fem accoustics

7/28/2019 FEM Accoustics

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Numerical Modeling in Acoustics

Abhijit Sarkar, V.R. Sonti

Facility for Research in Technical Acoustics

Department of Mechanical Engineering

Indian Institute of Science

Bangalore 560 012, India

This draft: June 16, 2005

1 Introduction

Numerical modeling is the watchword now in most areas of science and technology. With the rapidgrowth of computing power it has naturally found a place in research and development of large scalecommercial systems. Carrying the design iterations in a virtual-reality environment reduces the costincurred on experimentation. For acoustic simulation at large scales, techniques such as - FEM, BEM,SEA etc. are widely in use. Though commercial softwares are available for these purposes, blindfoldusage is not recomended. It is advisable for the practitioner to have a fair idea of the formulations

involved.It has been the humble experience of the authours that the academic literature pertaining to these

areas though abundantly available is not presented in a consolidated and concise form for a quick start.Our endevour is to achieve this objective through the present article. We try to explain in a lucidmanner the formulation of FEM and BEM in acoustic modeling. The reader who uses a commercialsoftware will be better-placed to understand what happens behind the scene. For the student startinghis research career this tutorial article should offer him an easy understanding of the central ideainvolved, which he/she may later complement by more advanced literature.

2 Finite Element MethodFinite Element Method(FEM) or Finite Element Analysis (FEA) is a widely used analysis tools forsimulation of structural dynamics. There are plentiful of good books available which introduces thissubject. Realizing its importance for the present and future, most graduate schools also offer coursesin this area. However, most of these books and courses emphasize too much on the structural equation.The fact, that FEM is a robust analysis methodology for a sufficiently large class of boundary valueproblems is perhaps missed (except by the few brighter students). Thus, FEA for acoustics seems tobe a different animal for the person well-versed in FEA for structures. We try to dissect this animalbefore the readers and also show it is essentially the same. The approach we follow in the presentation

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is to emphasize the aspects of FEA in acoustics which are seemingly different to FEA in structures.The portions which are identically same are outlined briefly partly due to constraints of space andtime and partly because there are well-written books on these concepts.

2.1 Governing EquationsThe first question to answer is what is it that we are trying to solve ? The answer is well-known - theacoustic wave equation given by 1

2p

t2= c22p (1)

In the above equation c is the acoustic velocity and p(x, t) is the elementary variable in acoustics-namely the acoustic pressure. It is the perturbation of pressure over the mean pressure due to theacoustic wave propagation. This variable is anlagous to displacement in structures. In structures thetwo most pertinent variables are displacement and force. Analogous to the later we have acousticvelocity u which is the oscillating velocity of the medium particles carrying the wave. This may seem

to be counter-intutive - pressures are not analogous to forces but to displacements and velocities arenot analogous to displacements but forces. The key is to relate the two acoustic variables i.e. pressureand velocity. The momentum conservation law applied to a differential acoustic element does the joband we have the following relation

u

t= p (2)

where is the density of the acoustic medium.Now, let us make a bold assumption that in real engineering environment we are interested in

steady state conditions. To illustrate, let us consider that we are interested in calculating the noiseemanated due to the airconditioner in the room. We understand the noise heard at any given moment

is the same as that heard 5 minutes later or 50 minutes later. This is in contrast to the transientsound we hear when we hit an object with a hammer. When we hit, a sound is heard but then it givesway to dead-calm. The key is under steady state, there is a definite periodicity i.e. the acoustic signalcollected over a certain time window is essentially the same if the the window is shifted in time. Dueto God-given Fourier Analysis, we can therefore represent the steady-state acoustic variables (both

p(x) and u(x)) as a linear combination of complex exponentials ejt . The new varaiable is refered toas frequency (more appropriately angular frequency). This also means that instead of analyzing p(t)(at a particular point x) we might choose to analyze each ejt frequency-by-frequency which make up

p(t). This approach is much simpler and hence is prefered. So let us try to analyze what happens ata particular frequency . This leads to the following simplifications

p(x, t) p(x)ejtu(x, t) u(x)ejt (3)

Putting back these simplifications in equation(2) we get

j u = p (4)

This equation relating acoustic velocity and acoustic pressure is infact anlogous to force-displacementrelation in structures. Consider a differential element of a 1D bar with a displacement field denoted

1full deduction available in any basic book on acoustics

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by w(x). At (x + dx) the displacement field is therefore w + dw. Thus the strain is dwdx

and hence if A& E be the cross-sectional area and modulus of elasticity respectively, the force is given by (A figurewill be given)

F = EAdw

dx(5)

Note equations(4) and (5) relate the primary variables through a spatial derivative relation. It showsstructural force is the spatial derivatives of structural displacement, wheras acoustic velocity is thespatial derivative of acoustic pressure. These hand-waving arguements should bring some peace to thereader regarding our claim between analogies of structural and acoustic variables. A more rigorousproof involving variational calculus is purposely avoided here to keep the matter simple.

