Code Aster Version 12 · Code_Aster Version 12 Titre : Éléments finis en acoustique Date : 23/10/2015 Page : 5/14 Responsable : DELMAS Josselin Clé : R4.02.01 Révision : 108905d27b80

Code_Aster Version 12

Titre : Éléments finis en acoustique Date : 23/10/2015 Page : 1/14Responsable : DELMAS Josselin Clé : R4.02.01 Révision :

108905d27b80

Finite elements in acoustics

Summary:

This document describes in low frequency stationary acoustics the equations used, the variational formulationswhich result from this as well as the corresponding translation in finite elements, for each of the two methodsused in Code_Aster : classical '' formulation” with an unknown factor p (acoustic pressure), and “mixed”formulation with two unknown factors p , v (pressure and speed acoustics).

Warning : The translation process used on this website is a "Machine Translation". It may be imprecise and inaccurate in whole or in part and isprovided as a convenience.Copyright 2017 EDF R&D - Licensed under the terms of the GNU FDL (http://www.gnu.org/copyleft/fdl.html)



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Contents1Introduction ............................................................................................................................................ 3

2Equations and boundary conditions of the problem .............................................................................. 4

2.1Equations and boundary conditions ................................................................................................ 4

3Classical formulation in pressure .......................................................................................................... 6

3.1Mathematical expression of the problem ........................................................................................ 6

3.2Discretization by finite elements ..................................................................................................... 6

3.2.1The matrix of rigidity .............................................................................................................. 7

3.2.2The matrix of mass ................................................................................................................ 7

3.2.3The matrix of damping .......................................................................................................... 7

3.2.4The vector source .................................................................................................................. 7

4Mixed formulation pressure-speed ........................................................................................................ 8

4.1Mathematical expression of the problem ........................................................................................ 8

4.1.1Local formulation ................................................................................................................... 8

4.1.2Mixed variational formulation ................................................................................................ 8

4.2Discretization by finite elements. .................................................................................................... 9

4.2.1The matrix of rigidity ............................................................................................................ 10

4.2.2The matrix of mass .............................................................................................................. 10

4.2.3The matrix of damping ......................................................................................................... 11

4.2.4The vector source ................................................................................................................ 11

5Orders specific to acoustic modeling ................................................................................................... 11

6Conclusion ........................................................................................................................................... 13

7Bibliography ......................................................................................................................................... 14

8Description of the versions of the document ....................................................................................... 14




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1 Introduction

Options of modeling were developed in Code_Aster, allowing to study the low frequency stationaryacoustic propagation, in closed medium, for fields of propagation to complex topology, i.e. to solvethere under the quoted conditions the equation of Helmholtz.

The solution by finite elements of this equation can be realized according to two methods:

• a first method consists in being fixed like unknown factors of the problem, only the nodalcomplex acoustic pressures, that is to say 1 degree of freedom per node [bib1]; it is thatwhich one describes as formulation to the finite elements “classical”,

• in the second method, called to the finite elements “mixed”, one sets like unknown factors atthe same time the nodal acoustic pressures and the 3 components nodal vibratory speed, ison the whole 4 degrees of freedom per node [bib5].

To know the ways of propagation of energy in the fluid, the acoustics expert has 2 sizes: activeacoustic intensity I and reactive acoustic intensity J ; these two sizes are defined like:

I=12Re [ p v

* ] and J=12Im [ pv

* ] éq 1-1

where v* indicate the combined complex one vibratory speed. The knowledge of these sizes brings avery important further information in the resolution of problems of all kinds, for example themeasurement of the powers radiated by the machines, the recognition and the localization of thesources.

The calculation of the acoustic intensity by the finite element method mixed must provide values moreprecise than the classical method; indeed in the mixed case one ensures the continuity of thederivative first of the pressure and not simply the continuity of the latter.

However if it is more precise, the mixed formulation spends on the other hand more size memory andtime CPU, while keeping the advantage of having, with equal number of degrees of freedom perwavelength, a relative error increasingly weaker on the calculation of the acoustic intensity.




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2 Equations and boundary conditions of the problem

2.1 Equations and boundary conditions

The equation to be solved is the equation of Helmholtz [bib2]:

k 2 p=s éq 2.1-1

• k indicate the number of wave of with the dealt problem; it can be complex or real,according to whether the propagation is carried out or not in a porous field [bib6]:

k=

céq 2.1-2

• c indicate the speed of sound, which can be complex in the case of a propagation in porousenvironment.

