Linkage between the unit logarithmic capacity in the theory of complex variables and the degenerate scale in the BEM/BIEMs

Kuo, Shyh‐Rong; Chen, Jeng‐Tzong; Kao, Shing‐Kai

doi:10.1016/j.aml.2013.04.011

Cited by 39 publications

(15 citation statements)

References 13 publications

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“…This causes troubles in seeking the true solution as well as the numerical solutions, which is called the degenerate scale problems (or degenerate scales). Recently, the study of degenerate scales becomes active, see [7,[9][10][11][13][14][15][16]18]. The singularity of the BIE and its numerical algorithms may be different.…”

Section: No Risk Of Degenerate Scalesmentioning

confidence: 97%

Boundary methods for Dirichlet problems of Laplace׳s equation in elliptic domains with elliptic holes

Zhang

Wei

et al. 2015

Engineering Analysis with Boundary Elements

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Section: No Risk Of Degenerate Scalesmentioning

confidence: 97%

Boundary methods for Dirichlet problems of Laplace׳s equation in elliptic domains with elliptic holes

Zhang

Wei

et al. 2015

Engineering Analysis with Boundary Elements

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“…However, the equivalence of the solution space in the BIEM and the partial differential equation (PDE) was not noticed. Degenerate scale [1], degenerate boundary [2], spurious eigenvalues and fictitious frequencies [3] …etc. only occurring in the BEM instead of the finite element method (FEM) indicate that the equivalence check is very important.…”

Section: Introductionmentioning

confidence: 99%

Revisit of the Dual Bem using SVD Updating Technique

et al. 2015

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The boundary element method (BEM) is easier than the finite element method (FEM) on the viewpoint of the discretization of one dimension reduction rather than the domain discretization of finite element method. The disadvantage of BEM is the rank deficiency in the influence matrix, e.g., degenerate boundary, degenerate scale, spurious eigenvalues and fictitious frequencies, which do not occur in the FEM. The conventional BEM can not be straightforward applied to solve a problem which contains a degenerate boundary without decomposing the domain to multi-regions. A hypersingular integral equation is used to ensure a unique solution for the problem containing a degenerate boundary. By combining the singular and hypersingular equations, it's termed the dual BEM due to its dual frame. Following the successful experience on the retrieval of information using the singular value decomposition (SVD) updating term and updating document, this technique is also used to extract out the degenerate-boundary information and the rigid-body information in the dual BEM. It is interesting to find that true information due to a rigid-body mode in physics is found in the right singular vector with respect to the corresponding zero singular value while the degenerate-boundary mode (geometry degeneracy) in mathematics is imbedded in the left singular vector with respect to the corresponding zero singular value. The role of the common right and left singular vectors of SVD for the four influence matrices in the dual BEM is also discussed in this paper. Two examples, a potential flow problem across a cutoff wall and a cracked bar under torsion were demonstrated to see the mathematical SVD structure of four influence matrices in the dual BEM.

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“…The proof why unit logarithmic capacity due to the Riemann conformal mapping in the theory of complex variables results in the degenerate scale was provided [23]. Based on the null-field integral approach the piezoelectricity problems with multiple elliptical inhomogeneities was solved [24].…”

Section: Introductionmentioning

confidence: 99%

Solution for Eshelby’s elliptic inclusion with polynomials distribution of the eigenstrains in plane elasticity

Chen

2014

Applied Mathematical Modelling

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Keywords:Eshelby's elliptic inclusion Plane elasticity Polynomials distribution of eigenstrains Complex variable method Closed form solution a b s t r a c t This paper provides a closed form solution for the Eshelby's elliptic inclusion in plane elasticity with the polynomials distribution of the eigenstrains. The complex variable method and the conformal mapping technique are used. The continuity conditions for the traction and displacement along the interface in the physical plane are reduced to a similar condition along the unit circle of the mapping plane. From those continuity conditions, we can obtain two sets of the complex potentials for the region outside of the unit circle and for the ring region, respectively. Further, we can obtain the complex potentials in the physical plane, or z-plane (z = x + iy). The mapping function maps the ring region in the mapping plane into a finite elliptic region with a crack on the real axis in the physical plane.

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Linkage between the unit logarithmic capacity in the theory of complex variables and the degenerate scale in the BEM/BIEMs

Cited by 39 publications

References 13 publications

Boundary methods for Dirichlet problems of Laplace׳s equation in elliptic domains with elliptic holes

Boundary methods for Dirichlet problems of Laplace׳s equation in elliptic domains with elliptic holes

Revisit of the Dual Bem using SVD Updating Technique

Solution for Eshelby’s elliptic inclusion with polynomials distribution of the eigenstrains in plane elasticity

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