Mixed least-squares formulations within the framework of the theory of porous media for modeling ionic polymer-metal composites

• Applicants: Professor Dr.-Ing. Joachim Bluhm; Professor Dr.-Ing. Jörg Schröder; Dr.-Ing. Alexander Schwarz
• Subject Area: Applied Mechanics, Statics and Dynamics ; Mechanics
• Term: from 2020 to 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 445534800

The objective of the proposed research project is the development of mixed Least-Squares finite element formulations for the analysis of the electromechanical behavior of ionic polymer-metal composites (IPMCs) within the framework of the theory of porous media (TPM). An incompressible four-phase model consisting of the phases polymer network, anions, cations and liquid is applied. The polymer network and the anions (fixed charges) have the same motion function, these two phases are combined to a solid phase. Furthermore, the same electrical potential is applied locally to all phases.A challenge in mixed Galerkin formulations is the robust approximations of the field quantities in space and time. Thus, the finite element ansatz spaces for the description of the coupled equations for the modeling of ionic polymer metal composites must satisfy certain stability conditions (LBB condition). Furthermore, simulations with real material parameters sometimes show large oscillations, e.g. in the fluid pressure.For these nonlinear, coupled boundary value problems the least squares method results in a minimization problem with symmetric positive semi-definite systems of equations. The ansatz spaces are not subjected to stability criteria, so that arbitrary conformal discretizations of the individual field quantities are possible and oscillation-free solution functions can be generated in principle.For ionic polymer metal composites, the temporal development of the concentration of the cations is described by a second order diffusion equation, which is transferred into a first order system within the LSFEM.The weighting of the residuals in the LSFEM is of particular importance with regard to the approximation quality and is systematically investigated. Furthermore, adaptive strategies in space and time are applied to achieve optimal convergence. We benefit from the fact that the LSFEM provides an a posteriori error estimator for mesh adaptivity as an inherent property of the method. The time adaptivity required for efficiency reasons is implemented by means of suitable Runge-Kutta methods.

• DFG Programme: Research Grants