Unconditionally Stable PressureCorrection Schemes for a Nonlinear FluidStructure Interaction Model
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Abstract
We consider in this paper numerical approximation of a nonlinear fluidstructure interaction (FSI) model with a fixed interface. We construct a new class of pressurecorrection schemes for the FSI problem, and prove rigorously that they are unconditionally stable. These schemes are computationally very efficient, as they lead to, at each time step, a coupled linear elliptic system for the velocity and displacement in the whole region and a discrete Poisson equation in the fluid region.
Keywords
Fluidstructure interaction Pressure correction Stability analysisMathematics subject classification:
74F10 76D05 65M12 35Q301 Introduction
Fluidstructure interaction (FSI) plays an important role in many scientific/engineering applications, e.g., design of engineering systems, blood flow in human arteries, etc. It has been extensively studied in recent years both analytically and computationally (cf. [6, 9, 11, 18] and the references therein).
There are mainly three approaches, monolithic, partitioned and semiimplicit projection, for solving FSI problems numerically. The partitioned approach (cf., for instance, [2, 4, 10, 24]) solves the fluid and structure dynamics separately with explicit interface conditions. While each subproblem can be solved efficiently by existing algorithms, the explicit treatment of the interface condition may lead to instability in the presence of strong addedmass effect [5] and requires very restrictive time step constraint. In contrast, the monolithic approach (cf., for instance, [19, 20, 23]) simultaneously solves the fluid and structure dynamics coupled by the implicit interface conditions. This type of schemes usually have good stability properties, but at each time step, a nonlinear coupled system has to be solved and, due to the presence of the pressure in the coupled system, it is usually difficult to design an effective iterative scheme to solve the nonlinear coupled system. On the other hand, the semiimplicit projection approach was first proposed in [12]. It decouples the computation of fluid velocity from that of the pressure and structure displacement by using a projection method. This method appears to have some computational advantage over the partitioned or monolithic approaches (cf., for instance, [1, 3, 12]).
In this paper, we shall construct a different class of semiimplicit projection schemes which decouple the computation of pressure from that of the velocity and structure displacement. Our schemes will be computationally very efficient. More precisely, in the first step of our schemes, we solve a coupled, but elliptic, system for an intermediate fluid velocity and the structure displacement; then in the second step, we solve a Poisson equation for the fluid pressure and obtain the fluid velocity with a simple correction. Furthermore, we shall also prove rigorously that these schemes are unconditionally stable.
For fluid problems, an effective approach to decouple the computation of the pressure from that of the velocity is to use a projectiontype method, originally proposed by Chorin and Temam in the late 1960s [7, 28]. A comprehensive review on various projectiontype methods can be found in [14]. However, a main difficulty in the design of a projection method for the FSI problem is to assign a boundary condition for the pressure at the interface. It is well known that a proper boundary condition for the pressure Poisson equation in a projectiontype method, at the Dirichlet part of the boundary, is the homogeneous Neumann boundary condition. Indeed, most existing projectiontype schemes for the FSI problem also use, explicitly or implicitly, the Neumanntype boundary condition for the pressure Poisson equation at the Dirichlet part of the boundary as well as at the interface. However, imposing a Neumanntype boundary condition for the pressure at the interface appears to affect, to a certain degree, the stability of the scheme, and we are not aware of any proof of unconditional stability for this type of projection schemes, only a conditional stability has been proved in [12] for a linear FSI problem. In a previous paper [17], the authors constructed an unconditionally stable scheme for a linear FSI problem. The aim of this paper is to extend it to a nonlinear FSI problem.
In [13], the authors proposed and analyzed pressurecorrection schemes for Navier–Stokes equations with open boundary where the usual stressfree boundary condition is applied. It is shown that the proper boundary condition at the open boundary is of Dirichlet type instead of Neumann type. Two schemes are constructed in [13], one is based on the standard pressure correction which leads to poor accuracy at the open boundary, while the other is based on the rotational pressure correction and with a proper Dirichlet boundary condition at the open boundary. It is shown in [13] that both the standard and rotational pressurecorrection projection schemes, when applied to the timedependent Stokes problem, are unconditionally stable, but the rotational version leads to much better accuracy. Since one of the matching interface conditions for the FSI problem is related to the stress, it makes sense to extend the approach in [13] for problems with open boundary to the FSI problem.
