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.

翻译：偏微分方程（PDE）在任意定义几何体内外的高精度高效模拟对许多应用领域至关重要。沉浸边界法（IBMs）可避免在复杂几何模型（由CAD或其他表示形式如STL、点云描述）周围生成贴体网格时通常繁琐且耗时的过程，尤其当需要高水平的网格自适应性时。本研究在近期发展的移位边界方法（SBM）框架下推进了沉浸边界法的领域。在SBM中，边界条件的施加位置从沉浸物体的实际边界偏移至附近的替代边界，并通过泰勒展开对边界条件进行修正。这一方法允许选择与笛卡尔网格一致的替代边界，同时不失精度或稳定性。本研究的主要贡献包括：（a）证明通过最优替代边界选择可显著降低SBM的数值误差；（b）数学证明在最优替代边界选择下SBM具有最优收敛性；（c）在超大规模并行八叉树网格上部署SBM，包括处理不完整八叉树的算法改进；（d）通过涉及复杂形状、尖锐转角及不同拓扑结构的多种仿真展示这些方法的适用性。重点聚焦于泊松方程和线弹性方程。