81177621 Computational Methods for Vibroacoustic Simulations

7/30/2019 81177621 Computational Methods for Vibroacoustic Simulations

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Computational Methods for Vibroacoustic Simulations

Frank Ihlenburg

November 11, 2011

Contents

1 Introduction 1

2 Models and Equations 2

2.1 Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Variational Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3 Finite Element Method 113.1 Finite Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 Discrete equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.3 Modal analysis of the forced response . . . . . . . . . . . . . . . . . . . . . . . 18

4 Statistical Energy Analysis 194.1 Energy content of vibrational subsystems . . . . . . . . . . . . . . . . . . . . 194.2 Power flow and dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.3 SEA balance equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.4 Example: rectangular plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

5 Boundary element method 225.1 Helmholtz integral equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.2 Boundary element method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1 Introduction

The word vibroacoustics stands for the interaction of mechanical vibrations with acoustic waves; thereaction of the eardrums to pressure changes in the air is a natural example. Using computationalsimulations of acoustic effects, mechanical engineers can evaluate the noise or comfort level of different

design at early stages of product development. In particular, one is interested to estimate the radiationof sound from engines or engine parts into free space or the level of cabin noise; i.e., the radiation,reflection and absorption of waves in closed cavities. The corresponding mathematical models of

Figure 1: Exterior and interior problem

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vibroacoustics lead to exterior or interior problems, respectively; see Fig. 1). Both types of fluid-solidinteraction are governed by the same system of coupled differential equations governing the vibrationsof the solid and the fluid domains, and the coupling between them. In practice, the coupled system issolved by numerical methods.

This text contains a brief introduction to analytical and computational models for coupled fluid-

structure vibrations. The differential equations of the coupled vibrations in a general three-dimensionalsetting are the starting point. The so-called strong formulation is then transformed into the weak or

variational formulation. Both formulations are illustrated on two simple examples. Section 3 contains

a brief review of the discretization procedure by the finite element method (FEM). This procedure

leads to a coupled system of linear algebraic equations that can be solved computationally. For large

systems and broad frequency bands it is advantageous to apply modal reduction. The basic ideas of

Statistical Energy Analysis (SEA) and the boundary element method (BEM) are contained in sections

4 and 5.

2 Models and Equations

The section starts from the differential equations of the coupled vibrations in a general three-dimensional setting. The so-called strong formulation is then transformed into the weak orvariational formulation.

2.1 Differential Equations

Consider a three-dimensional solid continuum s, which is in contact with a fluid continuumfalong a smooth wet surface . The vibrations of continua are governed by the equilibrium

Table 1: Differential equations for solid and fluid vibration

VIBRATIONS Solid Fluid

Equilibrium div + sU = 0 (1) P + fV = 0 (2)

Material Law = C : (3) P = BS (4)

Kinematics =1

2

(U)T + U

(5) S = divV (6)

condition for the forces of deformation and the forces of inertia. Elastic material behavior andsmall deformations are assumed. The corresponding systems of linear differential equationsare listed in Table 1. The following notations are used:

U(x, t) = Us(x, t) =

Ux(x, t)Uy(x, t)Uz(x, t)

- solid displacement vector;

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=

x xy xzxy y yz

xz yz z

, =

x xy xzxy y yz

xz yz z

- solid stress and strain tensors;

C - solid material tensor;

P(x, t) - fluid pressure;

V(x, t) = Uf(x, t) - fluid velocity vector;

S =dV

V= trf = divUf - fluid volume dilatation;

s, f, B - solid and fluid density, fluid bulk modulus.

The solid and the fluid volumes contain matter in two different states. Since a fluid mattercannot support shear stresses, the particle interaction is reduced to forces of compression.Thus the stress and strain tensors are diagonal,

f =

P 0 00 P 0

0 0 P

, f = 1

3

S 0 00 S 0

0 0 S

.

Since the fluid is assumed as ideal (frictionless), the tangential interaction between the solidand the fluid is neglected. The Equilibrium and kinematic compatability conditions holdtherefore only along the normal to the coupling surface. Hence, at any point of the couplingsurface, the exterior solid traction vector can be written in local coordinates as [0 , 0,P]T.On the other hand, by definition of the stress tensor, the traction vector at any (interior orexterior) surface with the surface normal ns is given by ns. The solid-fluid equilibriumequation can thus be written in arbitrary coordinates as ns = Pns , where ns is theexterior solid normal pointing into the fluid; cf. Fig. 2.

