Indirect Boundary Element Analysis For Non-homogeneous And Non-linear Problems

Many problems in physics, mechanics and engineering, can be summed up in solving a partial differential equation with the initial boundary value problem. However, apart from a few simple questions, most of the problems have to be turned to numerical methods to find their approximate solutions. At present, finite difference method(FDM), finite element method(FEM) and boundary element method(BEM) are the most important numerical methods in the field of engineering technology, and have been widely used in scientific research and engineering applications.Boundary element method is an effective numerical method. Compared to other numerical methods, the boundary element method has many advantages, such as reducing the dimension and discretization error coming from the boundary. However, in the boundary element method, the differential equation is transformed into an integral equation to be solved. If the fundamental solution of the differential equation is unknown, it is very difficult to solve the equation.Nonlinear science has become an important issue in numerous fundamental researchs and engineering applications. The quantitative study on the nonlinear problem depends on the quantitative solution of the nonlinear differential equation. It is different from the linear problem, and solving nonlinear differential equations is generally very difficult, so only a very few simple problems can find exact solutions. For nearly a century, although researchers in solving the nonlinear differential equations of the two main ways, analytical method and numerical method made great efforts, still lack of a direct access to all kinds of weak nonlinear and strongly nonlinear problems with high accuracy approximate solution of the general method.Based on the indirect boundary element method(IBEM) and radial basis functions(RBFs) theory, a novel method, called “source term iteration regularization”, is proposed for solving the non homogeneous and nonlinear of potential elasticity problems in this paper. The idea of its method is to assume that a simple operator is applied to the solution of the problem, obtaining a virtual source term, and then the virtual source term is expressed by radial basis functions(RBFs) series. But the key problem is to determine the coefficients in the virtual source series, because the determination of such coefficients needs to solve an ill-posed nonlinear system. In present work, the author presents a novel technique to solve such nonlinear system, referred to as “source term iteration regularization”. Numerical examples show that the FORTRAN program, based on this theory for solving the nonlinear and non homogeneous of elastic potential problems, has many advantages, such as high calculation speed, strong practicability, and the numerical solution and the exact solution agree fairly well.