failure in larger mine models. Instability identifiers are established for assessing the locations and relative magnitudes of unstable failure through measures of rapid dynamic motion. An energy balance is developed which calculates the excess energy released as a result of unstable equilibria in rock systems. These tools are validated through uniaxial and triaxial compressive strength tests and are extended to models of coal pillars and a simplified mining layout. The results of the finite difference simulations reveal that the instability identifiers and excess energy calculations provide a generalized methodology for assessing unstable failures within potentially complex
A finite-difference program for stresses in anisotropic, layered plates in bending
NASA Technical Reports Server (NTRS)
Salamon, N. J.
1975-01-01
The interlaminar stresses induced in a layered laminate that is bent into a cylindrical surface are studied. The laminate is modeled as a continuum, and the resulting elasticity equations are solved using the finite difference method. The report sets forth the mathematical framework, presents some preliminary results, and provides a listing and explanation of the computer program. Significant among the results are apparent symmetry relationships that will reduce the numerical size of certain problems and an interlaminar stress behavior having a sharp rise at the free edges.
The electromagnetic modeling of thin apertures using the finite-difference time-domain technique
NASA Technical Reports Server (NTRS)
Demarest, Kenneth R.
1987-01-01
A technique which computes transient electromagnetic responses of narrow apertures in complex conducting scatterers was implemented as an extension of previously developed Finite-Difference Time-Domain (FDTD) computer codes. Although these apertures are narrow with respect to the wavelengths contained within the power spectrum of excitation, this technique does not require significantly more computer resources to attain the increased resolution at the apertures. In the report, an analytical technique which utilizes Babinet's principle to model the apertures is developed, and an FDTD computer code which utilizes this technique is described.
Slat Noise Predictions Using Higher-Order Finite-Difference Methods on Overset Grids
NASA Technical Reports Server (NTRS)
Housman, Jeffrey A.; Kiris, Cetin
2016-01-01
Computational aeroacoustic simulations using the structured overset grid approach and higher-order finite difference methods within the Launch Ascent and Vehicle Aerodynamics (LAVA) solver framework are presented for slat noise predictions. The simulations are part of a collaborative study comparing noise generation mechanisms between a conventional slat and a Krueger leading edge flap. Simulation results are compared with experimental data acquired during an aeroacoustic test in the NASA Langley Quiet Flow Facility. Details of the structured overset grid, numerical discretization, and turbulence model are provided.
NASA Technical Reports Server (NTRS)
Doohovskoy, A.
1977-01-01
A change in MACSYMA syntax is proposed to accommodate the operator manipulators necessary to implement direct and indirect methods for the solution of differential equations, calculus of finite differences, and the fractional calculus, as well as their modern counterparts. To illustrate the benefits and convenience of this syntax extension, an example is given to show how MACSYMA's pattern-matching capability can be used to implement a particular set of operator identities which can then be used to obtain exact solutions to nonlinear differential equations.
A noniterative finite difference method for the compressible unsteady laminar boundary layer
NASA Astrophysics Data System (ADS)
Chang, K. S.; Kim, J. S.
1985-11-01
An investigation involving the determination of the friction drag and the rate of heat transfer at the surface of a body in dynamic motion must take into account details regarding the unsteady viscous flow. Difficulties concerning such an investigation are related to interaction effects due to aspects of increased dimensionality, nonlinearity, and compressibility. In the present study, the nonlinearity is eliminated by making use of approaches considered by Beam and Warming (1978) and Orlandi and Ferziger (1981). These approaches involve the employment of a technique of linearizing the general nonlinear implicit finite difference equations without sacrificing accuracy. A noniterative numerical formulation is developed to solve the unsteady compressible laminar boundary layer equations efficiently.
2D numerical simulation of the MEP energy-transport model with a finite difference scheme
Romano, V. . E-mail: romano@dmi.unict.it
2007-02-10
A finite difference scheme of Scharfetter-Gummel type is used to simulate a consistent energy-transport model for electron transport in semiconductors devices, free of any fitting parameters, formulated on the basis of the maximum entropy principle. Simulations of silicon n{sup +}-n-n{sup +} diodes, 2D-MESFET and 2D-MOSFET and comparisons with the results obtained by a direct simulation of the Boltzmann transport equation and with other energy-transport models, known in the literature, show the validity of the model and the robustness of the numerical scheme.
NASA Astrophysics Data System (ADS)
Adhikari, Achyut; Dev, Kapil; Asundi, Anand
2016-11-01
Wire grid polarizers (WGP), are sub-wavelength gratings with applications in display projection system due to their compact size, wide field of view and long-term stability. Measurement and testing of these structures are important to optimize their use. This is done by first measuring the Mueller matrix of the WGP using a Mueller matrix polarimeter. Next the finite difference time domain (FDTD) method is used to simulate a similar Mueller matrix thus providing the period and step height of the WGP. This approach may lead to more generic determination of sub-wavelength structures including diffractive optical structures.
NASA Technical Reports Server (NTRS)
Osher, S.
1984-01-01
The construction of a reliable, shock capturing finite difference method to solve the Euler equations for inviscid, supersonic flow past fighter and missile type configurations is highly desirable. The numerical method must have a firm theoretical foundation and must be robust and efficient. It should be able to treat subsonic pockets in a predominantly supersonic flow. The method must also be easily applicable to the complex topologies of the aerodynamic configuration under consideration. The ongoing approach to this task is described and for steady supersonic flows is presented. This scheme is the basic numerical method. Results of work obtained during previous years are presented.
A staggered mesh finite difference scheme for the computation of hypersonic Euler flows
NASA Technical Reports Server (NTRS)
Sanders, Richard
1991-01-01
A shock capturing finite difference method for systems of hyperbolic conservation laws is presented which avoids the need to solve Riemann problems while being competitive in performance with other current methods. A staggered spatial mesh is employed, so that complicated nonlinear waves generated at cell interfaces are averaged over cell interiors at the next time level. The full method combines to form a conservative version of the modified method of characteristics. The advantages of the method are discussed, and numerical results are presented for the two-dimensional double ellipse problem.
Polarization-current-based, finite-difference time-domain, near-to-far-field transformation.
Zeng, Yong; Moloney, Jerome V
2009-05-15
A near-to-far-field transformation algorithm for three-dimensional finite-difference time-domain is presented in this Letter. This approach is based directly on the polarization current of the scatterer, not the scattered near fields. It therefore eliminates the numerical errors originating from the spatial offset of the E and H fields, inherent in the standard near-to-far-field transformation. The proposed method is validated via direct comparisons with the analytical Lorentz-Mie solutions of plane waves scattered by large dielectric and metallic spheres with strong forward-scattering lobes.
NASA Technical Reports Server (NTRS)
Anderson, O. L.
1974-01-01
A finite-difference procedure for computing the turbulent, swirling, compressible flow in axisymmetric ducts is described. Arbitrary distributions of heat and mass transfer at the boundaries can be treated, and the effects of struts, inlet guide vanes, and flow straightening vanes can be calculated. The calculation procedure is programmed in FORTRAN 4 and has operated successfully on the UNIVAC 1108, IBM 360, and CDC 6600 computers. The analysis which forms the basis of the procedure, a detailed description of the computer program, and the input/output formats are presented. The results of sample calculations performed with the computer program are compared with experimental data.
Morshed, Monjur; Ingalls, Brian; Ilie, Silvana
2017-01-01
Sensitivity analysis characterizes the dependence of a model's behaviour on system parameters. It is a critical tool in the formulation, characterization, and verification of models of biochemical reaction networks, for which confident estimates of parameter values are often lacking. In this paper, we propose a novel method for sensitivity analysis of discrete stochastic models of biochemical reaction systems whose dynamics occur over a range of timescales. This method combines finite-difference approximations and adaptive tau-leaping strategies to efficiently estimate parametric sensitivities for stiff stochastic biochemical kinetics models, with negligible loss in accuracy compared with previously published approaches. We analyze several models of interest to illustrate the advantages of our method.
Computation of wing-vortex interaction in transonic flow using implicit finite difference algorithm
NASA Technical Reports Server (NTRS)
Srinivasan, G.; Steger, J. L.