• p is a complex size indicating the acoustic pressure and s , also complex, represents thesources terms of the problem.

• is a reality in all the cases, which indicates the pulsation:

=2 f éq 2.1-3

• f is the work frequency of the harmonic problem.

We represent on the figure [Figure 2.1-a] the unspecified confined field where applies the equation ofHelmholtz [éq 2.1-1] and the conditions to the borders.

• is open limited R3 of border ∂ regular, partitionnée in ∂v and ∂z ;

∂=∂v∪∂ z




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Fluide

z

yx

Frontière absorbante

d'impédance localisée

n

z

Z

vVn

Source vibratoire monochromatique

d'amplitude normale

Figure 2.1-a: Configuration of the problem

The equation [éq 2.1-1] is to be solved in a closed field . Boundary conditions to take into account

on the border ∂ field express themselves in their most general form like:

p∂ p∂ n

= éq 2.1-4

∂/∂ n appoint the operator of normal derivative.

, , are complex operators, who can be scalars, or integral operators according to whether theborder of application of the boundary condition is with local reaction or nonlocal reaction (case of theinteraction fluid-structure).

Developments currently carried out in Code_Aster concern only boundary conditions with localreaction, for which , , are scalars; the cases spécifiables are the following:

• =0,≠0, ≠0 who indicates a border of the field at imposed vibratory speed. Indeed,there exists a relation connecting the acoustic gradient of pressure complexes at the complexparticulate vibratory speed.

∂ p∂ n

=− j0V n éq 2.1-5

0 indicate the density of the fluid considered, and one imposes V n , normal vibratory

speed with the wall ( V n=v⋅n where n indicate the unit vector of the normal external with

the border ∂ ).

• 0 0 0, , relate to a border with acoustic impedance Z imposed. Acoustic

impedance Z is defined like the report of the pressure at the particulate vibratory speed inthe vicinity of the wall with imposed impedance:

ZVn

p

éq 2.1-6




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• 0 0 0, , represent the case where the acoustic pressure is imposed p at a

border (generally 0 , corresponding to p 0 ).

3 Classical formulation in pressure

3.1 Mathematical expression of the problem

The standard procedure aiming at posing the problem with the classical finite elements is the followingone:

• one supposes the sufficiently regular solution of the problem, pH2 . The equation is

multiplied:

k 2 0p éq 3.1-1

by a function test One integrates on and the formula of Green is used. The border

field , subdivides itself in 2 zones, a zone at imposed vibratory speed, v and one zone

with imposed acoustic impedance, z . The equation obtained can be rewritten in the form:

grad gradp . p. p. .

2

20

0 0c

dV jZ

dS j V dSz v

n

éq 3.1-2

• dV represent an element of differential volume in and dS represent an element ofsurface on .

• particulate vibratory speed is then determined by:

v gradj

0p éq 3.1-3

3.2 Discretization by finite elements

In the case of the classical finite elements, the elementary integrals are four K M C Ue e e e, , ,

according to the decomposition indicated in [éq 3.2-3] (K e is the matrix of rigidity, Me the matrix of

mass, Ce the matrix of damping and Ue the vector source). Two of them come from voluminal

integrals, the two others are the result of integrals respectively on a vibrating surface and a surfacewith imposed impedance.

It will be supposed that the total coordinates of an element can be written thanks to the data of melementary functions of form H i :

OM OM H i ii

m

1

éq 3.2-1

One is given moreover, of the basic functions N i , to describe the elementary pressure.Warning : The translation process used on this website is a "Machine Translation". It may be imprecise and inaccurate in whole or in part and isprovided as a convenience.Copyright 2017 EDF R&D - Licensed under the terms of the GNU FDL (http://www.gnu.org/copyleft/fdl.html)



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The pressure inside an element will be able to be written:

p , , N , ,ie

i

m

iex y z P

1

éq 3.2-2

where Pie is the pressure with the node i element e .

In the case as of isoparametric finite elements, the basic functions N i are equal to the functions of

form H i .

On each element of the field, the problem with the finite elements in pressure is written:

K M C P Ue e e e ej qq q

jq

2 1 1 éq 3.2-3

where Peq1 is the matrix column of the nodal values of the pressure on the element.

