Numerical Solution of Fluid-Structure Interaction by the Space-Time Discontinuous Galerkin Method

Abstract

This paper is devoted to the numerical solution of the interaction of compressible viscous flow with elastic structures. The flow in a time-dependent domain is described by the compressible Navier-Stokes equations written in the ALE formulation and the deformation of elastic structures is described by the dynamic linear elasticity system. For each individual problem we employ the discretization by the space-time discontinuous Galerkin finite element method (ST-DGM). The flow and elasticity problems are coupled via transmission conditions. The developed method is tested by numerical experiments.

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1 Formulation of the Problem

1.1 Flow Problem

We are concerned with the problem of compressible flow in a time-dependent bounded domain $\varOmega _t \subset \mathbb {R}^2$ with $t\in \left[ 0,T\right] .$ The boundary of $\varOmega _t$ is formed by three disjoint parts: $\partial \varOmega _t =\varGamma _I\cup \varGamma _O\cup \varGamma _{W_t},$ where $\varGamma _I$ is the inlet, $\varGamma _O$ is the outlet and $\varGamma _{W_t}$ represents impermeable time-dependent walls.

The time dependence of the domain $\varOmega _t$ is taken into account with the aid of the Arbitrary Lagrangian-Eulerian (ALE) method (see, e.g., [4]). It is based on a regular one-to-one ALE mapping of the reference configuration $\varOmega _0$ onto the current configuration $\varOmega _t : \mathcal {A}_t \,:\,\bar{\varOmega }_0 \longrightarrow \bar{\varOmega }_t ,\ \mathrm {i.e.}\ \varvec{X}\in \bar{\varOmega }_0 \longmapsto \varvec{x} = \varvec{x}(\varvec{X},t) = \mathcal {A}_t (\varvec{X})\in \bar{\varOmega }_t.$ Further, we define the domain velocity $\tilde{\varvec{z}}(\varvec{X},t) = \frac{\partial }{\partial t} \mathcal {A}_t(\varvec{X}),\,t \in \left[ 0,T\right] , \ \varvec{X}\in \varOmega _0$, $\varvec{z}(\varvec{x},t)=\tilde{\varvec{z}}(\mathcal {A}_{t}^{-1}(\varvec{x}),t),\ t\in \left[ 0,T\right] ,\ \varvec{x}\in \varOmega _t$ and the ALE derivative of the state vector function $\varvec{w}=\varvec{w}(\varvec{x},t)$ defined for $\varvec{x}\in \varOmega _t$ and $t\in \left[ 0,T\right]$: $\frac{D^{\mathcal {A}}}{Dt}\varvec{w}(\varvec{x},t) = \frac{\partial \tilde{\varvec{w}}}{\partial t}(\varvec{X},t)$, $\tilde{\varvec{w}}(\varvec{X},t)=\varvec{w}(\mathcal {A}_t(\varvec{X}),t),\ \varvec{X}\in \varOmega _0,\ \varvec{x} = \mathcal {A}_t(\varvec{X})$. Then the continuity equation, the Navier-Stokes equations and the energy equation can be written in the ALE form

$$\begin{aligned} \frac{D^{\mathcal {A}}\varvec{w}}{Dt}+\sum ^{2}_{s=1}\frac{\partial \varvec{g}_s (\varvec{w})}{\partial x_s}+\varvec{w}\text {div}\varvec{z}=\sum ^{2}_{s=1}\frac{\partial \varvec{R}_s (\varvec{w},\nabla \varvec{w})}{\partial x_s}, \end{aligned}$$

(1)

where $\varvec{w} = (\rho ,\rho v_1 ,\rho v_2 ,E)^T \in \mathbb {R}^4,$ $\varvec{g}_s (\varvec{w}) = \varvec{f}(\varvec{w})_s -z_s \varvec{w},$ $\varvec{f}_s = (\rho v_s ,\rho v_1 v_s +\delta _{1s}p,\rho v_2 v_s +\ \delta _{2s}p,(E+p)v_s )^T ,\quad \varvec{R}_s(\varvec{w},\nabla \varvec{w}) = (0,\tau ^{V}_{s1},\tau ^{V}_{s2}, \tau ^{V}_{s1}v_1 + \tau ^{V}_{s2}v_2 + k\frac{\partial \theta }{\partial x_s})^T ,\ s=1,2,$ $\tau ^{V}_{ij} = \lambda \delta _{ij}\text {div}\varvec{v}+2\mu d_{ij}(\varvec{v}), d_{ij}(\varvec{v})=\frac{1}{2}\left( \frac{\partial v_i}{\partial x_j} +\frac{\partial v_j}{\partial x_i}\right) ,\ i,j=1,2.$ We have $\varvec{R}_s(\varvec{w},\nabla \varvec{w}) = \sum _{k=1}^2 {\mathbb {K}}_{s,k} (\varvec{w})\frac{\partial \varvec{w}}{\partial x_k}$, where ${\mathbb {K}}_{s,k}(\varvec{w})$ are $4\times 4$ matrices depending on $\varvec{w}$, and $\varvec{f}_s(\varvec{w})= \mathbb {A}(\varvec{w})\varvec{w}$ with $\mathbb {A}(\varvec{w}) = D\varvec{f}_s(\varvec{w})/D\varvec{w}$.

