Condition number analysis and preconditioning of the finite cell method
Condition number analysis and preconditioning of the finite cell method
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DOI：
10.1016/j.cma.2016.07.006
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发表时间：
2017-04-01
影响因子：
7.2
通讯作者：
van Brummelen, E. H.
中科院分区：
文献类型：
--
作者：
de Prenter, F.;Verhoosel, C. V.;van Brummelen, E. H.
（等几何）有限单元法-其中一个域是沉浸在一个结构化的背景网格遭受空调问题时，细胞与小体积分数出现。在这方面的贡献，我们建立了一个严格的标度关系（I）FCM系统矩阵的条件数和最小的细胞体积分数。病态源于基函数在具有小体积分数的单元上很小，或者源于基函数几乎线性地依赖于这样的单元。基于这两个来源的病态，代数预处理技术，这是被称为对称不完全置换逆Cholesky（SIPIC）。一个详细的数值调查的有效性的SIPIC预处理器在改善（I）FCM条件数和提高收敛速度和精度的迭代求解器的泊松问题和二维和三维问题的线性弹性，其中Nitche的方法是适用于在法向或切向方向。预处理迭代求解器的准确性使得有限单元法的网格收敛研究成为可能。(C)© 2016 Elsevier B. V.版权所有。
The (Isogeometric) Finite Cell Method - in which a domain is immersed in a structured background mesh suffers from conditioning problems when cells with small volume fractions occur. In this contribution, we establish a rigorous scaling relation between the condition number of (I)FCM system matrices and the smallest cell volume fraction. Ill-conditioning stems either from basis functions being small on cells with small volume fractions, or from basis functions being nearly linearly dependent on such cells. Based on these two sources of ill-conditioning, an algebraic preconditioning technique is developed, which is referred to as Symmetric Incomplete Permuted Inverse Cholesky (SIPIC). A detailed numerical investigation of the effectivity of the SIPIC preconditioner in improving (I)FCM condition numbers and in improving the convergence speed and accuracy of iterative solvers is presented for the Poisson problem and for two- and three-dimensional problems in linear elasticity, in which Nitche's method is applied in either the normal or tangential direction. The accuracy of the preconditioned iterative solver enables mesh convergence studies of the finite cell method. (C) 2016 Elsevier B.V. All rights reserved.