3.2.1 The matrix of rigidity

The matrix of rigidity K e corresponds to the calculation of: e

dV

grad gradp .

She admits like term general:

K dVije

e

N Ni j

éq 3.2.1-1

3.2.2 The matrix of mass

The matrix of mass Me corresponds to the calculation of: 12c

dVe

p.


Mc

dVije

e

12

N Ni j éq 3.2.2-1

3.2.3 The matrix of damping

The matrix of damping Ce corresponds to the calculation of:

0

ZdS

ze

p.


CZ

dSije

ze

0

N Ni j éq 3.2.3-1

3.2.4 The vector source

The vector source Ue corresponds to the calculation of:

v

eV dSn

0




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He admits like term general:

U V dSie

nve

0 N i éq 3.2.4-1

4 Mixed formulation pressure-speed

4.1 Mathematical expression of the problem

4.1.1 Local formulationThe equation of Helmholtz [éq 1-1] with the boundary conditions [éq 2.1-3] result by way of localequations below:

i

i

ZV

z

n v

p div

p

. p

.

v

v grad

v n

v n

0

01

0

dans

dans

sur

sur

éq 4.1.1 1

éq 4.1.1 2

éq 4.1.1 3

éq 4.1.1 4

where 1 02/ c is the adiabatic coefficient of compressibility of the fluid.

The mathematical problem is the following: being given functions Z L Z and

Vn VH1

2 , to find functions p and v defined in and with values in C checking these

equations. They describe, in harmonic mode of pulsation , small fluctuations in pressure p andspeed v starting from at-rest state (c.à.d. acoustic pressure and acoustic particulate speed) of a fluidcompressible homogeneous, isotropic, nonviscous, confined in and subjected to a distribution of

normal velocity Vn on V .

0 , and c the density, the adiabatic coefficient of compressibility and the speed of sound represent

respectively relating to the fluid, in acoustic absence of disturbance; the coefficient 1/ Z is the

localised admittance of constituent material V with the pulsation considered.

To build a method of approximation by finite elements of this problem, it is necessary to put it in avariational form.

4.1.2 Mixed variational formulation

One takes the scalar product of the equation [éq 4.1.1-1] in L2 with an unspecified function q in

H 1 (it is the function-test).

The formula of Green and the fact that v check the boundary conditions [éq 4.1.1-3] and [éq 4.1.1 - 4]allow us to lead to:

i V

z v

n

pq* pq* . q* q*v grad éq 4.1.2-1




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One proceeds in the same way with the equation [éq 4.1.1-1] by taking his scalar product in L2

with a function-test u unspecified in L23

one obtains:

i 0 0v u grad u. * p. * éq 4.1.2-2

Now we multiply [éq 4.1.2-1] by j 0 and [éq 4.1.2-2] by j 0 , then we do it change offunction:

j v v

Thus we obtain the mixed variational formulation [éq 4.1.2-3]:

To find p, v X M such as:

02 2

0 0

02

0

1

0

v grad

v u grad u u

. q* / pq* pq* q* q

. * . *

c j j V X

p M

VZ

n

éq 4.1.2-3

where: X x ii H L L1 2 2 1 2 3 p ; p / , ,

and: M ii i L L2 3 212 3 v v , , ; v

4.2 Discretization by finite elements.The field and its borders V and Z is cut out in elementary fields and borders:

e eV

eZ, ,

on which are calculated elementary integrals.

To represent the fields of pe and of ve inside the element one uses the same basic functions N i .

Inside each element (comprising m nodes) one writes:

OM OM

v v

e

i

m

ie

e

i

m

ie

e

i

m

ie

p

N , ,

p N , ,

N , ,

i

i

i

1

1

1

, , are the curvilinear coordinates of a three-dimensional element;

OMie is the vector position of the node Mi element e with m nodes;




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N , ,i i m1 are the basic functions on the element e ;

vie ‘acceleration’ with the node is the vector Mi element e .