1981-01-01
An implicit delta form finite difference algorithm for Euler equations in conservation law form was used in preliminary calculations of three dimensional wing vortex interaction. Both steady and unsteady transonic flow wing vortex interactions are computed. The computations themselves are meant to guide upcoming wind tunnel experiments of the same flow field. Various modifications to the numerical method that are intended to improve computational efficiency are also described and tested in both two and three dimensions. Combination of these methods can reduce the overall computational time by a factor of 4.
Rotordynamic coefficients for labyrinth seals calculated by means of a finite difference technique
NASA Technical Reports Server (NTRS)
Nordmann, R.; Weiser, P.
1989-01-01
The compressible, turbulent, time dependent and three dimensional flow in a labyrinth seal can be described by the Navier-Stokes equations in conjunction with a turbulence model. Additionally, equations for mass and energy conservation and an equation of state are required. To solve these equations, a perturbation analysis is performed yielding zeroth order equations for centric shaft position and first order equations describing the flow field for small motions around the seal center. For numerical solution a finite difference method is applied to the zeroth and first order equations resulting in leakage and dynamic seal coefficients respectively.
Finite Differences and Collocation Methods for the Solution of the Two Dimensional Heat Equation
NASA Technical Reports Server (NTRS)
Kouatchou, Jules
1999-01-01
In this paper we combine finite difference approximations (for spatial derivatives) and collocation techniques (for the time component) to numerically solve the two dimensional heat equation. We employ respectively a second-order and a fourth-order schemes for the spatial derivatives and the discretization method gives rise to a linear system of equations. We show that the matrix of the system is non-singular. Numerical experiments carried out on serial computers, show the unconditional stability of the proposed method and the high accuracy achieved by the fourth-order scheme.
DNS of premixed turbulent V-flame: coupling spectral and finite difference methods
NASA Astrophysics Data System (ADS)
Hauguel, Raphael; Vervisch, Luc; Domingo, Pascale
2005-01-01
To allow for a reliable examination of the interaction between velocity fluctuations, acoustics and combustion, a novel numerical procedure is discussed in which a spectral solution of the Navier-Stokes equations is directly associated to a high-order finite difference fully compressible DNS solver (sixth order PADE). Using this combination of high-order solvers with accurate boundary conditions, simulations have been performed where a turbulent premixed V-shape flame develops in grid turbulence. In the light of the DNS results, a sub-model for premixed turbulent combustion is analyzed. To cite this article: R. Hauguel et al., C. R. Mecanique 333 (2005).
A 3-dimensional finite-difference method for calculating the dynamic coefficients of seals
NASA Technical Reports Server (NTRS)
Dietzen, F. J.; Nordmann, R.
1989-01-01
A method to calculate the dynamic coefficients of seals with arbitrary geometry is presented. The Navier-Stokes equations are used in conjunction with the k-e turbulence model to describe the turbulent flow. These equations are solved by a full 3-dimensional finite-difference procedure instead of the normally used perturbation analysis. The time dependence of the equations is introduced by working with a coordinate system rotating with the precession frequency of the shaft. The results of this theory are compared with coefficients calculated by a perturbation analysis and with experimental results.
Arbitrary Order Mixed Mimetic Finite Differences Method with Nodal Degrees of Freedom
Iaroshenko, Oleksandr; Gyrya, Vitaliy; Manzini, Gianmarco
2016-09-01
In this work we consider a modification to an arbitrary order mixed mimetic finite difference method (MFD) for a diffusion equation on general polygonal meshes [1]. The modification is based on moving some degrees of freedom (DoF) for a flux variable from edges to vertices. We showed that for a non-degenerate element this transformation is locally equivalent, i.e. there is a one-to-one map between the new and the old DoF. Globally, on the other hand, this transformation leads to a reduction of the total number of degrees of freedom (by up to 40%) and additional continuity of the discrete flux.
NASA Astrophysics Data System (ADS)
Yamamoto, Kaho; Iwai, Yosuke; Uchida, Yoshiaki; Nishiyama, Norikazu
2016-08-01
We numerically analyzed the light propagation in cholesteric liquid crystalline (CLC) droplet array by the finite-difference time-domain (FDTD) method. The FDTD method successfully reproduced the experimental light path observed in the complicated photonic structure of the CLC droplet array more accurately than the analysis of CLC droplets by geometric optics with Bragg condition, and this method help us understand the polarization of the propagating light waves. The FDTD method holds great promise for the design of various photonic devices composed of curved photonic materials like CLC droplets and microcapsules.
WONDY V: A one-dimensional finite-difference wave-propagation code
NASA Astrophysics Data System (ADS)
Kipp, M. E.; Lawrence, R. J.
1982-06-01
WONDY V solves the finite difference analogs to the Lagrangian equations of motion in one spatial dimension (planar, cylindrical, or spherical). Simulations of explosive detonation, energy deposition, plate impact, and dynamic fracture are possible, using a variety of existing material models. In addition, WONDY proves to be a powerful tool in the evaluation of new constitutive models. A preprocessor is available to allocate storage arrays commensurate with problem size, and automatic rezoning may be employed to improve resolution. A description of the equations solved, available material models, operating instructions, and sample problems are given.
WONDY V: a one-dimensional finite-difference wave-propagation code
Kipp, M.E.; Lawrence, R.J.
1982-06-01
WONDY V solves the finite difference analogs to the Lagrangian equations of motion in one spatial dimension (planar, cylindrical, or spherical). Simulations of explosive detonation, energy deposition, plate impact, and dynamic fracture are possible, using a variety of existing material models. In addition, WONDY has proven to be a powerful tool in the evaluation of new constitutive models. A preprocessor is available to allocate storage arrays commensurate with problem size, and automatic rezoning may be employed to improve resolution. This document provides a description of the equations solved, available material models, operating instructions, and sample problems.
One-dimensional transient finite difference model of an operational salinity gradient solar pond
NASA Technical Reports Server (NTRS)
Hicks, Michael C.; Golding, Peter
1992-01-01
This paper describes the modeling approach used to simulate the transient behavior of a salinity gradient solar pond. A system of finite difference equations are used to generate the time dependent temperature and salinity profiles within the pond. The stability of the pond, as determined by the capacity of the resulting salinity profile to suppress thermal convection within the primary gradient region of the pond, is continually monitored and when necessary adjustments are made to the thickness of the gradient zone. Results of the model are then compared to measurements taken during two representative seasonal periods at the University of Texas at El Paso's (UTEP's) research solar pond.
NASA Astrophysics Data System (ADS)
Liu, C.; Liu, Z.
1993-05-01
The fourth-order finite-difference scheme with fully implicit time-marching presently used to computationally study the spatial instability of planar Poiseuille flow incorporates a novel treatment for outflow boundary conditions that renders the buffer area as short as one wavelength. A semicoarsening multigrid method accelerates convergence for the implicit scheme at each time step; a line-distributive relaxation is developed as a robust fast solver that is efficient for anisotropic grids. Computational cost is no greater than that of explicit schemes, and excellent agreement with linear theory is obtained.
Computing interaural differences through finite element modeling of idealized human heads.
Cai, Tingli; Rakerd, Brad; Hartmann, William M
2015-09-01
Acoustical interaural differences were computed for a succession of idealized shapes approximating the human head-related anatomy: sphere, ellipsoid, and ellipsoid with neck and torso. Calculations were done as a function of frequency (100-2500 Hz) and for source azimuths from 10 to 90 degrees using finite element models. The computations were compared to free-field measurements made with a manikin. Compared to a spherical head, the ellipsoid produced greater large-scale variation with frequency in both interaural time differences and interaural level differences, resulting in better agreement with the measurements. Adding a torso, represented either as a large plate or as a rectangular box below the neck, further improved the agreement by adding smaller-scale frequency variation. The comparisons permitted conjectures about the relationship between details of interaural differences and gross features of the human anatomy, such as the height of the head, and length of the neck.
Computing interaural differences through finite element modeling of idealized human heads
Cai, Tingli; Rakerd, Brad; Hartmann, William M.
