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Numerical integration on regular background mesh

DOI：

 作者 单位 邮编 任林娟 北京理工大学 100081 周平章* 北京理工大学 100081

计算力学中的一类伪材料方法（如浸没边界法、切割单元法、有限胞元法等）通常需要在非贴体的背景网格上进行数值积分，此时被积函数在扩展积分域中会出现强间断，因此像高斯积分这样的传统方法表现不佳。本文列举并对比了两种解决该问题的积分方法，其中第二种方法是首次提出的，并且在精度方面表现优异。这两种方法都利用树状数据结构将背景网格细分为子域。第一种方法直接在子域中设置高斯点，而第二种方法先对边界进行重构，再将高斯点分布在重构的积分域内。文中的两个算例均展示出第二种方法（即综合利用细分与边界重构）具有快速且单调的收敛性，因此在大多数情况下更优越，并且通过一个计算力学问题验证了该方法的可靠性。由于本文的方法不要求积分域有解析表达式，它们可以广泛应用于工程实际中各种复杂几何体的数值积分求解。

In fictitious domain methods (e.g., the immersed boundary method, the cut cell method, and the finite cell method, etc.), it is usually required to carry out numerical integration over non-body-fitted background meshes. In this case, strong discontinuity occurs in the extended integration domain, so traditional methods like Gauss integration behave poorly. In this paper, two integration techniques to solve this problem are listed and compared, of which the second one is proposed for the first time and is shown to be promising in term of accuracy. Both techniques subdivide the background cell into smaller subcells using the tree data structure. The first technique directly distributes Gauss points in each subcell, while the second technique further reconstructs the boundary so that the Gauss points are distributed in the interior of the reconstructed integration domain. Two examples in this paper show that the second technique (i.e., subdivision with boundary reconstruction) has fast and monotone convergence, so is superior in most cases, and the reliability of the technique is verified by a computational mechanics problem. The proposed techniques are highly flexible and can be used in various applications since they do not require the integration domain to have closed-form expressions.
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