非侵入式方法在非拟合边界网格中施加强Dirichlet边界条件
A Non-intrusive Approach for the Imposition of Strong Dirichlet Boundary Conditions in Unfitted Boundary Meshes
- Technical University of Munich（慕尼黑工业大学）
- Universitat Politècnica de Catalunya（加泰罗尼亚理工大学）
- International Center for Numerical Methods in Engineering (CIMNE)（国际工程数值方法中心）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出一种非侵入式黑盒算法，在非拟合网格中施加强Dirichlet边界条件，仅需求解器支持脚本、节点级边界条件、单元停用及梯度访问，并在FEM和IGA中实现最优L2收敛。
AI中文摘要：
在非拟合边界方法中，强制施加本质边界条件是一个基本挑战。本文提出了一种非侵入式、黑盒策略，用于在非拟合网格中施加此类条件。该方法适用于用户无法访问求解器源代码或其数学公式的情况，这在商业软件中经常出现。所提出的算法允许原本为体拟合网格设计的求解器用于非拟合情形，前提是满足四个条件：（i）求解器必须支持通过脚本进行用户自定义；（ii）允许通过脚本在节点级别施加Dirichlet边界条件；（iii）允许停用物理域外的单元；以及（iv）提供对活动单元内解梯度的访问。最后一个条件也可以通过从节点值和连通性信息外部重建梯度来满足，前提是已知单元公式，因此在实际中是可选的。这些要求是非常合理的，并且绝大多数生产就绪的（可能是商业的）代码都满足这些要求。在当前工作中，我们展示了该非侵入式算法在有限元方法（FEM）和等几何分析（IGA）离散化中的应用，证明了最优的$L^2$范数误差收敛性。这通过Kratos Multiphysics代码（版本\texttt{v10.1}）从用户API演示，仅利用上述功能。
英文摘要：
The enforcement of essential boundary conditions is a fundamental challenge in unfitted boundary methods. This paper presents a non-intrusive, black-box strategy for imposing such conditions in unfitted meshes. The approach is intended for situations where the user does not have access to the solver's source code or its mathematical formulation, which is often the case when using commercial software. The proposed algorithm allows solvers originally designed for body-fitted meshes to be used in unfitted cases, provided that four conditions are satisfied: (i) the solver must support user customization by means of scripting, (ii) allow the imposition of Dirichlet boundary conditions at the node level through scripting, (iii) permit the deactivation of elements outside the physical domain, and (iv) provide access to the solution gradient within active elements. The last condition can also be satisfied by externally reconstructing the gradient from nodal values and connectivity information, provided the element formulation is known, making it optional in practice. These requirements are very fair demands and are satisfied by the vast majority of production-ready, possibly commercial, codes. In the current work, we show the application of this non-intrusive algorithm in the context of the Finite Element Method (FEM) and Isogeometric Analysis (IGA) discretizations, demonstrating optimal $L^2$-norm error convergence. This is demonstrated using the Kratos Multiphysics code (release \texttt{v10.1}) \emph{from the user API, simply leveraging the capabilities mentioned above.}
