Component-based reduced basis for parametrized symmetric eigenproblems

A component-based approach is introduced for fast and flexible solution of parameter-dependent symmetric eigenproblems. Considering a generalized eigenproblem with symmetric stiffness and mass operators, we start by introducing a “σ-shifted” eigenproblem where the left hand side operator corresponds to an equilibrium between the stiffness operator and a weighted mass operator, with weight-parameter σ > 0. Assuming that σ = λn > 0, the nth real positive eigenvalue of the original eigenproblem, then the shifted eigenproblem reduces to the solution of a homogeneous linear problem. In this context, we can apply the static condensation reduced basis element (SCRBE) method, a domain synthesis approach with reduced basis (RB) approximation at the intradomain level to populate a Schur complement at the interdomain level. In the Offline stage, for a library of archetype subdomains we train RB spaces for a family of linear problems; these linear problems correspond to various equilibriums between the stiffness operator and the weighted mass operator. In the Online stage we assemble instantiated subdomains and perform static condensation to obtain the “σ-shifted” eigenproblem for the full system. We then perform a direct search to find the values of σ that yield singular systems, corresponding to the eigenvalues of the original eigenproblem. We provide eigenvalue a posteriori error estimators and we present various numerical results to demonstrate the accuracy, flexibility and computational efficiency of our approach. We are able to obtain large speed and memory improvements compared to a classical Finite Element Method (FEM), making our method very suitable for large models commonly considered in an engineering context.