Geometrically accurate, efficient, and flexible quadrature techniques for the tetrahedral finite cell method

Abstract

We illustrate the importance of geometrically accurate volume quadrature for obtaining optimal accuracy with non-boundary-fitted finite element discretizations, when the problem domain is defined by sharp boundaries. We consider the tetrahedral finite cell method (TetFCM) and replace its recursive subdivision based integration approach with geometrically accurate quadrature rules that emanate from higher-order geometric parametrizations of cut tetrahedral elements. The elementwise parametrization procedure relies on the identification of the intersection topology and a series of higher-order mappings based on Lagrange polynomials. We demonstrate with several 3D examples that geometrically faithful local parametrization ensures optimal accuracy, while significantly reducing the number of quadrature points with respect to recursive subdivision. On the other hand, we highlight the strength of subdivision quadrature in the context of a patient-specific workflow for the simulation-based performance analysis of coupled bone/implant configurations. In particular, we show that accuracy, flexibility and computational efficiency of the TetFCM critically depends on flexibly applying the two different quadrature variants for fuzzy imaging data and sharp boundary representations, respectively.

Phase‐field boundary conditions for the voxel finite cell method: Surface‐free stress analysis of CT‐based bone structures

“…We start with a concise summary of the tetrahedral finite cell method in the context of linear elasticity and voxel geometries. For details on the tetrahedral finite cell method, we refer the interested reader to the recent contributions in previous studies . We note that the original variant of the finite cell method introduced by PARVIZIAN, DüSTER, and RANK has been based on non–boundary‐fitted Cartesian meshes with higher order approximation of the solution fields and adaptive quadrature of intersected elements on the basis of recursive subdivision.…”

The diffuse Nitsche method: Dirichlet constraints on phase‐field boundaries

“…On the one hand, both require special attention towards the numerical integration of elements cut by boundaries. In this context, a series of papers have recently highlighted the importance of geometrically faithful quadrature in embedded domain methods (see, for example, other studies [37][38][39][40][41][42][43] and the references therein).…”

Multi‐level hp‐finite cell method for embedded interface problems with application in biomechanics

“…More sophisticated integration schemes have been developed to efficiently evaluate the discontinuous domain integrals for the FCM. These schemes include the blended partitioning using the smart octree, moment fitting, adaptively weighted quadratures, and quadratic reparametrization for the tetrahedral FCM …”

“…These schemes include the blended partitioning using the smart octree, 50,51 moment fitting, 52 adaptively weighted quadratures, 53,54 and quadratic reparametrization for the tetrahedral FCM. 35 The other principal difference to conventional FEM is the enforcement of Dirichlet boundary conditions. As the boundary is no longer directly resolved by the finite elements, the Dirichlet boundary conditions cannot be enforced in the classical manner by manipulating the corresponding entries of the stiffness matrix and the load vector.…”