Research On Meshless Local Boundary Integral Equation Method: Algorithms And Applications

Conventional numerical methods, such as Finite Difference Method, Finite Element Method and Boundary Element Method, etc, have been widely adopted in the analysis of scientific and engineering problem, especially, many engineering softwares have also been developed based on FEM, which are general and available enough to form CAE industry. However, due to that the trail functions of finite element are based on the FE meshes, FEM meets limitations and difficulties when solving large deformation, moving boundary problem, and so on. The trail functions of meshless methods remove the dependence on the meshes, therefore the new algorithms have been widely investigated in the field of computational mechanics as soon as they arised. More than 30 schemes of meshless methods have been proposed for about one decade, and succeed in the impact with high velocity, hyper-large deformation, crack propagation, etc. As a whole, present meshless methods still require improving on whether theory or application research including the Local Boundary Integral Equation Method (LBIEM), which induces the idea of this paper.The current dissertation concentrates on the algorithm improvement and application research of the LBIEM, an important truly meshless method. The first part includes the choice of basic parameters of this algorithm, singularity treatment, adaptivity analysis, etc. the second part includes acoustic wave propagation and elasto-plastic problems.1,Improvement on the algorithmsIt is different at all that trail functions of meshless methods to approximate field variables are only based on the discrete nodes instead of meshes in FEM et al., which leads to more parameters to choose at the same time and also are these parameters the fundamental part of algorithms. The current paper presents a detailed observation on the compact weighted functions, radius of local sub-domain, radius of influence domain, companion solution, orthogonalized basis functions and boundary parameterization. Furthermore, more adaptive radii of local sub-domain and influence domain adaptively derived from node distribution are adopted in the implementation of LBIEM after neighbor nodes of every node are obtained with Delaunay triangulation.LBIEM can translate volume (or area) integral into area (or line) integral with the introduction of fundamental solution in BEM, while singularity will occur when integrating along the local boundary of boundary nodes. This dissertation presents a further investigation on the regularization treatment to overcome the singularity and expands it to treat strong singularity in the LBIE for Helmholtz problem. By the assessments of typical examples, the regularization treatment can tackle strongly singular integrals in the LBIEs of Laplace and Helmholtz equation problems. Furthermore, instead of LBIE for the boundary node, the moving least square approximation is introduced to enforce the boundary condition, then the so-called improved non-singular LBIEM is proposed. The improved LBIEM can avoid singular integral and reduce the approximation error in the boundary parameterization procedure. Numerical tests show that it is simple to implement but efficient.LBIEM is a truly meshless methods as its integrals are all performed over the local sub-domain and along the local boundary, therefore, adaptive analysis seems more promising. The dual error indicators are further studied and the potential derivatives are introduced to define the error. Making good use of post-processing technology that combines the moving least square approximation with Taylor expansion to obtain the potential derivatives which is more accurate than the original numerical solutions, then an a-posteriori error scheme is proposed. Numerical tests show that the proposed estimated error scheme can indicate the magnitude and distribution of real error efficiently.2,Extension on the applicationsEnough elements per wavelength to approximate the field variable are required when FEM is used to solve acoustic propagation problems, therefore, the computational scale augments accordingly with the wave number increasing. This paper extends LBIEM to analyze the acoustic wave propagation problems. Owing to the fitting advantage of MLSA, the frequency-dependent basis functions modified by the harmonic wave propagation solutions are easily adopted, and this treatment can highly improve the approximation accuracy and reduce the dispersion error without more additional nodes, which is promising in the acoustic analysis with high wave number cases.This dissertation gives a detailed introduction to LBIEM applications to two dimensional linear elastic problem, and assesses the feasibility of implementations. Then, after deriving the formulations for the plastic problem, the regularized hyper-singular LBIEs of displacement derivatives of boundary nodes are proposed to compute the strains directly, and further to obtain the more accurate stresses to iterate in one increment step.