# Computation of a fluid-structure Green's function using
a BEM-BEM coupling

Louise Pacaut, Jean-François Mercier, Stéphanie Chaillat, Gilles Serre

► To cite this version:

Louise Pacaut, Jean-François Mercier, Stéphanie Chaillat, Gilles Serre. Computation of a fluid-structure Green's function using a BEM-BEM coupling. WAVES 2024 - The 16th International Conference on Mathematical and Numerical Aspects of Wave Propagation, Jun 2024, Berlin, Germany. hal-04878298

HAL Id: hal-04878298

https://hal.science/hal-04878298v1

Submitted on 10 Jan 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
Computation of a fluid-structure Green's function using a BEM-BEM coupling

Louise Pacaut¹*, Jean-François Mercier¹, Stéphanie Chaillat¹, Gilles Serre²

¹POEMS, CNRS, Inria, ENSTA Paris, Institut Polytechnique de Paris, 91120 Palaiseau, France.

²Naval Group, 83190 Ollioules

*Email: louise.pacaut@ensta-paris.fr

**Abstract**

In order to determine the elasto-acoustic noise produced by a boat hull excited by a turbulent boundary layer, we propose a numerical method to compute the acoustic scattering by an elastic body surrounded by a fluid. To reduce the computational costs a Boundary Element Method (BEM) is used. Since the turbulent flow along the hull is known only statistically, a formulation combining the free field acoustic and elastic Green's functions is not adequate. A better suited choice is to determine a global Green's function satisfying the transmission conditions of the fluid-structure problem. The boundary integral representation of the scattered pressure is then simplified. This so-called "tailored Green's function" is determined by solving an acoustic/elastic coupled problem with a BEM. Here we focus on a particular difficulty: when the source is close to the surface, the numerical accuracy of the Green's function deteriorates. We describe a method to regularize our BEM scheme in this context. We validate the method for the problem of an elastic sphere in water.

**Keywords:** Tailored Green's function, coupled Boundary Element Method, Fluid-structure coupling

# 1 Integral representation of the total pressure for a fluid-structure interaction (FSI)

We consider an elastic body $\Omega_s$ of density $\rho_s$ embedded in a fluid $\Omega_f$ ($\rho_f$). The system is exited by a source $S$ in $\Omega_f$. The total pressure $p$ in $\Omega_f$ and the total displacement $\mathbf{u}$ in $\Omega_s$ are solution of:

Find $(p, \mathbf{u})$ satisfying the Sommerfeld condition and

$$ \left\{ \begin{array}{l} (\Delta + k_f^2)p(\mathbf{x}) = -S(\mathbf{x}), \quad \forall \mathbf{x} \in \Omega_f \\ \rho_s \omega^2 \mathbf{u} + \operatorname{div} \boldsymbol{\sigma}(\mathbf{u}) = 0, \quad \forall \mathbf{x} \in \Omega_s \\ \boldsymbol{\sigma}(\mathbf{u}) \cdot \mathbf{n} = -p\mathbf{n}, \quad \text{on } \Gamma \\ \\ \frac{\partial p}{\partial n} = \rho_f \omega^2 \mathbf{u} \cdot \mathbf{n} \quad \text{on } \Gamma, \end{array} \right. \quad (1) $$

with $\omega$ the angular frequency and $\boldsymbol{\sigma}$ the stress tensor. Introducing the Helmholtz Green's function $G_0$, the total pressure in the fluid domain can be computed through the boundary integral representation (BIR)

$$ p(\mathbf{x}) = \int_{\Omega_f} S(\mathbf{y}) G_0(\mathbf{x}, \mathbf{y}) d\mathbf{y} + \int_{\Gamma} p(\mathbf{Y}) \frac{\partial G_0(\mathbf{x}, \mathbf{Y})}{\partial n_{\mathbf{Y}}} - G_0(\mathbf{x}, \mathbf{Y}) \rho_f \omega^2 \mathbf{u} \cdot \mathbf{n} d\Gamma_{\mathbf{Y}}. \quad (2) $$

Therefore an integral equation, depending on the source $S$, must first be solved to find the pressure on the boundary. However, when the source $S$ is some turbulence, the usual statistical solution determined is the ensemble average $\langle pp^* \rangle$, which leads to high computational costs when considering (2). To avoid these unnecessary computational costs when using the free field Green's function, we suggest to introduce a Green's function tailored to the FSI that removes the boundary terms.

