4. Digital implementation#
4.1. Solving the Laplace equation by volume finite elements#
Let’s go back to the fluid Laplace problem with von Neumann-type boundary conditions:
\(\{\begin{array}{c}\begin{array}{c}\mathrm{\Delta }p=0\phantom{\rule{6em}{0ex}}\text{dans}\phantom{\rule{2em}{0ex}}{\mathrm{\Omega }}_{f}\\ (\frac{\partial p}{\partial \mathrm{n}}{)}_{{\mathrm{\Gamma }}_{1}}\text{=}-{\rho }_{f}\ddot{{\mathrm{x}}_{s}}\mathrm{.}\mathrm{n}\phantom{\rule{6em}{0ex}}\text{sur}\phantom{\rule{2em}{0ex}}{\mathrm{\Gamma }}_{1},\phantom{\rule{4em}{0ex}}{\mathrm{\Gamma }}_{1}=\phantom{\rule{2em}{0ex}}\underset{l=1,K}{\cup }{\gamma }_{l}^{\mathit{fs}}\end{array}\\ (\frac{\partial p}{\partial \mathrm{n}}{)}_{{\mathrm{\Gamma }}_{2}}=0\phantom{\rule{4em}{0ex}}\text{sur}\phantom{\rule{2em}{0ex}}{\mathrm{\Gamma }}_{2},\phantom{\rule{4em}{0ex}}{\mathrm{\Gamma }}_{2}=\partial \mathrm{\Omega }-{\mathrm{\Gamma }}_{1}\end{array}\)
Let’s write a variational formulation of this problem:
\({\int }_{{\mathrm{\Omega }}_{f}}v\mathrm{.}\mathrm{\Delta }pd\mathrm{\Omega }=0\phantom{\rule{2em}{0ex}}\forall v\)
Using Green’s formula with a normal that is assumed to be oriented from the structure to the fluid (therefore inside the fluid volume) and by asking \(\Gamma \mathrm{=}{\Gamma }_{1}\mathrm{\cup }{\Gamma }_{2}\):
\({\int }_{{\mathrm{\Omega }}_{f}}\mathrm{grad}v\mathrm{.}\mathrm{grad}pd\mathrm{\Omega }+{\int }_{\mathrm{\Gamma }}v\frac{\partial p}{\partial \mathrm{n}}\mathrm{.}\mathrm{n}dS=0\)
Either:
\({\int }_{{\mathrm{\Omega }}_{f}}\mathrm{grad}v\mathrm{.}\mathrm{grad}pd\mathrm{\Omega }={\rho }_{f}{\int }_{\mathrm{\Gamma }}v{\ddot{x}}_{n}dS\) eq 4.1-1
We consider a partition of fluid volume \({\mathrm{\Omega }}_{f}\) into a finite number of elements. On this discretization of the domain, we can write an approximate form of the hydrodynamic pressure field:
\({p}_{}\mathrm{=}\mathrm{\sum }_{i\mathrm{=}1}^{N}{N}_{i}(\mathrm{r}){p}_{i}\)
\({N}_{i}\) represents the nodal interpolation functions defined on the elements: they are equal to 1 at the node number \(i\), and \(0\) on all the others for finite elements P1.
Then, taking the nodal interpolation functions as test functions \(v\) in succession, we obtain a system of \(N\) equations by transferring to [éq 4.1-1]:
\(j=1,\mathrm{...},N;\phantom{\rule{2em}{0ex}}{\int }_{{\mathrm{\Omega }}_{f}}\sum _{i=1}^{N}{p}_{i}\mathrm{grad}{N}_{i}\left(\mathrm{r}\right)\mathrm{.}\mathrm{\Delta }{N}_{j}\left(\mathrm{r}\right)d\mathrm{\Omega }={\rho }_{f}{\int }_{{\gamma }_{l}^{\mathit{fs}}}{N}_{j}{\ddot{x}}_{n}\mathit{dS}\)
which can be written in the form of the linear system, whose solutions are the degrees of freedom of pressure \(\mathrm{P}\):
\(\mathrm{H}\mathrm{.}\mathrm{P}=\mathrm{\Phi }\) with \(\mathrm{\Phi }\) component vector \({\mathrm{\Phi }}_{j}={\rho }_{f}\underset{\mathrm{\Gamma }}{\int }{N}_{j}{\ddot{x}}_{n}\mathit{dS}\) |
Strictly speaking, this system is unique. It admits an infinity of solutions that differ from one constant. It is therefore necessary to impose pressure (Dirichlet-type boundary condition) at a point in the fluid to remove the uncertainty about the solution.
