Benchmark solution
=====================


Calculation method used for the reference solution
--------------------------------------------------------

The reference solution for the modal plate representation problem is given in [:ref:`1 <1>`].

The reference solution for the problem of the modal representation of the cavity is given in [:ref:`2 <2>`].

Integrating these two models and taking into account their mutual interaction leads to a coupled modal solution whose equations are described below (see also [:ref:`3 <3>`]).

The problem consists in evaluating the dynamic response of an elastic rectangular plate coupled to a parallelepipedic acoustic cavity, and subjected to a transverse point force. The geometry of the system under consideration is shown on the previous page.

The plate has a thickness :math:`t` and is made of a material characterized by the usual mechanical constants: Young's modulus :math:`E`, structural loss factor :math:`{\eta }_{S}`, Poisson's ratio :math:`\nu` and density :math:`{\rho }_{S}`. The fluid is characterized by a density :math:`{\rho }_{F}`, a speed of sound :math:`{c}_{F}`, and a loss factor of :math:`{\eta }_{F}`.

The point force (amplitude :math:`F`) is applied at position :math:`({x}_{F},{y}_{F})`.

The natural pulsations and the natural modes of the structure are calculated by the following expressions (:math:`m,n=\mathrm{1,2}\mathrm{,3},\mathrm{...}`):

:math:`{\omega }_{\mathit{mn}}^{S}={\left(\frac{D}{{\rho }_{S}t}\right)}^{1/2}\left({\left(\frac{m\pi }{{L}_{x}}\right)}^{2}+{\left(\frac{n\pi }{{L}_{y}}\right)}^{2}\right)` with :math:`D=\frac{E{t}^{3}}{12(1-{\nu }^{2})}`

:math:`{\Phi }_{\mathit{mn}}^{S}(x,y)={A}_{\mathit{mn}}^{S}\mathrm{sin}\left(\frac{m\pi x}{{L}_{x}}\right)\mathrm{sin}\left(\frac{n\pi x}{{L}_{y}}\right)`

The coefficient :math:`{A}_{\mathit{mn}}^{S}` is chosen so that the mode which is relative to it is orthonormalized with respect to the structural mass:

:math:`\underset{0}{\overset{L}{\int }}\underset{0}{\overset{L}{\int }}{\rho }_{s}{\left({\Phi }_{\mathit{mn}}^{S}(x,y)\right)}^{2}t\text{dx}\text{dy}=1`

Which leads to the following value for :math:`{A}_{\mathit{mn}}^{S}`: :math:`{A}_{\mathit{mn}}^{S}=\frac{2}{{({\rho }_{S}t{L}_{x}{L}_{y})}^{1/2}}`

The natural pulsations and the acoustic modes are calculated by assuming rigid walls for the cavity and lead to the following expressions (:math:`i,j,k=\mathrm{0,1}\mathrm{,2},\mathrm{...}`):

:math:`{\omega }_{\mathit{ijk}}^{F}=c{\left({\left(\frac{i\pi }{{L}_{x}}\right)}^{2}+{\left(\frac{j\pi }{{L}_{y}}\right)}^{2}+{\left(\frac{k\pi }{{L}_{z}}\right)}^{2}\right)}^{1/2}`

:math:`{\Phi }_{\mathit{ijk}}^{F}(x,y,z)={A}_{\mathit{ijk}}^{F}\mathrm{cos}(\frac{i\pi x}{{L}_{x}})\mathrm{cos}(\frac{j\pi y}{{L}_{y}})\mathrm{cos}(\frac{k\pi z}{{L}_{z}})`

The coefficient :math:`{A}_{\mathit{ijk}}^{F}` is evaluated by the normalization condition in relation to the acoustic mass:

:math:`\underset{0}{\overset{L}{\int }}\underset{0}{\overset{L}{\int }}\underset{0}{\overset{L}{\int }}\frac{1}{{\rho }_{F}{c}_{F}^{2}}{\left({\Phi }_{\mathit{ijk}}^{F}(x,y,z)\right)}^{2}\text{dx}\text{dy}\text{dz}=1`

and so :math:`{A}_{\mathit{ijk}}^{F}` is worth: :math:`{A}_{\mathit{ijk}}^{F}={\left(\frac{{\rho }_{F}{c}_{F}^{2}}{{L}_{x}^{\text{'}}{L}_{y}^{\text{'}}{L}_{z}^{\text{'}}}\right)}^{1/2}`

with:

