Element-Based Internal Variable Formulations for Finite Element Discretizations in FFT-Based Homogenization Methods

Although finite elements were made available for FFT-based computational homogenization methods, they are seldomly used for inelastic computations because traditionally the constitutive law is evaluated at each quadrature point of the element, making the storage of that many internal variables necessary, as well. Recently, an innovative discretization scheme based on tetrahedral finite differences (TET) was introduced, which may be interpreted as a finite element discretization with two “virtual” quadrature points. In this work, we devise a general way to formulate finite element discretizations with multiple quadrature points and only a single internal variable per element in a consistent manner. For the large class of generalized standard materials, we demonstrate that the natural variational formulation leads to a simple and compact scheme requiring only a single nonlinear material evaluation per element. We discuss the efficient implementation into displacement-based FFT codes and demonstrate the advantages and limitations of our element-based internal (EBI) approach when applied to the TET discretization, classical trilinear hexahedral finite elements (HEX8) and multi-quadrature point composite voxels. In particular, we compare the computational expense, the memory requirements, and the accuracy of the traditional discretizations, their EBI formulations, and a hybrid composition of the latter two to the results of the rotated staggered grid discretization.