2015-01-01
Acoustical interaural differences were computed for a succession of idealized shapes approximating the human head-related anatomy: sphere, ellipsoid, and ellipsoid with neck and torso. Calculations were done as a function of frequency (100–2500 Hz) and for source azimuths from 10 to 90 degrees using finite element models. The computations were compared to free-field measurements made with a manikin. Compared to a spherical head, the ellipsoid produced greater large-scale variation with frequency in both interaural time differences and interaural level differences, resulting in better agreement with the measurements. Adding a torso, represented either as a large plate or as a rectangular box below the neck, further improved the agreement by adding smaller-scale frequency variation. The comparisons permitted conjectures about the relationship between details of interaural differences and gross features of the human anatomy, such as the height of the head, and length of the neck. PMID:26428792
Transient analysis of printed lines using finite-difference time-domain method
Ahmed, Shahid
2012-03-29
Comprehensive studies of ultra-wideband pulses and electromagnetic coupling on printed coupled lines have been performed using full-wave 3D finite-difference time-domain analysis. Effects of unequal phase velocities of coupled modes, coupling between line traces, and the frequency dispersion on the waveform fidelity and crosstalk have been investigated in detail. To discriminate the contributions of different mechanisms into pulse evolution, single and coupled microstrip lines without (ϵ_{r} = 1) and with (ϵ_{r} > 1) dielectric substrates have been examined. To consistently compare the performance of the coupled lines with substrates of different permittivities and transients of different characteristic times, a generic metric similar to the electrical wavelength has been introduced. The features of pulse propagation on coupled lines with layered and pedestal substrates and on the irregular traces have been explored. Finally, physical interpretations of the simulation results are discussed in the paper.
Enhancing finite differences with radial basis functions: Experiments on the Navier-Stokes equations
NASA Astrophysics Data System (ADS)
Flyer, Natasha; Barnett, Gregory A.; Wicker, Louis J.
2016-07-01
Polynomials are used together with polyharmonic spline (PHS) radial basis functions (RBFs) to create local RBF-finite-difference (RBF-FD) weights on different node layouts for spatial discretizations that can be viewed as enhancements of the classical finite differences (FD). The presented method replicates the convergence properties of FD but for arbitrary node layouts. It is tested on the 2D compressible Navier-Stokes equations at low Mach number, relevant to atmospheric flows. Test cases are taken from the numerical weather prediction community and solved on bounded domains. Thus, attention is given on how to handle boundaries with the RBF-FD method, as well as a novel implementation for hyperviscosity. Comparisons are done on Cartesian, hexagonal, and quasi-uniform node layouts. Consideration and guidelines are given on PHS order, polynomial degree and stencil size. The main advantages of the present method are: 1) capturing the basic physics of the problem surprisingly well, even at very coarse resolutions, 2) high-order accuracy without the need of tuning a shape parameter, and 3) the inclusion of polynomials eliminates stagnation (saturation) errors. A MATLAB code is given to calculate the differentiation weights for this novel approach.
On the Definition of Surface Potentials for Finite-Difference Operators
NASA Technical Reports Server (NTRS)
Tsynkov, S. V.; Bushnell, Dennis M. (Technical Monitor)
2001-01-01
For a class of linear constant-coefficient finite-difference operators of the second order, we introduce the concepts similar to those of conventional single- and double-layer potentials for differential operators. The discrete potentials are defined completely independently of any notion related to the approximation of the continuous potentials on the grid. We rather use all approach based on differentiating, and then inverting the differentiation of a function with surface discontinuity of a particular kind, which is the most general way of introducing surface potentials in the theory of distributions. The resulting finite-difference "surface" potentials appear to be solutions of the corresponding continuous potentials. Primarily, this pertains to the possibility of representing a given solution to the homogeneous equation on the domain as a variety of surface potentials, with the density defined on the domain's boundary. At the same time the discrete surface potentials can be interpreted as one specific realization of the generalized potentials of Calderon's type, and consequently, their approximation properties can be studied independently in the framework of the difference potentials method by Ryaben'kii. The motivation for introducing and analyzing the discrete surface potentials was provided by the problems of active shielding and control of sound, in which the aforementioned source terms that drive the potentials are interpreted as the acoustic control sources that cancel out the unwanted noise on a predetermined region of interest.
Abdollahi, Amir; Jiang, Zhongwei; Arabshahi, Sayyed Alireza
2007-12-01
The mass sensitivity of the piezoelectric surface acoustic wave (SAW) sensors is an important factor in the selection of the best gravimetric sensors for different applications. To determine this value without facing the practical problems and the long theoretical calculation time, we have shown that the mass sensitivity of SAW sensors can be calculated by a simple three-dimensional (3-D) finite-element analysis (FEA) using a commercial finite-element platform. The FEA data are used to calculate the wave propagation speed, surface particle displacements, and wave energy distribution on different cuts of various piezoelectric materials. The results are used to provide a simple method for evaluation of their mass sensitivities. Meanwhile, to calculate more accurate results from FEA data, surface and bulk wave reflection problems are considered in the analyses. In this research, different cuts of lithium niobate, quartz, lithium tantalate, and langasite piezoelectric materials are applied to investigate their acoustic wave properties. Our analyses results for these materials have a good agreement with other researchers' results. Also, the mass sensitivity value for the novel cut of langasite was calculated through these analyses. It was found that its mass sensitivity is higher than that of the conventional Rayleigh mode quartz sensor.
Ground motion simulations in Marmara (Turkey) region from 3D finite difference method
NASA Astrophysics Data System (ADS)
Aochi, Hideo; Ulrich, Thomas; Douglas, John
2016-04-01
In the framework of the European project MARSite (2012-2016), one of the main contributions from our research team was to provide ground-motion simulations for the Marmara region from various earthquake source scenarios. We adopted a 3D finite difference code, taking into account the 3D structure around the Sea of Marmara (including the bathymetry) and the sea layer. We simulated two moderate earthquakes (about Mw4.5) and found that the 3D structure improves significantly the waveforms compared to the 1D layer model. Simulations were carried out for different earthquakes (moderate point sources and large finite sources) in order to provide shake maps (Aochi and Ulrich, BSSA, 2015), to study the variability of ground-motion parameters (Douglas & Aochi, BSSA, 2016) as well as to provide synthetic seismograms for the blind inversion tests (Diao et al., GJI, 2016). The results are also planned to be integrated in broadband ground-motion simulations, tsunamis generation and simulations of triggered landslides (in progress by different partners). The simulations are freely shared among the partners via the internet and the visualization of the results is diffused on the project's homepage. All these simulations should be seen as a reference for this region, as they are based on the latest knowledge that obtained during the MARSite project, although their refinement and validation of the model parameters and the simulations are a continuing research task relying on continuing observations. The numerical code used, the models and the simulations are available on demand.
Phase-field-based lattice Boltzmann finite-difference model for simulating thermocapillary flows.
Liu, Haihu; Valocchi, Albert J; Zhang, Yonghao; Kang, Qinjun
2013-01-01
A phase-field-based hybrid model that combines the lattice Boltzmann method with the finite difference method is proposed for simulating immiscible thermocapillary flows with variable fluid-property ratios. Using a phase field methodology, an interfacial force formula is analytically derived to model the interfacial tension force and the Marangoni stress. We present an improved lattice Boltzmann equation (LBE) method to capture the interface between different phases and solve the pressure and velocity fields, which can recover the correct Cahn-Hilliard equation (CHE) and Navier-Stokes equations. The LBE method allows not only use of variable mobility in the CHE, but also simulation of multiphase flows with high density ratio because a stable discretization scheme is used for calculating the derivative terms in forcing terms. An additional convection-diffusion equation is solved by the finite difference method for spatial discretization and the Runge-Kutta method for time marching to obtain the temperature field, which is coupled to the interfacial tension through an equation of state. The model is first validated against analytical solutions for the thermocapillary driven convection in two superimposed fluids at negligibly small Reynolds and Marangoni numbers. It is then used to simulate thermocapillary migration of a three-dimensional deformable droplet and bubble at various Marangoni numbers and density ratios, and satisfactory agreement is obtained between numerical results and theoretical predictions.
Candel, A.E.; Kabel, A.C.; Ko, Yong-kyu; Lee, L.; Li, Z.; Limborg-Deprey, C.; Ng, C.K.; Prudencio, E.E.; Schussman, G.L.; Uplenchwar, R.; /SLAC
2007-11-07
Over the past years, SLAC's Advanced Computations Department (ACD) has developed the parallel finite element (FE) particle-in-cell code Pic3P (Pic2P) for simulations of beam-cavity interactions dominated by space-charge effects. As opposed to standard space-charge dominated beam transport codes, which are based on the electrostatic approximation, Pic3P (Pic2P) includes space-charge, retardation and boundary effects as it self-consistently solves the complete set of Maxwell-Lorentz equations using higher-order FE methods on conformal meshes. Use of efficient, large-scale parallel processing allows for the modeling of photoinjectors with unprecedented accuracy, aiding the design and operation of the next-generation of accelerator facilities. Applications to the Linac Coherent Light Source (LCLS) RF gun are presented.
