2. Vibro-acoustic coupling#

2.1. Presentation#

Let’s say an elastic structure defined in a domain \({\Omega }_{s}\) that vibrates in the presence of a perfect, non-heavy, compressible fluid, in isentropic evolution defined in a domain \({\mathrm{\Omega }}_{f}\). \({\Sigma }_{f}\) being the edge of the fluid domain \({\Omega }_{f}\) and \({\Sigma }_{s}\) being the edge of the solid domain \({\Omega }_{s}\), we denote their common surface by \(\Sigma ={\Sigma }_{f}\cap {\Sigma }_{S}\). We note \({n}_{s}\) the normal outside the fluid domain \({\Omega }_{f}\).

_images/10000CAC000016390000110EEB1AD8C54E0F0BE2.svg

At a given moment, the state of the fluid is defined by its pressure field (scalar) \(p\) and that of the structure by its field of displacement (vector) \({u}_{s}\). It is considered that the coupled system is subject to small disturbances around its state of equilibrium where the fluid and the structure are at rest. The fluid-structure interaction problem then consists in solving two problems simultaneously:

  • one in the structure subjected, on \(\Sigma\), to a \(p\) pressure field imposed by the fluid;

  • the other in the fluid subjected to a field of displacement \(u\) of the wall \(\Sigma\).

Note: if the direction of normal fluid to structure or structure to fluid does not matter (this is a convention), it is nevertheless imperative that this normal always be oriented in the same direction. It is strongly recommended to keep the convention of orienting the structure towards the fluid for all fluid-structure interface models in order to maintain consistency with the other loads on the structure (see in particular the case of loads of type ONDE_PLANE).

2.2. Formulation of the vibro-acoustic problem#

2.2.1. Structure description#

Hypothesis: the structure is homogeneous and obeys the laws of linear elasticity. Given this hypothesis, we can write the following various equations governing the state of the structure [bib2].

2.2.1.1. Conservation of momentum equation#

The equation for the conservation of momentum is written, in the absence of volume forces other than inertia forces, as follows:

(2.1)#\[ {\ mathrm {\ rho}} _ {s} {\ ddot {u}} {\ ddot {u}}} _ {s}} _ {s} ({u} _ {s})) =0\ text {in} {s})) =0\ text {in} {\ mathrm {\ sigma}} _ {s}\]

With \({\rho }_{s}\) the density, \({u}_{s}\) the displacements and \({\mathrm{\sigma }}_{s}\) the stress tensor.

2.2.1.2. Compatibility relationship#

The classical compatibility relationship is established on the deformation tensor:

(2.2)#\[ {\ mathrm {\ epsilon}} _ {s} =\ frac {1} {2}\ left (\nabla {u} _ {s} +\nabla {u} _ {s} _ {s} _ {s} _ {s}} ^ {T}\ right)\]

Where \({\mathrm{\epsilon }}_{s}\) is the tensor for (small) deformations.

2.2.1.3. Isotropic law of linear elasticity behavior#

We assume that the solid is linear elastic, so:

(2.3)#\[ {\ mathrm {\ sigma}} _ {s} ({u} _ {s}) =C\ mathrm {:} {\ mathrm {\ epsilon}}} _ {s}} _ {s} ({u} _ {s})\]

\(C\) being the elasticity tensor, symmetric. For the isotropic case, we can also write:

\[\]

: label: eq-4

{mathrm {sigma}} _ {s} ({u} _ {s}) =mathrm {lambda}text {tr} ({mathrm {epsilon}}} _ {s}} _ {s}} _ {s}) _ {s} ({s})) I+2mathrm {mu} {mathrm {epsilon}} _ {s} ({u}) _ {s})

With \(\mathrm{\lambda }\) and \(\mathrm{\mu }\) the Lamé coefficients.

2.2.1.4. Boundary conditions#

There are boundary conditions on the surface of the structure such as:

\[\]

: label: eq-5

{begin {array} {c} {mathrm {sigma}}} _ {s}} _ {s} _ {s})cdot {n} _ {s} = {f} _ {s}text {on}text {on}text {on}}on}on}\ sigma}}} _ {s} _ {s} =overline {u}text {on}text {on}partial}on}partial {mathrm {Omega}} _ {U}end {array}

Where \(\partial {\mathrm{\Omega }}_{N}\) is the part of the surface of the structure on which the force (surface) conditions \({f}_{s}\) apply and \(\partial {\mathrm{\Omega }}_{U}\) is the part of the surface of the structure on which the moving conditions \(\overline{u}\) apply.

2.2.2. Fluid description#

Hypothesis: the fluid obeys the laws of linear acoustics.

2.2.2.1. Conservation of momentum equation#

The equation for the conservation of momentum is written, in the absence of sources:

(2.4)#\[ {\ mathrm {\ rho}} _ {0} {0} {\ ddot {u}} {\ ddot {u}}} _ {f}} _ {f}) =0\ text {in} {\ ddot {u}} {\ ddot {u}}} _ {f}) =0\ text {in} {\ ddot {u}} {u}} {\ ddot {u}}} _ {f}\]

With \({\mathrm{\sigma }}_{f}\) the fluid stress tensor, \({\rho }_{0}\) the density of the fluid in the rest state, and \({u}_{f}\) the fluid displacement field.

2.2.2.2. Mass conservation equation#

In the first order and in the absence of acoustic sources, the mass conservation equation is expressed by the relationship:

(2.5)#\[ {\ dot {\ mathrm {\ rho}}}} _ {f}} + {\ mathrm {\ rho}} _ {0}\ text {div}\ left ({\ dot {u}}}} _ {f}\ right) =0\]

2.2.2.3. Law of behavior#

Hypothesis: the fluid obeys the laws of linear acoustics, so:

(2.6)#\[ {\ mathrm {\ sigma}} _ {f} =-Pi\]

The fluid is assumed to be in barotropic evolution (pressure \(p\) is, for the given fluid, a known function of density alone):

(2.7)#\[ p= {\ mathrm {\ rho}}} _ {f} {c} _ {0} ^ {2}\]

where \({c}_{0}\) is the speed of sound in the fluid at rest.