Before proceeding let us make a simplification in the governing differential equation. Using theassumption (3) the wave equation (1) can be rewritten as

2

c2p(x) = 2p(x)

(2 + k2)p(x) = 0 (6)

where k = c

is called the wavenumber. This equation is known as the helmholtz equation and thevariable p(x) as the acoustic potential. In order to mantain consonance with the existing literature weswitch symbols and use (x) to denote acoustic potential at any spatial point (note time dependencehas anyway been done away with in equation(6)). Due to equation (4) the acoustic velocity is relatedto the acoustic potential as

ju(x) = (x) (7)

Thus, we will be looking to solve the helmholtz equation (which is equivalent to the wave equationunder steady state assumption) for the acoustic potential in a certain domain of interest . The pri-

mary variables i.e. acoustic pressure and acoustic velocity can always be recovered from the potentials.We prefer to think of acoustic potential as a mere mathematical entity and do not bother about itsphysical interpretation.

2.2 Integral Statement

For reasons that would be clear later, we need to transform the governing differential equation (6) intoan equivalent integral equation. This may seem something like a magic but let us see how we can goback and forth between the two representations. Since we claim the representations to be equivalent,given a representation we should be able to arrive at the other representation.

Definitely if equation (6) holds then for any arbitrary function w(x) the following holds

(w2 + k2w)d = 0 (8)

The above is definitely an integral representation, but is it equivalent ? That is can we get backequation (6) from equation (8) ? In the following paragraph we will show we can. Thus, the simpletrick worked and we arrived at an equivalent integral representation which we call weighted integralrepresentation; the function w(x) being the weights.

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To arrive at equation (6) from equation (8) we use the fact that w(x) is arbitrary. Let us choosew(x) to be the delta function (x x0) 2. Using this substitution in equation (8) we get

((x x0)

2 + k2(x x0))d = 0

2 + k2 = 0 at x = x0 (9)

By choosing such delta functions for each point x in we conclude equation(9) is infact satisfied forall points in . This is equivalent to the satisfaction of the governing differential equation (Helmholtzequation) within the domain.

2.3 Weak Form

From equation(8) we observe that must be twice differentiable spatial function. Also, the first termof the integrand is unsymmetric in w and . To gain some advantages that would be made clear indue course we will reduce the differentiability requirement on and obtain a symmetric expression in and w. To do this we need the divergence theroem.

Divergence theorem states the following (for non-believers look at Kreyszig)

d =

nd (10)

In the above formula is the boundary of the domain and the n represents the outward unit normalon the boundary. If = (w), then using the above we get

(w)d =

(w) nd

Using the vector calculus identity 3 (w) = (w) () + (w2) we get

() (w)d +

(w2)d =

(w) nd (11)

(w2)d =

() (w)d +

(w) nd (12)

Thus equation(8) may be alternatively expressed as

() (w) + k2w

d +

(w) nd = 0 (13)

The first integral in equation (13) is symmetric in w and (intutively it means the expressionremains unchanged if we switch w and ). This is in fact a great achievement as we will find out.Also the differentiability requirements on has been reduced.

A final remark about the justification of the weak form. The weak form entails no additionalassumptions, as the term weak may seem to imply. The weakness is due to lesser differntiabilityrequirement.

2delta function is actually not a function, however using distribution theory convolution type integrals are well-defined

which is what we will use3look at kreyszig for proof

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2.4 Variational formulation

Till now we had been trying to recast the differential equation alone and not the boundary condition.The time is now ripe to bring the boundary conditions in picture. Being a second order equation theboundary conditions for the helmholtz equation can be of the following types

1. Prescribed values of on p, by our previous discussion this implies boundary conditions withprescribed pressures. Boundary conditions of this type are referred to as essential boundaryconditions or Dirichlet boundary conditions. As an example, in studying underwater acousticwave propagation the air-water interface may be treated as having zero pressure.

2. Prescribed values of gradients of i.e. on u , by equation(7) this implies boundary conditionswith prescribed velocities. Boundary conditions of this type are referred to as natural boundaryconditions or Neumann boundary conditions. As an example, in studying sound radiation from avibrating body into the acoustic domain the boundary of the acoustic domain is given prescribedacoustic velocities (equal to the ambient structural velocity).