NASA Technical Reports Server (NTRS)
Stein, M.; Housner, J. D.
1978-01-01
A numerical analysis developed for the buckling of rectangular orthotropic layered panels under combined shear and compression is described. This analysis uses a central finite difference procedure based on trigonometric functions instead of using the conventional finite differences which are based on polynomial functions. Inasmuch as the buckle mode shape is usually trigonometric in nature, the analysis using trigonometric finite differences can be made to exhibit a much faster convergence rate than that using conventional differences. Also, the trigonometric finite difference procedure leads to difference equations having the same form as conventional finite differences; thereby allowing available conventional finite difference formulations to be converted readily to trigonometric form. For two-dimensional problems, the procedure introduces two numerical parameters into the analysis. Engineering approaches for the selection of these parameters are presented and the analysis procedure is demonstrated by application to several isotropic and orthotropic panel buckling problems. Among these problems is the shear buckling of stiffened isotropic and filamentary composite panels in which the stiffener is broken. Results indicate that a break may degrade the effect of the stiffener to the extent that the panel will not carry much more load than if the stiffener were absent.
Spin Squeezing and Entanglement via Finite-Dimensional Discrete Phase-Space Description
NASA Astrophysics Data System (ADS)
Marchiolli, Marcelo A.; Galetti, Diógenes; Debarba, Tiago
2013-02-01
We show how mapping techniques inherent to N2-dimensional discrete phase spaces can be used to treat a wide family of spin systems which exhibits squeezing and entanglement effects. This algebraic framework is then applied to the modified Lipkin-Meshkov-Glick (LMG) model in order to obtain the time evolution of certain special parameters related to the Robertson-Schrödinger (RS) uncertainty principle and some particular proposals of entanglement measure based on collective angular-momentum generators. Our results reinforce the connection between both the squeezing and entanglement effects, as well as allow to investigate the basic role of spin correlations through the discrete representatives of quasiprobability distribution functions. Entropy functionals are also discussed in this context. The main sequence correlations ↦ entanglement ↦ squeezing of quantum effects embraces a new set of insights and interpretations in this framework, which represents an effective gain for future researches in different spin systems.
NASA Technical Reports Server (NTRS)
Ko, William L.
1988-01-01
Accuracies of solutions (structural temperatures and thermal stresses) obtained from different thermal and structural FEMs set up for the Space Shuttle Orbiter (SSO) are compared and discussed. For studying the effect of element size on the solution accuracies of heat-transfer and thermal-stress analyses of the SSO, five SPAR thermal models and five NASTRAN structural models were set up for wing midspan bay 3. The structural temperature distribution over the wing skin (lower and upper) surface of one bay was dome shaped and induced more severe thermal stresses in the chordwise direction than in the spanwise direction. The induced thermal stresses were extremely sensitive to slight variation in structural temperature distributions. Both internal convention and internal radiation were found to have equal effects on the SSO.
NASA Technical Reports Server (NTRS)
Handschuh, Robert F.
1987-01-01
An exponential finite difference algorithm, as first presented by Bhattacharya for one-dimensianal steady-state, heat conduction in Cartesian coordinates, has been extended. The finite difference algorithm developed was used to solve the diffusion equation in one-dimensional cylindrical coordinates and applied to two- and three-dimensional problems in Cartesian coordinates. The method was also used to solve nonlinear partial differential equations in one (Burger's equation) and two (Boundary Layer equations) dimensional Cartesian coordinates. Predicted results were compared to exact solutions where available, or to results obtained by other numerical methods. It was found that the exponential finite difference method produced results that were more accurate than those obtained by other numerical methods, especially during the initial transient portion of the solution. Other applications made using the exponential finite difference technique included unsteady one-dimensional heat transfer with temperature varying thermal conductivity and the development of the temperature field in a laminar Couette flow.
Kilinç, Yeliz; Erkmen, Erkan; Kurt, Ahmet
2016-01-01
The aim of the current study was to comparatively evaluate the mechanical behavior of 3 different fixation methods following various amounts of superior repositioning of mandibular anterior segment. In this study, 3 different rigid fixation configurations comprising double right L, double left L, or double I miniplates with monocortical screws were compared under vertical, horizontal, and oblique load conditions by means of finite element analysis. A three-dimensional finite element model of a fully dentate mandible was generated. A 3 and 5 mm superior repositioning of mandibular anterior segmental osteotomy were simulated. Three different finite element models corresponding to different fixation configurations were created for each superior repositioning. The von Mises stress values on fixation appliances and principal maximum stresses (Pmax) on bony structures were predicted by finite element analysis. The results have demonstrated that double right L configuration provides better stability with less stress fields in comparison with other fixation configurations used in this study.
Chen, Aijie; Feng, Xiaoli; Zhang, Yanli; Liu, Ruoyu; Shao, Longquan
2015-01-01
To investigate the stress distribution in a maxillary canine restored with each of four different post systems at different levels of alveolar bone loss. Two-dimensional finite element analysis (FEA) was performed by modeling a severely damaged canine with four different post systems: CAD/CAM zirconia, CAD/CAM glass fiber, cast titanium, and cast gold. A force of 100 N was applied to the crown, and the von Mises stresses were obtained. FEA revealed that the CAD/CAM zirconia post system produced the lowest maximum von Mises stress in the dentin layer at 115.8 MPa, while the CAD/CAM glass fiber post produced the highest stress in the dentin at 518.2 MPa. For a severely damaged anterior tooth, a zirconia post system is the best choice while a cast gold post ranks second. The CAD/CAM glass fiber post is least recommended in terms of stress level in the dentin.
Kudryavtsev, Oleg
2013-01-01
In the paper, we consider the problem of pricing options in wide classes of Lévy processes. We propose a general approach to the numerical methods based on a finite difference approximation for the generalized Black-Scholes equation. The goal of the paper is to incorporate the Wiener-Hopf factorization into finite difference methods for pricing options in Lévy models with jumps. The method is applicable for pricing barrier and American options. The pricing problem is reduced to the sequence of linear algebraic systems with a dense Toeplitz matrix; then the Wiener-Hopf factorization method is applied. We give an important probabilistic interpretation based on the infinitely divisible distributions theory to the Laurent operators in the correspondent factorization identity. Notice that our algorithm has the same complexity as the ones which use the explicit-implicit scheme, with a tridiagonal matrix. However, our method is more accurate. We support the advantage of the new method in terms of accuracy and convergence by using numerical experiments.
NASA Astrophysics Data System (ADS)
Yu, Peicheng; Li, Fei; Dalichaouch, Thamine; Fiuza, Frederico; Decyk, Viktor; Davidson, Asher; Tableman, Adam; An, Weiming; Tsung, Frank; Fonseca, Ricardo; Lu, Wei; Vieira, Jorge; Silva, Luis; Mori, Warren
2016-10-01
we present a finite-difference-time-domain (FDTD) Maxwell solver for the particle-in-cell (PIC) algorithm, which is customized to effectively eliminate the numerical Cerenkov instability (NCI) which arises when a plasma (neutral or non-neutral) relativistically drifts on a grid when using the PIC algorithm. We control the EM dispersion curve in the direction of the plasma drift of a FDTD Maxwell solver by using a customized higher order finite difference operator for the spatial derivative along the direction of the drift (1& circ; direction). We show that this eliminates the main NCI modes with moderate | k1 | , while keeps additional main NCI modes well outside the range of physical interest with higher | k1 | . These main NCI modes can be easily filtered out along with first spatial aliasing NCI modes which are also at the edge of the fundamental Brillouin zone. The customized solver has the possible advantage of improved parallel scalability because it can be easily partitioned along 1& circ; which typically has many more cells than other directions for the problems of interest.
