Semihyper-Reduction for finite element structures with nonlinear surface loads on the basis of stress modes

Model reduction via projection is a common method to accelerate time integration of finite element (FE) structures by reducing the number of degrees-of-freedom (DOFs). However, nonlinear state-dependent surface loads are usually computed based on the nonreduced DOFs of the FE model. When a considerably high number of DOFs are involved in the nonlinear surface loads, their computation becomes a bottleneck. This paper presents a general approach for reduced time integration and reduced force computation for FE models. The required force trial vectors can be computed easily and systematically out of deformation trial vectors, commonly called “modes.” Those force trial vectors, which we call “stress modes,” can be determined a priori so that a nonlinear computation of the full system is not necessary. The new idea in this contribution is that stress recovery is used to decrease the number of equations for the force computation. A general framework for semihyper-reduction (SHR) is developed and its practical implementation is discussed. The term SHR is introduced because it is an intermediate approach between the straight-forward method of using the FE DOFs and pure hyper-reduction (HR) where the FE DOFs are omitted for computing state-depended surface loads. In order to demonstrate the proposed SHR approach practically, a numerical example of a planar crank drive is given, where a hydrodynamic lubrication film separates piston and cylinder. Thereby, very good result quality has been observed in comparison to a finite difference reference solution.