# 2 Computation of Green's function tailored to the FSI problem

The tailored Green's function is the solution of (1) for $S = \delta$. It is a couple $(G, \mathbf{u}_G)$ with $G$ the pressure in the fluid and $\mathbf{u}_G$ the induced displacement in the solid. $G$ and $\mathbf{u}_G$ are solutions of: Find $(G, \mathbf{u}_G)$ such that

$$ \left\{ \begin{array}{l} (\Delta + k_f^2)G = -\delta \quad \text{in } \Omega_f, \\ \rho_s \omega^2 \mathbf{u}_G + \operatorname{div} \boldsymbol{\sigma}(\mathbf{u}_G) = 0 \quad \text{in } \Omega_s, \\ \boldsymbol{\sigma}(\mathbf{u}_G) \cdot \mathbf{n} = -G\mathbf{n} \quad \text{on } \Gamma, \\ \\ \frac{\partial G}{\partial n} = \rho_f \omega^2 \mathbf{u}_G \cdot \mathbf{n} \quad \text{on } \Gamma. \end{array} \right. \quad (3) $$

The main interest of this Green's function is to reduce (2) to

$$ p(\mathbf{x}) = \int_{\Omega_f} S(\mathbf{y}) G(\mathbf{x}, \mathbf{y}) d\mathbf{y}, \quad (4) $$

which is now a direct integral representation and not an equation. Using the free space elastic Green's functions, we can write $\forall \mathbf{y} \in \Omega_s$

$$ (\mathbf{u}_G)_i(\mathbf{y}) = \int_{\Gamma} [\mathbf{U}^i(\mathbf{y}, \mathbf{Z}) \cdot [\boldsymbol{\sigma}(\mathbf{u}_G)(\mathbf{Z}) \cdot \mathbf{n}] - \mathbf{T}^i(\mathbf{y}, \mathbf{Z}) \cdot \mathbf{u}_G(\mathbf{Z})] d\Gamma_{\mathbf{Z}}, \quad (5) $$
where $\mathbf{U}^i$ is the fundamental displacement with

$$ \rho_s \omega^2 \mathbf{U}^i + \mathrm{div} \sigma(\mathbf{U}^i) = -\delta \mathbf{e}^i, \quad (6) $$

$\mathbf{e}^i$ the unit vector along the direction $i$ and $\mathbf{T}^i = \sigma(\mathbf{U}^i) \cdot \mathbf{n}$ [2]. Similarly, we can write $\forall (\mathbf{x}, \mathbf{y}) \in \Omega_f \times \Omega_f$ the BIR for $G$

$$ G(\mathbf{x}, \mathbf{y}) = G_0(\mathbf{x}, \mathbf{y}) + \int_{\Gamma} G(\mathbf{x}, \mathbf{Z}) \frac{\partial G_0(\mathbf{y}, \mathbf{Z})}{\partial n_{\mathbf{Z}}} - G_0(\mathbf{y}, \mathbf{Z}) \frac{\partial G(\mathbf{x}, \mathbf{Z})}{\partial n_{\mathbf{Z}}} d\Gamma_{\mathbf{Z}}. \quad (7) $$

We recognize the usual single and double layer potentials $S_f$ and $D_f$ in (7). We can also introduce in the solid the potentials $D_s^i$ and $S_s^i$ defined for $\phi \in \mathbb{R}^3$ by

$$ \begin{aligned} (D_s^i \phi)(\mathbf{y}) &= \int_{\Gamma} \mathbf{T}^i \cdot \phi(\mathbf{z}) \, d\Gamma_{\mathbf{z}}, \\ (S_s^i \phi)(\mathbf{y}) &= \int_{\Gamma} \mathbf{U}^i(\mathbf{y}, \mathbf{z}) \cdot \phi(\mathbf{z}) \, d\Gamma_{\mathbf{z}}. \end{aligned} \quad (8) $$

Using the traces on $\Gamma$ of each operator, we can deduce the coupled boundary integral equation system, verified by $G$ and $\mathbf{u}_G$ on the boundary $\Gamma$:

$$ \left\{ \begin{array}{l} \left( \frac{I}{2} - D_f \right) G + \rho_f \omega^2 S_f (\mathbf{u}_G \cdot \mathbf{n}) = G_0 \quad \text{on } \Gamma, \\ \left( \frac{(\mathbf{u}_G)_i}{2} + D_s^i \mathbf{u}_G \right) + S_s^i (G \mathbf{n}) = 0 \quad \text{on } \Gamma. \end{array} \right. \quad (9) $$

$G$ and $\mathbf{u}_G$ computed with (9) are then used in (7) to compute $G$ in $\Omega_f$.