3. In acoustics a third type of boundary condition frequently arises. This is the admittance ormixed boundary condition. Instead of prescribed velocities or pressures we enforce the conditionun = Z0p at the boundary where though Z0 is the known admittance both p & un are theunknown acoustic pressures and normal velocity on the boundary respectively. This type ofboundary condition frequently arises in case we need to model for the sound propagation insideacoustic enclosures. Using characterestic impedance value of the medium it is also possibleto model a non-reflecting boundary condition or anechoic termination. This feature is handyin determination for transmission loss for mufflers. From, the persepective of finite elementformulation a mixed boundary condition is a natural boundary condition as it contributes to thequadratic functional which is minimized. Admittance is also the reciprocal of impedance, thusboth impedance and admittance boundary conditions are interchangibly used in the literature.In the present article we refer only to admittance boundary condition. Intutively, admittanceboundary condition resembles a spring at the boundary with compliance proportional to theadmittance value.

With this background in place, let us pick up the thread back from the weak form i.e. equation(13).

() (w) + k2w

d +

p

(w) ndp +u

(w) ndu +z

(w) ndz = 0 (14)

where p, u and z denote the portions of the boundary with prescribed pressures, velocity and

impedance respectively. Virtually all problems in engineering also satisfy = p

u

z andp

u

z = . This implies all portions of the boundary falls in one and only one class of boundaryconditions. The weight function w of the weighted residual statement was stated to be arbitrary. Theycan therfore be treated as arbitrary variations of . The idea of variation of a function is illustratedthrough figure(). Should we clarify about what is meant by variation of a function ? To what extent?

At p since is prescribed, the variations along p need to be zero. Loosely speaking, at p since is constrained to have particular values there is no freedom for any variation in . Thus the secondintegral in equation(14) vanishes. Similarly at u the gradients of potential are prescribed through

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the imposed velocities. Thus, n = ju n = jun, where u is the presrcibed velocity andun the normal velocity at u. Thus, the third integral in equation(14) can be substituted as

u(w) ndu = j

u

wundu (15)

Finally the admittance boundary condition on z as un = Zp n = j Z gives the followingsimplification in the last integral of equation(14)

z(w) ndz = j

z

wZdz (16)

Using the above simplifications (15 & 16) we get from equation(14)

() (w) + k2w

d j

u

wundu jz

wZdz = 0 (17)

The key point to note in the above equation is that with un specified, it is linear in both arguements

w & . Also, we can interchange w and in the above expression, this implies a certian symmetry inthe above expression.

In opertor notation equation(17) can be expressed as a(w, ) = b(w), where the operators a(, )and b() are defined as follows

a(w, ) =

() (w) + k2w

d j

z

wZdz

b(w) = ju

wundu

Due to an advanced application involving variational calculus, when w is chosen to be the variationsin solution to a(w, ) = b(w) is equivalent to minimization of a(,)

2

b(). Thus, solution of theacoustic problem may be attained by minimizing the following integral

I =1

2

() () + k22

d j

u

undu j

2

z

Z2dz = 0 (18)

It might seem quite ad hoc that solution of some equation is equivalent to minimization of somethingelse. In fact a full-fledged derivation utilizing functional analysis machinery is beyond the scope of thepresent article. We may finish our job by citing some deeply mathematical textbooks which in theirlast chapter prove the above claim 4. So the average but inquistive engineer is forced to accept thisclaim without asking further questions. We try to give some hand-waving proof of the above claim tobring some peace to the poor fellow. This is illustrated in appendix A.

The moral of the long story has been that we have looked at different ways of solving the acousticequation. Starting from the most common brute force method of solving the differential equation atevery point in the domain we have arrived at an integral representation. With a simple applicationof divergence theorem, we have integrated by parts the weighted residual staement to arrive at theweak form. Finally we have transformed it to a minimization problem, thanks to some heavy-dutyfunctional analysis tools discovered by some genius mathematicians. Note all these solution procedureare perfectly equivalent and one can go from one step from the other. Each method is as general asany other. It is only due to numerical implementation we prefer one over the other.

4as most engineering articles do

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2.5 Approximations

The finite element approximation begins by approximation of the geometric domain. One of the mostuseful features of finite element method is that it is equally applicable to complex or simple geometries.This is because we have recast the differential equation into an expression involving integrals. We know

from high school calculus

()d =1

()d1 +2

()d2 + . . .n

()dn

where = 1 2 . . . n and 1 2 . . . n = (19)

This operation of decomposing into i, i = 1, . . . n is accomplished by meshing. Each i is consideredto be a simple geometry for ease of further processing. It is apt that a complicated geometry can neverbe exactly represented by such finite union of disjoint simplified geometries. However, what is mostcrucial is the discretization error can be made arbitrarily small with finer mesh. Thus, meshing thoughan approximation is a justified one. Within each i (considered sufficiently small), we interpolate the

field variables in terms of its nodal values. Again, with this scheme it would be possible to approximatethe actual functional form to arbitrary accuracy, provided off course we work with a sufficiently refinedmesh. The interpolation function used to interpolate the field variables from its nodal values are calledshape functions. In the following, we illustrate the interpolation methodology with a linear 1D element.

