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

拟议的研究项目的目标是开发混合最小二乘有限元公式，用于分析多孔介质理论（TPM）框架内离子聚合物-金属复合材料（IPMCs）的机电行为。采用不可压缩的四相模型，包括聚合物网络相、阴离子相、阳离子相和液相。聚合物网络和阴离子（固定电荷）具有相同的运动功能，这两相结合成固相。此外，相同的电势被局部地施加到所有相位。混合Galerkin公式的一个挑战是在空间和时间上的场量的鲁棒近似。因此，用于描述离子聚合物/金属复合材料的耦合方程的有限元分析空间必须满足一定的稳定性条件（LBB条件）。对于这些非线性、耦合的边值问题，最小二乘法的结果是一个对称半正定方程组的极小化问题。该方法不受稳定性准则的约束，因此可以对单个场量进行任意的共形离散，原则上可以生成无振荡的解函数。对于离子聚合物金属复合材料，阳离子浓度随时间的发展由二阶扩散方程描述，在LSFEM中，残差的加权对于逼近质量是特别重要的，并且被系统地研究。此外，在空间和时间的自适应策略，以实现最佳收敛。我们受益于这样一个事实，即LSFEM提供了一个后验误差估计网格自适应作为一个固有的属性的方法。效率原因所需的时间自适应性是通过合适的龙格-库塔方法来实现的。

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.