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Finite difference

About: Finite difference is a research topic. Over the lifetime, 19693 publications have been published within this topic receiving 408603 citations.


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TL;DR: In this article, the displacement discontinuity method (DSM) is proposed for solving complex boundary value problems in plane elastostatics, which is similar to integral equation or influence function techniques, and contrasts with finite difference and finite element procedures in that approximations are made only on the boundary contours, and not in the field.
Abstract: This paper is concerned with the development of a numerical procedure for solving complex boundary value problems in plane elastostatics. This procedure—the displacement discontinuity method—consists simply of placing N displacement discontinuities of unknown magnitude along the boundaries of the region to be analyzed, then setting up and solving a system of algebraic equations to find the discontinuity values that produce prescribed boundary tractions or displacements. The displacement discontinuity method is in some respects similar to integral equation or ‘influence function’ techniques, and contrasts with finite difference and finite element procedures in that approximations are made only on the boundary contours, and not in the field. The method is illustrated by comparing computed results with the analytical solutions of two boundary value problems: a circular disc subjected to diametral compression, and a circular hole in an infinite plate under a uniaxial stress field. In both cases the numerical results are in excellent agreement with the exact solutions.

505 citations

Journal ArticleDOI
TL;DR: In this article, a numerical method for predicting the sinusoidal steady-state electromagnetic fields penetrating an arbitrary dielectric or conducting body is described, which employs the finite-difference time-domain (FD-TD) solution of Maxwell's curl equations implemented on a cubic-unit-cell space lattice.
Abstract: A numerical method for predicting the sinusoidal steady-state electromagnetic fields penetrating an arbitrary dielectric or conducting body is described here. The method employs the finite-difference time-domain (FD-TD) solution of Maxwell's curl equations implemented on a cubic-unit-cell space lattice. Small air-dielectric loss factors are introduced to improve the lattice truncation conditions and to accelerate convergence of cavity interior fields to the sinusoidal steady state. This method is evaluated with comparison to classical theory, method-of-moment frequency-domain numerical theory, and experimental results via application to a dielectric sphere and acylindrical metal cavity with an aperture. Results are also given for a missile-like cavity with two different types of apertures illuminated by an axial-incidence plane wave.

504 citations

Journal ArticleDOI
TL;DR: In this paper, an eddy-viscosity based, subgrid-scale model for large eddy simulations is derived from the analysis of the singular values of the resolved velocity gradient tensor.
Abstract: An eddy-viscosity based, subgrid-scale model for large eddy simulations is derived from the analysis of the singular values of the resolved velocity gradient tensor. The proposed σ-model has, by construction, the property to automatically vanish as soon as the resolved field is either two-dimensional or two-component, including the pure shear and solid rotation cases. In addition, the model generates no subgrid-scale viscosity when the resolved scales are in pure axisymmetric or isotropic contraction/expansion. At last, it is shown analytically that it has the appropriate cubic behavior in the vicinity of solid boundaries without requiring any ad-hoc treatment. Results for two classical test cases (decaying isotropic turbulence and periodic channel flow) obtained from three different solvers with a variety of numerics (finite elements, finite differences, or spectral methods) are presented to illustrate the potential of this model. The results obtained with the proposed model are systematically equivalent or slightly better than the results from the Dynamic Smagorinsky model. Still, the σ-model has a low computational cost, is easy to implement, and does not require any homogeneous direction in space or time. It is thus anticipated that it has a high potential for the computation of non-homogeneous, wall-bounded flows.

502 citations

Journal ArticleDOI
TL;DR: Although there remain many unresolved problems, multigrid or multilevel schemes in the classical framework of finite difference and finite element discretizations exhibit by now a comparatively clear profile.
Abstract: More than anything else, the increase of computing power seems to stimulate the greed for tackling ever larger problems involving large-scale numerical simulation. As a consequence, the need for understanding something like the intrinsic complexity of a problem occupies a more and more pivotal position. Moreover, computability often only becomes feasible if an algorithm can be found that is asymptotically optimal. This means that storage and the number of floating point operations needed to resolve the problem with desired accuracy remain proportional to the problem size when the resolution of the discretization is refined. A significant reduction of complexity is indeed often possible, when the underlying problem admits a continuous model in terms of differential or integral equations. The physical phenomena behind such a model usually exhibit characteristic features over a wide range of scales. Accordingly, the most successful numerical schemes exploit in one way or another the interaction of different scales of discretization. A very prominent representative is the multigrid methodology; see, for instance, Hackbusch (1985) and Bramble (1993). In a way it has caused a breakthrough in numerical analysis since, in an important range of cases, it does indeed provide asymptotically optimal schemes. For closely related multilevel techniques and a unified treatment of several variants, such as multiplicative or additive subspace correction methods, see Bramble, Pasciak and Xu (1990), Oswald (1994), Xu (1992), and Yserentant (1993). Although there remain many unresolved problems, multigrid or multilevel schemes in the classical framework of finite difference and finite element discretizations exhibit by now a comparatively clear profile. They are particularly powerful for elliptic and parabolic problems.

489 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
2023153
2022411
2021722
2020679
2019678
2018708