An Il lustration

Consider a 1D element k as shown in figure with coordinate x. The ends are at x1 & x2 (x1 < x2).We try to linearly interpolate the field variable from its nodal values 1 & 2 resprectively. To arriveat the shape functions, let us assume that within k is given by = ax + b, where a & b is to bedetermined. At x = x1 & x = x2 we have 1 = ax1 + b & 2 = ax2 + b respectively. Solving these twoequations for a & b we get

a = 2 1

x2 x1

b =1x2 2x1

x2 x1

=(x2 x)1 + (x x1)2

x2 x1

=

N1 N2 1

2

[N]{} (20)

where N1

=x2 x

x2 x1and N

2=

x x1

x2 x1

The functions N1 & N2 are known as shape functions and is the vector of nodal values of .The above illustration shows that the interpolation function can be derived in the same way as

for a 1D structural element such as rod. This is true for even the more complicated elements. Thusinterpolation scheme for higher order elements and 2D/3D elements in acoustics can be developed inexactly the same way as in structures. More advanced techniques such as isoparametric formulationcan be applied with equal ease.

The shape function interpolation method allows us to relate a continous variable in terms of itsdiscrete nodal values. In general we have the two quantities related by a matrix relation = [N]{}

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(as shown in the illustration by equation 20 ). We may also express the gradient of in such discreteform as shown in the following

x

y

z

= N1x

N2x

Nnx

N1

y

N2

y Nn

zN1z

N2z

Nnz

12

...n

[B]{

} (21)

The above general equation relates the gradient within an element having n nodes. For the previousillustration, we have the following results

x=

1x2x1

1x2x1

12

[B]{} (22)

Using equation(20) & (21) we now have an approximation valid for & in each i i =1, 2, . . . n. Substituting this expression, each integral of the type

i()d is simplified to matrix

multiplying the vector of nodal values. The right hand side of equation(19) can the be evaluated as asummation. This summation procedure is elegantly done by the assembly process. We shall illustratethe assembly procedure through an example later. At this point we simply make a note that an integralexpression (of the form used in equation 18) can be approximately evaluated.

Using the discretised forms of & we have two equivalent directions for proceeding. The firstapproach uses the discretised expression in the variational formulation (equation 18) and minimizesthe resulting quadratic form. In the second approach, we use the weak form and instead of allowingarbitrary weight functions we restrict them to be the shape functions (already chosen). Putting weightfunction to be each of these shape function in equation(13), we get a linear system of equation. This iscalled the Galerkin method. We shall further elaborate on both these methodologies in the discussion

to follow.

2.6 Finite Element - Variational Method

As discussed in the previous section the discretisation procedure (equation 20 and 21) leads to thefollowing relation

2 T = {}T[N]T[N]{}

() () []T[] = {}T[B]T[B]{} (23)

The above relations can be used to obtain a discretised version of the integral used in equation(18).The discretised form is obtained for each i i = 1..N and using equation(19) the integral over isobtained. The discretised form for a certain element (k) is as follows

Ik = {}T1

2

k

[B]T[B]d

{} + k2{}T1

2

k

[N]T[N]d

{}

+j{}T

uk

[N]Tund

j{}T

1

2

Zk

Z[N]T[N]d

{} (24)

where uk and Zk represents portions of the boundary of k with specified velocities and admittances.

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To abrreviate the long expression we define the following symbols

K =k

[B]T[B]d

M = k2 k

[N]T[N]d

fu = juk

[N]Tund

Fz = jZk

Z[N]T[N]d (25)

In terms of these symbols, the equation(24) may be simplified as

Ik = 1

2{}T[K]{} +

k2

2{}T[M]{} + {}T{fu}

1

2{}T[Fz]{} (26)

In the above expression, {} is unknown and needs to be determined. The variational formulationrequires Ik to be minimum. This requires the gradient of Ik with respect to {} to be zero. It is shownin Appendix that this in turn leads us to the following system of linear equation applicable for a singleelement

[K] k2[M] + [Fz]

{} = {fu} (27)

2.7 Finite Element- Galerkin Method

Should we explain this or completely skip it ? The variational formulation has been derived in full,and this leads to the same thing anyway ....

2.8 Assembly & Constraints

The system of equation(27) is for a single element. I for the entire domain may be evaluated bysumming over all elements as indicated in equation(19). This summation is referred to as assemblyand to the full system equation. The assembly procedure is identical to that used in structures andwill not be described here except through a numerical example. Readers are referred to chapter 2 incook for this purpose.

The constraints i.e. the essential boundary conditions need to be imposed before solving the systemequations. This too is identical to that used in structures and hence will not be elaborated in thisarticle except in the numerical example. Readers are referred to Chadrupatla for this.

Should we explain these in more details ?

2.9 Numerical Examples

2.10 Conclusion

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