In this case the system of equations [éq 4.1.2-3] is written matriciellement for each element e :

pq

pq

j pq

jqe e e

e

ee e e

e

ee e e

e

ee

e

ev K

uv M

uv C

uSu

*

*

*

*

*

*

*

*

2 éq 4.2-1

Where:

pp

p v v v p v v ve ee

e

t

exe

ye

ze

me

mxe

mye

mzev

v

1 1 1 1, , , , , , is the vector solution in the

element e ; 4.2.1 The matrix of rigidity

K e is the matrix of elementary ‘rigidity’, corresponding to the calculation of the following part of

[éq 4.1.2 - 3]:

e

ee

0

02

0

v grad

v u grad u

. q*

. * p. *

One can write it by breaking up it into mxm under matrices K ije

dimensions 4 X 4 like Ci - below:

K

K K K

K K K

K K K

e

ej

em

e

ie

ije

ime

me

mje

mme

i j m

11 1 1

1

1

1

pour , , ,

with the following terms for K ije :

K ije

ij

ij

ij

ji i j

ji i j

ji i j

x y z

x

y

z

e e e

ee

ee

ee

0

0 0

0 0

0 0

0 0 0

0 02

0 02

0 02

NN

NN

NN

NN N N

NN N N

NN N N

4.2.2 The matrix of mass




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Me is the matrix of elementary ‘mass’, corresponding with the calculation of:

1 2/ pq*c

Its coefficients are the following:

M ci j r m

r mije

i j

11 4 3 4 3

12/ N N

, , , , ,pour

avec = , ,

The other terms are worthless

4.2.3 The matrix of damping

Ce is the matrix of elementary ‘damping’, coming from calculation from:

Ve

0 pq*

Its coefficients are the following:

Ci j r m

r mije

i j

Ve

0

1 4 3 4 3

1N N

, , , , ,pour

avec = , ,

The other terms are worthless.

4.2.4 The vector source

Se is the vector elementary ‘source’, representing the calculation of the terms:

Ze

Vn

0 q*

Its components are the following ones:

S Vi j r m

r mie

n i

ze

0

1 4 3 4 3

1N

, , , , ,pour

avec = , ,

The other terms are worthless.

After having obtained the field p,v on the field by resolution of the equation [éq 4.2-1] assembled,

one returns to the field p,v by opposite change of function; one can calculate the definite acousticintensities by [éq 1-1] which are in this case continuous in all the field

5 Orders specific to acoustic modeling

At the time of a study by modeling in acoustic finite elements with Code_Aster one uses generalorders and orders which are specific to acoustics, or whose keywords and options are particularwith this discipline; we present the list below of it.




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Definition of the characteristics of the propagation mediums

It is necessary to give the density (actual value) and the celerity of propagation (complex value); oneuses for that the order:

DEFI_MATERIAU with the following keywords:

keyword factor: FLUIDkeywords: RHO (density 0 )

CELE_C (celerity c )

Example: air = DEFI_MATERIAU (FLUIDE=_F (RHO= 1.3, CELE_C: IH 343. 0. ));

In this case 0=343. j0 .

Assignment of the model

It should obligatorily be specified that it is about phenomenon ‘acoustic’ and to choose one of the 3modelings possible of acoustics; the order is thus used:

AFFE_MODELE with the following keywords for which one specifies the possible values of assignment:

Keywords: PHENOMENON = ‘ACOUSTIC’MODELING = ' 3D' or ‘PLAN’

Boundary conditions

One must assign values normal vibratory speed per face (or edge into two-dimensional) to the meshsdefining the borders sources, and also values of acoustic impedance per face (edge into two-dimensional) with the meshs defining the borders in imposed impedance.

One uses the order specific to acoustics AFFE_CHAR_ACOU with the following keywords:

Keyword: MODEL

Keyword factor: VITE_FACEKeyword: MESH

GROUP_MAVNOR (normal vibratory speed Vn )

Keyword factor: IMPE_FACEKeyword: MESH

GROUP_MAIMPE (acoustic impedance Z )

Keyword factor: PRES_IMPONODE GROUP_NONEAR (pressure p imposed on the nodes)

Calculation of the elementary matrices

The various elementary matrices (rigidity, mass and damping) are calculated by specific options. Theorder is employed:

CALC_MATR_ELEM with the keyword OPTION for which one specifies the possible values ofassignment:

Keywords: OPTION ‘RIGI_ACOU’‘MASS_ACOU’




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‘AMOR_ACOU’

Note:

The assembled matrices can be obtained directly with the macro order ASSEMBLY and sameoptions.

