Partitioning strategies for the interaction of a fluid with a poroelastic material based on a Nitsche's coupling approach

We develop a computational model to study the interaction of a fluid with a poroelastic material. The coupling of Stokes and Biot equations represents a prototype problem for these phenomena, which feature multiple facets. On one hand it shares common traits with fluid-structure interaction. On the other hand it resembles the Stokes-Darcy coupling. For these reasons, the numerical simulation of the Stokes-Biot coupled system is a challenging task. The need of large memory storage and the difficulty to characterize appropriate solvers and related preconditioners are typical shortcomings of classical discretization methods applied to this problem. The application of loosely coupled time advancing schemes mitigates these issues because it allows to solve each equation of the system independently with respect to the others. In this work we develop and thoroughly analyze a loosely coupled scheme for Stokes-Biot equations. The scheme is based on Nitsche's method for enforcing interface conditions. Once the interface operators corresponding to the interface conditions have been defined, time lagging allows us to build up a loosely coupled scheme with good stability properties. The stability of the scheme is guaranteed provided that appropriate stabilization operators are introduced into the variational formulation of each subproblem. The error of the resulting method is also analyzed, showing that splitting the equations pollutes the optimal approximation properties of the underlying discretization schemes. In order to restore good approximation properties, while maintaining the computational efficiency of the loosely coupled approach, we consider the application of the loosely coupled scheme as a preconditioner for the monolithic approach. Both theoretical insight and numerical results confirm that this is a promising way to develop efficient solvers for the problem at hand

A Lagrange multiplier method for a Stokes-Biot fluid-poroelastic structure interaction model

We study a finite element computational model for solving the coupled problem arising in the interaction between a free fluid and a fluid in a poroelastic medium. The free fluid is governed by the Stokes equations, while the flow in the poroelastic medium is modeled using the Biot poroelasticity system. Equilibrium and kinematic conditions are imposed on the interface. A mixed Darcy formulation is employed, resulting in continuity of flux condition of essential type. A Lagrange multiplier method is employed to impose weakly this condition. A stability and error analysis is performed for the semi-discrete continuous-in-time and the fully discrete formulations. A series of numerical experiments is presented to confirm the theoretical convergence rates and to study the applicability of the method to modeling physical phenomena and the sensitivity of the model with respect to its parameters

A Computational model for fluid-porous structure interaction

This work utilizes numerical models to investigate the importance of poroelasticity in Fluid- Structure Interaction, and to establish a connection between the apparent viscoelastic behavior of the structure part and the intramural filtration flow. We discuss a loosely coupled computational framework for modeling multiphysics systems of coupled flow and mechanics via finite element method. Fluid is modeled as an incompressible, viscous, Newtonian fluid using the Navier-Stokes equations and the structure domain consists of a thick poroelastic material, which is modeled by the Biot system. Physically meaningful interface conditions are imposed on the discrete level via mortar finite elements or Nitsche's coupling. We further discuss the use of our loosely coupled non-iterative time-split formulation as a preconditioner for the monolithic scheme. We further investigate the interaction of an incompressible fluid with a poroelastic structure featuring possibly large deformations, where the assumption of large deformations is taken into account by including the full strain tensor. We use this model to study the influence of different parameters on energy dissipation in a poroelastic medium. The numerical results investigate the effects of poroelastic parameters on the pressure wave propagation, filtration of the incompressible fluid through the porous media, and the structure displacement

Multi-component model of intramural hematoma

A novel multi-component model is introduced for studying interaction between blood flow and deforming aortic wall with intramural hematoma (IMH). The aortic wall is simulated by a composite structure submodel representing material properties of the three main wall layers. The IMH is described by a poroelasticity submodel which takes into account both the pressure inside hematoma and its deformation. The submodel of the hematoma is fully coupled with the aortic submodel as well as with the submodel of the pulsatile blood flow. Model simulations are used to investigate the relation between the peak wall stress, hematoma thickness and permeability in patients of different age. The results indicate that an increase in hematoma thickness leads to larger wall stress, which is in agreement with clinical data. Further simulations demonstrate that a hematoma with smaller permeability results in larger wall stress, suggesting that blood coagulation in hematoma might increase its mechanical stability. This is in agreement with previous experimental observations of coagulation having a beneficial effect on the condition of a patient with the IMH

Rotation-Based Mixed Formulations for an Elasticity-Poroelasticity Interface Problem

In this paper we introduce a new formulation for the stationary poroelasticity equations written using the rotation vector and the total fluid-solid pressure as additional unknowns, and we also write an extension to the elasticity-poroelasticity problem. The transmission conditions are imposed naturally in the weak formulation, and the analysis of the effective governing equations is conducted by an application of Fredholm's alternative. We also propose a monolithically coupled mixed finite element method for the numerical solution of the problem. Its convergence properties are rigorously derived and subsequently confirmed by a set of computational tests that include applications to subsurface flow in reservoirs as well as to dentistry-oriented problems.Fondo Nacional de Desarrollo Científico y Tecnológico/[11160706]/FONDECYT/ChilePrograma Concurso Apoyo a Centros Científicos y Tecnológicos de Excelencia/[AFB170001]/PIA/ChileUCR::Sedes Regionales::Sede de OccidenteUCR::Vicerrectoría de Docencia::Ciencias Básicas::Facultad de Ciencias::Escuela de Matemátic

A space-time discontinuous Galerkin method for coupled poroelasticity-elasticity problems

This work is concerned with the analysis of a space-time finite element discontinuous Galerkin method on polytopal meshes (XT-PolydG) for the numerical discretization of wave propagation in coupled poroelastic-elastic media. The mathematical model consists of the low-frequency Biot's equations in the poroelastic medium and the elastodynamics equation for the elastic one. To realize the coupling, suitable transmission conditions on the interface between the two domains are (weakly) embedded in the formulation. The proposed PolydG discretization in space is then coupled with a dG time integration scheme, resulting in a full space-time dG discretization. We present the stability analysis for both the continuous and the semidiscrete formulations, and we derive error estimates for the semidiscrete formulation in a suitable energy norm. The method is applied to a wide set of numerical test cases to verify the theoretical bounds. Examples of physical interest are also presented to investigate the capability of the proposed method in relevant geophysical scenarios

A mathematical model for meniscus cartilage regeneration

We propose a continuous model for meniscus cartilage regeneration triggered by two populations of cells migrating and (de)differentiating within an artificial scaffold with a known structure. The described biological processes are influenced by a fluid flow and therewith induced deformations of the scaffold. Numerical simulations are done for the corresponding dynamics within a bioreactor which was designed for performing the biological experiments.Comment: GAMM2023, May 2023, Dresden (GERMANY), German