Virtual Element Methods for Coupled Biot-Kirchhoff Poroelasticity Problems on Polygonal Meshes

Conceitos Básicos

This paper presents conforming and nonconforming virtual element formulations of arbitrary polynomial degrees for the coupling of solid and fluid phases in deformable porous plates, with a focus on well-posedness, a priori error estimates, and residual-based a posteriori error analysis.

Resumo

The paper analyzes virtual element (VE) discretizations for a coupled Biot-Kirchhoff poroelasticity model, which describes the deformation of a fluid-saturated porous plate. The governing equations consist of a fourth-order equation for the transverse displacement of the middle surface coupled with a second-order equation for the pressure head relative to the solid.
The key highlights and insights are:

Novel enrichment operators are proposed that connect nonconforming VE spaces of general degree to continuous Sobolev spaces, satisfying additional orthogonal and best-approximation properties.

A priori error estimates in the best-approximation form are proved for both conforming and nonconforming VE formulations, showing robustness with respect to the main model parameters.

A residual-based reliable and efficient a posteriori error estimator is derived and analyzed, which can drive adaptive mesh refinement schemes.

The flexibility of VEMs in handling general polygonal meshes is exploited, and the proposed framework can serve as a fundamental building block for more complex mixed-dimensional poroelastic models.

Numerical experiments illustrate the performance of the suggested VE discretizations and confirm the theoretical findings on different polygonal meshes with mixed boundary conditions.

Estatísticas

None.

Citações

None.

Principais Insights Extraídos De

Virtual element methods for Biot-Kirchhoff poroelasticity

by Rekha Khot,D... às arxiv.org 05-01-2024

https://arxiv.org/pdf/2306.13890.pdf

Perguntas Mais Profundas

How can the proposed VE framework be extended to handle more complex coupled poroelastic models, such as those involving multi-layered structures or fluid-structure interaction

The proposed Virtual Element (VE) framework for Biot-Kirchhoff poroelasticity can be extended to handle more complex coupled poroelastic models by incorporating additional equations and variables that represent the interactions between different layers or components of the system. For multi-layered structures, the VE method can be adapted to discretize and solve the governing equations for each layer separately, considering the interactions and boundary conditions between the layers. This approach allows for a comprehensive modeling of the entire multi-layered poroelastic system.
In the case of fluid-structure interaction, the VE framework can be extended to include additional equations describing the interaction between the fluid and solid phases. This may involve coupling the equations for fluid flow with the equations for solid deformation, considering the mutual influence of the two components on each other. By incorporating appropriate coupling terms and boundary conditions, the VE method can provide a robust and accurate solution for fluid-structure interaction problems in poroelastic materials.

What are the potential challenges and limitations in applying the a posteriori error analysis to drive fully adaptive schemes for the coupled Biot-Kirchhoff problem

Applying a posteriori error analysis to drive fully adaptive schemes for the coupled Biot-Kirchhoff problem can face several challenges and limitations. One potential challenge is the computational cost associated with performing error estimation and adaptation at each iteration of the solution process. The complexity of the coupled poroelastic problem may require a significant amount of computational resources to accurately estimate errors and adapt the mesh or refine the solution.
Another challenge is the development of reliable and efficient error estimators that can accurately capture the error in the solution while considering the coupling between the solid and fluid phases. The presence of mixed boundary conditions, nonlinearity, and complex geometries in poroelastic problems can make it challenging to design error estimators that are both accurate and computationally feasible.
Furthermore, the implementation of adaptive strategies based on a posteriori error estimates may require careful tuning of parameters and thresholds to ensure the efficiency and effectiveness of the adaptive process. Balancing the trade-off between refinement and computational cost is crucial in driving adaptive schemes for complex coupled poroelastic problems.

Can the enrichment and companion operator techniques developed in this work be adapted to other types of coupled PDE systems beyond poroelasticity

The enrichment and companion operator techniques developed in this work for poroelasticity can be adapted to other types of coupled PDE systems beyond poroelasticity by modifying the enrichment strategies and companion operators to suit the specific characteristics of the new problem. These techniques are based on the idea of connecting nonconforming discrete spaces to conforming spaces of higher degree, which can be applied to various coupled PDE systems with different formulations and boundary conditions.
For example, in the context of fluid-structure interaction problems, the enrichment and companion operators can be tailored to handle the interactions between the fluid and solid components, ensuring compatibility and accuracy in the solution. By adjusting the enrichment functions and companion operators to account for the specific coupling terms and boundary conditions of the new system, the VE method can effectively address the challenges posed by different types of coupled PDE systems.