Besides the difficulty associated with the pressure boundary condition on the interface, another major difficulty is to prove the unconditional stability of the rotational pressurecorrection scheme for the nonlinear FSI problem. The original stability proof of the rotational pressurecorrection scheme in [15] was only valid for Stokes problems. An essential step of the proof was to take the “discrete time derivative” of the scheme. Unfortunately, this proof cannot be extended to the nonlinear case. In [8], the authors constructed an unconditionally stable rotational velocitycorrection scheme for the Navier–Stokes equations. However, they only provided a stability proof for the linear Stokes equations, while showing numerically that the scheme was unconditionally stable. In [25], the author proposed a Gauge–Uzawa approach for the rotational pressurecorrection scheme of the Navier–Stokes equations, and proved that the scheme was unconditionally stable. We shall extend the approach in [25] for the Gauge–Uzawa scheme of the NavierStokes equations to the rotational pressurecorrection schemes for the FSI problem.
To fix the idea, we consider in this paper a simple model of the FSI problem where the movement of the interface is assumed infinitesimal so the interface is treated as fixed. This nonlinear FSI problem captures many of the essential difficulties of the more general FSI problems with moving interface, and its wellposedness has been studied in [22].
The rest of the paper is organized as follows. In the next section, we describe the governing equations for our FSI model, formulate its weak form and the energy dissipation law. In Sect. 3, we construct a standard and rotational pressurecorrection scheme for the FSI problem and prove their unconditional stability. Then, in Sect. 4, we describe a generic approach for spatial discretization as well as a Fourier–Legendre method for a special case of a periodic channel. We present some numerical results in Sect. 5 to validate our numerical schemes and to demonstrate their temporal accuracy. Some concluding remarks are given in Sect. 6.
2 Governing Equations
We assume that the interface undergoes infinitesimal displacements, i.e., \(\Gamma _c\) is fixed. The more complicated situation with moving interface will be considered in a forthcoming paper.
For the wellposedness of the system (2.7), we refer to [22].
3 Time Discretization
For FSI problems, it is very important to design numerical schemes which have good, preferably unconditional, stability property. Usually, this is achieved by fully coupled, implicit schemes which require solving, at each time step, a coupled, nonlinear, saddlepoint system.
We construct in this section time discretization schemes based on the standard and rotational pressurecorrection approach for (2.7). These schemes are unconditionally stable and lead to, at each time step, a coupled, linear elliptic system in \(\Omega\) and a pressure Poisson equation in \(\Omega _f\), which can be efficiently solved by standard numerical methods. The stability analysis for each scheme is carried out in this section.
3.1 Standard PressureCorrection Scheme
We first construct a firstorder scheme for the FSI problem based on the standard pressurecorrection scheme for the Navier–Stokes problem with the open boundary condition [13]:
For the above scheme, we have the following result:
Theorem 3.1
Proof
To simplify the notations, we define, for any sequence \(\{u^{k}\}\), the discrete time derivatives \(\delta _tu^{n+1}:=\frac{u^{n+1}u^n}{\Delta t}\) and \(\delta ^2_{tt}u^{n+1}:=\frac{\delta _tu^{n+1}\delta _tu^n}{\Delta t}=\frac{u^{n+1}2u^n+u^{n1}}{\Delta t^2}\).
We recall that due to the artificial Dirichlet boundary condition for the pressure in (3.2c), a higherorder discretization for the velocity will not increase the accuracy. Hence, to obtain a higherorder scheme, one needs to resort to the rotational pressurecorrection (cf. [13]).
3.2 Rotational PressureCorrection Schemes
3.2.1 FirstOrder Scheme
We start by constructing a firstorder scheme.