Figure 2: Coupling surface at Solid-fluid boundary

Furthermore kinematic compatability (in normal direction) requires that the normal pro-jections of the solid and fluid displacement vectors must be equal. The coupling equations areshown in Table 2. The fluid normal, pointing out of the fluid, is related to the solid normalas nf = ns.

Figure 3: Longitudinal vibrations of a beam in contact with a fluid

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Table 2: Solid-fluid coupling conditions for ideal fluid

COUPLING Solid & Fluid

Equilibrium ns = Pnf (7)

Kinematic Compatibility U ns = V nf (8)

Example 1: Slab Consider a vibrating slab in contact with a compressible fluid as shown inFig. 3. The state variables are U = U(x, t) and P = P(x, t). The solid stress tensor, corresponding tothe uniaxial stress state, contains only one non-zero component = x. The stress divergence vectortherefore reduces to ,x =

x . Similar simplifications apply to the material and kinematic relations;

see Table 3.

Table 3: Differential equations for solid and fluid vibration: 1-D model problem

VIBRATIONS 1-D Solid Fluid

Equilibrium ,x +sU = 0 P,x +fV = 0

Material Law = E P = BS

Kinematics = U,x S = V,x

Table 4 contains the coupling conditions. The equilibrium of stress and pressure is illustrated in

Fig. 4. The differential equations and the coupling conditions can be written as a coupled set of

Figure 4: Static equilibrium

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equilibrium equations (1) reduces to the two equations

x,x +xy,y +sUx = 0

yx,x +y,y +sUy = 0 .

Assuming small deformations, these equations are neglected. However, the horizontal stresses lead tointerior bending moments which must be considered in the lateral equilibrium condition, viz.

mx,xx +2mxy ,xy +my,yy +sUz = P ,

where the interior bending moments are defined as mx =h/2h/2

xz dz etc. This equation expresses

the kinetic fluid-structure coupling normal to the midsurface.The elastic material law for plane stress is

xyxy

= E

1 2

1 0 1 0

0 0 12

xy

xy

The kinematic relations for small displacements are written as

x = Ux,x , y = Uy,y yx = Uy,x +Ux,y .

By the Bernoulli hypothesis there hold in addition the kinematic equations

Ux = zUz,x , Uy = zUz,y .

The kinematic coupling between plate and fluid is established as Uz = Vz where Vz denotes thez-component of the fluid particle velocity.

The fluid pressure satisfies the three-dimensional wave equation which is obtained by eliminatingS and V from the system (2, 4, 6). In the present geometrical setting, the solid normal ns coincideswith the unit vector ez of the z-axis. Multiplying equation (2) by ez, the kinematic coupling condition

(8) can be rewritten in terms of the pressure as

P

z= f

V ez

= fUz

The coupled differential model for the deflection U = Uz(x,y,t) and the pressure P then assumesthe form

Eh3

12(1 2)U + shU = P {x, y} [0, lx] [0, ly] , z = 0 (13)

P,z = fU {x, y} [0, lx] [0, ly] , z = 0 (14)

P,xx (x, t)1

c2P = 0 {x, y} [0, lx] [0, ly ] , z > 0 (15)

+ boundary conditions + initial conditions

2.2 Variational Equations

The variational or weak formulation of the differential equations is obtained from the differ-ential equations in two steps. First, multiply the differential equations with test functionsand integrate over the computational domains. Second, apply integration by parts in orderto shift the highest derivatives from the trial functions to the test functions. In mechanicalterms, the test functions represent virtual displacements or virtual forces, respectively.

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Solid: Multiplying the equilibrium equation (1) by a stationary vector function w(x) andintegratng over the solid domain s one obtains

s div w dV + s sU w dV = 0 .Now the Gauss integral theorem is used to integrate the first term by parts:

s

: w dV

( ns) w dA +

s

sU w dV = 0 .

Let us assume for simplicity that the fluid and the solid are coupled along the entire solidboundary . Then the coupling condition (7) can be introduced into the surface integralyielding

s

: w dV +

s

sU w dV +

(Pns) w dA = 0 .

Remarks:

1. By symmetry of the stress tensor, : w = : (w)T = : 12

(w)T + w

=

(U) : (w). Hence the first integral can be interpreted as work of the interior de-formation forces (stresses) due to the virtual strains that correspond to the virtualdisplacement w.

2. Using the solid material and kinematic equations (3, 5), the stresses (U) can bereplaced by a product of the material tensor and the gradient of the displacement

vector, = C : 12

(U)T + U

.