2.2.2.4. Pressure formulation#

We now have to combine these equations to get the equation of state of the fluid, there are several ways to do it. If we take the pressure \(p\) of the fluid as a state variable, by combining () and (), we get:

(2.8)#\[ {\ mathrm {\ rho}} _ {0}\ text {div}\ text {div}} ({\ ddot {u}} _ {f}) +\ mathrm {\ Delta} p=0\ text {in} {in} {\ text {in}} {in} {\ text {in} {in} {\ text {in} {in} {\ in} {\ mathrm {\ omega}}\]

By deriving equation () with respect to time and combining it with ():

(2.9)#\[ \ frac {1} {{c} _ {0}} ^ {2}}\ ddot {p}} + {\ mathrm {\ rho}} _ {0}\ text {div}\ left ({\ ddot {u}}} _ {f}\ right) =0\]

If we inject () into (), we get the d’Alembert equation:

(2.10)#\[ \ frac {1} {{c} _ {0} ^ {2}}}\ ddot {p}} -\ mathrm {\ Delta} p=0\ text {in} {\ mathrm {\ Omega}}} _ {f}\]

2.2.2.5. Formulation with two variables#

We associate with the pressure variable \(p\) an additional variable called fluid displacement potential and noted \(\mathrm{\varphi }\), we hypothesize the absence of a rotational term in the movements of the fluid such as:

(2.11)#\[ {u} _ {f} =\nabla\ mathrm {\ varphi}\]

See [bib1], [bib4], [bib5].

By taking the equation for the conservation of momentum () and injecting () into it after derivation, which is a rewrite of the Bernoulli equation with \({\dot{u}}_{f}^{2}=0\) (linearity hypothesis):

(2.12)#\[ \nabla\ left (p+ {\ mathrm {\ rho}}} _ {0}\ ddot {\ mathrm {\ varphi}}}\ right) =0\]

This gives us a direct relationship between pressure and the potential for fluid displacement:

(2.13)#\[ \ frac {1} {{\ mathrm {\ rho}}} _ {0} {rho}}} _ {0} {c}} _ {0} ^ {2}}} _ {0} ^ {2}}}\ ddot {\ mathrm {\ varphi}} =0\]

Hence the new form of wave propagation equation:

(2.14)#\[ \ frac {1} {{c} _ {0}} ^ {2}}\ ddot {p}} + {\ mathrm {\ rho}} _ {0}\ mathrm {\ Delta}\ ddot {\ mathrm {\ varphi}} =0\]

2.2.2.6. Formulation in speed potential#

It is also possible to introduce the fluid speed potential, noted \(\psi\) and such that:

(2.15)#\[ {\ dot {u}} _ {f} =\nabla\ mathrm {\ psi}\]

This gives us as a relationship with the \(p\) pressure of the fluid:

(2.16)#\[ p=- {\ mathrm {\ rho}}} _ {0}\ dot {\ mathrm {\ psi}}\]

The latter formulation is more natural when one has to impose a limit condition on the edge in speed, such as an incident wave.

2.2.3. Description of fluid-structure interaction#

At the fluid-structure interface \(\Sigma\), the fluid being non-viscous, it does not adhere to the wall. We therefore write the continuity of normal stresses:

(2.17)#\[ {\ mathrm {\ sigma}} _ {s}\ cdot {n} _ {n} _ {s} = -p {n} _ {s}\ text {on}\ mathrm {\ sigma}\]

And the continuity of normal trips:

(2.18)#\[ {u} _ {s}\ cdot {n} _ {s} _ {s} = {u} _ {f}\ cdot {n} _ {s}\ text {on}\ mathrm {\ Sigma}\]

As the fluid is initially considered to be at rest, the continuity of normal movements is equivalent to the continuity of normal velocities:

\[\]

: label: eq-21

{dot {u}} _ {s}cdot {n} _ {n} _ {s} _ {f}cdot {n} _ {s}text {on}text {on}text {on}mathrm {Sigma}

And the continuity of normal accelerations:

(2.19)#\[ {\ ddot {u}} _ {s}\ cdot {n} _ {n} _ {s} _ {f}\ cdot {n} _ {s}\ text {on}\ text {on}\ text {on}\ mathrm {\ Sigma}\]

By applying () in ():

(2.20)#\[ {\ ddot {u}} _ {f} =-\ frac {1} {{\ mathrm {\ rho}} _ {0}}}\nabla p\]

The equation for the continuity of normal accelerations () is written as:

(2.21)#\[ {\ ddot {u}} _ {s}\ cdot {n} _ {n} _ {s} _ {s} =-\ frac {1}} {{\ mathrm {\ rho}} _ {0}}\nabla p\ cdot {n}} _ {n} _ {n} _ {s}\ text {on}\ mathrm {\ sigma}\]

For the case where the equation of state of the fluid is expressed by the pressure and the displacement potential of the fluid, conditions IFS will be:

(2.22)#\[ \begin{align}\begin{aligned} {\ mathrm {\ sigma}} _ {s}\ cdot {n} _ {s} = {\ mathrm {\ rho}} _ {0}\ ddot {\ mathrm {\ varphi}} {n}\ varphi}} {n}\ cdot {n} _ {s}\ text {on}\ mathrm {\ sigma}}\\ {\ ddot {u}} _ {s}\ cdot {n} _ {s} _ {s} =\nabla\ ddot {\ mathrm {\ varphi}}\ cdot {n} _ {s} _ {s}\ text {n}\ s}\ text {on}\ text {on}\ text {on}\ mathrm {\ sigma}\end{aligned}\end{align} \]

And if we use the fluid’s speed potential:

(2.23)#\[ \begin{align}\begin{aligned} {\ mathrm {\ sigma}} _ {s}\ cdot {n} _ {s} = {\ mathrm {\ rho}} _ {0}\ dot {\ mathrm {\ psi}} {n}} {n}} {n}} _ {s}\ text {on}\ mathrm {\ sigma}}\\ {\ ddot {u}} _ {s}\ cdot {n} _ {s} _ {s} =\nabla\ mathrm {\ varphi}\ cdot {n} _ {s}\ text {on}\ text {on}\ text {on}\ text {on}\ text {on}\ on}\ mathrm {\ Sigma}\end{aligned}\end{align} \]

These writings show that the meaning of normal \({n}_{s}\) is really only a matter of writing convention. Whatever this orientation, the continuity equations will have exactly the same form.