The following notation is used: $\rho$—fluid density, $p$—pressure, $E$—total energy, $\varvec{v}=(v_1,v_2)$—velocity vector, $\theta$—absolute temperature, $c_v>0$—specific heat at constant volume, $\gamma >1$—Poisson adiabatic constant, $\mu >0,\lambda =-2\mu /3$—viscosity coefficients, $k>0$—heat conduction coefficient, $\tau ^V_{ij}$—components of the viscous part of the stress tensor. System (1) is completed by the thermodynamical relations $p = (\gamma -1)\left( E-\rho \frac{\left| \varvec{v}\right| ^{2}}{2}\right) , \quad \theta = \frac{1}{c_v}\left( \frac{E}{\rho } - \frac{\left| \varvec{v}\right| ^{2}}{2}\right)$ and equipped with the initial condition $\varvec{w}(\varvec{x},0) = \varvec{w}^{0}(\varvec{x}), \ \varvec{x}\in \varOmega _{0}$ and the boundary conditions:

$\rho =\rho _{D},\quad \varvec{v}=\varvec{v}_{D}, \quad \sum ^{2}_{j=1}\left( \sum ^{2}_{i=1}\tau ^V_{ij}n_{i}\right) v_{j} +k\frac{\partial \theta }{\partial \varvec{n}}=0$ on the inlet $\varGamma _{I}$,

$\varvec{v}=\varvec{z}_D(t)=\text {velocity of a moving wall},\quad \frac{\partial \theta }{\partial \varvec{n}}=0$, on the moving wall $\varGamma _{W_t}$,

$\sum ^{2}_{j=1}\tau ^V_{ij}n_{j}=0,\quad \frac{\partial \theta }{\partial \varvec{n}}=0,\quad i=1,2,$ on the outlet $\varGamma _{O}$,

with prescribed data $\rho _D ,\ \varvec{v}_D ,\ \varvec{z}_D .$ By $\varvec{n}$ we denote the unit outer normal.

1.2 Elasticity Problem

We consider an elastic body $\varOmega ^b \subset \mathbb {R}^2$, which has a common boundary $\varGamma _N^b$ with the reference domain $\varOmega _0$ occupied by the fluid at the initial time. Further, the boundary of $\varOmega ^b$ is formed by two disjoint parts $\partial \varOmega ^b = \varGamma _N^b \cup \varGamma _D^b,$ $\varGamma _N^b \subset \varGamma _{W_0}$ and $\varGamma _D^b$ is a fixed part of the boundary. Using the notation of the displacement of the body $\varvec{u}$ = $\varvec{u}(\varvec{X},t),\ \varvec{X}\in \varOmega ^b,\ t\in (0,T)$ we can write the equations describing the deformation of the elastic body $\varOmega ^b$ in the form

$$\begin{aligned} \rho ^b \frac{\partial ^2 \varvec{u}}{\partial t^2} + c_M \rho ^b \frac{\partial \varvec{u}}{\partial t} - \mathop {div}\varvec{\sigma (\varvec{u})} - c_K \frac{\partial }{\partial t} \mathop {div}\varvec{\sigma (\varvec{u})} = \varvec{f} \quad \text {in } \varOmega ^b\times (0,T), \end{aligned}$$

(2)

$$\begin{aligned} \varvec{u} = \varvec{u}_D \quad \text {in } \varGamma _D^b\times (0,T), \quad \varvec{\sigma } (\varvec{u}) \cdot \varvec{n} = \varvec{g}_N \quad \text {in } \varGamma _N^b\times (0,T), \end{aligned}$$

(3)

$$\begin{aligned} \varvec{u}(x, 0) = \varvec{u}_0(x), \quad x\in \varOmega ^b, \quad \frac{\partial \varvec{u}}{\partial t}(x,0) = \varvec{z}_0(x), \quad x\in \varOmega ^b. \end{aligned}$$

(4)

