基于单元子分法的结构多尺度边界单元法

Structural multi-scale boundary element method based on element subdivision technique

建立在基于单元子分法的一种有效自适应格式以及多区域边界元三步求解技术基础上提出了一种计算结构多尺度问题的多区域边界元法。首先,通过高斯积分误差分析公式确定边界单元在满足精度要求下所需要的高斯点数,当所需高斯点数超过规定数目时该单元就被自动划分成一定数量的子单元,从而消除结构多尺度所引起的近奇异性。在单元子分技术的基础上采用多区域边界元三步求解技术来处理材料非均质问题:第一步消除各子域的内部未知量,第二步消除各子域独自拥有的边界未知量,第三步根据位移相容性条件和面力平衡条件建立系统方程组并求解公共界面节点位移以及每个子域的其他未知量。数值算例结果表明本方法可以用较少的计算时间得到满意的结果,是处理结构多尺度问题的一种有效方法。

A boundary element method for solving structural multi-scale problems is described. The method is based on an efficient adaptive element subdivision technique and a three-step solution technique for multi-domain boundary element method. First, Gauss integration error formulate is used to ascertain the required number of Gauss points under given accuracy. Once the number exceeds the allowed number, the element is divided into a number of sub-elements, and consequently, near singularities caused by the multi-scales can be eliminated. A three-step solution technique for multi-domain boundary element method is adopted for solving non-homogeneous material problems based on the sub-element technique. The first step is to eliminate internal variables at the individual domain level. The second step is to eliminate boundary unknowns defined over nodes used only by a domain itself. And the third step is to establish the system of equations according to the compatibility of displacements and equilibrium of tractions at common interface nodes, and solve the system for displacements at common interface nodes as well as other unknowns within each sub-domain. Numerical examples are given to demonstrate that the presented method can obtain satisfied results with less computational time, and it is an efficient method for solving structural multi-scale problems.