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The automated finite element analysis of complex CAD models usingboundary-fitted meshes is rife with difficulties. Immersed finite elementmethods are intrinsically more robust but usually less accurate. In this work,we introduce an efficient, robust, high-order immersed finite element methodfor complex CAD models. Our approach relies on three adaptive structured grids:a geometry grid for representing the implicit geometry, a finite element gridfor discretising physical fields and a quadrature grid for evaluating thefinite element integrals. The geometry grid is a sparse VDB (Volumetric DynamicB+ tree) grid that is highly refined close to physical domain boundaries. Thefinite element grid consists of a forest of octree grids distributed overseveral processors, and the quadrature grid in each finite element cell is anoctree grid constructed in a bottom-up fashion. We discretise physical fieldson the finite element grid using high-order Lagrange basis functions. Theresolution of the quadrature grid ensures that finite element integrals areevaluated with sufficient accuracy and that any sub-grid geometric features,like small holes or corners, are resolved up to a desired resolution. Theconceptual simplicity and modularity of our approach make it possible to reuseopen-source libraries, i.e. openVDB and p4est for implementing the geometry andfinite element grids, respectively, and BDDCML for iteratively solving thediscrete systems of equations in parallel using domain decomposition. Wedemonstrate the efficiency and robustness of the proposed approach by solvingthe Poisson equation on domains given by complex CAD models and discretisedwith tens of millions of degrees of freedom.

A three-grid high-order immersed finite element method for the analysis of CAD models