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When modeling scientific and industrial problems, geometries are typicallymodeled by explicit boundary representations obtained from computer-aideddesign software. Unfitted (also known as embedded or immersed) finite elementmethods offer a significant advantage in dealing with complex geometries,eliminating the need for generating unstructured body-fitted meshes. However,current unfitted finite elements on nonlinear geometries are restricted toimplicit (possibly high-order) level set geometries. In this work, we introducea novel automatic computational pipeline to approximate solutions of partialdifferential equations on domains defined by explicit nonlinear boundaryrepresentations. For the geometrical discretization, we propose a novelalgorithm to generate quadratures for the bulk and surface integration onnonlinear polytopes required to compute all the terms in unfitted finiteelement methods. The algorithm relies on a nonlinear triangulation of theboundary, a kd-tree refinement of the surface cells that simplify the nonlinearintersections of surface and background cells to simple cases that arediffeomorphically equivalent to linear intersections, robust polynomialroot-finding algorithms and surface parameterization techniques. We prove thecorrectness of the proposed algorithm. We have successfully applied thisalgorithm to simulate partial differential equations with unfitted finiteelements on nonlinear domains described by computer-aided design models,demonstrating the robustness of the geometric algorithm and showing high-orderaccuracy of the overall method.

High order unfitted finite element discretizations for explicit boundary representations