The boundary contour method for two-dimensional linear elasticity: Applications in stress analysis and shape optimization

This dissertation, based on four papers, is involved with novel variants of the conventional boundary element method (BEM), called the boundary contour method (BCM) and hypersingular boundary contour method (HBCM), for two-dimensional (2- D) linear elasticity. A further development of the 2-D BCM is carried out whereas a general theory of the 2-D HBCM is introduced to the literature for the first time. Applications of these methods to stress analysis and structural shape optimization are the main objectives of this study. Several numerical examples having known analytical solutions are solved in order to show the advantages of both methods in these kinds of applications.;The BEM is a general purpose approach which starts from the boundary integral equation (BIE) (see a derivation of this equation in section 4.1 of chapter IV) in order to solve a given boundary value problem. The key idea of the new methods consists of using special displacement and stress shape functions in the domain of a body that satisfy the equilibrium and constitutive equations. As a result, the integrand vectors of the BIE and regularized hypersingular boundary integral equation (HBIE) (see a derivation of this equation in section 4.8 of chapter IV) are divergent-free, and thus the dimension of the usual integrals in the above equations can be reduced by one. In other words, surface integrals for three-dimensional (3-D) problems and line integrals in 2-D cases can be converted respectively into line integrals and the evaluation of analytical functions at boundary nodes.;This reduction in dimensionality offered by the BCM and HBCM, as well as the fact that these methods use special shape functions, are expected to make them competitive with the finite element method (FEM) and the BEM for some applications in computational mechanics. Due to these above features and especially, the absence of numerical integrations in the BCM and HBCM for 2-D problems, numerical results obtained for stress analysis are very accurate as it can be seen from the numerical examples presented in the papers.;Design sensitivities are coefficients required for numerically solving an optimization problem. Hence, the accuracy of these quantities plays a crucial role in shape optimization. As for stress analysis, the accuracy of numerical results for design sensitivity analysis can be well ensured by the BCM. This advantage and the fact that, as for the BEM, the BCM only needs boundary meshing, as opposed to domain meshing required by the FEM, make the BCM a very attractive method in shape optimal design.;The aforementioned advantages offered by the BCM and HBCM are remarkable. They are clearly shown through the papers and especially, from numerical results of the illustrative examples. The research presented in this dissertation aims to introduce the BCM and HBCM for 2-D linear elasticity, as well as a new successful approach for numerically solving shape optimization problems, into the world of computational and applied mechanics.