2.2.4. External interface radiation condition#

In the case where the structure is surrounded by the fluid, the propagation of waves in the fluid must respect the radiation conditions at the outer border of this fluid medium \(\partial {\mathrm{\Sigma }}_{R}\), this condition called Sommerfeld condition states that the waves are not reflected by the outer surface of the fluid. The radiation condition is written as:

\[\]

: label: eq-27

underset {rtomathrm {infty}}} {mathrm {lim}} {r} ^ {d-1/2}left (nabla pcdot {n} _ {f} +frac {dot {p}}} {dot {p}}} {{p}}}right) =0

Since this condition is asymptotic, its application to a finite element modelling is not direct. We will use an approximation of the condition, noted BGT (like the names of the authors, see [Bib6]). The zero order approximation is written as:

\[\]

: label: eq-28

nabla pcdot {n} _ {f} +frac {{mathrm {rho}} _ {0}} {{Z} _ {C}}dot {p} =0text {p} =0text {on} =0text {on}}text {on}text {on}}text {on}}partial {mathrm {Sigma}} _ {R}

textrm {with}

{Z} _ {C} = {mathrm {rho}}} _ {0} {c} _ {0}

This condition is exact for a plane wave. To asymptotically approach the infinity behavior of a spherical wave, for a given radius \(R\), a first-order approximation is used:

(2.24)#\[ \begin{align}\begin{aligned} \nabla p\ cdot {n} _ {f} + {\ mathrm {\ rho}}} _ {0}\ left (\ frac {1} {Z} _ {C}}\ dot {p} +\ frac {1} {p} +\ frac {1} {{Z}} _ {Z} _ {Z}} _ {R}} p\ right) =0\ text {on}\ partial {\ mathrm {\ Sigma}}} _ {R}}}\\ \ textrm {with}\\ {Z} _ {C} = {\ mathrm {\ rho}}} _ {0} {c} _ {0} `and:math:` {Z} _ {R} = {\ mathrm {\ rho}} _ {0} R\end{aligned}\end{align} \]

2.2.5. Fluid speed decomposition#

The speed of fluid \({\dot{u}}_{f}\) which can be written \({\dot{u}}_{f}=\nabla \mathrm{\psi }\) can be written as can be broken down into three components:

\[\]

: label: eq-30

mathrm {psi} = {mathrm {psi}}} ^ {mathit {inc}} + {mathrm {psi}} ^ {mathit {rad}}} + {mathrm {psi}}} ^ {mathit {ref}} + {mathit {rad}}} + {mathrm {rad}}} + {mathrm {psi}}

The first term is the term of incidence, the second term is the term of radiation, and the third term is the term of reflection. This decomposition corresponds to the same on the pressure of the fluid:

\[\]

: label: eq-31

p= {p} ^ {mathit {inc}}} + {p} ^ {mathit {rad}}} + {p} ^ {mathit {inc}}}

2.2.6. Formulation of the coupled problem#

In short, the formulation of the vibro-acoustic problem in terms of displacements for the structure and pressure in the fluid leads to the equations of the harmonic problem (\(P\)) in the solid medium:

\[\]

: label: eq-32

{C} _ {mathit {ijkl}}}cdot {u} _ {k,mathit {lj}}} + {omega} ^ {2} {rho} _ {S} {u}} _ {u} _ {u} _ {u} _ {u} _ {i} _ {i} =0text {in} {omega} _ {S}

The equation for the propagation of waves in the fluid medium:

(2.25)#\[ \ Delta p+ {k} ^ {2} p=0\ text {in} {\ Omega} _ {f}\]

With the two fluid-structure coupling equations. In constraints:

(2.26)#\[ {C} _ {\ mathit {ijkl}}\ cdot {u}} _ {u} _ {k, l}\ cdot {n} _ {i} =-p {\ delta} _ {\ delta} _ {ij}}\ cdot {ij}}}\ cdot {n} _ {i}\ text {on}\ Sigma\]

And in speeds:

(2.27)#\[ {u} _ {i}\ cdot {n} _ {i} =\ frac {1} =\ frac {1} {{\ mathrm {\ rho}} _ {0} {\ mathrm {\ omega}}} ^ {2}}\ frac {\ 2}}}\ frac {\ partial p} {\ partial p} {\ partial p} {\ partial p} {\ partial p} {\ partial n}\ text {on}\ mathrm {\ omega}}} ^ {2}}} ^ {2}}}\ frac {\ partial p} {\ partial p} {\ partial p} {\ partial p} {\ partial n}\ text {\]

Using decomposition (), we have the condition () at the interface that is written:

(2.28)#\[ \ {\ begin {array} {c}\nabla {\ psi} ^ {\ mathit {ref}} {\ mathit {ref}}\ cdot {n} _ {s} =- {v} ^ {\ mathit {inc}}}\ cdot {n}}}\ cdot {n} {inc}}}\ cdot {n} {inc}}}\ cdot {n} {inc}}}\ cdot {n} {inc}}}\ cdot {n} {inc}}}\ cdot {n}}}\ cdot {n} {inc}}}\ cdot {n} {inc}}}\ cdot {n}}}\ cdot {n} {inc}}}\ cdot {n}}}\ cdot {n} {inc}}}\} _ {s}\ cdot {n} _ {s}\ end {array}\ text {on}\ mathrm {\ Sigma}\]

2.3. Finite element models of the coupled problem#

The coupled problem is solved using the finite element method using the weak formulation of the problem.