Here $\varvec{\sigma }(\varvec{u})= \{\sigma _{ij}\}_{i,j=1}^2$, $\sigma _{ij}=\lambda ^b\mathrm{div}\varvec{u} \delta _{ij}+2\mu ^be_{ij}^b(\varvec{u})$ with $e_{ij}^b(\varvec{u})= (\partial u_i/\partial x_j + \partial u_j/\partial x_i)/2$. Further, $\varvec{f} : \varOmega ^b\times (0,T) \rightarrow \mathbb {R}^2$—outer volume force, $\varvec{u}_D : \varGamma _D^b \times (0,T)\rightarrow \mathbb {R}^2$—boundary displacement, $\varvec{g}_N : \varGamma _N^b \times (0,T)\rightarrow \mathbb {R}^2$—boundary normal stress, $\varvec{u}_0 : \varOmega ^b \rightarrow \mathbb {R}^2$—initial displacement, $\varvec{z}_0 : \varOmega ^b \rightarrow \mathbb {R}^2$—initial deformation velocity and $\rho ^b > 0$—material density are given functions. The expressions $c_M \rho ^b \frac{\partial \varvec{u}}{\partial t}$ and $c_K \frac{\partial }{\partial t} \mathop {div}\varvec{\sigma (\varvec{u})}$ represent the damping terms, with $c_M$, $c_K \ge 0$.

The flow and structural problems are coupled by the transmission conditions

$$\begin{aligned}&\varvec{v}=\frac{\partial \varvec{u}}{\partial t}, \quad \sum _{j=1}^2\sigma _{ij}(\varvec{X},t)n_j(\varvec{X})= -\sum _{j=1}^2\tau _{ij}^f(\varvec{x},t) n_j(\varvec{X}), \quad i=1, 2,\\&\quad \varvec{X}\in \ \varGamma _N^b, \ \varvec{x}= \varvec{X}+\varvec{u}(\varvec{X},t), \ \ \tau _{ij}^f=-p\, \delta _{ij} + \tau _{ij}^V.\nonumber \end{aligned}$$

(5)

2 Discrete Problem

2.1 Discretization of the Flow Problem

The problem will be discretized by the space-time discontinuous Galerkin method (ST-DGM). We construct a polygonal approximation $\varOmega _{ht}$ of the domain $\varOmega _t$. By ${\mathcal T}_{ht}$ we denote a partition of the closure $\overline{\varOmega }_{ht}$ of the domain $\varOmega _{t}$ into a finite number of closed triangles $K$ with mutually disjoint interiors such that $\overline{\varOmega }_{ht} = \bigcup _{K\in {\mathcal T}_{ht}} K.$

By ${\mathcal F}_{h}$, ${\mathcal F}_{h}^B, {\mathcal F}_{h}^I$ we denote the systems of all faces of all elements $K\in {\mathcal T}_{ht}$, boundary faces and inner faces, respectively. Further, we introduce the set of “Dirichlet” boundary faces ${{\mathcal F}_{h}^D}= \{\varGamma \in {\mathcal F}_{h}^B;\$a Dirichlet condition is prescribed on$\ \varGamma \}$. Each face $\varGamma$ is associated with a unit normal $\varvec{n}_{\varGamma }$, which has the same orientation as the outer normal on $\varGamma \in {\mathcal F}_{h}^B$. We set $h_{\varGamma }=$ length of $\varGamma \in {\mathcal F}_{h}$.

We introduce the space of piecewise polynomial functions $\varvec{S}_{ht}^r = \{v;v\vert _K\in P_{r}(K)\ \forall \,K\in {\mathcal T}_{ht}\}^4$, where $r > 0$ is an integer and $P_{r}(K)$ denotes the space of all polynomials on $K$ of degree $\le r$. A function $\varvec{\varphi }\in \varvec{S}_{ht}^r$ is, in general, discontinuous on interfaces $\varGamma \in {\mathcal F}_{h}^I$. By $\varvec{\varphi }_{\varGamma }^{(L)}$ and $\varvec{\varphi }_{\varGamma }^{(R)}$ we denote the values of $\varvec{\varphi }\in \varvec{S}_{ht}^r$ on $\varGamma$ from the side of the element $K_{\varGamma }^{(L)}$ and $K_{\varGamma }^{(R)}$ adjacent to $\varGamma$ lying in the opposite direction to $\varvec{n}_{\varGamma }$ and in the direction of $\varvec{n}_{\varGamma }$, respectively. Then we set ${\langle \varvec{\varphi }\rangle _{\varGamma }}=(\varvec{\varphi }_{\varGamma }^{(R)}+\varvec{\varphi }_{\varGamma }^{(L)})/2$ and ${[\varvec{\varphi }]_{\varGamma }}=\varvec{\varphi }_{\varGamma }^{(L)}-\varvec{\varphi }_{\varGamma }^{(R)}$.

