复杂表面问题的有限元计算与分析

碰撞、弹塑变形问题等交通运输器的安全试验通常采用实验室实物模型与计算机模型相结合进行模拟.对三维复杂区域的接触，国际标准方法采用接触变形算法模拟滑动控制.本课题采用流固耦合非牛顿流体方程初边值问题求解三维层结构特性，通过基于变分原理的摄动问题有限元方法，在高性能软件平台上实现数据的挖掘处理.由Sobolev空间嵌入原理，可将模型按接触区域进行分层单元剖分，将复杂区域剖分为若干相互连接、不重叠的六面体与空间平面四边形单元.同时，建立微观与宏观有限元双尺度计算模型进行模拟仿真对比，得到模型的能量与速度等一系列参数的变化曲线.此外，接触表面问题又可采用渐近摄动方法中的边界层理论进行研究，由此得到的微分方程特征函数空间，既可作为优化有限元基函数的解，又可用于建立一种新型的非线性特征值的渐近方法，也是估计材料特定参数的方法之一.最后，使用人工边界条件随机处理方法对求解结果的数据进行分析.

Finite element computation and analysis on complex contact boundary

The safety test of transportation devices, including impact test and plastoelastic deformation, is simulated both in laboratory and by computer. Contact deformation algorithm is an international standard method to simulate the skid control on complex 3-D areas. In this paper, coupled non-Newtonian fluid equations with initial boundary value are used to solve the 3-D layer structure. A finite element method (FEM) based on the variation principle is used to solve the perturbation problem, and data mining is processed by high performance software. According to the embedding principle, a stratified element division is processed, and the complex boundary is divided into several mutual connected yet not overlapped hexahedral and square elements. After building a FEM model of both macro scale and micro scale, varying curves of parameters including energy and velocity can be obtained. On the other hand, the boundary layer theory of asymptotic perturbation method is also a way to study the complex boundary problem. The characteristic function space gained can be used both to optimize the primary function of FEM, and to establish a new asymptotic method to solve the nonlinear eigen problem. It can be used to estimate the specific parameters of materials as well. The stochastic analysis of artificial boundary conditions is then used to study the resulting data.