Non-negative moment fitting quadrature for cut finite elements and cells undergoing large deformations

Abstract

AbstractFictitious domain methods, such as the finite cell method, simplify the discretization of a domain significantly. This is because the mesh does not need to conform to the domain of interest. However, because the mesh generation is simplified, broken cells with discontinuous integrands must be integrated using special quadrature schemes. The moment fitting quadrature is a very efficient scheme for integrating broken cells since the number of integration points generated is much lower as compared to the commonly used adaptive octree scheme. However, standard moment fitting rules can lead to integration points with negative weights. Whereas negative weights might not cause any difficulties when solving linear problems, this can change drastically when considering nonlinear problems such as hyperelasticity or elastoplasticity. Then negative weights can lead to a divergence of the Newton-Raphson method applied within the incremental/iterative procedure of the nonlinear computation. In this paper, we extend the moment fitting method with constraints that ensure the generation of positive weights when solving the moment fitting equations. This can be achieved by employing a so-called non-negative least square solver. The performance of the non-negative moment fitting scheme will be illustrated using different numerical examples in hyperelasticity and elastoplasticity.