Optimal surrogate boundary selection and scalability studies for the shifted boundary method on octree meshes
Optimal surrogate boundary selection and scalability studies for the shifted boundary method on octree meshes
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DOI：
10.1016/j.cma.2023.116686
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发表时间：
2023-12-20
影响因子：
7.2
通讯作者：
Ganapathysubramanian，Baskar
中科院分区：
文献类型：
--
作者：
Yang，Cheng-Hau;Saurabh，Kumar;Ganapathysubramanian，Baskar
对于许多应用领域来说，精确和高效地模拟任意定义的几何内和周围的偏微分方程组是至关重要的。浸没边界方法(IBM)可以减少在复杂几何模型(由CAD或其他表示，如STL、点云描述)周围创建贴体网格的通常费力和耗时的过程，特别是当需要较高级别的网格自适应时。在这项工作中，我们在最近发展的移位边界方法(SBM)的背景下推进了IBM的领域。在SBM中，施加边界条件的位置从浸入对象的实际边界转移到附近的代理边界，并且使用泰勒展开来校正边界条件。此方法允许选择符合笛卡尔网格的代理边界，而不会损失精度或稳定性。我们在这项工作中的贡献如下：(A)我们证明了SBM的数值误差可以通过代理边界的最优选择来大大减少，(B)我们在数学上证明了对于这种最优选择的代理边界的SBM的最优收敛，(C)我们将SBM部署在大规模并行八叉树网格上，包括处理不完整八叉树的算法改进，以及(D)我们通过涉及复杂形状、尖角和不同拓扑的各种模拟来展示这些方法的适用性。重点介绍了泊松方程和线弹性方程。
The accurate and efficient simulation of Partial Differential Equations (PDEs) in and around arbitrarily defined geometries is critical for many application domains. Immersed boundary methods (IBMs) alleviate the usually laborious and time-consuming process of creating body-fitted meshes around complex geometry models (described by CAD or other representations, e.g., STL, point clouds), especially when high levels of mesh adaptivity are required. In this work, we advance the field of IBM in the context of the recently developed Shifted Boundary Method (SBM). In the SBM, the location where boundary conditions are enforced is shifted from the actual boundary of the immersed object to a nearby surrogate boundary, and boundary conditions are corrected utilizing Taylor expansions. This approach allows choosing surrogate boundaries that conform to a Cartesian mesh without losing accuracy or stability. Our contributions in this work are as follows: (a) we show that the SBM numerical error can be greatly reduced by an optimal choice of the surrogate boundary, (b) we mathematically prove the optimal convergence of the SBM for this optimal choice of the surrogate boundary, (c) we deploy the SBM on massively parallel octree meshes, including algorithmic advances to handle incomplete octrees, and (d) we showcase the applicability of these approaches with a wide variety of simulations involving complex shapes, sharp corners, and different topologies. Specific emphasis is given to Poisson’s equation and the linear elasticity equations.