Remarks on a Fluid-Structure Interaction Scheme Based on the Least-Squares Finite Element Method at Small Strains

Abstract

The present contribution introduces a least-squares finite element method (LSFEM) based fluid-structure interaction (FSI) approach. The proposed method is based on the formulation of mixed finite elements in terms of stresses and velocities for both the fluid and the solid regime. The LSFEM offers the advantage of a flexibility to construct functionals with sophisticated physical quantities as e.g. stresses, velocities and displacements. The approximation of the stresses and velocities in suitable spaces, namely in the spaces $H(\text {div})$ and $H^1$, respectively, leads to the inherent fulfillment of the coupling conditions of a FSI method. A numerical example considering an incompressible, linear elastic material behavior at small deformations and the incompressible Navier–Stokes equations demonstrates the applicability of the LSFEM-FSI method.

Appendix

1.1 Algorithmic Implementation of Both Finite Element Formulations

Table 1 Algorithm for the SV formulation of the fluid part on element level

Table 2 Algorithm for the SV formulation of the solid part on element level

1.2 Raviart–Thomas Vector-Valued Interpolation Functions

Vectorial basis functions $\hat{\varvec{v}}^J_1$ of the reference element for the considered stress interpolation in the $RT_1$-case are given as

$$\begin{aligned} \hat{\varvec{v}}^1_1&= \left( \begin{array}{c} -8\eta \xi -8\xi ^2+6\xi \\ -8\eta ^2-8\eta \xi +12\eta +6\xi -4 \end{array}\right) \\ \hat{\varvec{v}}^2_1&= \left( \begin{array}{c} 8\xi ^2-4\xi \\ 8\xi \eta -2\eta -6\xi +2 \end{array}\right) \\ \hat{\varvec{v}}^3_1&= \left( \begin{array}{c} 8\xi ^2-4\xi \\ 8\xi \eta -2\eta \end{array}\right) \\ \hat{\varvec{v}}^4_1&= \left( \begin{array}{c} 8\xi \eta -2\xi \\ 8\eta ^2-4\eta \end{array}\right) \\ \hat{\varvec{v}}^5_1&= \left( \begin{array}{c} 8\eta \xi -6\eta -2\xi +2 \\ 8\eta ^2-4\eta \end{array}\right) \\ \hat{\varvec{v}}^6_1&= \left( \begin{array}{c} -8\eta \xi +6\eta -8\xi ^2+12\xi -4 \\ -8\eta ^2-8\eta \xi +6\eta \end{array}\right) \\ \hat{\varvec{v}}^7_1&= \left( \begin{array}{c} -8\eta \xi -16\xi ^2+16\xi \\ -8\eta ^2-16\eta \xi +8\eta \end{array}\right) \\ \hat{\varvec{v}}^8_1&= \left( \begin{array}{c} -16\eta \xi -8\xi ^2+8\xi \\ -16\eta ^2-8\eta \xi +16\eta \end{array}\right) \, . \end{aligned}$$

These functions are transformed to the basis functions of the physical space by

$$ \overline{\varvec{v}}^J_1= \frac{1}{\text {det } \varvec{T}} \varvec{T} \hat{\varvec{v}}^J_1 , $$

with the transformation matrix $\varvec{T} = \frac{\partial \varvec{x}}{\partial \varvec{\xi }}$, which is constant in here since straight-edged triangle meshes are used. Here $\varvec{x}$ denotes the coordinate vector of the element in the physical space and $\varvec{\xi }$ the coordinate vector of the reference element. Then the vector-valued Raviart–Thomas shape functions for $RT_1$ in two dimensions are obtained as

$$ \psi ^J_1 = \frac{l}{2} \overline{\varvec{v}}^J_1 \qquad \text {and} \qquad \text {div } \psi ^J_1 = \frac{l}{2} \text {div } \overline{\varvec{v}}^J_1 , $$

with l denoting the associated length of the edge of the interpolation sites $J = 1, \ldots ,6$. The normalization factor l / 2 is omitted for the interpolation sites inside the element, namely for $J = 7, 8$.