Second‐order, loosely coupled methods for fluid‐poroelastic material interaction

Abstract

This work focuses on modeling the interaction between an incompressible, viscous fluid and a poroviscoelastic material. The fluid flow is described using the time-dependent Stokes equations, and the poroelastic material using the Biot model. The viscoelasticity is incorporated in the equations using a linear Kelvin-Voigt model. We introduce two novel, noniterative, partitioned numerical schemes for the coupled problem. The first method uses the second-order backward differentiation formula (BDF2) for implicit integration, while treating the interface terms explicitly using a second-order extrapolation formula. The second method is the Crank-Nicolson and Leap-Frog (CNLF) method, where the Crank-Nicolson method is used to implicitly advance the solution in time, while the coupling terms are explicitly approximated by the Leap-Frog integration. We show that the BDF2 method is unconditionally stable and uniformly stable in time, while the CNLF method is stable under a CFL condition. Both schemes are validated using numerical simulations. Second-order convergence in time is observed for both methods. Simulations over a longer period of time show that the errors in the solution remain bounded. Cases when the structure is poroviscoelastic and poroelastic are included in numerical examples. KEYWORDSfluid-poroelastic structure interaction, partitioned methods, second-order convergence Numer Methods Partial Differential Eq. 2020;36:800-822. wileyonlinelibrary.com/journal/num

Decoupled modified characteristic finite element method for the time‐dependent Navier–Stokes/Biot problem

“…where 𝜇 p and 𝜆 p are the Lamé constants for the solid skeleton. Following [4,39], we use the following the interface conditions on Γ × (0, T]…”

“…. It follows from the Cauchy-Schwarz inequality, the inverse inequality (26), Young's inequality (27) and (39) that…”

“…Example 5.1 The exact solution is taken from References [4,39]. Let Ω f = (0, 1)×(0, 1) and Ω p = (0, 1)×(−1, 0), with Ω f and Ω p separated by Γ = (0, 1)×{0} ) .…”

“…The other is partitioned methods, which decouple the system into sub‐problems. Monolithic methods for FPSI problem have been studied in References [2–4, 6, 15, 48], and partitioned methods have been studied in References [6, 10, 11, 16, 35, 39].…”

Hybridizable discontinuous Galerkin methods for the coupled Stokes--Biot problem

“…Eliminating the Darcy velocity z, they consider stable Stokes elements for (u s , p s ) and (u b , p b ), where p b is the total pressure, and a piecewise continuous and polynomial space for p p . Finally, let us mention that a stress tensor based approach using a weakly symmetric mixed method for poroelasticity [29] and a conforming stable mixed method for Stokes was studied in [1,30], and that partitioned time discretization methods, focusing on efficient time stepping and stability of partitioned schemes, are studied in [16,15,33].…”