The discrete problem is derived in the following way: We multiply system (1) by a test function $\varvec{\varphi }_h\in \varvec{S}_{ht}^r$, integrate over $K\in \mathcal {T}_{ht}$, apply Green’s theorem, sum over all elements $K\in {\mathcal T}_{ht}$, use the concept of the numerical flux and introduce suitable terms mutually vanishing for a regular exact solution and linearize the resulting forms (see, e.g. [1, 3]). In this way we get the following forms:

$$\begin{aligned}&\hat{a}_h (\overline{{\varvec{w}}}_h, \varvec{w}_h, \varvec{\varphi }_h, t) = \sum _{K\in {\mathcal T}_{ht}} \int _{K} \sum _{s=1}^{2} \sum _{k=1}^{2} {\mathbb {K}}_{s,k} (\overline{\varvec{w}}_h)\ \frac{\partial \varvec{w}_h}{\partial x_k}\cdot \frac{\partial \varvec{\varphi }_h}{\partial x_s}\, \, \mathrm {d}{}\varvec{x}\\&\quad - \sum _{\varGamma \in {\mathcal F}^I_{ht}} \int _{\varGamma } \sum _{s=1}^{2} \left\langle \sum _{k=1}^{2} {\mathbb {K}}_{s,k} (\overline{\varvec{w}}_h) \frac{\partial \varvec{w}_h}{\partial x_k}\right\rangle (\varvec{n}_{\varGamma })_s\cdot [\varvec{\varphi }_h]\,\, \mathrm {d}S \nonumber \\&\quad -\sum _{\varGamma \in {\mathcal F}^D_{ht}}\int _{\varGamma } \sum _{s=1}^{2} \sum _{k=1}^{2} {\mathbb {K}}_{s,k} (\overline{\varvec{w}}_h) {\frac{\partial \varvec{w}_h}{\partial x_k}} (\varvec{n}_{\varGamma })_s\cdot {\varvec{\varphi }_h}\,\, \mathrm {d}S \nonumber \\&\quad -\varTheta \ \sum _{\varGamma \in {\mathcal F}^I_{ht}} \int _{\varGamma } \sum _{s=1}^2 \left\langle \sum _{k=1}^2 {\mathbb {K}}_{k,s}^T (\overline{\varvec{w}}_h) \frac{\partial \varvec{\varphi }_h}{\partial x_k}\right\rangle (\varvec{n}_{\varGamma })_s \cdot [\varvec{w}_h]\, \, \mathrm {d}S\nonumber \\&\quad -\varTheta \ \sum _{\varGamma \in {\mathcal F}^D_{ht}}\int _{\varGamma }\sum _{s=1}^2 \sum _{k=1}^2 {\mathbb {K}}_{k,s}^T ({\overline{\varvec{w}}_h}) {\frac{\partial \varvec{\varphi }_h}{\partial x_k}}(\varvec{n}_{\varGamma })_s \cdot {\varvec{w}_h}\, \, \mathrm {d}S,\nonumber \end{aligned}$$

(6)

$$\begin{aligned} d_h(\varvec{w}_h, \varvec{\varphi }_h, t) =\sum _{K\in {\mathcal T}_{ht}}\int _{K} (\varvec{w}_h\cdot \varvec{\varphi }_h)\,\mathrm {div}\varvec{z}\, \, \mathrm {d}x, \end{aligned}$$

(7)

$$\begin{aligned} J_h(\varvec{w}_h, \varvec{\varphi }_h, t)=\sum _{{\varGamma \in {\mathcal F}_{ht}^I}}\int _{\varGamma }\frac{\mu C_W}{h_{\varGamma }}[\varvec{w}_h]\cdot [\varvec{\varphi }_h]\,\, \mathrm {d}S +{\sum _{\varGamma \in {\mathcal F}_{ht}^D}\int _{\varGamma }\frac{\mu C_W}{h_{\varGamma }}\varvec{w}_h\cdot \varvec{\varphi }_h\,\, \mathrm {d}S},\nonumber \\ \end{aligned}$$

(8)

$$\begin{aligned}&\ell _h(\varvec{w}_h, \varvec{\varphi }_h, t) ={\sum _{\varGamma \in {\mathcal F}_{ht}^D}\int _{\varGamma }\frac{\mu C_W}{h_{\varGamma }}\varvec{w}_B\cdot \varvec{\varphi }_{h}\,\, \mathrm {d}S}\\&\quad -\ \varTheta \ \sum _{\varGamma \in {\mathcal F}^D_{ht}}\int _{\varGamma } \sum _{k=1}^2 {\mathbb {K}}_{k,s}^T ({\overline{\varvec{w}}_h}) {\frac{\partial \varvec{\varphi }_h}{\partial x_k}}(\varvec{n}_{\varGamma })_s \cdot \varvec{w}_B\, \, \mathrm {d}S, \nonumber \end{aligned}$$

