One-point quadrature of higher-order finite and virtual elements in nonlinear analysis

In the present article, a stability- and consistency-preserving integration scheme for polynomial Galerkin approaches of arbitrary order is presented. The basis is formed by Taylor series expansions of the stresses with respect to the strains, which in turn are expanded towards the spatial directions. With a split of the material and geometric nonlinearities and the assumption of a material behavior linearly variable within an element, the strain energy in elements of arbitrary shape and polynomial order can be evaluated exactly. Therefore, geometric moments have to be calculated in preprocessing, requiring only evaluations of derivatives at a single integration point during the analysis. The moments can be effectively integrated analytically over the boundary of the elements. As one of the manifold applications, the use in the context of second order virtual elements is elaborated for which the assembly time can be significantly reduced. The combination with the automatic differentiation and expression optimization software AceGen provides performant element routines. In the numerical examples, the integration scheme shows promising accuracy and makes the application in more complex material models up to computational homogenization attractive.