Finite element analysis on implicitly defined domains: An accurate representation based on arbitrary parametric surfaces

In this paper, we present some novel results and ideas for robust and accurate implicit representation of geometric surfaces in finite element analysis. The novel contributions of this paper are threefold: (1) describe and validate a method to represent arbitrary parametric surfaces implicitly; (2) represent arbitrary solids implicitly, including sharp features using level sets and boolean operations; (3) impose arbitrary Dirichlet and Neumann boundary conditions on the resulting implicitly defined boundaries. The methods proposed do not require local refinement of the finite element mesh in regions of high curvature, ensure the independence of the domain's volume on the mesh, do not rely on boundary regularization, and are well suited to methods based on fixed grids such as the extended finite element method (XFEM). Numerical examples are presented to demonstrate the robustness and effectiveness of the proposed approach and show that it is possible to achieve optimal convergence rates using a fully implicit representation of object boundaries. This approach is one step in the desired direction of tying numerical simulations to computer aided design (CAD), similarly to the isogeometric analysis paradigm.