Rigidity in Finite-Element Matrices: Sufficient Conditions for the Rigidity of Structures and Substructures
Rigidity in Finite-Element Matrices: Sufficient Conditions for the Rigidity of Structures and Substructures
复制标题
有限元矩阵中的刚度：结构和子结构刚度的充分条件
DOI：
10.1137/060650295
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发表时间：
2008
期刊：
影响因子：
--
通讯作者：
Sivan Toledo
中科院分区：
文献类型：
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作者：
Gil Shklarski;Sivan Toledo
我们提出了有限元矩阵的刚性代数理论。该理论提供了有限元矩阵的形式代数定义；有限元矩阵的刚性和两个此类矩阵之间的相互刚性的概念；以及刚性和互刚性的充分条件。我们还提出了一种新颖的有限元矩阵稀疏化技术，称为“fretsaw 扩展”。我们证明这种稀疏化技术生成的矩阵与原始矩阵相互刚性。我们还展示了一种用于钢丝锯扩展的特定构造算法生成的矩阵可以在基本上不填充的情况下进行因式分解。该算法可用于构造有限元矩阵的预处理器。我们的理论和算法都适用于广泛的有限元矩阵，包括由标量和矢量偏微分方程（例如静电和线性弹性）的有限元离散化产生的矩阵。理论和算法都是纯粹的代数组合。他们仅操纵元素矩阵，而忽略了基本连续问题的几何形状、材料属性和离散化细节。
We present an algebraic theory of rigidity for finite-element matrices. The theory provides a formal algebraic definition of finite-element matrices; notions of rigidity of finite-element matrices and of mutual rigidity between two such matrices; and sufficient conditions for rigidity and mutual rigidity. We also present a novel sparsification technique, called fretsaw extension, for finite-element matrices. We show that this sparsification technique generates matrices that are mutually rigid with the original matrix. We also show that one particular construction algorithm for fretsaw extensions generates matrices that can be factored with essentially no fill. This algorithm can be used to construct preconditioners for finite-element matrices. Both our theory and our algorithms are applicable to a wide range of finite-element matrices, including matrices arising from finite-element discretizations of both scalar and vector partial differential equations (e.g., electrostatics and linear elasticity). Both the theory and the algorithms are purely algebraic-combinatorial. They manipulate only the element matrices and are oblivious to the geometry, the material properties, and the discretization details of the underlying continuous problem.