Removing geometrical details from a complex domain is a classical operation in computer aided design. This procedure simplifies the meshing process, and it enables faster simulations with less memory requirements. However, depending on the partial differential equation that one wants to solve, removing some important geometrical features may greatly impact the solution accuracy. Unfortunately, the effect of geometrical simplification on the accuracy of the problem solution is often neglected or its evaluation is based on engineering expertise, only due to the lack of reliable tools. It is therefore important to have a better understanding of the effect of geometrical model simplification, also called defeaturing, to improve our control on the simulation accuracy along the design and analysis phases. In this work, we consider as a model problem the Poisson equation on a geometry with Neumann features, we consider some finite element discretization of it, and we build an adaptive strategy that is twofold. Firstly, it is able to perform geometrical refinements, that is, to choose at each iteration step which geometrical feature is important to obtain an accurate solution. Secondly, it performs standard mesh refinements; since the geometry changes at each iteration, the algorithm is designed to be used with an immersed method. To drive this adaptive strategy, we introduce an a posteriori estimator of the energy error between the exact solution defined in the exact fully-featured geometry, and the numerical approximation of the solution defined in the defeatured geometry. The reliability of the estimator is proven for very general (potentially trimmed multipatch) geometric configurations, and in particular for IGA with hierarchical B-splines. Finally, numerical experiments are performed to validate the presented theory and to illustrate the capabilities of the proposed adaptive strategy.

翻译：从复杂几何域中移除几何细节是计算机辅助设计中的经典操作。该过程可简化网格生成流程，并实现更低内存需求下的快速仿真。然而，根据待求解的偏微分方程类型，移除某些关键几何特征可能显著影响解的精度。遗憾的是，由于缺乏可靠工具，几何简化对问题求解精度的影响常被忽视，或仅能依赖工程经验进行评估。因此，深入理解几何模型简化（亦称去特征化）的影响，对于提升设计与分析阶段仿真精度的控制能力至关重要。本研究以带诺伊曼特征的几何域上的泊松方程为模型问题，采用有限元离散方法，构建了包含双重功能的自适应策略：其一，能够执行几何细化，即在每次迭代步中选择对获得精确解至关重要的几何特征；其二，执行标准网格细化。由于几何结构在每次迭代中发生变化，该算法设计为可与浸入式方法结合使用。为驱动此自适应策略，我们引入了一种后验估计器，用于度量精确全特征几何中定义的真解与去特征化几何中定义的数值近似解之间的能量误差。该估计器的可靠性在非常一般的几何构型下得到证明，尤其适用于具有分层B样条的等几何分析。最后，通过数值实验验证了所提理论，并展示了所构建自适应策略的实际效能。