A moving mesh finite difference method for equilibrium radiation diffusion equations
Yang, Xiaobo; Huang, Weizhang; Qiu, Jianxian
2015-10-01
An efficient moving mesh finite difference method is developed for the numerical solution of equilibrium radiation diffusion equations in two dimensions. The method is based on the moving mesh partial differential equation approach and moves the mesh continuously in time using a system of meshing partial differential equations. The mesh adaptation is controlled through a Hessian-based monitor function and the so-called equidistribution and alignment principles. Several challenging issues in the numerical solution are addressed. Particularly, the radiation diffusion coefficient depends on the energy density highly nonlinearly. This nonlinearity is treated using a predictor–corrector and lagged diffusion strategy. Moreover, the nonnegativity of the energy density is maintained using a cutoff method which has been known in literature to retain the accuracy and convergence order of finite difference approximation for parabolic equations. Numerical examples with multi-material, multiple spot concentration situations are presented. Numerical results show that the method works well for radiation diffusion equations and can produce numerical solutions of good accuracy. It is also shown that a two-level mesh movement strategy can significantly improve the efficiency of the computation.
DNS of Sheared Particulate Flows with a 3D Explicit Finite-Difference Scheme
NASA Astrophysics Data System (ADS)
Perrin, Andrew; Hu, Howard
2007-11-01
A 3D explicit finite-difference code for direct simulation of the motion of solid particulates in fluids has been developed, and a periodic boundary condition implemented to study the effective viscosity of suspensions in shear. The code enforces the no-slip condition on the surface of spherical particles in a uniform Cartesian grid with a special particle boundary condition based on matching the Stokes flow solutions next to the particle surface with a numerical solution away from it. The method proceeds by approximating the flow next to the particle surface as a Stokes flow in the particle's local coordinates, which is then matched to the finite difference update in the bulk fluid on a ``cage'' of grid points near the particle surface. (The boundary condition is related to the PHYSALIS method (2003), but modified for explicit schemes and with an iterative process removed.) Advantages of the method include superior accuracy of the scheme on a relatively coarse grid for intermediate particle Reynolds numbers, ease of implementation, and the elimination of the need to track the particle surface. For the sheared suspension, the effects of fluid and solid inertia and solid volume fraction on effective viscosity at moderate particle Reynolds numbers and concentrated suspensions will be discussed.
Development of an advanced finite-difference atmospheric general circulation model
Randall, D.A.
1992-03-01
We have proposed to provide and further develop an advanced finite-difference climate model for use in CHAMMP. The model includes advanced parameterizations of cumulus convection, boundary-layer processes, cloud formation, and land-surface vegetation, as well as parameterizations of radiative transfer and gravity wave drag. Postprocessing codes and a user's guide will also be provided. This research is being conducted in collaboration with Professors C.R. Mechoso and A. Arakawa at the University of California at Los Angeles (UCLA). The following research tasks are being carried out in support of CHAMMP: (1) Provide to CHAMMP a base-line finite-difference model and postprocessing codes for further development by the CHAMMP Science Team; (2) Provide to CHAMMP improved model physics to be developed in the course of our research project; (3) Provide to CHAMMP improved computational methods for use in the model; and, (4) Investigate the performance of current and to-be-developed physical parameterizations and computational methods at very high resolution.
Finite-Difference Schemes for a Scalar Reaction-Convection PDE.
NASA Astrophysics Data System (ADS)
Mickens, R. E.
1997-04-01
The scalar reaction-convection partial differential equation (SRCPDE) has been used as a model equation to investigate the properties of various numerical integration schemes.(R. J. LeVeque and H. C. Yee, J. Comput. Phys. 86), 187 (1990). In particular, these studies have provided information on the genesis of numerical instabilities and related issues such as stability and convergence of the solutions, the existence of ``false" fixed-points, bifurcation of solutions as the step-sizes are varied, etc.(H. C. Yee and P. K. Sweby, NASA Technical Memorandum 110398 (April 1996).) We consider a SRCPDE having a symmetric, nonlinear cubic source term in the dependent variable. A non-standard finite-difference scheme(R. E. Mickens, Nonstandard Finite Difference Models of Differential Equations) (World Scientific, 1994); Chapter 6. is constructed and its mathematical properties are studied and compared to those of the original PDE. We are also able to determine an exact scheme for the PDE. This research is supported by grants from ARO, DOE, NASA, and NIH-(MBRS).
A coarse-mesh nodal method-diffusive-mesh finite difference method
Joo, H.; Nichols, W.R.
1994-05-01
Modern nodal methods have been successfully used for conventional light water reactor core analyses where the homogenized, node average cross sections (XSs) and the flux discontinuity factors (DFs) based on equivalence theory can reliably predict core behavior. For other types of cores and other geometries characterized by tightly-coupled, heterogeneous core configurations, the intranodal flux shapes obtained from a homogenized nodal problem may not accurately portray steep flux gradients near fuel assembly interfaces or various reactivity control elements. This may require extreme values of DFs (either very large, very small, or even negative) to achieve a desired solution accuracy. Extreme values of DFs, however, can disrupt the convergence of the iterative methods used to solve for the node average fluxes, and can lead to a difficulty in interpolating adjacent DF values. Several attempts to remedy the problem have been made, but nothing has been satisfactory. A new coarse-mesh nodal scheme called the Diffusive-Mesh Finite Difference (DMFD) technique, as contrasted with the coarse-mesh finite difference (CMFD) technique, has been developed to resolve this problem. This new technique and the development of a few-group, multidimensional kinetics computer program are described in this paper.
Jia, X.; Mang, H.A.
2015-01-01
The consistently linearized eigenproblem (CLE) plays an important role in stability analysis of structures. Solution of the CLE requires computation of the tangent stiffness matrix K∼T and of its first derivative with respect to a dimensionless load parameter λ, denoted as K∼˙T. In this paper, three approaches of computation of K∼˙T are discussed. They are based on (a) an analytical expression for the derivative of the element tangent stiffness matrix K∼Te, (b) a load-based finite difference approximation (LBFDA), and (c) a displacement-based finite difference approximation (DBFDA). The convergence rate, the accuracy, and the computing time of the LBFDA and the DBFDA are compared, using the analytical solution as the benchmark result. The numerical investigation consists of the analysis of a circular arch subjected to a vertical point load at the vertex, and of a thrust-line arch under a uniformly distributed load. The main conclusion drawn from this work is that the DBFDA is superior to the LBFDA. PMID:25892827
Sheaffer, Jonathan; van Walstijn, Maarten; Fazenda, Bruno
2014-01-01
In finite difference time domain simulation of room acoustics, source functions are subject to various constraints. These depend on the way sources are injected into the grid and on the chosen parameters of the numerical scheme being used. This paper addresses the issue of selecting and designing sources for finite difference simulation, by first reviewing associated aims and constraints, and evaluating existing source models against these criteria. The process of exciting a model is generalized by introducing a system of three cascaded filters, respectively, characterizing the driving pulse, the source mechanics, and the injection of the resulting source function into the grid. It is shown that hard, soft, and transparent sources can be seen as special cases within this unified approach. Starting from the mechanics of a small pulsating sphere, a parametric source model is formulated by specifying suitable filters. This physically constrained source model is numerically consistent, does not scatter incoming waves, and is free from zero- and low-frequency artifacts. Simulation results are employed for comparison with existing source formulations in terms of meeting the spectral and temporal requirements on the outward propagating wave.
Hurrell, Andrew M
2008-06-01
The interaction of an incident sound wave with an acoustically impenetrable two-layer barrier is considered. Of particular interest is the presence of several acoustic wave components in the shadow region of this barrier. A finite difference model capable of simulating this geometry is validated by comparison to the analytical solution for an idealized, hard-soft barrier. A panel comprising a high air-content closed cell foam backed with an elastic (metal) back plate is then examined. The insertion loss of this panel was found to exceed the dynamic range of the measurement system and was thus acoustically impenetrable. Experimental results from such a panel are shown to contain artifacts not present in the diffraction solution, when acoustic waves are incident upon the soft surface. A finite difference analysis of this experimental configuration replicates the presence of the additional field components. Furthermore, the simulated results allow the additional components to be identified as arising from the S(0) and A(0) Lamb modes traveling in the elastic plate. These Lamb mode artifacts are not found to be present in the shadow region when the acoustic waves are incident upon the elastic surface.
On consistent boundary closures for compact finite-difference WENO schemes
NASA Astrophysics Data System (ADS)
Brehm, C.