We observe that the main difference of the rotational scheme (3.8)–(3.9) with the standard scheme (3.1)–(3.3) is the additional term \(\lambda \mu \,\text {div}\, \tilde{u}^{n+1}\) in (3.9a). This term replaces the artificial Dirichlet B.C. \(p^{n+1}_{\Gamma _c}=p^{n}_{\Gamma _c}\) by an improved B.C. \(p^{n+1}_{\Gamma _c}=(p^n\lambda \mu \text {div} \tilde{u}^{n+1})_{\Gamma _c}\). On the other hand, the numerical procedures for the two schemes are essentially identical.
The proof of unconditional stability for the rotational scheme is much more difficult. The original stability proof of the rotational pressurecorrection scheme in [15] was carried out only for Stokes problems, and an essential step of the proof was to take the “discrete time derivative” of the scheme. Unfortunately, this proof cannot be extended to the nonlinear case. However, we can prove that the above rotational scheme is unconditionally stable using a similar procedure to that in the proof below for the secondorder rotational scheme. We omit the details for the sake of brevity.
3.2.2 SecondOrder Scheme
We can now construct a secondorder rotational pressurecorrection scheme as follows:

One observes that all the terms, except the pressure, are discretized with a secondorder BDF or Adams–Bashforth formula. We recall that a firstorder treatment of the pressure term, coupled with secondorder treatment for other terms, can lead to secondorder accuracy for the velocity [14].

It is clear that, at each time step, the numerical procedure for solving (3.13)–(3.14) is essentially the same as for the firstorder scheme (3.1)–(3.3).
 In [25], the author proved the unconditional stability for a Gauge–Uzawa scheme of the Navier–Stokes equations. A useful idea in [25] is to introduce a sequence \(\{q^n\}\) defined byWe shall also use this sequence in our stability proof below.$$\begin{aligned} q^n=\lambda \mu \text {div}\; \tilde{u}^n+q^{n1}\text { with } q^{1}=q^0=0. \end{aligned}$$(3.15)
Theorem 3.2
Proof
Remark 3.3
With the stability results established in this section, it is also possible to derive similar error estimates for these schemes as in [13].
4 GalerkinType Spatial Discretization and Implementation
We briefly describe a general procedure to implement the time discretization schemes constructed in the last section. Let \(\mathbf{X}_h\subset {\mathbf{H}}^1_{0,\Gamma _f}(\Omega _f)\), \(M_h\subset {\mathbf{H}}^1(\Omega _f)\), \(M^0_h=\{q\in M_h:\, q_{\Gamma _c}=0\}\) and \(\mathbf{W}_h\subset \mathbf{H}^1_{0,\Gamma _s}(\Omega _s)\) be some finite dimensional approximation spaces, with \((\mathbf{X}_h,M_h)\) preferably satisfying the Babuska–Brezzi infsup condition. We also denote \(\mathbf{Y}_h=\mathbf{X}_h+\nabla M_h^0\). We note that one can generalize the stability proofs for the semidiscretized schemes in the last section to their full discretized versions using the above discrete settings; we refer to [14] for more detail in this regard.
To fix the idea, we take the scheme (3.8)–(3.9) as an example. The other schemes can be treated by using exactly the same procedure.
4.1 A General Setup
A Galerkin approximation of the scheme (3.8)–(3.9) is as follows:
We note that (4.5) is just a discrete Poisson equation in \(\Omega _f\) with the homogeneous Dirichlet boundary condition on \(\Gamma _c\), and (4.6) involves only a projection, so they can be efficiently solved.
4.2 An Example with a Fourier–Legendre Approximation
As an example, we consider a twodimensional periodic channel with \(\Omega _f=(0,2\pi )\times (0,1)\), \(\Omega _s=(0,2\pi )\times (1,0)\), so \(\Omega =(0,2\pi )\times (1,1)\), \(\Gamma _f=\{(x,y)\,x\in (0,2\pi ), y=1\}\), \(\Gamma _c=\{(x,y)\,x\in (0,2\pi ), y=0\}\) and \(\Gamma _s=\{(x,y)\,x\in (0,2\pi ), y=1\}\). We denote \(\mathbf{I^+}\,,\mathbf{{I^}}\,,\mathbf{{I}}\) by \(\mathbf{{I^+}}=[0,1]\), \(\mathbf{{I^}}=[1,0]\) and \(\mathbf{I}=[1,1]\). We assume that all functions are periodic in the xdirection.
