A high‐order immersed boundary discontinuous‐Galerkin method for Poisson's equation with discontinuous coefficients and singular sources
A high‐order immersed boundary discontinuous‐Galerkin method for Poisson's equation with discontinuous coefficients and singular sources
复制标题
具有不连续系数和奇异源的泊松方程的高阶浸入边界不连续Galerkin方法
DOI：
10.1002/nme.4835
复制
发表时间：
2015
影响因子：
2.9
通讯作者：
S. Govindjee
中科院分区：
文献类型：
--
作者：
G. Brandstetter;S. Govindjee
本文采用数值方法在固定网格上求解泊松方程，并在此基础上引入嵌入边界条件，重点研究了边界法向梯度的精确表示。边界上的梯度评估缺乏准确性是低阶嵌入边界方法的常见问题。虽然直接评估梯度是优选的，但通常使用后处理技术来提高梯度的质量。在这里，我们采用了一种新的方法的基础上的间断-伽辽金（DG）有限元法，灵感来自最近的工作[A. J. Lew和G.C.布斯卡利亚一种基于间断Galerkin的浸入边界方法。International Journal for Numerical Methods in Engineering，76：427 - 454，2008]。该方法在两个方面得到了改进：首先，我们用高阶几何图元局部逼近边界形状。其次，我们在单元中使用高阶形状函数。这些都是来自各种几何特征的边界的基础上分析解决方案的基本偏微分方程。发展包括三个基本的几何特征，在两个维度的泊松方程的解决方案：一个直的边界，一个圆形的边界，边界的不连续性。我们通过具有光滑圆形边界的分析基准示例以及由于凹角而存在的奇异性来证明该方法的性能。结果与低阶扩展有限元法以及[1]的DG方法进行了比较。我们报告了一个数量级的边界上的梯度的精度提高，以及在一个奇异源的存在下，提高收敛速度。原则上，该方法可以扩展到三维，更复杂的边界形状，和其他偏微分方程。版权所有© 2014约翰威利父子有限公司.
We adopt a numerical method to solve Poisson's equation on a fixed grid with embedded boundary conditions, where we put a special focus on the accurate representation of the normal gradient on the boundary. The lack of accuracy in the gradient evaluation on the boundary is a common issue with low‐order embedded boundary methods. Whereas a direct evaluation of the gradient is preferable, one typically uses post‐processing techniques to improve the quality of the gradient. Here, we adopt a new method based on the discontinuous‐Galerkin (DG) finite element method, inspired by the recent work of [A.J. Lew and G.C. Buscaglia. A discontinuous‐Galerkin‐based immersed boundary method. International Journal for Numerical Methods in Engineering, 76:427‐454, 2008]. The method has been enhanced in two aspects: firstly, we approximate the boundary shape locally by higher‐order geometric primitives. Secondly, we employ higher‐order shape functions within intersected elements. These are derived for the various geometric features of the boundary based on analytical solutions of the underlying partial differential equation. The development includes three basic geometric features in two dimensions for the solution of Poisson's equation: a straight boundary, a circular boundary, and a boundary with a discontinuity. We demonstrate the performance of the method via analytical benchmark examples with a smooth circular boundary as well as in the presence of a singularity due to a re‐entrant corner. Results are compared to a low‐order extended finite element method as well as the DG method of [1]. We report improved accuracy of the gradient on the boundary by one order of magnitude, as well as improved convergence rates in the presence of a singular source. In principle, the method can be extended to three dimensions, more complicated boundary shapes, and other partial differential equations. Copyright © 2014 John Wiley & Sons, Ltd.