Study On Mathematic Methods For The Free Surface Problem And The Patch Test Function

With the development of Mechanics, many mathematic problems were presented. Combination mathematics with mechanics propels them forward and new fields arise to meet requirements. How to solve Mechanical problems by using mathematical knowledge is a common object for both Mechanical and mathematical researchers. Moreover, using modern mathematical approaches to settle historical mathematical problems is also a noticeable subject. Based on the new knowledge and the development of mathematics, the thesis discusses the following Mechanical problems:Solving the seepage free surface problem: Seepage problems with free surface are usually met in engineering. The free surface satisfies the water head function as well as the pore pressure conditions, and its location is unknown beforehand. It is very difficult to identify seepage free Surface due to the strong nonlinearity. Based on the experience in action boundary and the development of nonsmooth analysis, the thesis establishes the nonsmooth equations model for seepage problem .At the same time, the mix fixed point method and the nonsmooth Newton method for finite element method are also presentedBased on the Gauss point method for seepage problem with free surface, the non-smooth equations model for seepage problem is proposed as well as the mix fixed point method which belongs to the fixed mesh method. The seepage free surface is determined by using computer numerical computation with only once mesh plotting and no other approximate processing for the data. This thesis also discusses the existence of the solution of non-smooth equations and the convergence of the proposed fixed point method. Moreover, the free surface of seepage is plotted through interpolation of pressure intensity on the nodes. Numerical simulation shows that the new method is simple and possesses rapid convergence rate. It also offers a theoretical base for convergence analysis of iterative method.For the previous nonsmooth equations model and fixed mesh method, a new nonsmooth damped Newton method is given based on the definition of the generalized derivative. The derivatives of the nonsmooth equations are computed to make the generalized derivative matrix nonsingular. Then the nonsmmoth damped Newton method is used to solve them.Analytic solutions of the KdV equation with variable coefficients: Shallow water wave problems belong to problems of free surface wave, and can be molded by the nonsmooth equations model given above in principle. After a long research of shallow water wave, a set of nonlinear partial differential equation theory has been established base on perturbation expanding. Aiming at the analytic solution of partial differential equation of shallow water wave, this thesis firstly studies the analytic solution of the KdV equation with space variable. The nonlinear transformation is introduced during solving the KdV equation with variable depth, and a class of analytic solution of KdV equations for variable depth is solved.Enhanced patch test of finite element methods: There is no complete patch test proposed for the finite element analysis on Mindlin plate and thin cylindrical shell before. In this thesis, an enhanced patch test function for Mindlin plate and thin cylindrical shell elements is proposed. It also proves that patch test function for axisymmetric element can not contain constant shear. Shear of constant stess patch test must be zero. The patch test function for C~1 axisymmetric couple stress is quadratic function. And it proves that the quadratic patch test function does not contain constant shear strain term either.