2017-04-01
The accuracy of compact finite-difference schemes can be degraded by inconsistent domain or box boundary treatments. A consistent higher-order boundary closure is especially important for block-structured Cartesian AMR solvers, where the computational domain is generally decomposed into a large number of boxes containing a relatively small number of grid points. At each box boundary, a consistent higher-order boundary closure needs to be applied to avoid a reduction of the formal order-of-accuracy of the numerical scheme. This paper presents such a boundary closure for the fifth-order accurate compact finite-difference WENO scheme by Ghosh and Baeder [1]. The accuracy of the new boundary closure is validated by employing the method of manufactured solutions. A comparison of the new compact boundary closure with the original explicit boundary closure demonstrates the improved accuracy for the new compact boundary closure, while the behavior of the scheme across discontinuities appears unaffected. The linear stability analysis results indicate that a linearly stable compact WENO boundary closure is achieved.
2013-01-01
In the paper, we consider the problem of pricing options in wide classes of Lévy processes. We propose a general approach to the numerical methods based on a finite difference approximation for the generalized Black-Scholes equation. The goal of the paper is to incorporate the Wiener-Hopf factorization into finite difference methods for pricing options in Lévy models with jumps. The method is applicable for pricing barrier and American options. The pricing problem is reduced to the sequence of linear algebraic systems with a dense Toeplitz matrix; then the Wiener-Hopf factorization method is applied. We give an important probabilistic interpretation based on the infinitely divisible distributions theory to the Laurent operators in the correspondent factorization identity. Notice that our algorithm has the same complexity as the ones which use the explicit-implicit scheme, with a tridiagonal matrix. However, our method is more accurate. We support the advantage of the new method in terms of accuracy and convergence by using numerical experiments. PMID:24489518
2015-01-01
PURPOSE The objective of this study was to evaluate the influence of various cement types on the stress distribution in monolithic zirconia crowns under maximum bite force using the finite element analysis. MATERIALS AND METHODS The models of the prepared #46 crown (deep chamfer margin) were scanned and solid models composed of the monolithic zirconia crown, cement layer, and prepared tooth were produced using the computer-aided design technology and were subsequently translated into 3-dimensional finite element models. Four models were prepared according to different cement types (zinc phosphate, polycarboxylate, glass ionomer, and resin). A load of 700 N was applied vertically on the crowns (8 loading points). Maximum principal stress was determined. RESULTS Zinc phosphate cement had a greater stress concentration in the cement layer, while polycarboxylate cement had a greater stress concentration on the distal surface of the monolithic zirconia crown and abutment tooth. Resin cement and glass ionomer cement showed similar patterns, but resin cement showed a lower stress distribution on the lingual and mesial surface of the cement layer. CONCLUSION The test results indicate that the use of different luting agents that have various elastic moduli has an impact on the stress distribution of the monolithic zirconia crowns, cement layers, and abutment tooth. Resin cement is recommended for the luting agent of the monolithic zirconia crowns. PMID:26816578
He, Lihui; Liu, Lijie; Gao, Bei; Gao, Shang; Chen, Yifu; Zhihui, Liu
2013-08-01
To establish three-dimensional finite element model of two-piece post crown to the mandibular first molar residual roots, and analyze the stress distribution characteristic to the residual roots with different adhesives, so as to get the best combination under different conditions. The complete mandibular first molar in vitro was selected, the crown was removed along the cemento-enamel junction, then the residual roots were scanned by CT. CT images were imported into a reverse engineering software, and the three-dimensional finite element model of the mandibular first molar residual roots was reconstructed. Titanium two-piece post crown of the mandibular first molar residual roots was produced, then was scanned by CT. The model was reconstructed and assembled by MIMICS. The stress distribution of the root canal and root section under the vertical load and lateral load with different bonding systems were analyzed. Three-dimensional finite element model of two-piece post crown to the mandibular first molar residual roots was established. With the increasing of elastic modulus of the adhesives, the maximum stress within the root canal was also increasing. Elastic modulus of zinc phosphate was the biggest, so the stress within the root canal was the biggest; elastic modulus of Superbond C&B was the smallest, so the stress within the root canal was the smallest. Lateral loading stress was much larger than the vertical load. Under vertical load, the load on the root section was even with different bonding systems. Under lateral load, the maximum stress was much larger than the vertical load. The stress on the root section was minimum using zinc phosphate binder, and the stress on the root section was maximum using Superbond C&B. In two-piece post crown restorations, there is significant difference between different adhesives on tooth protection. When the tooth structure of the root canal orifices is weak, in order to avoid the occurrence of splitting, the larger elastic
NASA Technical Reports Server (NTRS)
Nordstrom, Jan; Carpenter, Mark H.
1998-01-01
Boundary and interface conditions for high order finite difference methods applied to the constant coefficient Euler and Navier-Stokes equations are derived. The boundary conditions lead to strict and strong stability. The interface conditions are stable and conservative even if the finite difference operators and mesh sizes vary from domain to domain. Numerical experiments show that the new conditions also lead to good results for the corresponding nonlinear problems.
Chang, Weng-Long; Ren, Ting-Ting; Feng, Mang
2015-01-01
In this paper, it is shown that the proposed quantum algorithm for implementing Boolean circuits generated from the DNA-based algorithm solving the vertex-cover problem of any graph G with m edges and n vertices is the optimal quantum algorithm. Next, it is also demonstrated that mathematical solutions of the same biomolecular solutions are represented in terms of a unit vector in the finite-dimensional Hilbert space. Furthermore, for testing our theory, a nuclear magnetic resonance (NMR) experiment of three quantum bits to solve the simplest vertex-cover problem is completed.
Takemoto, Hironori; Mokhtari, Parham; Kitamura, Tatsuya
2010-12-01
The vocal tract shape is three-dimensionally complex. For accurate acoustic analysis, a finite-difference time-domain method was introduced in the present study. By this method, transfer functions of the vocal tract for the five Japanese vowels were calculated from three-dimensionally reconstructed magnetic resonance imaging (MRI) data. The calculated transfer functions were compared with those obtained from acoustic measurements of vocal tract physical models precisely constructed from the same MRI data. Calculated transfer functions agreed well with measured ones up to 10 kHz. Acoustic effects of the piriform fossae, epiglottic valleculae, and inter-dental spaces were also examined. They caused spectral changes by generating dips. The amount of change was significant for the piriform fossae, while it was almost negligible for the other two. The piriform fossae and valleculae generated spectral dips for all the vowels. The dip frequencies of the piriform fossae were almost stable, while those of the valleculae varied among vowels. The inter-dental spaces generated very small spectral dips below 2.5 kHz for the high and middle vowels. In addition, transverse resonances within the oral cavity generated small spectral dips above 4 kHz for the low vowels.
Kleinstreuer, C.; Patterson, M.R.
1980-05-01
A two- or three-dimensional finite difference mesh generator capable of discretizing subrectangular flow regions (planar coordinates) with arbitrarily shaped bottom contours (vertical dimension) was developed. This economical, interactive computer code, written in FORTRAN IV and employing DISSPLA software together with graphics terminal, generates first a planar rectangular grid of variable element density according to the geometry and local kinematic flow patterns of a given fluid flow problem. Then subrectangular areas are deleted to produce canals, tributaries, bays, and the like. For three-dimensional problems, arbitrary bathymetric profiles (river beds, channel cross section, ocean shoreline profiles, etc.) are approximated with grid lines forming steps of variable spacing. Furthermore, the code works as a preprocessor numbering the discrete elements and the nodal points. Prescribed values for the principal variables can be automatically assigned to solid as well as kinematic boundaries. Cabinet drawings aid in visualizing the complete flow domain. Input data requirements are necessary only to specify the spacing between grid lines, determine land regions that have to be excluded, and to identify boundary nodes. 15 figures, 2 tables.
Finite Difference modeling of VLF Propagation in the Earth-Ionosphere Waveguide
NASA Astrophysics Data System (ADS)
Marshall, R. A.; Wallace, T.; Turbe, M.
