Acoustics AACTx R150 L02 Modal Analyses

2011 ANSYS, Inc. March 4, 2014 1

Modal Analyses

Acoustics ACTx R150


Modal analyses are used to model the acoustic modes of an acoustic cavity (standing waves) or to compute the mode shapes of vibro-acoustic systems.

Man can then tell which acoustic modes are causing problem or identify vibration cause.

Ability to include impedance and interaction with structure

Block Lanczos, Damped and unsymmetric eigensolvers available

Modal Analyses

Image on the right

shows standing

wave patterns in

an acoustic cavity


For pure acoustics modal analysis the acoustic modes are computed using the following equation:

For fluid structure interaction problems the acoustic and the structural coupled modes are computed using the following equation:

Modal Analyses

02 pKCjM fff

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Applying Impedance Boundary


Acoustic impedance indicates how much sound pressure is generated by the vibration of molecules of a particular acoustic medium at a given frequency.

The specific acoustic impedance z of an acoustic component (in Pas/m) is the ratio of sound pressure p to particle velocity v at its connection point:

Impedance Boundary

v

pZ


In general, a phase relation exists between the pressure and the particle velocity. So the impedance becomes complex:

Where:

R is the resistance in Pa.s/m

X is the reactance in Pa.s/m

The resistive part represents the various loss mechanisms an acoustic wave experiences. For resistive effects, energy is removed from the wave and converted into other forms. This energy is said to be 'lost from the system'.

The reactive part represents the ability of air to store the kinetic energy of the wave as potential energy since air is a compressible medium.

Impedance Boundary

iXRZ


Acoustic admittance Y can also be defined. The admittance corresponds to the inverse of the impedance:

Where:

G is the conductance in m/(Pa.s)

B is the susceptance in m/(Pa.s)

The conductance and the susceptance can converted from the impedance using the following relations:

Impedance & Admittance

iXRiBG

ZY

11

22

22

XR

XB

XR

RG


Impedance or admittance boundary can be applied using the Impedance Boundary object on free surfaces or on FSI interface (Viscoelastic material with a thin layer of a porous fabric separating both media to reduce the sound).

Boundary condition on FSI interface

FEM formulation

Impedance Boundary

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The Impedance Boundary object allows you to define the resistance and the reactance parts of the impedance (MAPDL command: (SF,,IMPD, Resistance, Reactance)):

In modal analyses you have to use the admittance form so input the conductance and the product of the capacitive susceptance and the angular velocity (MAPDL command: (SF,,IMPD, - Conductance, Susceptance*)):

Note: The impedance boundary condition would create a damping matrix, so we would need to use the damped eigensolver to correctly include this damping matrix.

Impedance Boundary

G

*B=2f*B

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It is also possible to define frequency varying impedance by setting Frequency Dependency property to Yes.

Note: MAPDL command: SF,,IMPD, %value1%, %value2%

Impedance Boundary

Add a row

Use RMB to delete rows

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The attenuation coefficient is the ratio of the absorbed sound power density to the incident sound power density. The ratio is expressed as follows:

The impedance (Im(Z)=0) is defined by

where:

= attenuation coefficient

= absorbed sound power density

= incident sound power density

is the sound impedance of the acoustic media

Attenuation Surface

Inc

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The attenuation coefficient can be defined using the Attenuation Surface object available in the Boundary Conditions menu:

Note: MAPDL command: SF,,ATTN, alpha

Attenuation Surface

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It is also possible to define frequency varying attenuation surface by setting Frequency Dependency property to Yes.

Note: MAPDL command: SF,,ATTN, %value1%

Attenuation Surface

Add a row

Use RMB to delete rows

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In harmonic (discussed later) well see that having an impedance equal to 0*c corresponds to a non reflective boundary condition for a planar wave condition (anechoic termination) but it is not completely exact for modal analysis.

Please note that if you're doing a modal analysis, there isn't really a "mode" associated with an anechoic termination. The analogy in structural analyses is having a 'semi-infinite beam' - there isn't a "mode" for an infinite or semi-infinite beam.

Consequently, you may see your frequencies change a bit, depending on how "long" you make the inlet/outlet length since this anechoic termination makes sense in harmonic response or transient analyses but not so much in modal analyses.

Plane Wave Absorption

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Fluid Structure Interaction

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Structural loads and boundary conditions can be applied as usual using standard objects:

Note that the Fixed Support set all degrees of freedom to zero. So if a Fixed Support is applied on nodes connected to acoustic elements the pressure dof will also be set to 0.

Thus its recommended to use Displacement support instead.

Structural Loads & BC

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For fluid structure interaction problem two different coupling algorithm are available and can be chosen at the Acoustics Body object level:

Unsymmetric formulation

Recently implemented Symmetric formulation

Acoustic Structure Coupling

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Putting structure and fluid fields together we end up with the unsymmetric coupled (u, p) formulated FSI matrix system:

This unsymmetric formulation requires a large amount of memory because we need to store the full matrix and not only the upper triangular half.

