A semi-decoupled MAC scheme for the coupled fluid-poroelastic material interaction
Introduction
The interaction between the incompressible viscous flow and the flow in saturated porous media with small deformable is important in many engineering and biological applications, such as reservoir engineering, groundwater flow, blood flow through vessels [1], [2], [3], [4]. We consider a fluid-poroelastic structure interaction (FPSI) problem, in which the fluid is described by the Stokes equations and the structure is provided by the fully dynamic Biot model, that is, coupled Stokes-Biot problem. The model features two different couplings, the Stokes-Darcy (SD) problem [5], [6], [7], [8], [9], [10] and the fluid-structure interaction (FSI) problem [11], [12], [13], [14]. It carries on all difficulties involving SD and FSI problems. Due to complexity of interface conditions, the analysis in SD cannot directly be used in FPSI. Thus, numerical methods for solving FPSI system paly a significant role.
The Stokes-Biot problem has been analyzed to be well-posed [15]. Concerning the computational perspective, different numerical approximates have attracted much attention of scholars. There are mainly two ways for solving the coupled Stokes-Biot flow model, including the monolithic method [16], [17], [18], [19], [20], [21], [22], [23] and partitioned approach [24], [25], [26], [27], [28], [29]. For example, Ambartsumyan et al. focused on the monolithic scheme for Stoke-Biot problem with the approximation of the mass conservation via a Lagrange multiplier [18]. Wen et al. constructed a strongly conservative discrete scheme [21], and proposed the interior penalty discontinuous finite element method [22]. Due to the monolithic approach requires solving a large linear system, it is computationally complex; obviously, to develop an effective and stable partitioned method to save computer storage and reduce executive time is natural and necessary. Buskač et al. developed a loosely coupled finite element scheme based on Lie operator splitting method to separate the Navier-Stokes equations from the Biot problem [24] and studied an alternative partitioned strategies for Stokes-Biot problems based on a Nitsche's method [25]. Moreover, Buskač [26] introduced a kinematically β-coupled scheme to split the fluid problem from the dynamic poroelasticity problem, and derived its stability. Recently, Kunwar [27] discussed the second-order time discretization optimization-based decoupling algorithms for Stokes-Biot system. Oyekole and Buskač [28] constructed two types of second-order loosely coupled schemes for Stokes/Biot problem, and proved the stability of them. Guo et al. [29] performed the stability and error estimates for a decoupled modified characteristic finite element scheme for Navier-Stokes/Biot problem. These partitioned methods are mainly based on the finite element methods where the interface conditions are subtly embedded in the weak form, which is not feasible in all applications. In this paper, we will concentrate on the partitioned approach based on the MAC finite difference scheme.
The MAC scheme has been a hot research topic for its fascinating properties such as satisfying the discrete incompressibility constraint, as well as locally conserving of the mass, momentum and energy [30], [31]. In recent years, MAC method has been used to solve Darcy-Forchheimer equation [32], Maxwell's equation [33], Stokes problem [34], linear elasticity problem [35], Stokes-Darcy problem [5], [7]. Rui et al. [5] constructed the second order MAC scheme for the SD system by using the techniques of integration by parts on some controlling volumes near the interface. We constructed a locking free finite difference method using the similar techniques for Stokes-Biot problem, established the stability and the error estimates [23]. However, it needs solve a large system which costs a lot of CPU time. The aim of this work is to develop and analyze an efficient semi-decoupled numerical solver for the Stokes-Biot system. More precisely, we split the displacement of structure from the whole system by designing a time-lagging scheme at each time step, then the residual system composed of velocity and pressure is similar to the coupled SD model. However, we need deal with the more complex interface conditions. In this paper, we rigorously and carefully derive the stability and error estimates by choosing the proper interface conditions. We obtain the second order convergence in the discrete norm for the velocity, pressure and displacement and the second order superconvergence for some terms of the norm for the velocity and displacement.
The paper is organized as follows. Section 2 is devoted to showing the Stokes-Biot equations and the semi-decoupled MAC scheme. The analysis of stability is derived in Section 3. An error estimate is carried out in Section 4. The numerical simulations are shown in Section 5. Throughout the paper we use C, whether it has subscrip or not, to represent a positive constant. It is independent of the spatial mesh size or the time step.
Section snippets
Model, notations and semi-decoupled MAC scheme
In this section, we consider the coupled Stokes-Biot model. Without loss of generality, we restrict the model to a two-dimensional (2D) geometrical model . The fluid domain and poroelastic material region in our problem are given, respectively, by and the boundary , the interface .
Let T be the final time. We model the free flow using the Stokes equations:
The analysis of stability
In this section, we consider the discrete LBB condition and the stability analysis for semi-decoupled Stokes-Biot MAC scheme. Before proceeding, let us address the following property that will serve as auxiliary result for the stability and error analysis in Section 3.1. For simplicity, we present the preliminaries of discrete LBB condition in the Appendix. Here, we just show the result of discrete LBB condition. Then we derive the stability of scheme (2.23a), (2.23b), (2.23c), (2.23d), (2.23e)
The error estimates
In this section, we consider the error estimate of the semi-decoupled MAC scheme (2.23a)-(2.25d). First, by using Taylor's expansion, we can easily obtain the following results (Lemma 4.1, Lemma 4.3) for the pressure, velocity and displacement, which will be used in establishing the error analysis. In this section, the subscripts will be dropped out, the grid point still be denoted by γ. For some special points, we keep the subscripts. We use the standard Sobolev spaces that are
The numerical examples
In the numerical examples, we investigate accuracy of the proposed numerical scheme. We test Example 1, Example 2 to verify the convergence rates for the errors of velocity, pressure and displacement using the semi-decoupled MAC scheme. Here the results on non-uniform grids are similar to on uniform grids, so they are omitted.
Example 1 The exact solution is taken from Reference [21]. We set , and . The domain , the Stokes domain , poroelastic domain
Acknowledgements
This work is supported by the National Natural Science Foundation of China Grant No. 12131014.
References (41)
- et al.
Stable loosely-coupled-type algorithm for fluid-structure interaction in blood flow
J. Comput. Phys.
(2009) - et al.
A monolithic approach to fluid-structure interaction
Comput. Fluids
(2004) Analysis of the coupled Navier-Stokes/Biot problem
J. Math. Anal. Appl.
(2017)- et al.
A Nitsche-based cut finite element method for the coupling of incompressible fluid flow with poroelasticity
Comput. Methods Appl. Mech. Eng.
(2019) - et al.
Coupling Biot and Navier-Stokes equations for modelling fluid-poroelastic media interaction
J. Comput. Phys.
(2009) - et al.
A strongly conservative finite element method for the coupled Stokes-Biot model
Comput. Math. Appl.
(2020) - et al.
Partitioning strategies for the interaction of a fluid with a poroelastic material based on a Nitsche's coupling approach
Comput. Methods Appl. Mech. Eng.
(2015) A loosely-coupled scheme for the interaction between a fluid, elastic structure and poroelastic material
J. Comput. Phys.
(2016)Conservation properties of unstructured staggered mesh schemes
J. Comput. Phys.
(2000)- et al.
An energy-preserving MAC-Yee scheme for the incompressible MHD equation
J. Comput. Phys.
(2001)