Efficient reference configuration formulation in fully nonlinear poroelastic media

TL;DR

This work addresses the challenge of computing a reference configuration for fully nonlinear poroelastic media by treating the reference porosity $Φ$ as an unknown initial condition whose stationary state defines the reference configuration. The authors reformulate the problem as a time-dependent porous medium system and employ Anderson acceleration to dramatically reduce the iterations needed to reach the stationary state, achieving reductions up to about 80%. To overcome primal inconsistency arising from second-order derivatives of displacement, they introduce consistent mixed formulations in both pressure and velocity. The framework is demonstrated with energy-based constitutive models inspired by cardiac tissue (Usyk and Bruinsma energies) and validated in 2D/3D benchmarks and a realistic left-ventricle perfusion scenario, highlighting improved robustness, boundary-condition consistency, and computational efficiency. The results enable reliable reference-configuration computations for large-deformation poroelastic simulations in biomedical contexts and set the stage for extensions to multi-phase media and cardiac ablation modeling.

Abstract

Typical pipelines for model geometry generation in computational biomedicine stem from images, which are usually considered to be at rest, despite the object being in mechanical equilibrium under several forces. We refer to the stress-free geometry computation as the reference configuration problem, and in this work we extend such a formulation to the theory of fully nonlinear poroelastic media. The main steps are (i) writing the equations in terms of the reference porosity and (ii) defining a time dependent problem whose steady state solution is the reference porosity. This problem can be computationally challenging as it can require several hundreds of iterations to converge, so we propose the use of Anderson acceleration to speed up this procedure. Our evidence shows that this strategy can reduce the number of iterations up to 80\%. In addition, we note that a primal formulation of the nonlinear mass conservation equations is not consistent due to the presence of second order derivatives of the displacement, which we alleviate through adequate mixed formulations. All claims are validated through numerical simulations in both idealized and realistic scenarios.

Figures (7)

• Figure 1: (a) the computed reference configuration with respect to the initial geometry (black contour). (b) the forward problem solution as computed starting from the reference configuration computed in (a).
• Figure 2: Evolution of the average spatial porosity $\frac{1}{|\Omega|}\int_\Omega\phi\,dx$ along the iterations of both the reference configuration and forward problems. The given value $\overline \phi$ is shown for comparison.
• Figure 3: (a) the computed reference configuration with respect to the initial geometry (black contour). (b) the forward problem solution as computed starting from the reference configuration in (a).
• Figure 4: Evolution of the average spatial porosity $\frac{1}{|\Omega|}\int_\Omega\phi\,dx$ along the iterations of both the reference configuration and forward problems in both phases for the brick 3D geometry. The given value $\overline \phi$ is shown for comparison.
• Figure 5: Evolution of the average spatial porosity $\frac{1}{|\Omega|}\int_\Omega\phi\,dx$ along the iterations of both the reference configuration and forward problems. We show the evolution obtained using the (a) primal formulation, (b) the mixed pressure formulation and (c) the mixed velocity formulation.

Theorems & Definitions (6)

• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Remark 5
• Remark 6