(9)

$$\begin{aligned}&\hat{b}_{h}(\overline{\varvec{w}}_{h}, \varvec{w}_{h}, \varvec{\varphi }_{h}, t)=\\&\quad -\sum _{K\in {\mathcal T}_{ht_{k+1}}} \int _{K}\sum ^{2}_{s=1}((\mathbb {A}_{s}(\overline{\varvec{w}}_h(x))-z_{s}(x)\mathbb {I})\varvec{w}_h(x))\cdot \frac{\partial \varvec{\varphi }_{h}(x)}{\partial x_{s}}dx\nonumber \\&\quad +\sum _{\varGamma \in {\mathcal F}^I_{ht}}\int _{\varGamma }\Big ({\mathbb {P}}_g^{+}\big (\big \langle \overline{\varvec{w}}_{h}\big \rangle _{\varGamma },\varvec{n}_{\varGamma }\big )\varvec{w}^{(L)}_{h} +{\mathbb {P}}_g^{-}\big (\big \langle \overline{\varvec{w}}_{h} \big \rangle _{\varGamma },\varvec{n}_{\varGamma }\big )\varvec{w}^{(R)}_{h}\Big ) \cdot [\varvec{\varphi }_{h}]\, \, \mathrm {d}S\nonumber \\&\quad +\sum _{\varGamma \in {\mathcal F}^B_{ht}}\int _{\varGamma }\Big ({\mathbb {P}}_g^{+}\big (\big \langle \overline{\varvec{w}}_{h}\big \rangle _{\varGamma },\varvec{n}_{\varGamma }\big )\varvec{w}^{(L)}_{h} +{\mathbb {P}}_g^{-}\big (\big \langle \overline{\varvec{w}}_{h}\big \rangle _{\varGamma },\varvec{n}_{\varGamma }\big ){\overline{\varvec{w}}^{(R)}_{h}}\Big )\cdot \varvec{\varphi }_{h}\, \, \mathrm {d}S,\nonumber \end{aligned}$$

(10)

$C_W>0$ is a sufficiently large constant. We set $\varTheta =1$ or $\varTheta =0$ or $\varTheta =-1$ and get the so-called symmetric version (SIPG) or incomplete version (IIPG) or nonsymmetric version (NIPG), respectively, of the discretization of viscous terms. The symbols $\mathbb {P}^+_g(\varvec{w},\varvec{n})$ and $\mathbb {P}^-_g(\varvec{w},\varvec{n})$ denote the “positive” and “negative” parts of the matrix $\mathbb {P}_g(\varvec{w},\varvec{n})= \sum _{s=1}^2 (\mathbb {A}_s(\varvec{w})-z_s\mathbb {I}) n_s$ defined, e.g., in [2]. The boundary state $\varvec{w}_B$ is defined on the basis of the prescribed Dirichlet boundary conditions and extrapolation.

For the space-time discretization we consider a partition $0\ =\ t_0\ <\ t_1\ <\ \ldots \ <\ t_M=T$ of the time interval $[0,T]$ and denote $I_m=(t_{m-1},t_m),\ \tau _m = t_m - t_{m-1},$ for $\ m=1,\ldots ,M.$ We define the space $\,\mathbf{S}_{h\tau }^{rq} = \left\{ \phi \ ;\left. \phi \right| _{I_m} = \sum _{i=0}^q \zeta _i\phi _i, \mathrm{where}\, \phi _i\in S_{ht}^r,\right.$ $\left. \zeta _i \in P^q(I_m) \right\} ^2$ with integers $r, q \ge 1$. $P^q(I_m)$ denotes the space of all polynomials in $t$ on $I_m$ of degree $\le q$. For $\varvec{\varphi }\in \mathbf{S}_{h\tau }^{rq}$ we set $\varvec{\varphi }_m^{\pm } = \varvec{\varphi }(t_m^{\pm }) = \lim _{t\rightarrow t_{m\pm }} \varvec{\varphi }(t),\quad \{ \varvec{\varphi }\}_m = \varvec{\varphi }_m^+ -\ \varvec{\varphi }_m^-.$ The initial state $\varvec{w}_{h\tau }(0-)\in \mathbf S _{h0}^p$ is defined as the $L^2(\varOmega _{h0})$-projection of $\varvec{w}^0$ on $\mathbf S _{h0}^r$. Moreover, we introduce the prolongation $\overline{\varvec{w}}_{h\tau }(t)$ of $\varvec{w}_{h\tau }|_{I_{m-1}}$ on the interval $I_m$. By $(\cdot ,\cdot )_t$ we denote the $L^2(\varOmega _{ht})$-scalar product.

