A component-based hybrid reduced basis/finite element method for solid mechanics with local nonlinearities

The SCRBE (Static-Condensation Reduced-Basis-Element) method is a component-to-system model order reduction approach for efficient many-query and real-time treatment of linear partial differential equations characterized by many spatially distributed constitutive, geometry, and topology parameters. In this paper we incorporate the SCRBE approach into a framework for analysis of problems in solid mechanics which are largely linear with the exception of local nonlinearities. In particular, we exploit a linear–nonlinear domain decomposition to develop a hybrid formulation: we consider a SCRBE approximation over the (assumed predominant) part of the domain associated to a linear elasticity model; we revert to a full finite element (FE) approximation over the part of the domain associated to the locally nonlinear model. We adapt the SCRBE port training procedures to anticipate the behavior of the field over the linear–nonlinear interface and hence ensure an accurate solution over the entire domain. We choose a globally conforming approximation which exploits the intrinsic port structure of the SCRBE method and yields a decoupled “non-invasive” formulation of the respective linear and nonlinear blocks of the residual vector and (Newton) Jacobian matrix. We present numerical results for several local nonlinearities – elastic contact with and without friction, plastic (contact) – which demonstrate substantial savings relative to standard FE approximation over the entire domain, in particular in the regime in which the linear part of the domain represents the majority of the (FE) degrees of freedom.