A general finite difference method for arbitrary meshes

Perrone, Nicholas; Kao, Robert

doi:10.1016/0045-7949(75)90018-8

Cited by 264 publications

(125 citation statements)

References 5 publications

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“…Let us now introduce some notation to allow us to write the Taylor series in matrix notation before we proceed with our discussion of solving (13). Let…”

Section: Weighted Least Squares Approximationsmentioning

confidence: 99%

“…Some earlier examples of least-squares approximations include [13] and [14], which attempted to improve the conditioning of interpolation-based GFD. More recently, the least-squares-based GFD have been analyzed more systematically by Benito et al [15,27], and it been successfully applied to the solutions of parabolic and hyperbolic PDEs [15,28,27] and of advection-diffusion equation [29].…”

Section: Generalized Finite Difference and Weighted Least Squaresmentioning

confidence: 99%

“…Examples of such alternative methods include the diffuse element or element-free Galerkin methods [9,10], least-squares FEM [11], generalized or meshless finite different methods [12,13,14,15,16], generalized or extended FEM [17,18], and partition-of-unity FEM [19]. To reduce the dependency on mesh quality, these methods avoid the use of the piecewise-polynomial Lagrange basis functions found in the classical FEM.…”

Section: Introductionmentioning

confidence: 99%

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Overcoming element quality dependence of finite elements with adaptive extended stencil FEM (AES‐FEM)

Conley

Delaney

Jiao

2016

Numerical Meth Engineering

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The finite element methods (FEM) are important techniques in engineering for solving partial differential equations, but they depend heavily on element shape quality for stability and good performance. In this paper, we introduce the Adaptive Extended Stencil Finite Element Method (AES-FEM) as a means for overcoming this dependence on element shape quality. Our method replaces the traditional basis functions with a set of generalized Lagrange polynomial (GLP) basis functions, which we construct using local weighted least-squares approximations. The method preserves the theoretical framework of FEM, and allows imposing essential boundary conditions and integrating the stiffness matrix in the same way as the classical FEM. In addition, AES-FEM can use higher-degree polynomial basis functions than the classical FEM, while virtually preserving the sparsity pattern of the stiffness matrix. We describe the formulation and implementation of AES-FEM, and analyze its consistency and stability. We present numerical experiments in both 2D and 3D for the Poisson equation and a time-independent convection-diffusion equation. The numerical results demonstrate that AES-FEM is more accurate than linear FEM, is also more efficient than linear FEM in terms of error versus runtime, and enables much better stability and faster convergence of iterative solvers than linear FEM over poor-quality meshes.

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“…Let us now introduce some notation to allow us to write the Taylor series in matrix notation before we proceed with our discussion of solving (13). Let…”

Section: Weighted Least Squares Approximationsmentioning

confidence: 99%

Section: Generalized Finite Difference and Weighted Least Squaresmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Overcoming element quality dependence of finite elements with adaptive extended stencil FEM (AES‐FEM)

Conley

Delaney

Jiao

2016

Numerical Meth Engineering

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“…In addition, cubic splines are selected in both SPH, CSP, and GFD methods as the weighting function. A comparison of different criterions of particle selection and different weighting functions in the GFD method is given in [47]. In the following equations, we use W for the weighting function.…”

Section: Spatial Derivative Approximation By Gfdmentioning

confidence: 99%

“…Perrone and Kao [47] also contributed to the development of this method at that time. Subsequently, a variation of the GFD method using the moving least squares approximation was proposed by Lizska and Orkisz [48].…”

Section: Introductionmentioning

confidence: 99%

A Lagrangian Approach for Computational Acoustics with Meshfree Method

Zhang

Smith

Zhang

et al. 2017

Preprint

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Abstract:Although Eulerian approaches are standard in computational acoustics, they are less effective for certain classes of problems like bubble acoustics and combustion noise. A different approach for solving acoustic problems is to compute with individual particles following particle motion. In this paper, a Lagrangian approach to model sound propagation in moving fluid is presented and implemented numerically, using three meshfree methods to solve the Lagrangian acoustic perturbation equations (LAPE) in the time domain. The LAPE split the fluid dynamic equations into a set of hydrodynamic equations for the motion of fluid particles and perturbation equations for the acoustic quantities corresponding to each fluid particle. Then, three meshfree methods, the smoothed particle hydrodynamics (SPH) method, the corrective smoothed particle (CSP) method, and the generalized finite difference (GFD) method, are introduced to solve the LAPE and the linearized LAPE (LLAPE). The SPH and CSP methods are widely used meshfree methods, while the GFD method based on the Taylor series expansion can be easily extended to higher orders. Applications to modeling sound propagation in steady or unsteady fluids in motion are outlined, treating a number of different cases in one and two space dimensions. A comparison of the LAPE and the LLAPE using the three meshfree methods is also presented. The Lagrangian approach shows good agreement with exact solutions. The comparison indicates that the CSP and GFD method exhibit convergence in cases with different background flow. The GFD method is more accurate, while the CSP method can handle higher Courant numbers.

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A Finite Point Method in Computational Mechanics. Applications to Convective Transport and Fluid Flow

Oñate

Idelsohn

Zienkiewicz

et al. 1996

Int. J. Numer. Meth. Engng.

710

346

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The paper presents a fully meshless procedure for solving partial differential equations. The approach termed generically the 'finite point method' is based on a weighted least square interpolation of point data and point collocation for evaluating the approximation integrals. Some examples showing the accuracy of the method for solution of adjoint and non-self adjoint equations typical of convective-diffusive transport and also to the analysis of a compressible fluid mechanics problem are presented.

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A general finite difference method for arbitrary meshes

Cited by 264 publications

References 5 publications

Overcoming element quality dependence of finite elements with adaptive extended stencil FEM (AES‐FEM)

Overcoming element quality dependence of finite elements with adaptive extended stencil FEM (AES‐FEM)

A Lagrangian Approach for Computational Acoustics with Meshfree Method

A Finite Point Method in Computational Mechanics. Applications to Convective Transport and Fluid Flow

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