Integration within Polygonal Finite Elements

Abstract

Engineering mechanics formulations of aerospace industry problems overwhelmingly rely upon spatial averaging techniques. Crucial applications in the area of dynamic response analysis and stochastic estimation of material degradation can be cited as important cases. Integration procedures on finite domains underlie physically acceptable averaging processes. Unlike one-dimensional cases, integrals within arbitrary areas and volumes cannot be approximated by a Gaussian form of numerical quadrature. Here the divergence theorem is applied once and twice, respectively, for polygonal and polyhedral integration domains, to construct integrals on boundary wireframes. The sum of Gaussian quadrature values on linear segments of the wireframe yields the final result of numerical integration on a finite element.