Efficient integration method for fictitious domain approaches
Efficient integration method for fictitious domain approaches
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DOI：
10.1007/s00466-015-1197-3
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发表时间：
2015-10-01
影响因子：
4.1
通讯作者：
Gabbert, Ulrich
中科院分区：
文献类型：
--
作者：
Duczek, Sascha;Gabbert, Ulrich
在本文中，我们提出了一种有效而精确的数值方法来求解虚域方法中系统矩阵的积分，如有限单元法（FCM）。在FCM的框架中，物理域嵌入到一个几何上更大的简单形状域中，该简单形状域使用规则的笛卡尔网格进行离散化。因此，通常采用基于空间树的自适应正交技术来解析结构的几何形状。根据所研究结构的复杂程度，这种方法占了大部分的计算量。为了减少系统矩阵的计算量，提出了一种基于散度定理（Gau-Ostrogradsky定理）的高效正交方案。利用这个定理，积分的维数减少了1，即不需要求解整个域的积分，只需要考虑其轮廓。在本文中，我们介绍了积分法的一般原理及其实现。对几个二维基准问题的计算结果突出了它的性质。将该方法与传统的空间树集成技术进行了比较。
In the current article, we present an efficient and accurate numerical method for the integration of the system matrices in fictitious domain approaches such as the finite cell method (FCM). In the framework of the FCM, the physical domain is embedded in a geometrically larger domain of simple shape which is discretized using a regular Cartesian grid of cells. Therefore, a spacetree-based adaptive quadrature technique is normally deployed to resolve the geometry of the structure. Depending on the complexity of the structure under investigation this method accounts for most of the computational effort. To reduce the computational costs for computing the system matrices an efficient quadrature scheme based on the divergence theorem (Gau-Ostrogradsky theorem) is proposed. Using this theorem the dimension of the integral is reduced by one, i.e. instead of solving the integral for the whole domain only its contour needs to be considered. In the current paper, we present the general principles of the integration method and its implementation. The results to several two-dimensional benchmark problems highlight its properties. The efficiency of the proposed method is compared to conventional spacetree-based integration techniques.