The Research Of Biharmonic Equation Of Variational Inequality And Boundary Element Method

The frictional problem in elastic plate is one of the most familiar problems in mechanics. Its solution is to found a mathematical model of fourth-order variational inequality. The key and difficulty of solving this problem is that founding its variational functional and solving method. Variational inequality method, having been developed in the recent years, provides a united frame and powerful facility for the frictional problems. The main numerical methods of variational inequality problem contain finite element method and boundary element method, each has its advantages and disadvantages. But when solving boundary variational inequality and fourth-order elliptic equation boundary value problem, the boundary element method is more superior to the finite element method. Because it can achieve the effect of dimension reducing and calculation decreasing by BEM.This paper includes five chapters. Chapter 1 mainly introduces the development of BEM, variational inequality and solving biharmonic equation by BEM; and research dynamic of scholar of country and foreign.In chapter 2, we establish Sobolev space based on the desire of the fourth-order variational inequality of the second kind and biharmonic equation boundary value problem to the unknown function. Then introducing a complete set of theories in this space, such as: generalized solution, generalized function, generalized (feebleness) derivative, trace theorem and Brezzi theory, equivalent norm theorem and so on.In chapter 3, it is presented an attestation on equality between a fourth-order variational inequality of the second kind for frictional problem in elastic plate and the corresponding biharmonic equation boundary value problem. Regularization for the nondifferentiable item of this kind of variational inequality.In chapter 4, applying boundary element method and multiple reciprocity method to receive the approximate solution of the bihamonic equation boundary value problem. Finally the numerical example shows that this method has faster convergence and higher precision.In chapter 5, the astringency analysis of the approximate solution and accurate solution for the boundary element method is presented.