2016-12-01
Very-low-frequency (VLF, 3—30 kHz) waves can propagate efficiently in the waveguide formed by the Earth and the D-region ionosphere. vVariation in the signals monitored by a stationary receiver can be attributed to variations in the lower ionosphere. As such, these signals are used to monitor the D-region ionosphere in daytime and nighttime. However, the use of VLF transmitter signals to quantitatively diagnose the D-region ionosphere is complicated by i) the propagation of many modes in the waveguide, and their interference, and ii) the effect of the ionosphere along the entire path on the receiver signal at a single location. In this paper, we compare the modeled phase and amplitude of VLF signals using three methods: a Finite-Difference Time-Domain (FDTD) model, a Finite-Difference Frequency-Domain (FDFD) model, and the Long-Wave Prediction Capability (LWPC) model, which has been the method de rigueur since the 1970s. While LWPC solves mode propagation and coupling in the anisotropic waveguide, the FD methods directly solve for electric and magnetic fields from Maxwell's equations on a finite-difference grid. Thus, FD methods provide greater freedom to vary the physical inputs of the model, limited only by the spatial resolution, but at the expense of computation time. We compare the simulated amplitude and phase of these models by running them with identical physical inputs. In this work we compare both i) the absolute amplitude and phase trends as a function of distance, and ii) the magnitude of amplitude and phase variations for given ionosphere changes. Modeling results show that FDTD and FDFD simulations track the amplitude and phase as a function of distance very closely when compared to LWPC. Phase drift due to numerical dispersion is observed at large distances, of a few tens of degrees per 1000 km. These phase drifts increase quadratically with frequency, as expected from numerical dispersion in FD methods. In fact, the phase drift can be mostly
NASA Astrophysics Data System (ADS)
Vincenti, H.; Vay, J.-L.
2016-03-01
Very high order or pseudo-spectral Maxwell solvers are the method of choice to reduce discretization effects (e.g. numerical dispersion) that are inherent to low order Finite-Difference Time-Domain (FDTD) schemes. However, due to their large stencils, these solvers are often subject to truncation errors in many electromagnetic simulations. These truncation errors come from non-physical modifications of Maxwell's equations in space that may generate spurious signals affecting the overall accuracy of the simulation results. Such modifications for instance occur when Perfectly Matched Layers (PMLs) are used at simulation domain boundaries to simulate open media. Another example is the use of arbitrary order Maxwell solver with domain decomposition technique that may under some condition involve stencil truncations at subdomain boundaries, resulting in small spurious errors that do eventually build up. In each case, a careful evaluation of the characteristics and magnitude of the errors resulting from these approximations, and their impact at any frequency and angle, requires detailed analytical and numerical studies. To this end, we present a general analytical approach that enables the evaluation of numerical errors of fully three-dimensional arbitrary order finite-difference Maxwell solver, with arbitrary modification of the local stencil in the simulation domain. The analytical model is validated against simulations of domain decomposition technique and PMLs, when these are used with very high-order Maxwell solver, as well as in the infinite order limit of pseudo-spectral solvers. Results confirm that the new analytical approach enables exact predictions in each case. It also confirms that the domain decomposition technique can be used with very high-order Maxwell solvers and a reasonably low number of guard cells with negligible effects on the whole accuracy of the simulation.
NASA Technical Reports Server (NTRS)
Chan, S. T. K.; Lee, C. H.; Brashears, M. R.
1975-01-01
A finite element algorithm for solving unsteady, three-dimensional high velocity impact problems is presented. A computer program was developed based on the Eulerian hydroelasto-viscoplastic formulation and the utilization of the theorem of weak solutions. The equations solved consist of conservation of mass, momentum, and energy, equation of state, and appropriate constitutive equations. The solution technique is a time-dependent finite element analysis utilizing three-dimensional isoparametric elements, in conjunction with a generalized two-step time integration scheme. The developed code was demonstrated by solving one-dimensional as well as three-dimensional impact problems for both the inviscid hydrodynamic model and the hydroelasto-viscoplastic model.
Serpentine: Finite Difference Methods for Wave Propagation in Second Order Formulation
Petersson, N A; Sjogreen, B
2012-03-26
second order system is significantly smaller. Another issue with re-writing a second order system into first order form is that compatibility conditions often must be imposed on the first order form. These (Saint-Venant) conditions ensure that the solution of the first order system also satisfies the original second order system. However, such conditions can be difficult to enforce on the discretized equations, without introducing additional modeling errors. This project has previously developed robust and memory efficient algorithms for wave propagation including effects of curved boundaries, heterogeneous isotropic, and viscoelastic materials. Partially supported by internal funding from Lawrence Livermore National Laboratory, many of these methods have been implemented in the open source software WPP, which is geared towards 3-D seismic wave propagation applications. This code has shown excellent scaling on up to 32,768 processors and has enabled seismic wave calculations with up to 26 Billion grid points. TheWPP calculations have resulted in several publications in the field of computational seismology, e.g.. All of our current methods are second order accurate in both space and time. The benefits of higher order accurate schemes for wave propagation have been known for a long time, but have mostly been developed for first order hyperbolic systems. For second order hyperbolic systems, it has not been known how to make finite difference schemes stable with free surface boundary conditions, heterogeneous material properties, and curvilinear coordinates. The importance of higher order accurate methods is not necessarily to make the numerical solution more accurate, but to reduce the computational cost for obtaining a solution within an acceptable error tolerance. This is because the accuracy in the solution can always be improved by reducing the grid size h. However, in practice, the available computational resources might not be large enough to solve the problem with a
Constructing space difference schemes which satisfy a cell entropy inequality
NASA Technical Reports Server (NTRS)
Merriam, Marshal L.
1989-01-01
A numerical methodology for solving convection problems is presented, using finite difference schemes which satisfy the second law of thermodynamics on a cell-by-cell basis in addition to the usual conservation laws. It is shown that satisfaction of a cell entropy inequality is sufficient, in some cases, to guarantee nonlinear stability. Some details are given for several one-dimensional problems, including the quasi-one-dimensional Euler equations applied to flow in a nozzle.
Constructing space difference schemes which satisfy a cell entropy inequality
NASA Technical Reports Server (NTRS)
Merriam, Marshal L.
1989-01-01
A numerical methodology for solving convection problems is presented, using finite difference schemes which satisfy the second law of thermodynamics on a cell-by-cell basis in addition to the usual conservation laws. It is shown that satisfaction of a cell entropy inequality is sufficient, in some cases, to guarantee nonlinear stability. Some details are given for several one-dimensional problems, including the quasi-one-dimensional Euler equations applied to flow in a nozzle.
Development of the Finite Difference Time Domain Method on a Lebedev Grid for Anisotropic Materials
NASA Astrophysics Data System (ADS)
Nauta, Marcel D.
The finite-difference time-domain (FDTD) method is derived on a Lebedev grid, instead of the standard Yee grid, to better represent the constitutive relations in anisotropic materials. The Lebedev grid extends the Yee grid by approximating Maxwell's equations with tensor constitutive relations using only central differences. A dispersion relation with stability criteria is derived and it is proven that the Lebedev grid has a consistent calculus. An integral derivation of the update equations illustrates how to appropriately excite the grid. This approach is also used to derive the update equations at planar material interfaces and domain edge PEC. Lebedev grid results are compared with analytical and Yee grid solutions using an equal memory comparison. Numerical results show that the Lebedev grid suffers greater dispersion error but better represents material interfaces. Focus is given to generalizing the concepts that make the Yee grid robust for isotropic materials. Keywords: FDTD, anisotropic materials, Lebedev grid, collocated grids.
NASA Astrophysics Data System (ADS)
Székely, Ferenc
2008-04-01
SummaryThe iterative composite mesh simulation (CMS) technique operates a coupled system of point centered finite difference groundwater flow models. It allows for high lateral and vertical mesh resolution at the sites of interest, whereas a coarser mesh may be applied in other parts of the formation. The boxed spatial zooming method utilizes a "box-in-box" architecture to build the coupled 3D system of embedded (nested) parent and child meshes exhibiting different vertical and lateral extensions and resolutions. A mesh interface simulator is used to equate the heads and balance the fluxes along the common vertical plains and shared lateral layers (interfaces) linking the meshes. This multi mesh simulation method is used by software FLOW and has been successfully tested in multi-aquifer systems against selected unsteady and steady state well flow problems with available analytical solutions. The base compatibility with MODFLOW databases supports wider use of the FLOW simulator.