Nodal unknowns are noted \({U}_{s}\), \(P\),, \(\mathrm{\Phi }\), and \(\mathrm{\Psi }\).

Let \(({N}_{i}^{f})\) be the finite element shape functions for the fluid and \(({N}_{i}^{s})\) the finite element shape functions for the structure. The same shape functions will be used for \(P\), \(\mathrm{\Phi }\), and \(\mathrm{\Psi }\).

2.3.1. U-P formulation#

This training is the most « natural » because it uses the physical variables of the problem. The set to be solved is therefore:

(2.29)#\[\begin{split} \ {\ begin {array} {c} {\ mathrm {\ rho}}} _ {s} {\ ddot {u}} _ {s} -\ text {div} ({\ mathrm {\ sigma}}} _ {s}} _ {s}} _ {s}) _ {s} _ {s} -\ {\ mathrm {\ sigma}} _ {s} _ {s}} _ {s} _ {s} _ {s} _ {s}} _ {s} _ {s} _ {\ mathrm}} _ {s} _ {s} _ {\ mathrm}} _ {s} _ {\ mathrm}} _ {s} _ {\ mathrm}} _ {s}} _ {s} _ {s}} _ {s} m {\ sigma}} _ {s} ({u} _ {s})\ cdot {n} _ {s})\ cdot {n} _ {s} = {f} _ {s}\ text {on}\\ mathrm {\ Omega}}} _ {s} _ {s})\ cdot {n} _ {s})\ cdot {n}\\ {u} _ {s} =\ overline {u} _ {s} =\ overline {u} _ {s} =\ overline {u}\ text {on}\ partial {\ mathrm {\ Omega}}} _ {Omega}} _ {U}\\ {\ mathrm {\ sigma}} _ {s}\ cdot {n} _ {n} _ {s} = -p {n} _ {s}\ text {on}\ mathrm {\ Sigma}\\ {\ ddot {u}}} _ {u}} _ {s} _ {s} =-\ frac {1} {{\ mathrm {\ sigma}}\\ {\ ddot {u}}} _ {0}}\nabla p\ cdot {n} _ {s}\ text {on}\ mathrm {\ Sigma}\\ frac {1} {{c} _ {0} ^ {2}}}\ ddot {p}}\ ddot {p}} -\ ddot {p} -\ mathrm {p} -\ mathrm {p} -\ mathrm {p} -\ mathrm {\ Delta} p=0\ text {in} {\ mathrm {\ Omega}}} _ {f}}\ ddot {p}} -\ ddot {p} -\ mathrm {p} -\ mathrm {p} -\ mathrm {\ Delta} p=0\ text {in} {\ mathrm {\ Omega}}\end{split}\]

The finite element approximation of the complete problem leads to the following system:

(2.30)#\[\begin{split} \ left [\ begin {array} {cc} {M} {M} _ {s} _ {s} & 0\\ - {\ mathrm {\ rho}}} _ {T} & {M} _ {f}\ end {array}\ right]\ end {array}\ right]\ left\ {\ array}\ right]\ left\ {\ begin {array}\ right]\ left\ {\ begin {array}\ right]\ left\ {\ begin {array} {right]\ left\ {\ begin {array} {right]\ left\ {\ begin {array} {c}} {\ ddot {U}}} _ {s}\\ ddot {P}\ end {array}\ right\} +\ left [\ begin {array} {\ array} {cc} {k} _ {s} & C\\ 0& {K} _ {f}\ end {array}\ right]\ left\ {\ begin {array} {array} {array} {array}\\ begin {array}\ right\ right]]\ left\ {\ begin {array} {c} {array} {array} {c} {array} {F} _ {s}\\ 0\ end {array}\ right\}\end{split}\]

Quantities \({M}_{s}\), \({K}_{s}\) and \({F}_{s}\) relate to the modeling of the structure and we refer the reader to the appropriate documentation. For the fluid, we have:

(2.31)#\[ \begin{align}\begin{aligned} {({M} _ {f})}} _ {\ mathit {ij}}} = {\ int} _ {{\ mathrm {\ Omega}} _ {f}}\ frac {1} {{1} {{c}} _ {{c} _ {0}} ^ {2}}\ mathrm {.} {N} _ {i} ^ {f} {N} {N} _ {j}\ mathrm {.} d\ mathrm {\ omega}\\ \ textrm {and}\\ {({K} _ {f})} _ {\ mathit {ij}}} = {\ int}} = {\ int} _ {\ mathrm {\ Omega}} _ {f}}\nabla {N} _ {i} ^ {f} ^ {f} {f}\ cdot\nabla {n} _ {n} _ {j}}\ mathrm {.} d\ mathrm {\ omega}\end{aligned}\end{align} \]

And for fluid-structure coupling:

(2.32)#\[ {(C)} _ {\ mathit {ij}}} = {\ int}} = {\ int} _ {\ int} _ {j} ^ {f}\ left ({N} _ {i} ^ {s} ^ {s} ^ {s} ^ {s} ^ {s}}} = {s}\ right)\ mathrm {.} d\ mathrm {\ Sigma}}\ left ({N} _ {i} ^ {s} ^ {f}\ left ({N} _ {i} ^ {s} ^ {s} ^ {f}\ left ({N} _ {i} ^ {s} ^ {s} ^ {f}\ left ({N} _ {i} ^\]

The matrices \({M}_{s}\) and \({M}_{f}\) are positive definite, the matrices \({K}_{s}\) and \({K}_{f}\) are positive semi-defined (the displacement and pressure limit conditions must be added). The resulting system () is not symmetric.