Now the space-time DG approximate solution is defined as a function $\varvec{w}_{h\tau }\in \mathbf S _{h\tau }^{rq}$ satisfying the following relation for $m=1,\ldots ,M$:

$$\begin{aligned}&\int _{I_m} \left( \left( \frac{D^\mathcal {A}\varvec{w}_{h\tau } }{D t}(t),\varvec{\varphi }_{h\tau } \right) _{t} + \hat{a}_h(\overline{\varvec{w}}_{h\tau }, \varvec{w}_{h\tau } ,\varvec{\varphi }_{h\tau },t) \right) \,\, \mathrm {d}t\\&\quad +\int _{I_m}\left( \hat{b}_h(\overline{\varvec{w}}_{h\tau }, \varvec{w}_{h\tau } ,\varvec{\varphi }_{h\tau },t) + \int _{I_m} J_h(\varvec{w}_{h\tau },\varvec{\varphi }_{h\tau },t)\right) \, \, \mathrm {d}t\nonumber \\&\quad +(\{\varvec{w}_{h\tau }\}_{m-1},\varvec{\varphi }_{h\tau }(t_{m-1}+))= \int _{I_m} \ell _h (\varvec{w}_{hD},\varvec{\varphi }_{h\tau },t)\,\, \mathrm {d}t, \quad \forall \varvec{\varphi }_{h\tau } \in \mathbf{S}_{h\tau }^{rq}. \nonumber \end{aligned}$$

(11)

2.2 Discretization of the Elasticity Problem

The elasticity problem will also be discretized by the ST-DGM. To this end, the problem is reformulated as a couple of equations of the first order in time: find functions $\varvec{u}$ and $\varvec{z} : \varOmega ^b\times [0, T] \rightarrow \mathbb {R}^2$ such that

$$\begin{aligned}&\rho ^b \frac{\partial \varvec{z}}{\partial t} + c \rho ^b \varvec{z} - \mathop {div}\varvec{\sigma (\varvec{u})} = \varvec{f} \quad \text {in } \varOmega ^b\times (0,T), \end{aligned}$$

(12)

$$\begin{aligned}&\frac{\partial \varvec{u}}{\partial t} - \varvec{z} = 0 \quad \text{ in } \varOmega ^b\times (0,T), \end{aligned}$$

(13)

$$\begin{aligned}&\varvec{u} = \varvec{u}_D \quad \text{ in } \ \varGamma _D^b\times (0,T),\quad \varvec{\sigma } (\varvec{u}) \cdot \varvec{n} = \varvec{g}_N \quad \text{ in } \ \varGamma _N^b\times (0,T), \end{aligned}$$

(14)

$$\begin{aligned}&\varvec{u}(x,0) = \varvec{u}_0(x), \quad \varvec{z}(x, 0) = \varvec{z}_0(x), \quad x\in \varOmega ^b. \end{aligned}$$

(15)

Now we proceed in a similar way as in Sect. 2.1. By $\varOmega ^b_h$ we denote a polygonal approximation of the domain $\varOmega ^b$. The sets $\varGamma _{Dh}^b$, $\varGamma _{Nh}^b \subset \partial \varOmega ^b_h$ will approximate $\varGamma _D^b$ and $\varGamma _N^b$. Let ${\mathcal T}_{h}^b$ be a partition of the closure $\overline{\varOmega }^b_h$ We define the finite dimensional space $\varvec{S}^b_{hs} = \left\{ v\in L^2(\varOmega ^b_h); v|_K \in P_s (K), K \in {\mathcal T}_{h}^b \right\} ^2$, where $s > 0$ is an integer. By ${\mathcal F}_{h}^{b}, {\mathcal F}_{h}^{bD}, {\mathcal F}_{h}^{bN}, {\mathcal F}_{h}^{bI}$ we denote the system of all faces of all elements $K \in {\mathcal T}_{h}^b$, boundary Dirichlet, Neumann faces and inner faces. If we introduce the forms

$$\begin{aligned}&a_h^b(\varvec{u}, \varvec{v}) = \sum _{K \in {\mathcal T}_{h}^b} \int _K \varvec{\sigma }(\varvec{u}): \varvec{e}(\varvec{v}) \, \mathrm {d}x- \sum _{\varGamma \in {{\mathcal F}}^{bI}_h} \int _\varGamma \left( \langle \varvec{\sigma }(\varvec{u}) \rangle \cdot \varvec{n} \right) \cdot \left[ \varvec{v} \right] \, \mathrm {d}S\\&- \sum _{\varGamma \in {{\mathcal F}}^{bD}_h} \int _\varGamma \left( \varvec{\sigma }(\varvec{u}) \cdot \varvec{n} \right) \cdot \varvec{v} \, \mathrm {d}S- \varTheta \sum _{\varGamma \in {{\mathcal F}}^{bI}_h} \int _\varGamma \left( \langle \varvec{\sigma }(\varvec{v}) \rangle \cdot \varvec{n} \right) \cdot \left[ \varvec{u} \right] \, \mathrm {d}S\nonumber \\&- \varTheta \sum _{\varGamma \in {{\mathcal F}}^{bD}_h} \int _\varGamma \left( \varvec{\sigma }(\varvec{v}) \cdot \varvec{n} \right) \cdot \varvec{u} \, \mathrm {d}S,\nonumber \end{aligned}$$

