A necessary and sufficient boundary integral formulation for plane elasticity problems

He, Wei; Ding, Hu; Hai-Chang, Hu

doi:10.1002/(sici)1099-0887(199607)12:7<413::aid-cnm988>3.0.co;2-2

Cited by 18 publications

(7 citation statements)

References 9 publications

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“…For a degenerate-scale problem containing an elliptical boundary, two different degenerate scales are given in equation (15) and equation (16). The exact solution of displacement field is given by equation (4) and equation (5), subjected to the boundary conditions of equation (6), where the angle of rotation ( r  ), the Lamé constants G and  are given as 30°, 1.0 and 0.25, respectively.…”

Section: An Illustrative Examplementioning

confidence: 96%

“…According to the above analytical result, it is found that only the coefficients of the constant term could not be determined due to the zero denominator when the size of the domain is a degenerate scale of equation (15) and equation (16). Range deficiency by a constant term for the solution space appears.…”

Section: Necessary and Sufficient Bem/biems For 2-d Elasticity Problemsmentioning

confidence: 99%

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A necessary and sufficient BEM/BIEM for two-dimensional elasticity problems

Chen

Huang

Lee

et al. 2015

WIT Transactions on Modelling and Simulation

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In the development of finite element method (FEM), the patch test is required. We may wonder whether we need any special test for the boundary element method (BEM). A sufficient and necessary boundary integral equation method (BIEM) to ensure a unique solution is our concern. In this paper, we revisit this issue for the interior two-dimensional (2-D) elasticity problem and investigate the equivalence of solution space between the integral equation and the partial differential equation. Based on the degenerate kernel and the eigenfunction expansion, the range deficiency of the integral operator of the single-layer potential for the solution space in the degenerate-scale problem for the 2-D elasticity in the BIEM is analytically studied. Following the Fichera's idea, we enrich the conventional BIEM by adding constants and corresponding constraints. In addition, we introduce the concept of modal participation factor (MPF) to examine whether the adding term of the rotation is required for interior simply-connected problems. Finally, a simple example of the degenerate-scale problem containing an elliptical boundary subjected to various boundary conditions of the rigid body translation and rotation for 2-D elasticity problems was demonstrated by using the necessary and sufficient BIEM.

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Section: An Illustrative Examplementioning

confidence: 96%

Section: Necessary and Sufficient Bem/biems For 2-d Elasticity Problemsmentioning

confidence: 99%

A necessary and sufficient BEM/BIEM for two-dimensional elasticity problems

Chen

Huang

Lee

et al. 2015

WIT Transactions on Modelling and Simulation

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“…He found that the conventional boundary integral equations for 2D potential and elasticity problems are not equivalent to the boundary value problem of corresponding partial differential equation. He and his co-authors at Peking University and Zhejiang University, presented the sufficient and necessary boundary integral equation equivalent to the partial differential equation for the boundary value problems [34,35] .…”

Section: Some Representative Bem Investigations By Other Groups In Chinamentioning

confidence: 99%

Some Aspects of the BEM Research in China

Yao

2007

EJBE

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In this paper, a short historical review of the BEM research in China is given first. The investigation on the BEM in China was started in 1978 at the Tsinghua University, based on the pioneering work published by Frank Rizzo [F. J. Rizzo, An integral equation approach to boundary value problems of classical elastostatics, Quart. Appl. Math., 25, 83-95 (1967)] and others. During the last 25 years, we have organized six national conferences on BEM, two international conferences and eight China-Japan/Japan-China symposia on BEM. The BEM investigations in the authors' group at Tsinghua are briefly introduced next. Some representative investigations by other groups in Mainland China are also introduced. The related papers and other publications are listed in the References. Historical ReviewIn China, we started the investigation on boundary element methods in 1978 at Tsinghua University. At first, the second author, Du, introduced the related papers published by Frank Rizzo, T. A. Cruse and other BEM pioneers to the researchers in China, and suggested to them to pay attentions on the development of the boundary integral equation (BIE) method. In the authors' group, we started with the elastic stress concentration problems using boundary integral equation -boundary element method, including axi-symmetric body with symmetric and asymmetric loading, and also three-dimensional elasticity problems. Another topic is the plate bending problems, which related to higher order differential operator. At that time, the paper of Frank Rizzo published in 1967 was the first literature on BEM available to us. In the first several years, we learnt that the well-known kernel functions can be generalized by superposition, integration and differentiation to obtain some useful new kernel functions for different problems, such as the torsion of circular shaft with variable diameters, axi-symmetric problems, plate bending problems and crack problems. In 1982, we published our work at Acta Mechanica Solida Sinica in Chinese [1] , and gave a workshop to more than 100 audiences organized by the editorial board of that journal.

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“…The degenerate scale problems (nonuniqueness) in BEM for potential problem [23] and plane elasticity [2,17,21,22] have been done even for the plate problem (biharmonic equation) [18,24] and numerical experiments have been performed [9]. Chen et al [9,12] have determined the degenerate scale for Laplace and Navier operators by using circulant and series expansion in terms of degenerate kernel for fundamental solutions [20].…”

Section: Introductionmentioning

confidence: 99%

Degenerate scale for the analysis of circular thin plate using the boundary integral equation method and boundary element methods

Chen

et al. 2005

Comput Mech

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In this paper, the degenerate scale for plate problem is studied. For the continuous model, we use the null-field integral equation, Fourier series and the series expansion in terms of degenerate kernel for fundamental solutions to examine the solvability of BIEM for circular thin plates. Any two of the four boundary integral equations in the plate formulation may be chosen. For the discrete model, the circulant is employed to determine the rank deficiency of the influence matrix. Both approaches, continuous and discrete models, lead to the same result of degenerate scale. We study the nonunique solution analytically for the circular plate and find degenerate scales. The similar properties of solvability condition between the membrane (Laplace) and plate (biharmonic) problems are also examined. The number of degenerate scales for the six boundary integral formulations is also determined.

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A necessary and sufficient boundary integral formulation for plane elasticity problems

Cited by 18 publications

References 9 publications

A necessary and sufficient BEM/BIEM for two-dimensional elasticity problems

A necessary and sufficient BEM/BIEM for two-dimensional elasticity problems

Some Aspects of the BEM Research in China

Degenerate scale for the analysis of circular thin plate using the boundary integral equation method and boundary element methods

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