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基于行消元回代法的多域边界元分析方法

ISSN号：0459-1879
期刊名称：力学学报
时间：2012.3.18
页码：361-368
分类：O241.83[理学—计算数学;理学—数学]
作者机构：[1]大连理工大学工业装备结构分析国家重点实验室,大连116024
相关基金：国家自然科学基金（10872050）和中央高校基本科研基金（DUT11ZD（G）01）资助项目.
相关项目：主值隔离径向积分边界单元法及其在非线性问题中的应用

关键词：多域边界元法, 高斯消元法, 行消元回代法, 稀疏方程, multi-domain boundary element method, Gaussian elimination method, row elimination-back-substitution method, sparse system of equations

中文摘要：

提出了一种用多域边界元技术求解大型工程问题的新算法．首先，采用三步变量凝聚技术，将由内部点、边界点和公共结点表述的每一子域的基本边界元代数方程表述成只有公共结点变量为未知量的代数方程，然后，根据公共结点的平衡方程和协调条件组集具有稀疏系数特征的总体系统方程组．为了有效求解该系统方程组，首次在边界元法中引进一种能有效求解大型非对称稀疏系数矩阵方程组的行消元回代法（REBSM），该方法可在方程的每一行组形成时进行消元和回代，当方程组组集完毕后即可得到方程的解，不需要最后的回代过程．因为一些项的重复计算在每一行的处理中合并掉，因此REBSM要比传统的高斯消元法需要较少的内存，而且计算速度具有数量级的提高，可为边界元法求解大型工程问题提供有力的方程求解器．

英文摘要：

A novel multi-domain boundary element method （MDBEM） analysis technique is presented to solve large-scale engineering problems. Firstly, the basic integral equations of each domain formulated in terms of internal, boundary and interface nodal variables are reduced to the algebraic equations in terms of interface nodal variables only by the three-step variable condensing technique. Then, a sparse system of equations formulated in terms of interface nodal quantities is assembled using the equilibrium equation and consistence condition at interface nodes. To solve the system of equations efficiently, this paper, for the first time, introduces a robust linear equation solution method, called the row elimination-back-substitution method （REBSM）, to solve the non-symmetric sparse system of equations. REBSM performs both the elimination and back-substitution procedures when each row of the system is formed. When the last row is finished for assembling, the solutions of the system are obtained at the same time, without the need of the last back-substitution procedure. Since some repeated terms are incorporated, REBSM needs less storage than the Gaussian elimination method, has an improvement in computational speed by orders of magnitude, and provide BEM a robust equation solver for solving large engineering problems.

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