Variational three-field reduced order modeling for nearly incompressible materials

This study presents an innovative approach for developing a reduced-order model (ROM) tailored specifically for nearly incompressible materials at large deformations. The formulation relies on a three-field variational approach to capture the behavior of these materials. To construct the ROM, the full-scale model is initially solved using the finite element method (FEM), with snapshots of the displacement field being recorded and organized into a snapshot matrix. Subsequently, proper orthogonal decomposition is employed to extract dominant modes, forming a reduced basis for the ROM. Furthermore, we efficiently address the pressure and volumetric deformation fields by employing the k-means algorithm for clustering. A well-known three-field variational principle allows us to incorporate the clustered field variables into the ROM. To assess the performance of our proposed ROM, we conduct a comprehensive comparison of the ROM with and without clustering with the FEM solution. The results highlight the superiority of the ROM with pressure clustering, particularly when considering a limited number of modes, typically fewer than 10 displacement modes. Our findings are validated through two standard examples: one involving a block under compression and another featuring Cook's membrane. In both cases, we achieve substantial improvements based on the three-field mixed approach. These compelling results underscore the effectiveness of our ROM approach, which accurately captures nearly incompressible material behavior while significantly reducing computational expenses.