Legendre 级数多极边界元法理论研究

It is a complicated boundary element problem to apply Boundary Element Method(BEM) and Fast Multipole Method(FMM) in the solution of some mathematical and mechanical engineering practical problems. To study the mathematical and mechanical problems in Fast Multipole Boundary Element Method (FM-BEM) is much meaningful in theory. At present, the study of these problems mainly concentrates several boundary element systems upon numerical simulation. In comparison, the development of theoretical research is slow. Seldom reports are about theoretical approximate solution and truncation error for the FM-BEM in some mathematical and mechanical problems. In this project, theory of Legendre series will be used to find the fundamental solutions and errors for some FM-BEM problems. Important research subjects include potential problem, elasticity and elasto-plasticity. The project will make a study of solution methods for the analytical solutions for the FM-BEM problems, and bring regulations for the solution of BEM problems by method of Legendre series to light. This method considers the influence of Legendre series expansion for fundamental solutions on the solution precision and efficiency for a system. Different from pure analysis of numerical examples, this method will give some analysis expressions for the truncation errors and a kind of theoretical computation method for the FM-BEM problems. This project includes some innovative work, for example, to theoretically find the error expressions of FM-BEM fundamental solutions and the analytical solutions for the whole computation system, which establishes foundation for theoretical solution of mathematical and mechanical BEM problems.

边界元法与多极展开法相结合求解数学和力学等工程实际问题是一个复杂的边界元问题，深入研究多极边界元法的数学和力学问题具有重要的理论意义。目前，国内外对该问题的研究主要集中在几种边界元系统的数值模拟方面，相比之下，理论研究的发展较为 缓慢，系统地求解数学和力学问题的多极边界元理论近似解及其误差分析方法的报道更为少 见。本项目采用Legendre 级数理论求解多极边界元问题的基本解及其误差，重点研究位势、 弹性和弹塑性问题，探索多极边界元问题的解析解求解方法，揭示Legendre 级数方法求解 边界元问题的规律。该方法考虑了基本解的Legendre 级数展开对系统求解精度和效率的影 响，区别于单纯的数值算例分析情况，给出截断误差分析表达式和多极边界元问题的理论计 算方法。本项目具有理论求解多极边界元基本解误差及系统解析解的创新性，为理论求解数 学和力学边界元问题奠定基础。

深入研究多极边界元法的数学和力学理论对于复杂的大规模边界元问题具有重要的理论意义。本项目采用Legendre 级数理论求解多极边界元问题的基本解及其误差，探索多极边界元问题的解析解求解方法，揭示Legendre 级数方法求解边界元问题的规律。本项目完成了以下研究内容：位势问题位势基本解和位势梯度基本解的Legendre级数相关展开式及截断误差估计；弹性问题位移基本解和面力基本解Legendre级数相关展开式及截断误差估计；有关截断指标对计算精度和计算效率的影响；边界元问题近似解的计算方法和误差控制方法；变参数GMRES(m) 高效数值算法及收敛性理论；弹性接触问题多极边界元法罚因子数学规划模型；有关参数对系统求解的影响；数值模拟三维轧制问题。此外，对多极边界元法中奇异积分的计算问题进行了探索。项目的研究成果为进一步理论求解数学、力学和工程等领域的边界元问题奠定坚实的理论基础，为大规模科学与工程计算提供高效的数值计算方法。