The approach we propose is based on the Model Order Reduction (MOR) after Implicit Schur method [1] which is used for the Krylov subspace reduction of piezoelectric devices. While their work uses the knowledge of the loading applied to the model to generate a Krylov subspace reduction basis, we propose to build a reduction basis with a priori unknown loading by modal synthesis. The basis is built from the eigenvectors of the problem after the static condensation by Schur complement of one of the physics. Typically, the Schur complement matrix is computed explicitly and it leads to dense operators [2] which limit the problem scales that can be studied due to large memory requirements and costly computations for the eigensolver used afterward.

For Krylov-based eigensolvers, the most computationally difficult step is to obtain a basis spanning the eigenspace of the problem on the considered eigenvalue range. By generalizing the MOR after Implicit Schur method, this basis can be constructed by an iterative procedure using the original sparse operators instead of the dense condensed operators. The original model may be significantly larger compared to the condensed model for typical cases. However, keeping the sparsity is a critical computational advantage for the considered problems. This method is minimally intrusive for the eigensolvers that only require the implementation of a matrix-vector product. Comparing this implicit Schur complement approach to the explicit Schur complement approach shows large computational cost reductions. It also underlines the problem scale limitations of the explicit approach even on high performance computing hardware.