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CivilComp Proceedings
ISSN 17593433 CCP: 85
PROCEEDINGS OF THE FIFTEENTH UK CONFERENCE OF THE ASSOCIATION OF COMPUTATIONAL MECHANICS IN ENGINEERING Edited by: B.H.V. Topping
Paper 47
Solving TwoDimensional Orthotropic Potential Cauchy Problems of Thin Bodies using the Boundary Element Method H.L. Zhou^{1}^{2}, Z.W. Guan^{1} and Z.R. Niu^{2}
^{1}Department of Engineering, The University of Liverpool, United Kingdom
H.L. Zhou, Z.W. Guan, Z.R. Niu, "Solving TwoDimensional Orthotropic Potential Cauchy Problems of Thin Bodies using the Boundary Element Method", in B.H.V. Topping, (Editor), "Proceedings of the Fifteenth UK Conference of the Association of Computational Mechanics in Engineering", CivilComp Press, Stirlingshire, UK, Paper 47, 2007. doi:10.4203/ccp.85.47
Keywords: inverse problem, boundary element method, nearly singular integral, analytical integral, TSVD, orthotropic potential, undulatingcurve method.
Summary
It is well known that there are singular integrals and nearly singular integrals arising
in the boundary element method (BEM). Unlike singular integrals, nearly singular integrals are not singular in
the sense of mathematics. Conventional Gauss numerical quadrature is invalid to
evaluate these integrals. Nearly singular integrals usually exist in two situations.
One is known as boundary layer effect problem; the other is named as the thin body
effect problem.
It is often encountered that all the potentials and fluxes are known on a part of the boundary and no boundary data can be directly measured on the rest of boundary in engineering for potential problems. This is the Cauchy inverse problem. The truncated singular value decomposition (TSVD) [1] and the boundary element method are applied to solve the Cauchy inverse problem in linear elasticity [2]. The completely analytical integral algorithm proposed for 2D direct orthotropic potential problems [3] is applied to evaluate the nearly singular integrals in the inverse problems. Thin body problems are considered. The analytical integral formulas are directly derived with integration by parts. The present algorithm applies the new analytical formulas to deal with the nearly singular integrals. The system equation is solved by truncated singular value decomposition technique. The condition number and singular value characteristics of the system equation matrix A are discussed. In particular, for some thin body problems, matrix A appears to have a wellconditioned behaviour. However, some thin body Cauchy problems are rank deficient. For rank deficient problems, the second order physical field variables, namely flux solutions are dominant to determine the optimal number k associated with the useful singular values. If the flux solutions are accurate, the corresponding first order physical field variables potentials are always accurate. As the small singular value leads to the undulation of the numerical solutions, an undulatingcurve method is proposed to select the truncated number k. The unknown potential and flux boundary conditions for thin body problems with very small thicknesstolength ratios are accurately calculated. References
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