(16)

$$\begin{aligned}&J_h^b(\varvec{u}, \varvec{v}) = \sum _{\varGamma \in {{\mathcal F}}^{bI}_h} \int _\varGamma \frac{C_W^b}{h_\varGamma } \left[ \varvec{u} \right] \cdot \left[ \varvec{v} \right] \, \mathrm {d}S+ \sum _{\varGamma \in {{\mathcal F}}^{bD}_h}\int _{\varGamma }\frac{C_W^b}{h_{\varGamma }} \varvec{u}\cdot \varvec{v}\, \, \mathrm {d}S,\end{aligned}$$

(17)

$$\begin{aligned}&\ell _h^b(\varvec{v})(t) = \sum _{K \in {\mathcal T}_{h}^b} \int _K \varvec{f}(t) \cdot \varvec{v} \, \mathrm {d}x+ \sum _{\varGamma \in {{\mathcal F}}^{bN}_h} \int _\varGamma \varvec{g}_N(t) \cdot \varvec{v} \, \mathrm {d}S\\&- \varTheta \sum _{\varGamma \in {{\mathcal F}}^{bD}_h} \int _\varGamma \left( \varvec{\sigma }(\varvec{v}) \cdot \varvec{n} \right) \cdot \varvec{u}_D(t) \, \mathrm {d}S+ \sum _{\varGamma \in {{\mathcal F}}^{bD}_h} \int _\varGamma \frac{C_W^b}{h_\varGamma } \varvec{u}_D(t) \cdot \varvec{v} \, \mathrm {d}S, \nonumber \end{aligned}$$

(18)

$$\begin{aligned}&\left( \varvec{u}, \varvec{v} \right) _{\varOmega ^b_h} = \int _{\varOmega ^b_h} \varvec{u} \cdot \varvec{v} \, \mathrm {d}x= \sum _{K \in {\mathcal T}_{h}^b} \int _K \varvec{u} \cdot \varvec{v} \, \mathrm {d}x, \end{aligned}$$

(19)

where $C_W^b>0$ is a sufficiently large constant, $\varTheta =1$, $\varTheta =0$ or $\varTheta =-1$ and $\varvec{S}_{h\tau }^{b, s q} = \left\{ v\in L^2(\varOmega ^b_h\times (0,T); v|_{I_m}= \sum _{i=0}^q t^i\varphi _i \ \mathrm{with}\ \varphi _i\in S^b_{hs},\ m = 1, \dots , M \right\} ^2$, the ST-DG approximate solution can be defined as a couple $\varvec{u}_{h\tau }, \varvec{z}_{h\tau }\in \varvec{S}_{h\tau }^{b, s q}$ such that

$$\begin{aligned}&\mathrm{(a)}\ \int _{I_m} \Big ( \rho ^b \big ( \frac{\partial \varvec{z}_{h\tau }}{\partial t}, \varvec{v}_{h\tau } \big )_{\varOmega ^b_h} + C \left( \rho ^b \varvec{z}_{h\tau }, \varvec{v}_{h\tau } \right) _{\varOmega ^b_h} + a_h^b(\varvec{u}_{h\tau },\varvec{v}_{h\tau })\\&\qquad +J_h^b(\varvec{u}_{h\tau },\varvec{v}_{h\tau }) \Big ) \, \mathrm {d}t+ (\{\varvec{u}_{h\tau }\}_{m - 1}, \varvec{v}_{h\tau }(t_{m-1}+))_{\varOmega ^b_h}\nonumber \\&\qquad = \int _{I_m} \ell (\varvec{v}_{h\tau }) \, \mathrm {d}t\ \forall \varvec{v}_{h\tau }\in \varvec{S}_{h\tau }^{b, s q},\nonumber \\&\mathrm{(b)}\ \int _{I_m} \left( \left( \frac{\partial \varvec{u}_{h\tau }}{\partial t}, \varvec{w}_{h\tau } \right) _{\varOmega ^b_h} - \left( \varvec{z}_{h\tau }, \varvec{w}_{h\tau } \right) _{\varOmega ^b_h} \right) \, \mathrm {d}t\nonumber \\&\qquad + (\{\varvec{u}_{h\tau }\}_{m - 1}, \varvec{w}_{h\tau }(t_{m-1}+))_{\varOmega ^b_h} = 0 \quad \forall \varvec{w}_{h\tau } \in \varvec{S}_{h\tau }^{b, s q},\nonumber \\&\qquad \qquad m=1, \ldots , M. \nonumber \end{aligned}$$