Calculation of the elementary vector source

The elementary vector is calculated by one specific option; the loading should obligatorily beindicated. The order is employed:

CALC_VECT_ELEM with the keyword OPTION for which one specifies the only value of possibleassignment:

Keywords: OPTION ‘CHAR_ACOU’Keywords: LOAD

Calculation of the solution

After assembly of the elementary matrices and vector the harmonic solution can be calculated directlywith the order: DYNA_LINE_HARM.

Postprocessings

Starting from the result of the resolution of the matric transcription of the equations [éq 3.1-2] or[éq 4.1.2 - 3], orders of postprocessing make it possible to obtain the nodal fields of following acousticsizes:

• level Lp of acoustic pressure P in dB LP

p: log.

202 0 1010 5

• real part of the acoustic pressure• imaginary part of the acoustic pressure

• active acoustic intensity I v1

2Re p *

• reactive acoustic intensity J v1

2Im p *

These fields are calculated by the use of the ordering of postprocessing CALC_CHAMP (the concept ofthe result is of the type ‘ACOU_HARMO‘or’MODE_ACOU‘):

CALC_CHAMP with the keywords RESULT and OPTION for which one specifies the possible values ofassignment:

Keyword: RESULTKeyword: ACOUSTICS ‘PRAC_ELNO’ (pressure level in dB )

(real part of the pressure) (imaginary part of the pressure)‘INTE_ELNO’ (intensity activates) (intensity reactivates)

6 Conclusion

Modules were thus integrated in Code_Aster, allowing to do calculations of interior acoustics in lowfrequencies for complex geometries by two methods: finite elements acoustic classics and mixedacoustic finite elements. The two formulations were validated by comparison with the same analyticalsolution; cases tests are presented in the handbook of V7 validation under coding AHLV100. As it wasenvisaged, the precision, with identical grid, is higher in the mixed case; if account of the obstruction




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memory is taken this superiority is advantageous only if we want to obtain the field of intensity: oneshould use the mixed finite elements only in this case there.

7 Bibliography

1) A. BOUIZI: ‘Put in work of a computer code of finite elements for treatment of the equation ofHelmholtz in closed space’ - Work of end of studies, E.C.L. 1986.

2) A. BOUIZI: ‘Spectral analysis of the equation of Helmholtz.’ - Report of DEA, Central Schoolof Lyon, 1986.

3) A. BOUIZI: ‘Mixed finite elements in stationary linear acoustics: Development of codeAIRMEF’ - Acoustic Department. DER - EDF. HE - 24/88.02. 1988.

4) A. BOUIZI, MR. COURTADE, D. JEANDEL, E. LUZZATO, A. MIGNOT, C. SURRY. :‘Conditions of compatibility of Brezzi-Babuska for methods finite elements mixed inconformity in Mechanics and Acoustic’ AUM, Acts of the 8th French Congress of Mechanics,Nantes, 1987.

5) A. BOUIZI, MR. COURTADE, D. JEANDEL, E. LUZZATO, C. SURRY: ‘Treatment of theequation of Helmholtz by a code of mixed finite elements in closed space’ GAMI, ConferenceVibrations Shocks, 1988, ECL, 1988.

6) A. BOUIZI: ‘Resolution of the equations of linear Acoustics by a method finite elementsmixed’. Thesis presented in front of the Central School of Lyon - Speciality: Mechanics -.Supported the 3/2/89.

7) C. HABASQUE: ‘Experimental validation of the code computer of acoustics internal, Lowfrequency’. Report of internship of DEA. ECL 1986 (+ Project of end of studies).

8) A.ADOBES: ‘Digital and experimental study of the fields of standing waves’ Ratio DER/EDF -HE-2287.22

9) F. STIFKENS, A.ADOBES: ‘Assessment of the integration of the classical finite elements inAster’ - Ratio DER/EDF - HP-64/91.149

10) F. STIFKENS: ‘Integration of the mixed acoustic finite elements in Aster' Ratio DER/EDF –HP-61/92.081

8 Description of the versions of the document

Version Aster

Author (S) Organization(S)

Description of the modifications

3 F.STIFKENSEDF-R&D/AMV

Initial text


Documents

Code Aster Version 12 · Code_Aster Version 12 Titre : Éléments finis en acoustique Date : 23/10/2015 Page : 5/14 Responsable : DELMAS Josselin Clé : R4.02.01 Révision : 108905d27b80