It is clear that (4.5) will reduce to a sequence of onedimensional problems in \(\mathbf{I}^+\) which can be easily solved by a Legendrespectral method.
5 Numerical Results
We employ the Fourier–Legendre method presented in the last section and choose (M, N) large enough so that the errors are dominated by the time discretization. In the following examples, we choose \(\lambda =0.5\), which is a preselected parameter introduced in (3.9a) and (3.14a).
6 Conclusions
We constructed in this paper standard and rotational pressure correction schemes for the FSI problem with a fixed interface and proved rigorously that they are unconditionally energy stable. These schemes are new and fundamentally different from existing schemes for the FSI problem. Besides the unconditional stability, they are also computationally very efficient: at each time step, they lead to (i) a coupled linear elliptic system for the velocity and displacement, with the coupling condition at the interface between the fluid and solid regions, which can be efficiently solved by using a standard domain decomposition (with two domains) approach; and (ii) a discrete Poisson equation in the fluid region.
We validated these schemes by using a Fourier–Legendre spatial discretization for the FSI problem in a periodic channel. In particular, our numerical results indicate that the convergence rates of the secondorder rotational scheme for the velocity, pressure and displacement in \(L^2\)norm are close to 3/2order.
Although we only considered the FSI problem with fixed interface, we believe that the essential approaches used here in constructing our numerical schemes can be extended to the FSI problem with moving interface [21], which we plan to address in a future endeavor.
References
 1.Badia, S., Codina, R.: On some fluidstructure iterative algorithms using pressure segregation methods. Application to aeroelasticity. Int. J. Numer. Methods Eng. 72(1), 46–71 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
 2.Badia, S., Nobile, F., Vergara, C.: Fluidstructure partitioned procedures based on Robin transmission conditions. J. Comput. Phys. 227(14), 7027–7051 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
 3.Badia, S., Quaini, A., Quarteroni, A.: Splitting methods based on algebraic factorization for fluidstructure interaction. SIAM J. Sci. Comput. 30(4), 1778–1805 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
 4.Burman, E., Fernández, M.A.: Stabilized explicit coupling for fluidstructure interaction using Nitsche’s method. C. R. Math. Acad. Sci. Paris 345(8), 467–472 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
 5.Causin, P., Gerbeau, J.F., Nobile, F.: Addedmass effect in the design of partitioned algorithms for fluidstructure problems. Comput. Methods Appl. Mech. Eng. 194(42–44), 4506–4527 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
 6.Chakrabarti, S.K., Hernandez, S., Brebbia, C.A.: Fluid structure interaction and moving boundary problems, volume 43 of Advances in Fluid Mechanics. WIT Press, Southampton, 2005. Edited papers from the 3rd International Conference on Fluid Structure Interaction and the 8th International Conference on Computational Modelling and Experimental Measurements of Free and Moving Boundary Problems held in La Coruna, September 19–21 (2005)Google Scholar
 7.Chorin, A.J.: Numerical solution of the Navier–Stokes equations. Math. Comp. 22, 745–762 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
 8.Dong, S., Shen, J.: An unconditionally stable rotational velocitycorrection scheme for incompressible flows. J. Comput. Phys. 229(19), 7013–7029 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
 9.Farhat, C., Lesoinne, M., LeTallec, P.: Load and motion transfer algorithms for fluid/structure interaction problems with nonmatching discrete interfaces: momentum and energy conservation, optimal discretization and application to aeroelasticity. Comput. Methods Appl. Mech. Eng. 157(1–2), 95–114 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