Simulating incompressible flow on moving meshfree grids using General Finite Differences (GFD)
NASA Astrophysics Data System (ADS)
Vasyliv, Yaroslav; Alexeev, Alexander
2016-11-01
We simulate incompressible flow around an oscillating cylinder at different Reynolds numbers using General Finite Differences (GFD) on a meshfree grid. We evolve the meshfree grid by treating each grid node as a particle. To compute velocities and accelerations, we consider the particles at a particular instance as Eulerian observation points. The incompressible Navier-Stokes equations are directly discretized using GFD with boundary conditions enforced using a sharp interface treatment. Cloud sizes are set such that the local approximations use only 16 neighbors. To enforce incompressibility, we apply a semi-implicit approximate projection method. To prevent overlapping particles and formation of voids in the grid, we propose a particle regularization scheme based on a local minimization principle. We validate the GFD results for an oscillating cylinder against the lattice Boltzmann method and find good agreement. Financial support provided by National Science Foundation (NSF) Graduate Research Fellowship, Grant No. DGE-1148903.
On the computational noise of finite-difference schemes used in ocean models
NASA Technical Reports Server (NTRS)
Batteen, M. L.; Han, Y.-J.
1981-01-01
Different distributions of variables over the horizontal array of grid points in an ocean circulation model are investigated, using the shallow water equations as a guide in the choice of finite-difference schemes for use in ocean modeling. It is shown that the scheme with diffusive dissipation, in which the horizontal velocity is carried at the center and the height field is carried at each corner of a rectangular grid, successively suppresses numerical noise in a coarse (greater than 100 km) grid ocean model. For resolutions smaller than 50 km, it is shown that the scheme in which zonal velocity is carried at points to the east and west of the point of a rectangular grid where the height is carried, with meridional velocity carried to the north and south of the height point, can be free of noise for the gravest mode.
Samak, M. Mosleh E. Abu; Bakar, A. Ashrif A.; Kashif, Muhammad; Zan, Mohd Saiful Dzulkifly
2016-01-01
This paper discusses numerical analysis methods for different geometrical features that have limited interval values for typically used sensor wavelengths. Compared with existing Finite Difference Time Domain (FDTD) methods, the alternating direction implicit (ADI)-FDTD method reduces the number of sub-steps by a factor of two to three, which represents a 33% time savings in each single run. The local one-dimensional (LOD)-FDTD method has similar numerical equation properties, which should be calculated as in the previous method. Generally, a small number of arithmetic processes, which result in a shorter simulation time, are desired. The alternating direction implicit technique can be considered a significant step forward for improving the efficiency of unconditionally stable FDTD schemes. This comparative study shows that the local one-dimensional method had minimum relative error ranges of less than 40% for analytical frequencies above 42.85 GHz, and the same accuracy was generated by both methods.
Finite Difference Time Domain Electromagnetic Scattering from Frequency-Dependent Lossy Materials
NASA Technical Reports Server (NTRS)
Luebbers, Raymond J.; Beggs, John H.
1991-01-01
During this effort the tasks specified in the Statement of Work have been successfully completed. The extension of Finite Difference Time Domain (FDTD) to more complicated materials has been made. A three-dimensional FDTD code capable of modeling interactions with both dispersive dielectric and magnetic materials has been written, validated, and documented. This code is efficient and is capable of modeling interesting targets using a modest computer work station platform. However, in addition to the tasks in the Statement of Work, a significant number of other FDTD extensions and calculations have been made. RCS results for two different plate geometries have been reported. The FDTD method has been extended to computing far zone time domain results in two dimensions. Finally, the capability to model nonlinear materials has been incorporated into FDTD and validated. The FDTD computer codes developed have been supplied, along with documentation, and preprints describing the other FDTD advances have been included with this report as attachments.
NASA Astrophysics Data System (ADS)
Hu, Zeming; Chen, Xuechun; Wu, Yulin
The block-implicit finite-difference method is used to calculate 3D incompressible turbulent flows in the body-fitted coordinate system. In the numerical discretization the hybrid difference scheme is used to treat Reynolds-averaged Navier-Stokes equations. The iterative solution of velocities and pressure on the flow field is obtained by solving simultaneously the Reynolds-averaged N-S equations and continuity equation for each cell. In the iterative process the Gauss-Seidel method is used to solve nonlinear algebraic equations. The turbulent flow is simulated by the k-epsilon turbulence modeling in conjunction with Reynolds equations. The turbulent flow of a curved duct with square cross sections is treated in detail.
Exact finite-size corrections for the spanning-tree model under different boundary conditions
NASA Astrophysics Data System (ADS)
Izmailian, N. Sh.; Kenna, R.
2015-02-01
We express the partition functions of the spanning tree on finite square lattices under five different sets of boundary conditions in terms of a principal partition function with twisted-boundary conditions. Based on these expressions, we derive the exact asymptotic expansions of the logarithm of the partition function for each case. We have also established several groups of identities relating spanning-tree partition functions for the different boundary conditions. We also explain an apparent discrepancy between logarithmic correction terms in the free energy for a two-dimensional spanning-tree model with periodic and free-boundary conditions and conformal field theory predictions. We have obtained corner free energy for the spanning tree under free-boundary conditions in full agreement with conformal field theory predictions.
NASA Technical Reports Server (NTRS)
Doherty, Michael P.; Dalsania, Vithal
1990-01-01
An analysis was performed to predict the thermal distortion of the solar dynamic concentrator for Space Station Freedom in low earth orbit and to evaluate the effects of that thermal distortion on concentrator on-orbit performance. The analysis required substructural finite element modeling of critical concentrator structural subsystems, structural finite element modeling of the concentrator, mapping of thermal loading onto the structural finite element model, and the creation of specialized postprocessors to assist in interpreting results. Concentrator temperature distributions and thermally induced displacements and slope errors and the resulting receiver flux distribution profiles are discussed. Results determined for a typical orbit indicate that concentrator facet rotations are less than 0.2 mrad and that the change in facet radius due to thermal flattening is less than 5 percent. The predicted power loss due to thermal distortion effects is less than 0.3 percent. As a consequence the thermal distortions of the solar dynamic concentrator in low earth orbit will have a negligible effect on the flux distribution profiles within the receiver.
Modeling of piezoelectric transducers with combined pseudospectral and finite-difference methods.
Filoux, E; Callé, S; Certon, D; Lethiecq, M; Levassort, F
2008-06-01
A new hybrid finite-difference (FD) and pseudospectral (PS) method adapted to the modeling of piezoelectric transducers (PZTs) is presented. The time-dependent equations of propagation are solved using the PS method while the electric field induced in the piezoelectric material is determined through a FD representation. The purpose of this combination is to keep the advantages of both methods in one model: the adaptability of FD representation to model piezoelectric elements with various geometries and materials, and the low number of nodes per wavelength required by the PS method. This approach is implemented to obtain an accurate algorithm to simulate the propagation of acoustic waves over large distances, directly coupled to the calculation of the electric field created inside the piezoelectric material, which is difficult with classical algorithms. These operations are computed using variables located on spatially and temporally staggered grids, which attenuate Gibbs phenomenon and increase the algorithm's accuracy. The two-dimensional modeling of a PZT plate excited by a 50 MHz sinusoidal electrical signal is performed. The results are successfully compared to those obtained using the finite-element (FE) algorithm of ATILA software with configurations spatially and temporally adapted to the FE requirements. The cost efficiency of the FD-PS time-domain method is quantified and verified.
Transfer-matrix approach for finite-difference time-domain simulation of periodic structures.
Deinega, Alexei; Belousov, Sergei; Valuev, Ilya
2013-11-01
Optical properties of periodic structures can be calculated using the transfer-matrix approach, which establishes a relation between amplitudes of the wave incident on a structure with transmitted or reflected waves. The transfer matrix can be used to obtain transmittance and reflectance spectra of finite periodic structures as well as eigenmodes of infinite structures. Traditionally, calculation of the transfer matrix is performed in the frequency domain and involves linear algebra. In this work, we present a technique for calculation of the transfer matrix using the finite-difference time-domain (FDTD) method and show the way of its implementation in FDTD code. To illustrate the performance of our technique we calculate the transmittance spectra for opal photonic crystal slabs consisting of multiple layers of spherical scatterers. Our technique can be used for photonic band structure calculations. It can also be combined with existing FDTD methods for the analysis of periodic structures at an oblique incidence, as well as for modeling point sources in a periodic environment.