THE MIXED-CELL-COMPLEX, PARTITION-OF-UNITY METHOD
THE MIXED-CELL-COMPLEX, PARTITION-OF-UNITY METHOD
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DOI：
10.1016/j.cma.2008.04.026
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发表时间：
2009-03
影响因子：
7.2
通讯作者：
C. Riker;S. Holzer
中科院分区：
文献类型：
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作者：
C. Riker;S. Holzer
本文提出了一种求解偏微分方程的伽辽金方法，它将基于粒子的方法与传统的有限元方法相结合。本文将该方法命名为混合单元复合分割统一方法（mixed-cell-complex partitioning of unity method， MCCPUM）。它可以被任意地认为是基于区域内及其边界上的一组散射粒子，或者是基于区域的Delaunay细胞分解。与无单元伽辽金方法和其他无网格技术相比，统一的划分不是建立在圆形或矩形的支撑上，而是建立在由Voronoi/Delaunay空间分解生成的重叠多面体支撑上：混合细胞复合体。该方法继承了真正无网格格式的大部分优点，同时极大地促进了Galerkin近似中弱形式的数值积分。离散化完全由节点的选择和与节点相关的局部近似阶数决定。这里使用了任意阶的勒让德多项式。给出了二维泊松问题的数值算例，并讨论了该方法的有效性。
We present a Galerkin method for solving partial differential equations which is a blend of ideas from particle-based methods on the one side and traditional finite element methods on the other side. The method is here named mixed-cell-complex partition of unity method (MCCPUM). It can be arbitrarily considered as being based on a set of scattered particles in the domain and on its boundary, or on a Delaunay cell decomposition of the domain. In contrast to the element-free Galerkin method and other meshless techniques, the partition of unity is not constructed on circular or rectangular supports, but rather on overlapping polyhedral supports generated from a Voronoi/Delaunay decomposition of space: the mixed-cell-complex. This approach inherits most of the advantages of truly meshless schemes, while it greatly facilitates the numerical integration of the weak forms required in Galerkin approximations. The discretization is exclusively governed by the selection of nodes and the approximation orders associated to the nodes locally. Here Legendre polynomials of arbitrary orders are used. The mixed-cell-complex and the corresponding Galerkin discretization are explained, numerical examples for the Poisson problem in two dimensions are presented, and the efficiency of the method is discussed.