Comparison of a Minimization and a Saddle Point Formulation of a Coupled Mechanics‐Diffusion Problem Implemented in deal.II

• Published in: Proceedings in applied mathematics and mechanics Vol. 25; no. 1
• Main Authors: Jeeja, Akshay Balachandran, Kiefer, Bjoern, Prüger, Stefan, Rheinbach, Oliver, Röver, Friederike
• Format: Journal Article
• Language: English
• Published: 01.03.2025
• ISSN: 1617-7061; 1617-7061
• DOI: 10.1002/pamm.202400188

For a problem in chemo‐mechanics, stated as a rate‐type variational principle, a minimization and a saddle point formulation are considered. Both formulations couple mechanical balance equations to mass diffusion and are intended to accurately model the diffusion‐induced swelling of hydrogels. The variational formulation is discretized by finite elements and implemented using the deal.II software library. The linearized systems in the incremental‐iterative solution procedure are solved using the unsymmetric multifrontal package (UMFPACK) sparse direct solver. The two formulations, employing three different, conforming finite element types, are considered for the case of mechanically‐induced diffusion and compared with respect to the evolution of the reaction force, mesh convergence, and the solution time. This initial step, comparing the performance of sparse direct solvers, sets the stage for future assessments involving domain decomposition, where sparse direct solvers are a computational kernel.