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A Stabilized Finite Element Method for Morpho-Visco-Poroelastic Model

Sabia Asghar, Duncan den Bakker, Etelvina Javierre, Qiyao Peng, Fred J. Vermolen

TL;DR

This work develops a morpho-viscoporoelastic model that couples elastic, viscous and porous tissue mechanics with microstructural growth. It establishes symmetry of the strain tensor and analyzes linear stability of steady states in both continuous and semi-discrete settings, deriving conditions under which the equilibria are stable for nonnegative growth parameter $\alpha$. The authors design a stabilized Galerkin FEM by perturbing the pressure equation, obtaining explicit M-matrix based criteria and a threshold for the stabilization parameter $\beta$ to ensure monotonic, oscillation-free pressure solutions. Numerical experiments on a unit square corroborate the stability bounds and demonstrate TVD behavior of the pressure field as $\beta$ increases. Overall, the paper provides a rigorous stability framework and a practical stabilization strategy for reliable simulations of morphoelastic growth in poroelastic tissues, with clear paths to higher-dimensional extensions.

Abstract

We propose a mathematical model that combines elastic, viscous and porous effects with growth or shrinkage due to microstructural changes. This phenomenon is important in tissue or tumor growth, as well as in dermal contraction. Although existence results of the solution to the problem are not given, the current study assesses stability of the equilibria for both the continuous and semi-discrete versions of the model. Furthermore, a numerical condition for monotonicity of the numerical solution is described, as well as a way to stabilize the numerical solution so that spurious oscillations are avoided. The derived stabilization result is confirmed by computer simulations. In order to have a more quantitative picture, the total variation has been evaluated as a function of the stabilization parameter.

A Stabilized Finite Element Method for Morpho-Visco-Poroelastic Model

TL;DR

This work develops a morpho-viscoporoelastic model that couples elastic, viscous and porous tissue mechanics with microstructural growth. It establishes symmetry of the strain tensor and analyzes linear stability of steady states in both continuous and semi-discrete settings, deriving conditions under which the equilibria are stable for nonnegative growth parameter . The authors design a stabilized Galerkin FEM by perturbing the pressure equation, obtaining explicit M-matrix based criteria and a threshold for the stabilization parameter to ensure monotonic, oscillation-free pressure solutions. Numerical experiments on a unit square corroborate the stability bounds and demonstrate TVD behavior of the pressure field as increases. Overall, the paper provides a rigorous stability framework and a practical stabilization strategy for reliable simulations of morphoelastic growth in poroelastic tissues, with clear paths to higher-dimensional extensions.

Abstract

We propose a mathematical model that combines elastic, viscous and porous effects with growth or shrinkage due to microstructural changes. This phenomenon is important in tissue or tumor growth, as well as in dermal contraction. Although existence results of the solution to the problem are not given, the current study assesses stability of the equilibria for both the continuous and semi-discrete versions of the model. Furthermore, a numerical condition for monotonicity of the numerical solution is described, as well as a way to stabilize the numerical solution so that spurious oscillations are avoided. The derived stabilization result is confirmed by computer simulations. In order to have a more quantitative picture, the total variation has been evaluated as a function of the stabilization parameter.

Paper Structure

This paper contains 15 sections, 7 theorems, 56 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. The mathematical model
  3. Stability of the model
  4. Symmetry of the strain tensor
  5. Stability of the steady state
  6. Stability of steady states in the continuous problem
  7. Stability of steady states in the semi-discrete problem
  8. The numerical scheme
  9. Galerkin approximation
  10. Monotonicity requirements
  11. Stabilized finite-element method
  12. Numerical results
  13. Pressure profiles and stabilization
  14. Total Variation (TV) for pressure field
  15. Conclusions

Key Result

Lemma 3.1

Let ${\bf v}, {\bf L} \in \mathbb{R}^{d \times d}$ be d-dimensional tensors. Suppose that ${\bf L}$ is skew-symmetric (${\bf L}^T = -{\bf L}$), then for all ${\bf v} \in \mathbb{R}^{d \times d}$ the tensorial scalar product satisfies

Figures (3)

  • Figure 1: Pressure fields for different pairs of $\kappa$ and $h$.
  • Figure 2: Pressure fields for $\kappa = 10^{-6}$ and $h=0.07$ and different values of the stabilization parameter $\beta$.
  • Figure 3: TV for pressure fields for $\kappa = 10^{-6}$ and $h=0.07$ versus different values of the stabilization parameter $\beta$ for one time step $\Delta t=0.1$.

Theorems & Definitions (11)

  • Lemma 3.1
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 3.4
  • Remark 3.1
  • Remark 3.2
  • Remark 4.1
  • Theorem 4.1
  • Remark 4.2
  • ...and 1 more