Machine Computation Using the Exponentially Convergent Multiscale Spectral Generalized Finite Element Method
Machine Computation Using the Exponentially Convergent Multiscale Spectral Generalized Finite Element Method
复制标题
使用指数收敛多尺度谱广义有限元方法进行机器计算
DOI：
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发表时间：
2014
期刊：
影响因子：
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通讯作者：
R. Lipton
中科院分区：
文献类型：
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作者：
I. Babuska;Xuehai Huang;R. Lipton
提出了一种多尺度谱广义尼特元法（MS-GFEM），用于求解非均质介质内部的大型二维和三维应力分析问题。它可以用来解决大到无法直接用有限元技术解决的问题，并设计用于在大规模并行机器上实现。该方法本质上是多尺度的，并在粗网格上使用最优谱化局部基函数族。证明了该方法具有指数级的收敛速度。为了更好地理解这种方法，我们描述了它在复合材料内部二维平面应变问题中的实现。在这里，它们被一个最小的距离隔开，但是没有对它们的结构做出特殊的假设，比如周期性或遍历性。MS-GFEM的实现使用与计算域内ber数量相同的操作顺序提供离散解算子。这种实现是最优的，因为解决方案的操作数量与问题的输入数据的顺序相同。用于表示离散逆算子的MS-GFEM矩阵的大小由粗网格的尺度和谱基的收敛速度控制，其数量级可以远小于ber的数目。该策略具有通用性，可应用于求解与椭圆偏微分方程离散解相关的大型有限元系统。
A multiscale spectral generalized nite element method (MS-GFEM) is presented for the solution of large two and three dimensional stress analysis problems inside heterogeneous media. It can be employed to solve problems too large to be solved directly with FE techniques and is designed for implementation on massively parallel machines. The method is multiscale in nature and uses an optimal family of spectrally dened local basis functions over a coarse grid. It is proved that the method has an exponential rate of convergence. To x ideas we describe its implementation for a two dimensional plane strain problem inside a ber reinforced composite. Here bers are separated by a minimum distance however no special assumption on the ber conguration such as periodicity or ergodicity is made. The implementation of MS-GFEM delivers the discrete solution operator using the same order of operations as the number of bers inside the computational domain. This implementation is optimal in that the number of operations for solution is of the same order as the input data for the problem. The size of the MS-GFEM matrix used to represent the discrete inverse operator is controlled by the scale of the coarse grid and the convergence rate of the spectral basis and can be of order far less than the number of bers. This strategy is general and can be applied to the solution of very large FE systems associated with the discrete solution of elliptic PDE.