To obtain a matrix of mass and fluid stiffness that is strictly zero, by convention, it suffices to put \({\mathrm{\rho }}_{0}={c}_{0}=0\) (i.e. the coefficients RHO and CELE_R in DEFI_MATERIAU).

For condition BGT, the system is modified and is equal to:

(2.33)#\[\begin{split} \ left [\ begin {array} {cc} {M} {M} _ {s} _ {s} & 0\\ - {\ mathrm {\ rho}}} _ {T} & {M} _ {f}\ end {array}\ right]\ end {array}\ right]\ left\ {\ array}\ right]\ left\ {\ begin {array}\ right]\ left\ {\ begin {array}\ right]\ left\ {\ begin {array} {right]\ left\ {\ begin {array} {right]\ left\ {\ begin {array} {c}} {\ ddot {U}}} _ {s}\\ ddot {P}\ end {array}\ right\} +\ left [\ begin {array} {c} {cc} 0& 0\\ 0&\ frac {{\ mathrm {\ rho}} _ {0}} {{Z} _ {C}} Q\ end {array}}}} Q\ end {array} {\ array} {c}}} {\ Z} _ {Z} _ {C}}} {C}}} Q\ end {array}}} Q\ end {array}\ {C}}} Q\ end {array}\\\ dot {P}}} end {array}\ right\} +\ left [\ begin {array} {\ array} {cc} {K} _ {s} & C\\ 0& {K} _ {f} +\ frac {{\ mathrm {\ rho}}} _ {\ rho}}} _ {0}}} _ {0}}} {0}}} {0}}} {0}}} {{Z} _ {R}}} Q\ end {array}\ right]\ left\ {\ begin {array} {\ array} {\ rho}}} _ {\ rho}}} _ {0}}} _ {0}}} {0}}} {0}}} {{Z} _ {R}} {s}\\ P\ end {array}\ right\} =\ left\ {\ begin {array} {c} {F} _ {s}\\ 0\ end {array}\ right\}\end{split}\]

We therefore add a term in amortization with the matrix \(Q\) such that:

(2.34)#\[ {(Q)} _ {\ mathit {ij}}} = {\ int}} = {\ int} _ {\ mathrm {\ Sigma}} _ {R}} {N} _ {f} {N} {N} {N} _ {n} _ {n} _ {j}} _ {j}} ^ {f} ^ {f}\ mathrm {.} d\ mathrm {\ Gamma}\]

2.3.2. Formulation in u-p-phi#

Since the previous system is non-symmetric, the search for eigenmodes requires a modal solver QEP, which is generally less robust than symmetric modal solvers. This is why you can also use a U-P- PHI formulation.

The set to be solved is therefore:

(2.35)#\[\begin{split} \ {\ begin {array} {c} {\ mathrm {\ rho}}} _ {s} {\ ddot {u}} _ {s} -\ text {div} ({\ mathrm {\ sigma}}} _ {s}} _ {s}} _ {s}) _ {s} _ {s} -\ {\ mathrm {\ sigma}} _ {s} _ {s}} _ {s} _ {s} _ {s} _ {s}} _ {s} _ {s} _ {\ mathrm}} _ {s} _ {s} _ {\ mathrm}} _ {s} _ {\ mathrm}} _ {s} _ {\ mathrm}} _ {s}} _ {s} _ {s}} _ {s} m {\ sigma}} _ {s} ({u} _ {s})\ cdot {n} _ {s})\ cdot {n} _ {s} = {f} _ {s}\ text {on}\\ mathrm {\ Omega}}} _ {s} _ {s})\ cdot {n} _ {s})\ cdot {n}\\ {u} _ {s} =\ overline {u} _ {s} =\ overline {u} _ {s} =\ overline {u}\ text {on}\ partial {\ mathrm {\ Omega}}} _ {Omega}} _ {U}\\ {\ mathrm {\ sigma}} _ {s}\ cdot {n} _ {s} = {\ mathrm {\ rho}} _ {0}\ ddot {\ mathrm {\ varphi}}} {n}} {n}} {n} _ {s} _ {s} _ {s} _ {s} _ {s} _ {s} _ {s} _ {s} _ {s} _ {s} _ {s} _ {s} _ {s} _ {s} _ {s} _ {s} _ {s} _ {s} _ {s} _ {s} _ {s}\ cdot {s} {s}\ cdot {s} {s}\ cdot {s} {s} {s}\ cdot {s} {s}} _ {s} =\nabla\ ddot {\ mathrm {\ varphi}}}\ cdot {n} _ {s}\ text {on}\ mathrm {\ Sigma}\\ frac {1} {\ frac {1} {1} {1} {1} {1} {1} {1} {\ mathrm {\ rho}}}} _ {0} {c}} _ {0} {2}} p+\ frac {1} {1} {{c} _ {0} ^ {2}}\ ddot {\ mathrm {\ varphi}} =0\ text {in} {\ mathrm {\ Omega}} _ {f}\\ frac {1} {{c} _ {{c} _ {0}} _ {0} _ {0} _ {0}\ mathrm {\ delta} _ {0} _ {0}\ mathrm {\ delta} _ {\ delta} _ {0}\ mathrm {\ delta} _ {0}\ mathrm {\ delta} _ {0}\ mathrm {\ delta} _ {\ delta} _ {0}\ mathrm {\ delta} _ {\ delta} _ {0}\ mathrm {\ delta} _ {\ delta} _ {0}\ mathrm {\ delta} _ {\ mathrm {\ varphi}} =0\ text {in} {\ mathrm {\ Omega}} _ {f}\ end {array}\end{split}\]

The finite element approximation of the complete problem leads to the following system:

