Hamidreza Dehghani

The governing equations and numerical solution strategy to solve porohyperelastic problems as multiscale multiphysics media are provided in this contribution. The problem starts from formulating and non-dimensionalising a Fluid-Solid Interaction (FSI) problem using Arbitrary Lagrangian-Eulerian (ALE) technique at the pore level. The resultant ALE-FSI coupled systems of PDEs are expanded and analysed using the asymptotic homogenisation technique which yields three partially novel systems of PDEs, one governing the macroscopic/effective problem supplied by two microscale problems (fluid and solid). The latter two provide the microscopic response fields whose average value is required in real-time/online form to determine the macroscale response. This is possible efficiently by training an Artificial Neural Network (ANN) as a surrogate for the Direct Numerical Solution (DNS) of the microscale solid problem. The present methodology allows for solving finite strain (multiscale) porohyperelastic problems accurately using the direct derivative of the strain energy, for the first time. Furthermore, a simple real-time output density check is introduced to achieve an optimal and reliable training dataset from DNS. A Representative Volume Element (RVE) is adopted which is followed by performing a microscale (RVE) sensitivity analysis and a multiscale confined consolidation simulation showing the importance of employing the present method when dealing with finite strain poroelastic/porohyperelastic problems.

This contribution introduces and discusses a formulation of poro-hyperelasticity at finite strains. The prediction of the time-dependent response of such media requires consideration of their characteristic multi-scale and multi-physics parameters. In the present work this is achieved by formulating a non-dimensionalised fluid–solid interaction problem (FSI) at the pore level using an arbitrary Lagrange–Euler description (ALE). The resulting coupled systems of PDEs on the reference configuration are expanded and analysed using the asymptotic homogenisation technique. This approach yields three partially novel systems of PDEs: the macroscopic/effective problem and two supplementary microscale problems (fluid and solid). The latter two provide the microscopic response fields whose average value is required in real-time/online form to determine the macroscale response (a concurrent multi-scale approach). In order to overcome the computational challenges related to the above multi-scale closure, this work introduces a surrogate approach for replacing the direct numerical simulation with an artificial neural network. This methodology allows for solving finite strain (multi-scale) porohyperelastic problems accurately using direct automated differentiation through the strain energy. Optimal and reliable training data sets are produced from direct numerical simulations of the fully-resolved problem by including a simple real-time output density check for adaptive sampling step refinement. The data-driven approach is complemented by a sensitivity analysis of the RVE response. The significance of the presented approach for finite strain poro-elasticity/poro-hyperelasticity is shown in the numerical benchmark of a multi-scale confined consolidation problem. Finally, to show the robustness of the method, the system response is dimensionalised using characteristic values of soil and brain mechanics scenarios.

Mechanical modelling of poroelastic media under finite strain is usually carried out via phenomenological models neglecting complex micro-macro scales interdependency. One reason is that the mathematical two-scale analysis is only straightforward assuming infinitesimal strain theory. Exploiting the potential of ANNs for fast and reliable upscaling and localisation procedures, we propose an incremental numerical approach that considers rearrangement of the cell properties based on its current deformation, which leads to the remodelling of the macroscopic model after each time increment. This computational framework is valid for finite strain and large deformation problems while it ensures infinitesimal strain increments within time steps. The full effects of the interdependency between the properties and response of macro and micro scales are considered for the first time providing more accurate predictive analysis of fluid-saturated porous media which is studied via a numerical consolidation example. Furthermore, the (nonlinear) deviation from Darcy’s law is captured in fluid filtration numerical analyses. Finally, the brain tissue mechanical response under uniaxial cyclic test is simulated and studied.