多体接触问题的非匹配网格的区域分解算法

It is well known that the multi-body contact problems have always happened and occupied a position of special importance in the mechanics of solids. So it got particular attention of engineers and computational experts. However, due to the complex nature of contact problems, Up to present, there are still many fundmental unsolved problems for thenumerical solutions of their finite methods . In particular, it becomes.very difficult to solve large scale contact problems. Up to present, the.work in this field have seldom been found in the literature. In order to solve these problems, we study the multi-body contact problems using domain decomposition methods with nonmatching grids so that they can been become to solve in finite subdomains. In other words, it is solving some normal variational equations on domains which their boundaries are not happened to contact and solving some problems of the restrained variational inequality on domains which their boundaries are happened to contact, and the large scale contact problems are changed to solve small scale problems in parallel on subdomains.Therefore, we have studied.several kinds of contact problems. Firstly, we have reseached the finite.element approximation of the Signorini problem using Lagrange multiplier methods with piecewise constant elements. Optimal error bounds are obtained and iterative algorithm and its convergence are given..Furthermore, global superconvergences are proved. Secondly, we have studied domain decomposition methods with nonmatching grids for the Signorini's problem and devoted to the construction of domain.decomposition methods with nomatching grids based on mixed finite element.methods for the Signorini's problem with a rigid frictionless foundation..Optimal error bounds and the convergences are proved. Furthermore the global superconvergences estimates are given. And we have showed that the error estimates of the energy norm are half order higher than usual error estimates. Finally, we have discussed domain decomposition methods with nonmatching grids for the unilateral problem and devoted to the construction of domain decomposition methods with nonmatching grids based on mixed finite element methods for the unilateral problem. Optimal error bounds are proved. Furthermore, global superconvergence estimates are given. These results are the best and the newest in this field up to now. The work have provided the new methods and guidances for the study further of the other multi-body contact problems. Our papers have published in.the J.Comput.Math. In order to study the global superconvergences of the decomposition methods with nomatching grids for the contact problems, we have considered the solution of the biharmonic equation using Adini nonconforming finite.element and report new results of the asymptotic error expansions of the.interpolation error functionals and nonconforming remainder.These expansions were used to develop two extrapolation formulas and a series of sharp error estimates.Further we have given the superconvergences and the asymptotic error expansions of the Adini's element for the second order imhomogeneous Neumann problem. Besides, we have presented a class of Petrov-Galerkin finite element methods for nonlinear Volterra integro-differential equations. These methods have global optimal.convergence rates, and have certain global and local uperconvergencefeatures. Several post-processing techniques are proposed to obtain globally superconvergent approximations.These results are the best and the newest in this field up to now. In particular,the superconvergences of the Adini nonconforming finite element for the biharmonic equation are the most important results in the field of the nonconforming finite element and answered the theoretical problems which exists.superconvergences of the high order equation.Our papers have published.in journal of J.Comput.Math. and Dynamic of Continuous, Discrete and Impulsive Systems and so on.We have published 6 papers in journal of J.Comput.Ma

我们用非匹配网格区域分解算法研究多体接触问题，使该问题转化为在有限个子区域上求解，即用拉格朗日乘子法研究无摩擦及带摩擦两类问题的多体接触问题的有限元逼近，探讨解的存在唯一性，算法的收敛性及误差分析，进一步研究它的超收敛性以及Ｕｚａｗａ算法中系数的最佳选择。为解大型接触问题提供并行算法，丰富变分不等式的算法理论。