Numerical integration approaches for finite element simulations

Abstract

This thesis presents a comprehensive study on the development and application of novel numerical integration methods, particularly focusing on Gaussian-type cubature rules and their implications in Finite Element Method (FEM). The research is structured into three pivotal segments, each targeting different aspects of numerical integration to enhance the precision and efficiency of cubature rules within computational geometries.

The first segment addresses the derivation of explicit consistency conditions for constructing optimal fully symmetric cubature rules for tetrahedra. Utilizing a novel non-monomial fully symmetric polynomial basis, this work successfully defines the consistency conditions necessary for determining the most efficient rule structures, thereby minimizing the number of integration points required without compromising the accuracy.

In the second segment, the focus shifts to exploring rotational symmetry and multisymmetric polynomials in the moment equations for cubature rules. This includes the development of a new rotationally symmetric monomial basis, which simplifies the complicated system of moment equations. The resultant novel cubature rules, particularly for tetrahedra, demonstrate fewer integration points compared to existing rules, thus enhancing computational efficiency.

The final segment investigates the formulation of FEM and the patch test for new elements. It introduces a groundbreaking framework that leverages the established theory of cubature formulas to devise rules that not only pass the patch test but do so with fewer integration points. This framework is pivotal for advancing the blending of numerical methods into practical engineering applications, ensuring both reliability and efficiency.

Overall, the thesis encapsulates significant advancements in the field of numerical integration, presenting new methodologies and algorithms that refine the creation of cubature rules. These innovations provide substantial contributions to the domains of computational mathematics and engineering, particularly in the optimization of FEM. The results published within this thesis highlight the potential of these new approaches to set a foundation for future research in numerical integration methods.