Coupling of fluid and structure theory

In a coupled vibro-acoustic analysis, the fluid pressure on the structure boundary causes surface tractions on the structure:

\left{ {{F_s}} \right} = ;\int_S {N_s^T\left{ \varphi \right};dS}

where:

• {N_S} is a shape function for the structure \varphi = - p{n_S}{n_S} is the outward normal to the structure at the fluid boundary interface.

Substituting equation \varphi = - p{n_S} in equation \left{ {{F_s}} \right} = ;\int_S {N_s^T\left{ \varphi \right};dS} gives:

\left{ {{F_s}} \right} = ; - \int_S {N_s^T;p;\left{ {{n_s}} \right}dS} = ; - \int_S {N_s^T{N_f};\left{ {{n_s}} \right};dS;\left{ p \right}} = ; - \left[ A \right]\left{ p \right}

The equation of motion for the structure then becomes:

\left[ {{M_s}} \right]\left{ {{{\ddot u}_s}} \right} + \left[ {{B_s}} \right]\left{ {{{\dot u}_s}} \right} + \left[ {{K_s}} \right]\left{ {{u_s}} \right} = ;\left{ {{P_s}} \right} - \left[ A \right]\left{ p \right}

where:

• \left[ A \right] = \int_{{S_2}} {N_s^T{N_f};\left{ {{n_s}} \right}dS} or \left[ A \right] = - \int_{{S_2}} {N_s^T{N_f};\left{ {{n_f}} \right}dS}{N_S} are structure shape functions.{N_f} are fluid shape functions.{n_S} = {n_f}

At the fluid structure interface {n_f} \cdot {\ddot u_f} + {n_s} \cdot {\ddot u_s} = 0 equation \left[ {{M_f}} \right]\left{ {\ddot p} \right} + \left[ {{B_f}} \right]\left{ {\dot p} \right} + \left[ {{K_f}} \right]\left{ p \right} + \left[ {{A_f}} \right]\left{ {{{\ddot u}_f}} \right} = \left{ {{P_f}} \right} then becomes:

\left[ {{M_f}} \right]\left{ {\ddot p} \right} + \left[ {{B_f}} \right]\left{ {\dot p} \right} + \left[ {{K_f}} \right]\left{ p \right} - \left[ {{A^T}} \right]\left{ {{{\ddot u}_s}} \right} = \left{ {{P_f}} \right}

The combined fluid structure equations then become:

\left[ {\begin{array}{{20}{c}} {{M_s}}&0\ { - {A^T}}&{{M_f}} \end{array}} \right]\left{ {\begin{array}{{20}{c}} {{{\ddot u}_s}}\ {\ddot p} \end{array}} \right} + \left[ {\begin{array}{{20}{c}} {{B_s}}&0\ 0&{{B_f}} \end{array}} \right]\left{ {\begin{array}{{20}{c}} {{{\dot u}_s}}\ {\dot p} \end{array}} \right} + \left[ {\begin{array}{{20}{c}} {{K_s}}&A\ 0&{{K_f}} \end{array}} \right]\left{ {\begin{array}{{20}{c}} {{u_s}}\ p \end{array}} \right} = \left{ {\begin{array}{*{20}{c}} {{P_s}}\ {{P_f}} \end{array}} \right}

where:

• M is the mass matrix.

• B is the damping matrix.

• K is the stiffness matrix.

• P is the loading vector.

• Subscript s represents the partitions of the structure.

• Subscript f represents the fluid degrees of freedom.

• Degrees of freedom are displacements, u for the structure and pressure.

• Degrees of freedom are displacements, p for the fluid.

• [A] matrix is the coupling between the fluid and structure degrees of freedom at the wetted interface.

This equation is unsymmetric. It is symmetrized by using a variable transformation as follows:

Let a velocity potential q be defined such that p = \dot q.

Substituting equation p = \dot q in equation

\left[ {\begin{array}{{20}{c}} {{M_s}}&0\ { - {A^T}}&{{M_f}} \end{array}} \right]\left{ {\begin{array}{{20}{c}} {{{\ddot u}_s}}\ {\ddot p} \end{array}} \right} + \left[ {\begin{array}{{20}{c}} {{B_s}}&0\ 0&{{B_f}} \end{array}} \right]\left{ {\begin{array}{{20}{c}} {{{\dot u}_s}}\ {\dot p} \end{array}} \right} + \left[ {\begin{array}{{20}{c}} {{K_s}}&A\ 0&{{K_f}} \end{array}} \right]\left{ {\begin{array}{{20}{c}} {{u_s}}\ p \end{array}} \right} = \left{ {\begin{array}{*{20}{c}} {{P_s}}\ {{P_f}} \end{array}} \right},

assuming harmonic time dependence, and then integrating the fluid equation in time gives:

\left[ {\begin{array}{{20}{c}} {{M_s}}&0\ 0&{ - {M_f}} \end{array}} \right]\left{ \begin{array}{l} {{\ddot u}_s}\ {\ddot q} \end{array} \right} + \left[ {\begin{array}{{20}{c}} {{B_s}}&A\ {{A^T}}&{ - {B_f}} \end{array}} \right]\left{ {\begin{array}{{20}{c}} {{{\dot u}_s}}\ {\dot q} \end{array}} \right} + \left[ {\begin{array}{{20}{c}} {{K_s}}&0\ 0&{ - {K_f}} \end{array}} \right]\left{ {\begin{array}{{20}{c}} {{u_s}}\ q \end{array}} \right} = \left{ {\begin{array}{{20}{c}} {{P_s}}\ {\frac{{{P_f}}}{{i\omega }}} \end{array}} \right}

Also, because of Euler's momentum equation {\rho _0}{\ddot u_f} + \nabla p = 0, it follows that:

{\dot u_f} = - {\frac{1}{\rho }_0}\nabla q

where:

• {\rho _0} is the density of the ambient medium.

• {\ddot u_f} is the acoustic particle acceleration.

• {p} is the acoustic pressure.

The pressure and velocities can therefore be recovered from equation p = \dot q and equation {\dot u_f} = - {\frac{1}{\rho }_0}\nabla q respectively.

Defining the fluid-structure interface boundary condition

Depending on the solver, you can define the vibro-acoustic coupling interface in one of two ways:

• Two-way, strong coupling — That is, the vibration of the structure excites the fluid which in turn causes pressure loading on the structure.

• One-way, weak coupling — That is, the effect of the fluid on the structure is assumed to be negligible. However, the vibration of the structure on the fluid is assumed to be significant.

Two-way coupling

Because the Noise and Vibration solver environment supports only one-way coupling, see documentation of solvers that support two-way coupling.

One-way coupling

In one-way, weak coupling, the equation

\left[ {\begin{array}{{20}{c}} {{M_s}}&0\ { - {A^T}}&{{M_f}} \end{array}} \right]\left{ {\begin{array}{{20}{c}} {{{\ddot u}_s}}\ {\ddot p} \end{array}} \right} + \left[ {\begin{array}{{20}{c}} {{B_s}}&0\ 0&{{B_f}} \end{array}} \right]\left{ {\begin{array}{{20}{c}} {{{\dot u}_s}}\ {\dot p} \end{array}} \right} + \left[ {\begin{array}{{20}{c}} {{K_s}}&A\ 0&{{K_f}} \end{array}} \right]\left{ {\begin{array}{{20}{c}} {{u_s}}\ p \end{array}} \right} = \left{ {\begin{array}{*{20}{c}} {{P_s}}\ {{P_f}} \end{array}} \right}

becomes

\left[ {\begin{array}{{20}{c}} {{M_s}}&0\ { - {A^T}}&{{M_f}} \end{array}} \right]\left{ {\begin{array}{{20}{c}} {{{\ddot u}_s}}\ {\ddot p} \end{array}} \right} + \left[ {\begin{array}{{20}{c}} {{B_s}}&0\ 0&{{B_f}} \end{array}} \right]\left{ {\begin{array}{{20}{c}} {{{\dot u}_s}}\ {\dot p} \end{array}} \right} + \left[ {\begin{array}{{20}{c}} {{K_s}}&0\ 0&{{K_f}} \end{array}} \right]\left{ {\begin{array}{{20}{c}} {{u_s}}\ p \end{array}} \right} = \left{ {\begin{array}{*{20}{c}} {{P_s}}\ {{P_f}} \end{array}} \right}

and can be split into two sets of equations:

{M_s}{\ddot u_s} + {B_s}{\dot u_s} + {K_s}{u_s} = ;{P_s};;;;;;;;{M_f}\ddot p + {B_f}\dot p + {K_f}p = {P_f} + {A^T}{\ddot u_s}

This allows the first equation to be solved independently of the second equation. The resulting acceleration {\ddot u_s} then becomes the load vector for the second set of equations. Because the size of the equations to be solved is now considerably smaller, solve times are faster.

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Define fluid-structure interface modeling parameters (Noise and Vibration)

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One-way fluid-structure interface coupling

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