(2.36)#\[\begin{split} \ left [\ begin {array} {ccc} {M} {M} _ {s} & 0& - {\ mathrm {\ rho}}} _ {0} C\\ 0& {M} _ {f}\\\\\ - {\ mathrm {\ rho}}\ - {\ mathrm {\ rho}}} _ {\ rho}}} _ {0} {C}} _ {f} & - {\ mathrm {\ rho}}} _ {f} & - {\ mathrm {\ rho}}} _ {f} & - {\ mathrm {\ rho}}} _ {f} & - {\ mathrm {\ rho}}} _ {f} & - {\ mathrm {\ rho}}} o}} _ {0} {K} _ {f}\ end {array}\ right]\ left\ {\ begin {array} {c} {\ ddot {U}} _ {s}\\ ddot {P}\\ ddot {P}\\ ddot\\ ddot {\ ddot {\ p}\\ ddot {P}\\\ ddot {P}\\ ddot {P}\\ ddot {P}\\ ddot {P}\\ ddot {P}\\ ddot {P}\\ ddot {P}\\ ddot {P}\\ ddot {P}\\ ddot {P}\\ ddot {P}\\ ddot {P}\\ ddot {P}\\ ddot {P}\\ ddot {P}\\ {s} & 0& 0\\ 0&\ 0&\ frac {1} {{\ mathrm {\ rho}}} _ {0}} {M} _ {f} & 0\\ 0& 0\ end {array}\ 0& 0\ end {array}\\\\\ {array}\\\ 0& 0& 0\ end {array}\ right]\ left]\ left\ {\ begin {array} {right]\ left]\ left\ {\ begin {array} {\ right]\ left\ {\ begin {array} {c} {U} _ {s}\\ P\\\ mathrm {\ Phi}\ end {array}\ right]\ left]\ left\ {\ begin {array} {\ right]\ left\ {\ begin {array} {\ right]\ right\} =\ left\ {\ begin {array} {c} {c} {F} _ {s}\\ 0\\ 0\ end {array}\ right\}\end{split}\]

The matrices are the same as in the previous paragraph. We can see that the system is symmetric but that it contains zeros on the diagonal. In particular, the mass term relating to the variable \(p\) is problematic because it will make modal calculations more difficult. Indeed, if we consider that the eigenvalues of the problem are of the form \(\sqrt{(\frac{K}{M})}\), we see that the term null will require:

  • or to calculate the eigenvalues in a frequency band excluding zero, which will require in particular to have a good a priory idea of this frequency band (in general, a pre-calculation without coupling with the fluid is used for this);

  • or to modify the global system by adding a value on the diagonal that is large enough to make the system invertible, and not too big so as not to disturb the results

In addition, the presence of the new variable \(\mathrm{\Phi }\) requires defining boundary conditions on this variable.

For condition BGT, the system is modified and is equal to:

(2.37)#\[\begin{split} \ left [\ begin {array} {ccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& -\ frac {{\ mathrm {\ rho}} _ {0} ^ {2}}} {{Z}}} {{Z}}} {{Z} _ {C}}} {{Z} _ {C}}} {{Z} _ {C}}} Q\ end {array}\ right]\ left\ {\ begin {array} {c}} {\ stackrel {} {U}} {\ stackrel {} {2}}} {{Z} _ {2}}} {{Z} _ {C}}} {{Z} _ {C}}} {{Z} _ {C}}} {{}} _ {s}\\\ stackrel {} {P} {P}\\\ stackrel {} {\ mathrm {\ Phi}}\ end {array}\ right\} +\ left [\ begin {array} {} {{array} {} {P} {ccc} {ccc}} {ccc} {M} {M} _ {0}} _ {f}\\ - {\ mathrm {\ rho}}} _ {0} {rho}}} _ {\ rho}} _ {\ rho}} _ {0} {K} _ {K} _ {K} _ {K} _ {f} _ {f} -\ frac {{\ mathrm {\ rho}}} _ {\ rho}} _ {R}} _ {R}} Q\ end {array}\ right]\ left\ {\ begin {array} {array} {c} {\ ddot {U}} _ {s}\\ ddot {P}\\ ddot {\ mathrm {\ Phi}}\ end {\ Phi}}\ end {array}}}\ end {array}\\ Phi}}}\ end {array}\ Phi}}}\ end {array}\ Phi}}}\ end {array}\ right\}} +\ left [\ begin {array} {U}}} _ {k} _ {s} & 0& 0\\ 0&\ 0&\ frbag {1} {{\ mathrm {\ rho}} _ {0}} {0}} {M}} {M} _ {f} & 0\\ 0& 0\ end {array}\ right]\ left\ {\ begin {array} {c} {c} {U} _ {U} _ {U} _ {U} _ {s} _ {s} _ {s}\ s}\\ s}\\ P\\\ mathrm {\ Phi}\ end {array}\ right\} =\ left\ {\ begin {array} {array} {U} _ {U} _ {U} _ {s} _ {s} _ {s}\ s}\ s}\\ P\\ mathrm {\ phi}\ end {array}\ right\} =\ left\ {\ begin {array {F} _ {s}\\ 0\\ 0\\ 0\ end {array}\ right\}\end{split}\]

We can see that the addition of this condition makes the system of order three. However, there is no robust schema for this order, in code_aster, we therefore modify the system by making it of order two by a damping term and using the relationship defined in the system of equations, that is to say \(\frac{1}{{\rho }_{0}{c}_{0}^{2}}p+\frac{1}{{c}_{0}^{2}}\ddot{\varphi }=0\text{dans}{\mathrm{\Omega }}_{f}\). The system then becomes:

