Multi-patch discontinuous Galerkin isogeometric analysis for wave propagation: Explicit time-stepping and efficient mass matrix inversion
Multi-patch discontinuous Galerkin isogeometric analysis for wave propagation: Explicit time-stepping and efficient mass matrix inversion
复制标题
DOI：
10.1016/j.cma.2018.01.022
复制
发表时间：
2017-08
影响因子：
7.2
通讯作者：
Jesse Chan;John A. Evans
中科院分区：
文献类型：
--
作者：
Jesse Chan;John A. Evans
我们提出了一类样条有限元方法的时域波传播，特别是服从显式时间步进。所提出的方法利用一个不连续的Galerkin离散化，以强制连续性的解决方案领域的几何补丁在多补丁设置，这产生了一个方便的块对角结构的质量矩阵。在每个补丁，我们展示了如何准确和有效地反演质量矩阵中存在的弯曲的几何形状，通过使用重量调整近似的质量矩阵逆。这种近似恢复了张量积结构，同时保持可证明的高阶精度和半离散能量稳定性。我们还估计最大稳定的时间步长样条为基础的有限元，并表明，使用样条空间的结果在不太严格的CFL限制比等价C 0或不连续的有限元空间。最后，我们探讨了使用最佳节点向量的基础上L2 n-宽度。我们展示了如何使用最佳节点向量可以提高逼近性能和最大稳定时间步长，并提出了一个简单的启发式方法近似的最佳节点位置。数值实验证实了所提出的方法的准确性和稳定性。
We present a class of spline finite element methods for time-domain wave propagation which are particularly amenable to explicit time-stepping. The proposed methods utilize a discontinuous Galerkin discretization to enforce continuity of the solution field across geometric patches in a multi-patch setting, which yields a mass matrix with convenient block diagonal structure. Over each patch, we show how to accurately and efficiently invert mass matrices in the presence of curved geometries by using a weight-adjusted approximation of the mass matrix inverse. This approximation restores a tensor product structure while retaining provable high order accuracy and semi-discrete energy stability. We also estimate the maximum stable timestep for spline-based finite elements and show that the use of spline spaces results in less stringent CFL restrictions than equivalent C 0 or discontinuous finite element spaces. Finally, we explore the use of optimal knot vectors based on L 2 n-widths. We show how the use of optimal knot vectors can improve both approximation properties and the maximum stable timestep, and present a simple heuristic method for approximating optimal knot positions. Numerical experiments confirm the accuracy and stability of the proposed methods.