The Method Of Fundamental Solutions For Solving The Inverse Problem Of Three-dimensional Linear Elasticity, As Well As Several Typical Movement Of The Functional Equation On The Volume Of Fluid Interface Tracking Simulation

Numerical methods for inverse problems raise much interest in recent years. This thesis considers an inverse problem in linear elasticity: the Cauchy problem associated with the Navier system. There are several studies devoted to the two-dimensional case, such as the boundary element method, an alternative iterative regularization method implemented using the BEM, the BEM in conjunction with the conjugate gradient method, and the BEM coupled with Tikhonov regularization and truncated singular value decomposition etc. Due to the inherent drawback with the BEM, it has not been applied to the 3D case.In Chapter 2, we apply the method of fundamental solution to the 3D inverse problem. The matrix equations arising from the MFS are highly ill-conditioned, and standard methods, such as the least squares method, fail to yield satisfactory results. Two regularization methods, i.e., Tikhonov regularization and truncated singular value decomposition, are employed to solve the resulting matrix equations, while the regularization parameters for the regularization methods are determined by the L-curve criterion. Numerical results show that the proposed method can solve the problem effectively, and the two regularization methods give results of comparable accuracy. Numerical calculations and simulations involving front-tracking, and free boundary arise in numerous scientific and engineering contexts, and play roles of increasing importance. The famous Methods of Volume of Fluid (VOF) and Level Set Method (LSM) are two basic numerical schemes for such problems, and have been applied to solve practical problems. Chapter 3 presents our preliminary investigation of volume-tracking methods for fluid volume function's moving-interfaces. Chapter 3 presents numerical results for several typical volume-tracking methods for fluid volume function's moving-interfaces. Based on existing numerical schemes, a novel numerical scheme is constructed. The proposed scheme is based on the method of CICSAM using new high-resolution difference scheme instead of ULTIMATE-QUICKEST scheme. Newly constructed dissipative high resolution difference method can be denoted by:...