Fast Method of Particular Solutions for Solving Partial Differential Equations

Method of particular solutions (MPS) has been implemented in many science and engineering problems but obtaining the closed-form particular solutions, the selection of the good shape parameter for various radial basis functions (RBFs) and simulation of the large-scale problems are some of the challenges which need to overcome. In this dissertation, we have used several techniques to overcome such challenges. The closed-form particular solutions for the Matérn and Gaussian RBFs were not known yet. With the help of the symbolic computational tools, we have derived the closed-form particular solutions of the Matérn and Gaussian RBFs for the Laplace and biharmonic operators in 2D and 3D. These derived particular solutions play an important role in solving inhomogeneous problems using MPS and boundary methods such as boundary element methods or boundary meshless methods. In this dissertation, to select the good shape parameter, various existing variable shape parameter strategies and some well-known global optimization algorithms have also been applied. These good shape parameters provide high accurate solutions in many RBFs collocation methods. Fast method of particular solutions (FMPS) has been developed for the simulation of the large-scale problems. FMPS is based on the global version of the MPS. In this method, partial differential equations are discretized by the usual MPS and the determination of the unknown coefficients is accelerated using a fast technique. Numerical results confirm the efficiency of the proposed technique for the PDEs with a large number of computational points in both two and three dimensions. We have also solved the time fractional diffusion equations by using MPS and FMPS

The method of fundamental solutions for solving wave equations

In recent years, the method of fundamental solutions (MFS) has emerged as a novel meshless method in the scientific computing community. In the past, the MFS was essentially restricted to solving homogeneous elliptic equations. Recently, the MFS has gradually extended to solving various types of elliptic and time-dependent problems through the uses of radial basis functions (RBFs); In this thesis, we focus on solving wave equations through the MFS. Currently, there are two major approaches to solve the wave equation: (i) elimination of the time dependence by using the Laplace transform and (ii) discretization in time to approximate the time derivative. We propose to reduce the given wave equations to a series of inhomogeneous modified Helmholtz equations. The solution can then be split into evaluating both homogeneous and particular solutions. To evaluate the homogeneous solution, the MFS is adopted. Furthermore, a closed form particular solution is required for the proposed method. Intensive numerical tests are performed to compare the advantages and disadvantages of these two approaches

Radial Basis Function Differential Quadrature Method for the Numerical Solution of Partial Differential Equations

In the numerical solution of partial differential equations (PDEs), there is a need for solving large scale problems. The Radial Basis Function Differential Quadrature (RBFDQ) method and local RBF-DQ method are applied for the solutions of boundary value problems in annular domains governed by the Poisson equation, inhomogeneous biharmonic equation, and the inhomogeneous Cauchy-Navier equations of elasticity. By choosing the collocation points properly, linear systems can be obtained so that the coefficient matrices have block circulant structures. The resulting systems can be efficiently solved using matrix decomposition algorithms (MDAs) and fast Fourier transforms (FFTs). For the local RBFDQ method, the MDAs used are modified to account for the sparsity of the arrays involved in the discretization. An adjusted Fasshauer estimate is used to obtain a good shape parameter value in the applied radial basis functions (RBFs) for the global RBF-DQ method while the leave-one-out cross validation (LOOCV) algorithm is employed for the local RBF-DQ method using a sample of local influence domains. A modification of the kdtree algorithm is used to select the nearest centers for each local domain. In several numerical experiments, it is shown that the proposed algorithms are capable of solving large scale problems while maintaining high accuracy

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

Micro-macro Modeling of Advanced Materials by Hybrid Finite Element Method

Advanced composite materials are increasingly used in a variety of fields due to their desirable properties. The use of these advanced materials in different applications requires a thorough understanding of the effect of their complex microstructures and the effect of the operating environment on the materials. This requires an efficient, robust and powerful tool that is able to predict the behavior of composites under a variety of loading conditions. This research addresses this problem and develops a new convenient numerical method and framework for users to perform such analyses of composites. In this thesis, the hybrid fundamental solution based finite element method (HFS-FEM) is developed and applied to model composite materials across microscale and macroscale and from single field to multi-field. The basic idea and detailed formulations of the HFS-FEM for elasticity and potential problems are first presented. Then this method is extended to solve general three-dimensional (3D) elasticity problems with body forces and to model anisotropic materials encountered in composite analysis. Standard tests for proposed elements are carried out to assess their performance. Further, an efficient numerical homogenization method based on HFS-FEM is applied to predict the macroscopic elasticity properties and thermal conductivity of heterogeneous composites in micromechanical analysis. The effect of material parameters, such as fiber volume fractions, inclusion shapes and arrangements on the effective coefficients of composites are investigated by means of the proposed micro. mechanical models. Meanwhile, special elements are also proposed for mesh reduction and efficiency improvement in the analyses. Finally, the HFS-FEM method is developed for modeling two-dimensional (2D) and 3D thermoelastic problems. The particular solutions related to the body force and temperature change are approximated using the radial basis function interpolation. The new HFS-FEM is also developed for modeling plane piezoelectric materials in two different formulations: Lekhnitskii formalism and Stroh formalism. Numerical examples are provided for each kind of problems to demonstrate the accuracy, efficiency and versatility of the proposed method