 10.Felippa, C.A., Park, K., Farhat, C.: Partitioned analysis of coupled mechanical systems. Comput. Methods Appl. Mech. Eng. 190, 3247–3270 (2001)CrossRefzbMATHGoogle Scholar
 11.Fernández, M .A.: Coupling schemes for incompressible fluidstructure interaction: implicit, semiimplicit and explicit. SeMA J 55, 59–108 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
 12.Fernández, M.A., Gerbeau, J.F., Grandmont, C.: A projection semiimplicit scheme for the coupling of an elastic structure with an incompressible fluid. Int. J. Numer. Methods Eng. 69(4), 794–821 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
 13.Guermond, J.L., Minev, P., Shen, J.: Error analysis of pressurecorrection schemes for the timedependent Stokes equations with open boundary conditions. SIAM J. Numer. Anal. 43(1), 239–258 (2005). (electronic)MathSciNetCrossRefzbMATHGoogle Scholar
 14.Guermond, J.L., Minev, P., Shen, J.: An overview of projection methods for incompressible flows. Comput. Methods Appl. Mech. Eng. 195(44–47), 6011–6045 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
 15.Guermond, J.L., Shen, J.: On the error estimates for the rotational pressurecorrection projection methods. Math. Comput. 73(248), 1719–1737 (2004). (electronic)MathSciNetCrossRefzbMATHGoogle Scholar
 16.He, Y., Nicholls, D.P., Shen, J.: An efficient and stable spectral method for electromagnetic scattering from a layered periodic structure. J. Comput. Phys. 231(8), 3007–3022 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
 17.He, Y., Shen, J.: Unconditionally stable pressurecorrection schemes for a linear fluidstructure interaction problem. Numer. Math. Theory Methods Appl. 7(4), 537–554 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
 18.Hou, G., Wang, J., Layton, A.: Numerical methods for fluidstructure interaction–a review. Commun. Comput. Phys. 12(2), 337–377 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
 19.Hron, J., Turek, S.: A Monolithic FEM/Multigrid Solver for an ALE Formulation of FluidStructure Interaction with Applications in Biomechanics. Springer, Berlin, Heidelberg (2006)CrossRefzbMATHGoogle Scholar
 20.Hübner, B., Walhorn, E., Dinkler, D.: A monolithic approach to fluidstructure interaction using spacetime finite elements. Comput. Methods Appl. Mech. Eng. 193(23), 2087–2104 (2004)CrossRefzbMATHGoogle Scholar
 21.Ignatova, M., Kukavica, I., Lasiecka, I., Tuffaha, A.: On wellposedness for a free boundary fluidstructure model. J. Math. Phys. 53(11), 115624 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
 22.Kukavica, I., Tuffaha, A., Ziane, M.: Strong solutions to a nonlinear fluid structure interaction system. J. Differ. Equ. 247(5), 1452–1478 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
 23.Küttler, U., Wall, W.A.: Fixedpoint fluidstructure interaction solvers with dynamic relaxation. Comput. Mech. 43(1), 61–72 (2008)CrossRefzbMATHGoogle Scholar
 24.Matthies, H.G., Steindorf, J.: Partitioned strong coupling algorithms for fluidstructure interaction. Comput. Struct. 81, 805–812 (2003)CrossRefGoogle Scholar
 25.Pyo, J.H.: Error estimates for the second order semidiscrete stabilized GaugeUzawa method for the NavierStokes equations. Int. J. Numer. Anal. Model. 10(1), 24–41 (2013)MathSciNetzbMATHGoogle Scholar
 26.Quarteroni, A., Valli, A.: Domain decomposition methods for partial differential equations. Numerical Mathematics and Scientific Computation. The Clarendon Press Oxford University Press, New York (1999)zbMATHGoogle Scholar
 27.Shen, J.: Efficient spectralGalerkin method. I. Direct solvers of second and fourthorder equations using Legendre polynomials. SIAM J. Sci. Comput. 15(6), 1489–1505 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
 28.Temam, R.: Sur l’approximation de la solution des équations de Navier–Stokes par la méthode des pas fractionnaires i. Arch. Rat. Mech. Anal. 32, 135–153 (1969)CrossRefzbMATHGoogle Scholar
 29.Toselli, A., Widlund, O.: Domain decomposition methods—algorithms and theory. Springer Series in Computational Mathematics, vol. 34. Springer, Berlin (2005)CrossRefzbMATHGoogle Scholar