Using an unsymmetric algorithm also requires to use the unsymmetric or damped eigensolvers:


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Using unsymmetric algorithm its possible to use both unsymmetric and uncoupled formulations. The best solution here in terms of number of DOF to compute is to create a single of layer of elements using unsymmetric algorithm at the FSI boundary and use uncoupled algorithm for all other elements.


Structural body

Acoustic body using Uncoupled algorithm

Acoustic body with a single element layer using Unsymmetric algorithm

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Symmetric formulation

In V13 we introduced a more efficient symmetry formulation.

With unsymmetric matrices we required twice of much memory because we need to store the full matrix and not only the upper triangular half so the memory required doubled and also the CPU time increases maybe about 1.5 time. So with the symmetric formulation this allow to maintain the symmetric nature of the matrices so the memory requirement doesnt double and the CPU time doesnt increase.

Cores Solver Option Speed-up

1 Sparse Unsym 1.00

1 Sparse Sym 1.64

2 Sparse Unsym 1.00

2 Sparse Sym 1.56

4 Sparse Unsym 1.00

4 Sparse Sym 1.50

The table on the right compares the overall solution time speed-up for 275k DOF solved on dual quad-core Intel Xeon E5530.

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The symmetric coupling algorithm is based on the introduction of a displacement potential :

This causes an increase of the matrices size but leads to symmetric matrices and then reduces the elapsed time of the simulation.

All the elements in the model must use the symmetric formulation (impossible to mix unsymmetric and symmetric formulation).

Symmetric formulation

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Eigenvalues of steel cavity with water

Acoustic Structure Coupling

mode Unsymmetric Arnoldi (Hz) Symmetric Lanczos(Hz)

1st 63.1713 63.1713

2nd 224.299 224.299

3rd 364.550 364.550

4th 515.053 515.053

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Eigensolvers

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Block Lanczos, Unsymmetric and Damped solvers can used depending on the model:

Block Lanczos: pure acoustic or FSI with symmetric algorithm without viscosity, impedance or attenuation surface

Unsymmetric: FSI with unsymmetric algorithm

Damped: pure acoustic or FSI with viscosity, impedance, attenuation surface or absorbing elements.

Eigensolvers

Block Lanczos Unsymmetric Damped

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The stiffness-coupled symmetric formulation may lead to the divergence for FSI eigen problem with zero beginning frequency. So it is recommend to assign the beginning frequency (e.g. 1.E-02).

Frequency range

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Sloshing

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In fluid dynamics, slosh refers to the movement of liquid inside another object (which is, typically, also undergoing motion). Strictly speaking, the liquid must have a free surface to constitute a slosh dynamics problem, where the dynamics of the liquid can interact with the container to alter the system dynamics significantly.

Important examples include propellant slosh in spacecraft tanks and rockets (especially upper stages), and cargo slosh in ships and trucks transporting liquids (for example oil and gasoline).

Sloshing

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Assuming that the actual surface is at an elevation relative to the mean surface in z-direction, the pressure for a sloshing (free) surface is given by:

The acoustic fluid matrix equation with sloshing effect is expressed as:

Where:

is the acoustic sloshing mass matrix

Sloshing

gp F

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As previously seen an acceleration has to be defined to take into account sloshing effects.

An acceleration load is available in Boundary Conditions (Note: the standard Acceleration load isnt available in Modal analyses).

The component of the acceleration are expressed in the global coordinate system.

Acceleration

Note: MAPDL Command : ACEL,CompX,CompY,CompZ

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Man can define the free surfaces using the Free Surface object available in the Boundary Conditions menu:

Note: MAPDL Command : SF,,FREE

Free Surface

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Postprocessing

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The pressure is the degree of freedom of the acoustic problem resolution.

It can be post processed using the Acoustics Pressure object available in the Results menu.

Man can then choose the mode to display and the scoped geometry in the detail view.

The corresponding frequency is also reported in the detail view.

The result can also be animated.

Acoustic Pressure

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The acoustics velocity is calculated as follow:

It is possible to post-process the acoustics velocity in X, Y, Z direction or the sum using the corresponding objects available in Results menu. Displaying the acoustic velocity vectors is also possible.

Note: By default velocities arent stored in the result file. You need to modify the output controls of the analysis and ask to store the stresses to make it available

Acoustic Velocity

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It can be post processed using one the Acoustics Velocity object available in the Results menu.

Man can then choose the mode to display and the scoped geometry in the detail view.

The corresponding frequency is also reported in the detail view.

The result can also be animated.

Acoustic Velocity

Acoustics AACTx R150 L02 Modal Analyses

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