These precautions taken, by inverting the [éq 4.1-2] system, we obtain the pressure field throughout the \({\mathrm{\Omega }}_{f}\) volume of fluid, including at the fluid/structure interface, where we are obviously interested in it.
4.2. Calculation of the coefficients of the added mass matrix on a modal basis#
The value of the integral must be estimated numerically:
\({m}_{\mathit{il}\text{jk}}=\underset{{\gamma }_{l}^{\mathit{fs}}}{\int }{p}_{\text{jk}}{\mathrm{X}}_{\mathit{il}}\mathrm{.}\mathrm{n}\mathit{dS}\) eq 4.2-1
from a field with pressure nodes represented by a column vector noted \({\mathrm{P}}_{\mathit{jk}}\) and from a field with movement nodes corresponding to a modal deformation of an air structure and represented by the column vector \({\mathrm{X}}_{\mathit{il}}\).
However, on the fluid/structure interface, the approximate pressure field \({p}_{\mathrm{jk}}\) due to the discretization of the interface into \(N\) edge elements can be written:
\({p}_{\text{jk}}={\sum ^{N}}_{m=1}{N}_{m}\left(\mathrm{r}\right){p}_{{\text{jk}}_{m}}\)
while the « modal » displacement field is written on this same discretization:
\({\mathrm{X}}_{\mathit{il}}=\sum _{n=1}^{N}{N}_{n}\left(\mathrm{r}\right){\mathrm{X}}_{{\mathit{il}}_{n}}\)
So, by bringing these two expressions into the integral [éq 4.2-1], we get:
\(\begin{array}{c}{m}_{\mathit{il}\text{jk}}\simeq \underset{{\gamma }_{l}^{\mathit{fs}}}{\int }(\sum _{m=1}^{N}{N}_{m}\left(\mathrm{r}\right){p}_{{\text{jk}}_{m}})\left[\begin{array}{c}\sum _{n=1}^{N}{N}_{n}\left(\mathrm{r}\right){X}_{{\mathit{ilx}}_{n}}\mathrm{{\rm A}}{n}_{x}+\sum _{n=1}^{N}{N}_{n}\left(\mathrm{r}\right){X}_{{\mathit{ily}}_{n}}\mathrm{{\rm A}}{n}_{y}\end{array}\right]\mathit{dS}\\ {m}_{\mathit{il}\text{jk}}\simeq \sum _{m=1}^{N}\sum _{n=1}^{N}{p}_{{\text{jk}}_{m}}(\underset{{\gamma }_{l}^{\mathit{fs}}}{\int }{N}_{m}\left(\mathrm{r}\right){N}_{n}\left(\mathrm{r}\right){n}_{x}\mathit{dS}){X}_{{\mathit{ilx}}_{n}}+\sum _{m=1}^{N}\sum _{n=1}^{N}{p}_{{\text{jk}}_{m}}(\underset{{\gamma }_{l}^{\mathit{fs}}}{\int }{N}_{m}\left(\mathrm{r}\right){N}_{n}\left(\mathrm{r}\right){n}_{y}\mathit{dS}){X}_{{\mathit{ily}}_{n}}\end{array}\)
It was assumed in the above demonstration that the problem is two-dimensional.
This can be in the form of a dot product, involving a vector matrix product:
\({m}_{\mathit{il}\text{jk}}={\mathrm{P}}_{\text{jk}}^{T}{\mathrm{A}}_{x}{\mathrm{X}}_{\mathit{ilx}}+{\mathrm{P}}_{\text{jk}}^{T}{\mathrm{A}}_{y}{\mathrm{X}}_{\mathit{ily}}\) with \({\mathrm{A}}_{x}\) coefficient matrix \(\underset{{\gamma }_{l}^{\mathit{fs}}}{\int }{N}_{i}{N}_{j}{n}_{x}\mathit{dS}\) |
and \({\mathrm{A}}_{y}\) coefficient matrix \(\underset{{\gamma }_{l}^{\mathit{fs}}}{\int }{N}_{i}{N}_{j}{n}_{y}\mathit{dS}\) |