An Analytical Approach for the Green's Functions of Biharmonic Problems with Circular and Annular Domains

Chen, Jeng‐Tzong; Liao, Hsiang-Chou; Lee, W. M.

doi:10.1017/s1727719100003609

Cited by 6 publications

(2 citation statements)

References 14 publications

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“…Similar to the Laplace equation, degenerate scales for the 2D biharmonic equation results from the fundamental solution r 2 ln(r). Chen et al (2006a, 2005b, 2007a, 2007b, 2009d) had studied the biharmonic equation by using the null-field BIEM or Green’s function. Costabel and Dauge (1996) proposed the theory that a 4 by 4 matrix of the eigenproblem was derived for the biharmonic equation.…”

Section: Introductionmentioning

confidence: 99%

Degenerate-scale problem of the boundary integral equation method/boundary element method for the bending plate analysis

Chen

Kuo

Chang

et al. 2017

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Purpose The purpose of this paper is to detect the degenerate scale of a 2D bending plate analytically and numerically. Design/methodology/approach To avoid the time-consuming scheme, the influence matrix of the boundary element method (BEM) is reformulated to an eigenproblem of the 4 by 4 matrix by using the scaling transform instead of the direct-searching scheme to find degenerate scales. Analytical degenerate scales are derived from the boundary integral equation (BIE) by using the degenerate kernel only for the circular case. Numerical results of the direct-searching scheme and the eigen system for the arbitrary shape are also considered. Findings Results using three methods, namely, analytical derivation, the direct-searching scheme and the 4 by 4 eigen system, are also given for the circular case and arbitrary shapes. Finally, addition of a constant for the kernel function makes original eigenvalues (2 real roots and 2 complex roots) of the 4 by 4 matrix to be all real. This indicates that a degenerate scale depends on the kernel function. Originality/value The analytical derivation for the degenerate scale of a 2D bending plate in the BIE is first studied by using the degenerate kernel. Through the reformed eigenproblem of a 4 by 4 matrix, the numerical solution for the plate of an arbitrary shape can be used in the plate analysis using the BEM.

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Section: Introductionmentioning

confidence: 99%

Degenerate-scale problem of the boundary integral equation method/boundary element method for the bending plate analysis

Chen

Kuo

Chang

et al. 2017

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“…[18][19][20][21][22][23][24][25]). Recently, Chen et al [26] have analytically derived the Green's functions of biharmonic problems with circular and annular domains. Among pertinent works in this regard, the fundamental solution proposed by Pan and Chou [3] is particularly chosen and implemented in BEM for our present study.…”

Section: Introductionmentioning

confidence: 99%

On the Efficiency of Analyzing 3D Anisotropic, Transversely Isotropic, and Isotropic Bodies in BEM

Shiah¹,

Sun²

2010

J. mech.

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Due to a lack of closed-form solutions for three dimensional anisotropic bodies, the computational burden of evaluating the fundamental solutions in the boundary element method (BEM) has been a research focus over the years. In engineering practice, transversely isotropic material has gained popularity in the use of composites. As a degenerate case of the generally anisotropic material, transverse isotropy still needs to be treated separately to ease the computations. This paper aims to investigate the computational efficiency of the BEM implementations for 3D anisotropic, transversely isotropic, and isotropic bodies. For evaluating the fundamental solutions of 3D anisotropy, the explicit formulations reported in [1,2] are implemented. For treating transversely isotropic materials, numerous closed form solutions have been reported in the literature. For the present study, the formulations presented by Pan and Chou [3] are particularly employed. At the end, a numerical example is presented to compare the computational efficiency of the three cases and to demonstrate how the CPU time varies with the number of meshes.

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Fourier Expansion to Elastic Vector Wave Functions and Applications of Wave Bases to Scattering in Half-Space

2012

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This paper proposes a complete basis set for analyzing elastic wave scattering in half-space. The half-space is an isotropic, linear, and homogeneous medium except for a finite inhomogeneity. The wave bases are obtained by combining buried source functions and their reflected counter-waves generated from the infinite-plane boundary. The source functions are the vector wave functions of infinite-space. Based on the source functions expressed in the Fourier expansion form, the reflected counter-waves are easily obtained by solving the infinite-plane boundary conditions. Few representations adopt Wely's integration, but the Fourier expansion is developed from it and applied to decouple the angular-differential terms of the vector wave functions. In addition to the scattering of the finite inhomogeneity, the transition matrix method is extended to express the surface boundary conditions. For the numerical application in this paper, the P- and the SV- waves are assumed as the incoming fields. As an example, this paper computes stress concentrations around a cavity. The steepest-descent path method yielding the optimum integral paths is used to ensure the numerical convergence of the wave bases in the Fourier expansion. The resultant patterns from these approaches are compared with those obtained from numerical simulations.

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An Analytical Approach for the Green's Functions of Biharmonic Problems with Circular and Annular Domains

Cited by 6 publications

References 14 publications

Degenerate-scale problem of the boundary integral equation method/boundary element method for the bending plate analysis

Degenerate-scale problem of the boundary integral equation method/boundary element method for the bending plate analysis

On the Efficiency of Analyzing 3D Anisotropic, Transversely Isotropic, and Isotropic Bodies in BEM

Fourier Expansion to Elastic Vector Wave Functions and Applications of Wave Bases to Scattering in Half-Space

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