(20)

The initial states $\varvec{u}_h(0-), \varvec{z}_h(0-)\in \varvec{S}_{hs}^b$ are defined by $(\varvec{u}_h(0-), \varvec{v}_h)_{\varOmega _h^b} = (\varvec{u}^0, \varvec{v}_h)_{\varOmega _h^b}), (\varvec{z}_h(0-), \varvec{v}_h)_{\varOmega _h^b} = (\varvec{z}^0, \varvec{v}_h)_{\varOmega _h^b}$ for all $\varvec{v}_h\in \varvec{S}_{hs}^b$.

In the FSI problem the coupling of the discrete flow problem (11) and structural problem (20) are realized via the discrete version of transmission conditions (5). The coupled problem is solved with the aid of the following coupling procedure.

• 1. Assume that the approximate solution of the flow problem on the time level $t_{k}$ is known as well as the deformation of the structure $\varvec{u}_{h,k}$.

• 2. Set $\varvec{u}_{h,k+1}^{0}:=\varvec{u}_{h,k}, \ l:=1$ and apply the iterative process:

  • a. Compute the stress tensor $\tau ^f_{ij}$ and the aerodynamical force acting on the structure and transform it to the interface $\varGamma ^b_{Nh}$.

  • b. Solve the elasticity problem, compute the deformation $\varvec{u}_{h,k+1}^{l}$ at time $t_{k+1}$ and approximate the domain $\varOmega _{ht_{k+1}}^{l}$.

  • c. Determine the ALE mapping $\mathcal{A}_{t_{k+1}h}^{l}$ and approximate the domain velocity $\varvec{z}_{h,k+1}^{l}$.

  • d. Solve the flow problem on the approximation of $\varOmega _{ht_{k+1}}^{l}$.

  • e. If the variation of the displacement $\varvec{u}_{h,k+1}^{l}$ and $\varvec{u}_{h,k+1}^{l-1}$ is larger than the prescribed tolerance, go to (a) and $l:=l+1$. Else $k:=k+1$ and goto (2).

This represents the so-called strong coupling. If in the step (e) we set $k:=k+1$ and go to (2) already in the case when $l=1$, then we get the weak (loose) coupling.

3 Numerical Results

We consider a 2D model of gas flow past an elastic airfoil. For testing our method we assume that the material of the airfoil is very soft. It is characterized by the Lamè parametres $\lambda ^b = 2\cdot 10^7\,\text {Pa}$ and $\mu ^b = 5\cdot 10^6\,\text {Pa}$. The structural damping coefficients are chosen as $c_M = 0.1\,\text {s}^{-1}$ and $c_K = 0.1\,\text {s}$ and the material density is given by $\rho ^b = 10^4\,\text {kg m}^{-3}$.

The fluid flow simulation was carried out using the following data: $\mu = 1.72\cdot 10^{-5}\,\text {kg m}^{-1}.\text {s}$, far-field pressure $p = 101250\,\text {Pa}$, far-field density $\rho =1.225\,\text {kg m}^{-3}$, Poisson adiabatic constant $\gamma = 1.4$, specific heat $c_v = 721.428\,\text {m}^2\, \text {s}^{-2}$ $\text {K}^{-1}$, heat conduction coefficient $k = 2.428\cdot 10^{-2}\,\text {kg m . s}^{-2}\,\text {K}^{-1}$. The far-field velocity was $40\,\text {m s}^{-1}$. Figure 1 shows the triangulation at the initial time $t=0$.

Fig. 1

Triangulation at time $t=0$ used for the computation of fluid flow and triangulation for the elasticity problem

Fig. 2

Visualization of velocity vectors and of the deformed elastic airfoil at time $t=0.15$ s

Fluid flow is solved by the ST-DGM with quadratic polynomials in space and linear polynomials in time. For the elasticity problem we also used the ST-DGM, but with linear polynomials in space and constant polynomials in time. For both problems the non-symmetric version (NIPG) was used. For flow problem we set $C_W=1000$ on the interior elements and $C_W=10000$ on the boundary elements in order to keep the prescribed Dirichlet boundary conditions, particularly in the boundary layer. For elasticity we set $C_W^b = 10^{10}$ in order to match the magnitude of the Lamè parametres. We used the time step $\tau = 2.25\cdot 10^{-6}\,\text {s}$. The strong coupling was used for the FSI process. The accuracy $10^{-6}$ was achieved with at most 5 iteration on each time level. Figure 2 shows the visualization of the deformed airfoil and the velocity vectors.