### 3 Regularization for a source close to the surface

When the source $S$ corresponds to the turbulence, $\mathbf{y}$ is naturally in the vicinity of the hull. However, it is known [1] that when the source is close to the surface, the BIR (7) becomes singular since $\mathbf{y}$ and $\mathbf{Z}$ are close. Adapted from [1], we introduce $\mathbf{Y}_0$ the closest point on the boundary mesh to $\mathbf{y}$ and a function $\psi$ solution of the homogenous Helmholtz equation such that

$$ \psi(\mathbf{x}, \mathbf{Y}_0) = G(\mathbf{x}, \mathbf{Y}_0), \quad \text{and} \quad \partial_{n_y} \psi(\mathbf{x}, \mathbf{Y}_0) = \partial_{n_y} G(\mathbf{x}, \mathbf{Y}_0). \quad (10) $$

$\psi$ can be determined as

$$ \psi(\mathbf{x}, \mathbf{y}) = G(\mathbf{x}, \mathbf{Y}_0)g(\mathbf{y}) + \partial_{n_y} G(\mathbf{x}, \mathbf{Y}_0)f(\mathbf{y}), \quad (11) $$

with functions $f$ and $g$ determined explicitly. It is shown in [1] that the tailored Green's function is computed thanks to the robust BIR

$$ \begin{aligned} G(\mathbf{x}, \mathbf{y}) - \psi(\mathbf{x}, \mathbf{y}) ={}& G_0(\mathbf{x}, \mathbf{y}) + I_{\infty}^1 - I_{\infty}^2 \\ & + \int_{\Gamma} [G(\mathbf{x}, \mathbf{Z}) - \psi(\mathbf{x}, \mathbf{Z})] \left[ \frac{\partial G_0(\mathbf{y}, \mathbf{Z})}{\partial n_{\mathbf{Z}}} - \frac{\partial G_0(\mathbf{Y}_0, \mathbf{Z})}{\partial n_{\mathbf{Z}}} \right] d\Gamma_{\mathbf{Z}} \\ & - \int_{\Gamma} \left[ \frac{\partial G(\mathbf{x}, \mathbf{Z})}{\partial n_{\mathbf{Z}}} - \frac{\partial \psi(\mathbf{x}, \mathbf{Z})}{\partial n_{\mathbf{Z}}} \right] [G_0(\mathbf{y}, \mathbf{Z}) - G_0(\mathbf{Y}_0, \mathbf{Z})] d\Gamma_{\mathbf{Z}} \end{aligned} \quad (12) $$

<figure><img src="image_2.png" /><figcaption>Figure 1: Comparison between the imaginary part of the analytical and numerical tailored Green's functions computed without (a) and with (b) the close surface regularization (frequency $f = \omega/(2\pi)$).</figcaption></figure>

with $I_{\infty}^1$ and $I_{\infty}^2$ are integrals on a sphere of infinite radius that usually don't vanish for simple choices of $f$ and $g$ but that are explicitly calculated [1]. The main idea of this regularization is that, when $\mathbf{Z} \to \mathbf{Y}_0$, $G(\mathbf{x}, \mathbf{Z}) - \psi(\mathbf{x}, \mathbf{Z})$ and $\partial_{n_{\mathbf{Z}}} G(\mathbf{x}, \mathbf{Z}) - \partial_{n_{\mathbf{Z}}} \psi(\mathbf{x}, \mathbf{Z})$ tend to 0. Thus, the growing of $G_0(\mathbf{y}, \mathbf{Z})$ and $\partial_{n_{\mathbf{Z}}} G_0(\mathbf{y}, \mathbf{Z})$ when $\mathbf{Z} \to \mathbf{Y}_0$ is smoothed. To validate the close surface regularization (12), we consider a unit sphere of glass fibre reinforced plastic (GRP) embedded in water. The observer $\mathbf{x}$ is located at $(0,0,10 \, m)$ and the source $\mathbf{y}$ is at $(0,0,1.01 \, m)$. Figure 1 shows a comparison between the analytical and numerical computations of the tailored Green's functions. It emphasizes the need of the regularization for real life applications, and shows its good implementation.

### References

[1] SUN, Qiang, et al. 2015 Boundary regularized integral equation formulation of the Helmholtz equation in acoustics. R. Soc. open sci. 2: 140520.

[2] BONNET, Marc. 1999 Boundary integral equation methods for solids and fluids. Springer