Robust numerical integration of embedded solids described in boundary representation

Embedded and immersed methods have become essential tools in computational mechanics, as they allow discretizing arbitrarily complex geometries without the need for boundary-fitted meshes. One of their main challenges is the accurate numerical integration of cut elements. Among the various integration schemes developed for this purpose, moment fitting has proven to be a powerful technique that provides highly efficient and accurate integration rules. This publication presents a framework for the robust and efficient numerical integration of embedded solids described by oriented boundary meshes using moment fitting. The developments include an intersection algorithm that aims to drastically accelerate the computation of the necessary moments while achieving high accuracy. A closed surface parameterization of each cut domain is computed to facilitate the direct application of the divergence theorem. The algorithm is subject to a single quality criterion that guarantees the accurate evaluation of boundary integrals. At the same time, it allows to disregard classical mesh criteria, such as high aspect ratios, strongly varying angles, etc., resulting in extremely fast runtimes. In addition, an existing robust flood fill-based element classification scheme is further developed to initiate filling from arbitrary seed elements and to enable parallel execution, increasing its flexibility and efficiency. The successful application of all proposed algorithms to 4948 valid and flawed STLs from the Thingi10K database (Zhou and Jacobson, 2016) demonstrates their extraordinary robustness. In all cases, the wall-clock time scales at most linearly with the number of elements in the background mesh. We show that higher-order quadrature rules on the boundary elements enable efficient computation of the moments via the divergence theorem with near-machine precision. Finally, the presented methodologies are used to perform direct FE analyses on clean and flawed B-Rep models. All proposed algorithms are publicly available in the open-source C++ framework QuESo – Quadrature for Embedded Solids (https://github.com/manuelmessmer/QuESo), where the moment fitting equations are assembled and solved.