(2.38)#\[\begin{split} \ left [\ begin {array} {ccc} {M} {M} _ {s} & 0& - {\ mathrm {\ rho}}} _ {0} C\\ 0& {M} _ {f}\\\\\ - {\ mathrm {\ rho}}\ - {\ mathrm {\ rho}}} _ {\ rho}}} _ {0} {C}} _ {f} & - {\ mathrm {\ rho}}} _ {f} & - {\ mathrm {\ rho}}} _ {f} & - {\ mathrm {\ rho}}} _ {f} & - {\ mathrm {\ rho}}} _ {f} & - {\ mathrm {\ rho}}} o}} _ {0} {K} _ {f} -\ frac {{\ mathrm {\ rho}} _ {0} ^ {2}} {{Z} _ {R}} Q\ end {array}\\\\\\\ right]\ right]\ left]\ left\ {f} -\\ ddot {right]]\ left]\\\ ddot {right]]\ left\ {\ begin {array} {\ array} {right]\ left\ {\ begin {array} {c} {\ begin {array} {c} {\ ddot {U}}} _ {s}\\ ddot {P}\\ ddot {P}\\ ddot {P}\\ ddot {P}\\ ddot {P}\\ ddot {\ mathrm {\ Phi}}\ end {array}\ right\}\ right\} +\ left [\ begin {array} {ccc} 0& 0& 0\\ 0&\\ 0&\ frac {{\ mathrm {\ mathrm {\ rho}}} _ {0}}} {0}} {{Z} _ {C}} Q& 0\ end {array}\ 0& 0\\\ 0& 0\\\ 0&\\ 0&\\ 0&\\ 0&\\ 0&\\ 0&\\ 0&\ frac {{\ mathrm {\ rho}}} _ {0}} _ {0}} {0}} {{Z} _ {C}} Q& 0\ end {array}\ 0& 0\\\ 0} {c} {\ dot {U}} _ {s}\\\ dot {P}\\ dot {P}\\ dot {\ mathrm {\ Phi}}\ end {array}\ right\} +\ left [\ begin {array} {s}} {ccc} {K} {K} _ {K} _ {K} _ {K} _ {K} _ {0} & 0& 0\\ 0&\ array}\ right\}}\ right\}}\ right\}} +\ left [\ begin {array}} {\ array}}\ right\}} +\ left [\ begin {array}} {\ array}} {\ array}}\ right\}} +\ left [\ begin {array} {s}} {\ array}}\ right\}} +\ left} {M} _ {f} & 0\\ 0& 0& 0\ end {array}\ right]\ left\ {\ begin {array} {c} {U} _ {s}\\ P\\ mathrm {\ Phi}\ end {array}\ right\}\ right\} =\ left\ {\ array}\ right\} =\ left\ {\ begin {array}\ right\} =\ left\ {\ begin {array}\ right\} =\ left\ {\ begin {array}\ right\}\end{split}\]

Note that the new first-order matrix becomes non-symmetric.

2.3.3. Formulation in u-psi#

The third formulation proposed in code_aster is the one using the variable \(\psi\) (speed potential) for the fluid. The set to be solved is:

(2.39)#\[\begin{split} \ {\ begin {array} {c} {\ mathrm {\ rho}}} _ {s} {\ ddot {u}} _ {s} -\ text {div} ({\ mathrm {\ sigma}}} _ {s}} _ {s}} _ {s}) _ {s} _ {s} -\ {\ mathrm {\ sigma}} _ {s} _ {s}} _ {s} _ {s} _ {s} _ {s}} _ {s} _ {s} _ {\ mathrm}} _ {s} _ {s} _ {\ mathrm}} _ {s} _ {\ mathrm}} _ {s} _ {\ mathrm}} _ {s}} _ {s} _ {s}} _ {s} m {\ sigma}} _ {s} ({u} _ {s})\ cdot {n} _ {s})\ cdot {n} _ {s} = {f} _ {s}\ text {on}\\ mathrm {\ Omega}}} _ {s} _ {s})\ cdot {n} _ {s})\ cdot {n}\\ {u} _ {s} =\ overline {u} _ {s} =\ overline {u} _ {s} =\ overline {u}\ text {on}\ partial {\ mathrm {\ Omega}}} _ {Omega}} _ {U}\\ {\ mathrm {\ sigma}} _ {s}\ cdot {n} _ {s} = {\ mathrm {\ rho}} _ {0}\ dot {\ mathrm {\ psi}} {n}} {n}} {n}} _ {s}} {n} _ {s}} {n} _ {s}} {n} _ {s}} {n} _ {s}} {n} _ {s}} {n} _ {s}} {n} _ {s}} {n} _ {s}} {n} _ {s}} {n} _ {s}} {n} _ {s}} {n} _ {s}} {n} _ {s}} {n} _ {s}} {n} _ {s}} s} =\nabla\ mathrm {\ psi}\ cdot {n} _ {s}\ text {on}\ mathrm {\ Sigma}\\ frac {1} {{c} _ {0}} _ {0} ^ {2}}\ cdot {n} _ {2}}}\ ddot {\ mathrm {\ psi}}} -\ mathrm {\ Delta}\ mathrm {\ delta}\ mathrm {\ delta}\ mathrm {\ delta}\ mathrm {\ psi}} = 0\ text {in} {\ mathrm {\ omega}} _ {f}\ end {array}\end{split}\]

The finite element approximation of the complete problem leads to the following system:

