Adaptive analysis-aware defeaturing: the case of Neumann boundary conditions

Abstract

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.