A general hyper-reduction strategy for finite element structures with nonlinear surface loads based on the calculus of variations and stress modes

The presence of nonlinear and state-dependent surface loads within structural dynamic applications often causes an increased computational effort for time integration. Although the degrees of freedom (DOFs) of finite element (FE) models can be reduced significantly via model order reduction (MOR), the determination of nonlinear effects usually depends on the full physical domain. This paper introduces a hyper-reduction (HR) approach, which allows the computation of nonlinear surface loads by a limited number of integration points. One of the fundamental underlying concepts is the creation of mode bases for both the MOR and the reduced load computation, whereby the latter ones are denoted as stress modes. These can be determined a priori from linear FE analyses, for which no fully nonlinear dynamic computations are required. Another relevant step includes the formulation of the contact law based on the calculus of variations, combined with the use of these stress modes. The resulting relationships are in the dimension of the stress modes subspace but still rely on the full physical domain, which is why in this paper, the empirical cubature method (ECM) is proposed to obtain a reduction of the integration points. Based on these theoretical concepts, a general HR framework for nonlinear and state-dependent surface loads is outlined. For a demonstration of the proposed procedure, a planar crank drive with an elastohydrodynamic (EHD) contact between the piston and the cylinder is presented. The quality of the results and the potential for saving computing time are demonstrated by comparing the HR approach with conventional solution techniques.