固体力学中快速多极边界元法研究进展

和快速多极方法相结合，使边界元法处理大规模工程与科学问题变得十分有效,首先概括介绍了快速多极边界元法，接着介绍其精度和效率的验证以及与常规边界元法的比较，并给出在微机机群上的快速多极边界元并行算法,给出了快速多极边界元法的一些应用，其中包括：复合材料的二维、三维模拟，含大量裂纹的二维弹性固体及其疲劳裂纹扩展的模拟．此外还介绍了用于弹塑性问题的快速多极边界元新方法．     

用于大规模断裂分析的快速多极边界元法

将快速多极算法(FMM)应用到边界元法(BEM)中,对断裂力学问题进行大规模计算.基于对偶边界积分方程(DBIE)构造代数方程组,采用广义极小残值迭代法(GMRES)求解.利用自适应四叉树结构执行快速多极算法,系数矩阵不需要显式存储,与未知量向量的乘积通过树结构的递归操作获得,计算复杂度与存储需求均缩减为O(N)(N为问题的自由度数).此外,该文提出了一种改进的预条件方案,使GMRES的求解时间与内存消耗进一步降低.数值算例表明: 该方案在保证精度的前提下,使计算规模与计算效率有可观的提高;算例的最大规模达到了300万自由度.     

FM-BEM Evaluation for Effective Elastic Moduli of Microcracked Solids

A fast multipole boundary element method (FM-BEM) was applied for the analysis of microcracked solids. Both the computational complexity and memory requirement are reduced to O(N), where N is the number of degrees of freedom. The effective elastic moduli of a 2-D solid containing thousands of randomly distributed microcracks were evaluated using the FM-BEM. The results prove that both the differential method and the method proposed by Feng and Yu provide satisfactory estimates to such problems. The effect of a non-uniform distribution of microcracks has been studied using a novel model. The numerical results show that the non-uniform distribution induces a small increase in the global stiffness.     

Fast Multipole BEM for Simulation of 2-D Solids Containing Large Numbers of Cracks

The fast multipole method was used to solve the traction boundary integral equation for 2-D crack analysis, The use of both multipole and local expansions reduces both the computational complexity and the memory requirement to O(N). The multipole expansion uses a complex Taylor series expansion to reduce the number of multipole moments, The generalized minimum residual method solver (GMRES) was selected as the iterative solver, An improved preconditioner for GMRES was developed which uses less CPU time and less memory. A new initial candidate vector for the iterative solver was developed to further improve the efficiency, The numerical examples apply the method to the analysis of cracks in infinite 2-D space with the largest model having 900 000 degrees of freedom.     

模拟含随机分布椭圆形夹杂弹性体的边界元法

为了模拟含随机分布椭圆形夹杂固体的等效力学特性 ,根据椭圆形夹杂积分区域的相似性 ,提出了求解此类问题的边界元计算方案。通过利用著名的有限元商用软件MSC.Marc和边界元程序对含有两个非常接近椭圆夹杂板材的对比计算分析 ,表明该文提出的边界元方案比有限元法具有更高的计算精度和计算效率。最后 ,对含有 5 0个随机分布椭圆夹杂的板材进行了计算分析 ,从而为长纤维状复合材料宏观等效力学特性的研究提供了可靠的数值模拟方法。