:math:`\begin{array}{c}{L}_{x}^{\text{'}}\text{}={L}_{x}\text{si}i=0\\ \text{}=0\text{si}i>0\end{array}` :math:`\begin{array}{c}{L}_{y}^{\text{'}}\text{}={L}_{y}\text{si}j=0\\ \text{}=0\text{si}j>0\end{array}` :math:`\begin{array}{c}{L}_{z}^{\text{'}}\text{}={L}_{z}\text{si}k=0\\ \text{}=0\text{si}k>0\end{array}`

The coupled system is described by the following equations:

:math:`\left[\begin{array}{cc}{\Lambda }_{S}& {\rm E}\\ {\omega }^{2}{{\rm E}}^{T}& {\Lambda }_{F}\end{array}\right]\left(\begin{array}{c}{X}_{S}\\ {X}_{F}\end{array}\right)=\left(\begin{array}{c}{F}^{S}\\ 0\end{array}\right)`

with:


* :math:`{X}_{S}` and :math:`{X}_{F}` are the structural and acoustic modal participation factors respectively;


* :math:`{\Lambda }_{S}=\text{diag}\left\{{\left({\omega }_{\mathit{mn}}^{S}\right)}^{2}-{\omega }^{2}\right\}`


* :math:`{\Lambda }_{F}=\text{diag}\left\{{\left({\omega }_{\mathit{ijk}}^{F}\right)}^{2}-{\omega }^{2}\right\}`


* :math:`{F}^{S}={\Phi }_{\mathit{mn}}^{S}({x}_{F},{y}_{F})\cdot F`


* :math:`{{\rm E}}_{\{\mathit{mn}\}\{\mathit{ijk}\}}=\underset{0}{\overset{L}{\int }}\underset{0}{\overset{L}{\int }}{\Phi }_{\mathit{mn}}^{S}(x,y){\Phi }_{\mathit{ijk}}^{F}(x,y\mathrm{,0})\text{dx}\text{dy}={A}_{\mathit{mn}}^{S}{A}_{\mathit{ijk}}^{F}{c}_{x}{c}_{y}`


* :math:`{c}_{x}=-\frac{{L}_{x}}{\pi }\left(\frac{\mathrm{cos}(m-i)\pi }{2(m-i)}+\frac{\mathrm{cos}(m+i)\pi }{2(m+i)}\right)`


* :math:`{c}_{y}=-\frac{{L}_{y}}{\pi }\left(\frac{\mathrm{cos}(n-j)\pi }{2(n-j)}+\frac{\mathrm{cos}(n+j)\pi }{2(n+j)}\right)`

The coupled dynamic response (displacement :math:`\text{u}` for the structure and acoustic pressure :math:`\text{p}` for the fluid) can then be calculated by:

:math:`u(x,y)=\sum _{m,n}{\Phi }_{\mathit{mn}}^{S}(x,y){X}_{\mathit{mn}}^{S}`

:math:`p(x,y,z)=\sum _{i,j,k}{\Phi }_{\mathit{ijk}}^{F}(x,y,z){X}_{\mathit{ijk}}^{F}`


Benchmark results
----------------------

We calculate the pressure at points :math:`\mathit{N1}`, :math:`\mathit{N2}`, :math:`\mathit{N3}`.


Uncertainty about the solution
---------------------------


* Analytical solution.


Bibliographical references
---------------------------


* [:ref:`1 <1>`] Acoustic radiation of structures, Vibroacoustics, Fluid-structure interaction, p. 27 by Claude Lesueur — Collection of the Department of Studies and Research of EDF.

[:ref:`2 <2>`] Acoustic radiation of structures, Vibroacoustics, Fluid-structure interaction, p. 80 by Claude Lesueur — Collection of the Department of Studies and Research of EDF.

[:ref:`3 <3>`] Derivation of a reference solution for a Cavity-Plate problem by J.-P. Coyette from Numerical Integration Technologies — Internal note.


*