力学中变分不等式的边界元与有限元耦合方法

Consider solving the Dirichlet problem of Helmholtz equation on unbounded region with boundary a open smooth curve in the plane.国家自然科学基金资助项目结题报告4We use single-layer potential to construct a solution. This leads to the solution of a logarithmic integral equation of the first.kind for the Helmholtz equation. This equation if reformulated using a special change of variable, leading to a new first kind equation with a smooth solution function. This new equation is split into three parts. Then a quadrature method that takes special advantage of the splitting of the integral equation is used to solve the equation numerically. An error nalysis in a Sobolev space setting is given. And numerical results show that fast convergence is clearly exhibited.... In second paper, we propose a new numerical method to compute the ground state solution of trapped interacting Bose-Einstein condensation (BEC) at zero or very low temperature by minimizing.a functional. As preparatory steps we begin with the 3d Gross-Pitaevskii equation (GPE), scale it to get a one- parameter model and show how to reduce it to 2d and 1d GPEs. The ground state.solution is formulated via minimizing a functional under a constraint. The finite element approximation for 1d, 2d with.radial symmetry and 3d with spherical symmetry are presented and approximate ground state solutions, which will be used as initial guess in our practical numerical computation of the minimizing.problem, of the GPE in two extreme regimes: very weak interactions and strong repulsive interactions are provided. Numerical results in 1d, 2d with radial symmetry and 3d with spherical symmetry for atoms ranging up to millions in the condensation are reported to demonstrate the novel numerical method. .. In this paper, a exterior Signorini problem is reduced to a variational inequality on a bounded inner region with the help of.a coupling of boundary integral and finite element methods. We established a equivalence between the original exterior Signorini problem and the variational inequality on the bounded inner region coupled with two integral equations on an auxiliary boundary. We also introduce a finite element approximation of the variational inequality and a boundary element approximation of the integral equations. Furthermore, the optimal error estimates are given.

本项目对力学中出现的各种带磨擦边界的弹性，弹性-塑性，平板弯曲的单側现象，粘弹性雀骼啾浞植坏任侍饨惺捣椒ㄑ芯俊６晕藿缜蚣按蠓段颍捎帽呓缭胗邢拊詈戏椒ㄑ芯扛髦直浞植坏仁降氖当平