(2.40)#\[\begin{split} \ left [\ begin {array} {cc} {M} _ {M} _ {s} _ {s} & 0\\ 0& - {\ mathrm {\ rho}}} _ {f}\ end {array}\ right]\ left\ {\ array}\ right]\ left\ {\ begin {M}\\ array}\ right]\ left\ left\ {\ array}\ right]\ left\ {\ begin {array}\ right]\ left\ {\ begin {array}\ right]\ left\ {\ begin {array} {\ right]\ left\ {\ begin {array} {\ right]\ left\ {\ begin {array} {s}}\ end {array}\ right\} +\ left [\ begin {array} {cc} 0& - {\ mathrm {\ rho}}} _ {0} C\\ - {\ mathrm {\ rho}}} _ {0} {\ rho}}}} _ {0} {C}}} _ {\ dot {U}} ^ {T} ^ {T} & 0\ end {array}\ right]\ left\ {\ begin {array} {c}} {c}}} _ {s}\\\ dot {\ mathrm {\ Psi}}}\ end {array}\ right\}} +\ left [\ begin {array} {cc} {K} _ {s} & 0\\ 0& - {\ mathrm {\ rho}}} - {\ mathrm {\ rho}}}} _ {0} {K}} _ {f}\ end {array}\ right]\ left\ {\ begin {array} {0& - {\ mathrm {\ rho}}}} - {\ mathrm {\ rho}}}} _ {0} {K} _ {f}\ end {array}\ right]\ left\ {\ begin {array} {\ 0& - {\ mathrm {\ rho}}} - {\ mathrm} _ {s}\\ mathrm {\ Psi}\ end {array}\ right\} =\ left\ {\ begin {array} {c} {F} _ {s}\\\ 0\ end {array}\ right\}\end{split}\]

This system is symmetric but it contains a speed term, which in particular requires a quadratic modal solver.

For condition BGT, the system is modified and is equal to:

(2.41)#\[\begin{split} \ left [\ begin {array} {cc} {M} _ {M} _ {s} _ {s} & 0\\ 0& - {\ mathrm {\ rho}}} _ {f}\ end {array}\ right]\ left\ {\ array}\ right]\ left\ {\ begin {M}\\ array}\ right]\ left\ left\ {\ array}\ right]\ left\ {\ begin {array}\ right]\ left\ {\ begin {array}\ right]\ left\ {\ begin {array} {\ right]\ left\ {\ begin {array} {\ right]\ left\ {\ begin {array} {s}}\ end {array}\ right\} +\ left [\ begin {array} {cc} 0& - {\ mathrm {\ rho}}} _ {0} C\\ - {\ mathrm {\ rho}}} _ {0} {\ rho}}} _ {0} {\ rho}} _ {\ mathrm {\ rho}}} _ {\ mathrm {\ rho}}} _ {\ mathrm {\ rho}}} _ {\ mathrm {\ rho}}} _ {\ mathrm {\ rho}}} _ {\ mathrm {\ rho}}} _ {\ mathrm {\ rho}}} _ {0} ^ {2}}} {{Z}} _ {0} ^ {2}} {C}} Q\ end {array}\ right]\ right]\ left\ {\ begin {array} {c} {\ dot {U}} _ {s}\\ dot {\ mathrm {\ Psi}}\ end {array}}\ end {array}\ right\}\ right\}}\ right\}}} +\ right\}} +\ right\}} +\ left [\ begin {array}}} +\ right\}} +\ left [\ begin {array}}} +\ left [\ begin {array}} {\ array} {\ rho} +\ left [\ begin {array}} {c}} {\ dot {U}}\\ dot {\ mathrm {\ Psi}}}}\ end {array}\ right]}\ right\}}\ left} _ {0} ({K} _ {f} +\ frac {{\ mathrm {\ rho}}} _ {0}} {{Z} _ {C}} Q)\ end {array}\ right]\ right]\ left\ left\ {\ begin {f} +\ frac {\ begin {array} +\ frac {\ mathrm {\ rho}}} _ {0}} {0}} {Z} _ {C}} Q)\ end {array}\ right]\ left\ left\ {\ begin {array}}\ left\ {\ begin {array} {array} {\ array} {\ array} {\ array} { right\} =\ left\ {\ begin {array} {c} {f} _ {s}\\ 0\ end {array}\ right\}\end{split}\]

2.3.4. Finite element discretization#

If limit conditions such as imposed speed or wall impedance imposed on the fluid are imposed, the following matrix system is solved:

\[\]

: label: eq-50

left [begin {array} {ccc} K& 0& 00\ 0&0&frac {{M} _ {f}} {{rho} _ {0} {c} ^ {2}}} & 0\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 00end {array}right] - {omega} ^ {2}left [begin {array} {ccc}} {ccc}} & 00& 0& 0& 0& 0& 00end {array}right] - {omega} ^ {2}left [begin {array} {ccc}} {ccc}}} & 00& 0& 0& 0& 0& 0rho} _ {0} {M} _ {Sigma}\ 0&0&frac {{M} _ {mathrm {fl}}} {{c} ^ {2}}\ {rho}} _ {0} {rho} _ {0} {M}} _ {mathrm {fl}}}\ {rho} _ {0} {rho} _ {0} {rho} _ {0} {M}} _ {mathrm {fl}}}\ {rho} _ {0} {rho} _ {0} {rho} _ {0} {M}} _ {0} {rho} _ {0} {rho} _ {0} {rho} _ {{c} ^ {2}} & {rho} _ {0} Hend {array}right] +i {omega} ^ {3}left [begin {array} {ccc} 0& 0& 0\ 0\ 0& 0\ 0& 0& 0& 0\ 0& 00&0&0&0&0& 0&0& 0&frac {{rho} _ {0} ^ {2}}} {Z} Qend {array}} {Z} Qend {array}\ 0array}right] =left [begin {array} {} {} 0\ iomega\ Vend {array}right]

\(\mathrm{Q}\) being the matrix obtained from the bilinear form \({\mathrm{\int }}_{{\Sigma }_{z}}{\Phi }^{2}\mathit{dS}\) and \(\mathrm{V}\), the vector obtained from \({\mathrm{\int }}_{{\Sigma }_{V}}{\rho }_{0}{v}_{0}\Phi \mathit{dS}\).

\(\mathrm{Q}\) being the matrix obtained from the bilinear form \({\mathrm{\int }}_{{\Sigma }_{z}}{\Phi }^{2}\mathit{dS}\) and \(\mathrm{V}\), the vector obtained from \({\mathrm{\int }}_{{\Sigma }_{V}}{\rho }_{0}{v}_{0}\Phi \mathit{dS}\).