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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 paper, implicit boundary condition procedures are presented for use with implicit finite difference schemes for the unsteady Euler equations, based on the mathematical theory of characteristics for hyperbolic systems of equations.
Abstract: Implicit boundary condition procedures are presented for use with implicit finite difference schemes for the unsteady Euler equations. This new boundary point treatment is based on the mathematical theory of characteristics for hyperbolic systems of equations. Along with the theoretical background, the practical application of the method to several types of boundaries is also explained using several examples. The specific boundary conditions covered include subsonic inflow and outflow, surface tangency, and shock waves. The example problems include one-dimensional Laval nozzle flow, dual-throat rocket engine nozzle flow, and supersonic flow past a sphere. The implicit boundary treatment permits the use of large time steps allowing the finite difference algorithm to converge to the asymptotic steady state much faster than schemes that use explicitly applied boundary conditions. At least an order of magnitude increase in computational speed is demonstrated in the examples shown.' Background T HE growing popularity of solutions to the Euler equations in transonics and their continued application in supersonics have increased the need for quicker solutions. The potential of implicit schemes in this direction has not been fully exploited for want of correct, implicit application of boundary conditions. The predominant use of implicit algorithms for the Navier-Stokes equations has partly been responsible for the neglect of implicit boundary point treatment for the Euler equations. Thus, there is a need for correct and stable procedures for the easy implicit application of boundary conditions. Such methods will serve the two purposes of 1) reaching time-asymptotic steady state faster and 2) permitting a time step for truly unsteady flow that is not necessarily restricted by the CFL stability criterion but is based upon the magnitude of the transients. For clues and information on how to construct such boundary condition procedures, one must turn to the mathematical theory of characteristi cs for hyperbolic systems of equations. The unsteady Euler equations belong to this category. The theory for hyperbolic systems is rich with information on signal propagation directions. The characteristics theory clearly points to the number of boundary conditions that may and need be prescribed without overdetermining the solution. Boundary condition procedures based on this theory have been known and applied for several years by Kentzer, 1 Porter and Coakley,2 de Neef, 3 and others. In earlier work by this author,4'5 easily understood and implementable methods for boundary point treatment were presented. However, all of the above techniques were developed for explicit finite difference schemes. It seems that it must be easy to extend such methodologies based on mathematical theory for hyperbolic systems to implicit finite difference schemes, and indeed, it is simple enough. The rest of this paper describes such implicit boundary condition procedures. The given examples illustrate in detail the application of the proposed methodology to specific types of boundaries and demonstrate the merits of the new scheme.

101 citations

Journal ArticleDOI
TL;DR: In this article, the analysis of static deformations, free vibrations and buckling loads on laminated composite plates is performed by local collocation with radial basis functions in a finite differences framework.

101 citations

Journal ArticleDOI
TL;DR: In this article, a new Rapid Expansion Method (REM) is proposed for the time integration of the acoustic wave equation and the equations of dynamic elasticity in two spatial dimensions, which is applicable to spatial grid methods such as finite differences, finite elements or the Fourier method.
Abstract: We present a new rapid expansion method (REM) for the time integration of the acoustic wave equation and the equations of dynamic elasticity in two spatial dimensions. The method is applicable to spatial grid methods such as finite differences, finite elements or the Fourier method. It is based on a Chebyshev expansion of the formal solution to the appropriate wave equation written in operator form. The method yields machine accuracy yet it is faster than methods based on temporal differencing. Its disadvantages are that it does not apply to all types of material rheology, and it can also require much storage when many snapshots and time sections are desired. Comparisons between numerical and analytical solutions for simple acoustic and elastic problems demonstrate the high accuracy of the REM.

101 citations

Journal ArticleDOI
TL;DR: In this article, a high-order Boussinesq-type finite difference model was extended to simulate wave propagation out to a large number of times depth in a high order infinite difference model.

101 citations

Journal ArticleDOI
01 Jan 1994
TL;DR: In this article, a method of finite differences is used to solve the forward problem for a given piston configuration; some nontrivial issues arise in determining boundary conditions, and the finite difference equations are then rearranged into a linear system of equations which formulates the inverse problem.
Abstract: In elasticity imaging, a surface deformation is applied to an object using small pistons, and the resulting induced strains in the interior of the object are measured using ultrasonic imaging. Two important problems are considered: (1) the forward problem of determining the strains induced by a known deformation of an object with known elasticity; and (2) the inverse problem of reconstructing elasticity from measured strains and the equations of equilibrium. The method of finite differences is used to solve the forward problem for a given piston configuration; some nontrivial issues arise in determining boundary conditions. The finite difference equations are then rearranged into a linear system of equations which formulates the inverse problem; this system can be solved for the unknown elasticities. This formulation of the inverse problem is completely consistent with the forward problem; this is useful for iterative methods in which the deformation is adaptively changed. A comparison between simulated and actual measured results demonstrate the feasibility of the proposed procedure. >

101 citations


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