Benefited from the reduced dimensionality and the fundamental solutions, the boundary element method (BEM) has been firmly established as an effective method for solving linear engineering problems with complex 3-D geometries. For non-homogeneous and/or nonlinear problems, a major difficulty for the BEM to be competitive in solving nonlinear or inhomogeneous problems is to find an efficient and accurate way to evaluate the volume integrals containing the nonhomogeneous/nonlinear term in the integral formulation'. Various methods have been proposed and developed. However, these methods either loss the boundary-only essence of BEM or not accurate due to approximation of source terms by radial basis function. In this thesis work, an efficient and accurate boundary integral approach...[ Read more ]

Benefited from the reduced dimensionality and the fundamental solutions, the boundary element method (BEM) has been firmly established as an effective method for solving linear engineering problems with complex 3-D geometries. For non-homogeneous and/or nonlinear problems, a major difficulty for the BEM to be competitive in solving nonlinear or inhomogeneous problems is to find an efficient and accurate way to evaluate the volume integrals containing the nonhomogeneous/nonlinear term in the integral formulation'. Various methods have been proposed and developed. However, these methods either loss the boundary-only essence of BEM or not accurate due to approximation of source terms by radial basis function. In this thesis work, an efficient and accurate boundary integral approach is developed for nonlinear problems based on the grid-based direct integration method. In this method, the original volume integral is replaced by an integral of which the source function is continuous over the entire grid. As such the volume integration can be evaluated on the Cartesian grid directly without the need to identify and discretize the problem domain embedded inside the grid. The advantages of this method is that no volume fitting mesh is needed, it is also very accurate and efficient due to acceleration algorithms. In this thesis, we firstly present the complexity and accuracy analysis of the BEM coupled with the the grid-based direct-volume integration BEM, for solving quasilinear problems. Various numerical examples are employed to verify the theoretical findings. Then the extension of the grid-based direct volume integration BEM to 3-D geometrically nonlinear elasticity will be investigated with the introduction of Galerkin vectors for elasticity. The regularizations of strong singular and hypersingular boundary integrals, as well as strongly singular domain integrals are also presented. Numerical examples have been performed to demonstrate the effectiveness of the three-dimensional formulas and numerical implementation.