3. It is customary to reshape the symmetric stress and strain tensors as 6 1 vectors, e.g.

= [x, y, z, xy, xz, yz ]T. The material law can then be denoted in matrix-vectorform as = C, and the kinematic equations become (w) = Dw where

D =

x 0 0

0 y 0

0 0 z

2y

2x 0

2z 0 2x

0 2z

2y

Using this notation, we obtain : w = (Dw)T C (DU).

Fluid: The equilibrium equation (2) is multiplied by a stationaryvector function q whereq is a scalar test function with the physical units of pressure.

fP q dV +

ffV q dV = 0 .

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Now the Gauss integral theorem is applied to the second integral yieldingf

P q dV

f

fdivVq dV +

fV (qnf) dA = 0 .

It follows from the fluid material and kinematic equations (4,6) that the divergence of the fluidvelocity can be replaced by the pressure as divV = P /B. Similarly, the normal velocity onthe boundary is replaced by the solid acceleration using kinematic compatability. The resultis written as

f

P q dV +

f

fB

Pq dV

f

U ns

q dA = 0 .

Solid-Fluid Interaction: Assuming that the acoustic medium is homogeneous, the mate-rial parameters can be taken out of the integrals over the fluid domain. The coupled systemof variational equations then assumes the form

s

(Dw)T C (DU) +

s

swTU dV +

wT (Pns) dA = 0 , (16)

1

f

f

(q)TP dV 1

B

f

qP dV +

(qns)T U dA = 0 . (17)

Example 1: Slab The solid equilibrium equation is multiplied by a stationary test function w(x)and integrated over the solid domain [l, 0]:

0l

,x w dx + s0l

U w dx = 0

The first term is then integrated by parts0l

w,x dx + (l)w(l) (0)w(0) + s

0l

U w dx = 0

Assume that the slab is constrained at x = l, then w(l) = 0. Inserting = E and = U,x intothe first integral, and using the coupling equilibrium condition to replace (0), one obtains

E

0l

U,x w,x dx + s

0l

Uwdx + P(0)w(0) = 0 .

Similarly, the fluid equilibrium equation is multiplied by a test function q,x and integrated overthe fluid domain,

l0

P,x q,x dx + f

l0

V q,x dx = 0 .

The second term is now integrated by parts,

l0

P,x q,x dx + f

V(l)q(l) V(0)q(0)

f

l0

V ,x q dx = 0 .

Using the material law and the kinematic relation, the velocity V is replaced by the pressure in thesecond integral. Similarly the kinetic coupling condition is applied to replace V(0). A rigid termina-tion of the fluid is assumed at x = l such that V(l) = 0. Then the coupled system of free undamped

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vibrations is written in the form

E0

l

U,x w,x dx + s 0

l

U w dx + P(0)w(0) = 0 , (18)

1

f

l0

P,x q,x dx 1

B

l0

P q dx + U(0)q(0) = 0 , (19)

where the second equation has been divided by f in order to write both equations in the same phys-ical dimensions. This system is not yet completely symmetric, due to the different orders of timederivation in the coupling term. Deriving hence the first equation in time and replacing U = V leadsto the symmetrised version:

E

0

l

V,x w,x dx + s 0

l

V w d x + P(0)w(0) = 0 (20)

1

f

l

0

P,x q,x dx1

B

l

0

P q dx + V(0)q(0) = 0 (21)

Example 2: Plate. The variational equation of a rectangular plate loaded by fluid pressurefollows directly from the principle of virtual displacements. According to this principle, the virtualwork of the interior forces (of deformation and inertia) is at all times equal to the virtual work of theexternal pressure. Since the pressure is directed against the positive lateral displacement, the externalwork is taken with a negative sign. Denoting as above the virtual displacement by w one gets

lx

0 ly

0 h/2

h/2

(U) (w) dxdydz + s lx

0 ly

0 h/2

h/2

U w dxdydz = lx

0 ly

0

(Pez) w dxdydz

Assuming that the thickness remains constant during deformation, the vector notation of the straintensor reduces to = [x, y, xy, xz, yz ]

T . Assuming further that the Bernoulli hypothesis holds, the

shear strain across the thickness must vanish, and hence the strain vector reduces to = [x, y, xy ]T .

Furthermore its components are related to the second derivates of the lateral displacement via

(w) = (wz) = z [wz,xx , wz,yy , 2wz,xy ]T .

Since Ux, Uy Uz the in-plane contributions of the inertia term are neglected, U w Uzwz. Writingshort U = Uz, w = wz and carrying out the integration over the thickness leads to

Eh3

12(1 2

)

lx

0

ly

0

w,xxw,yy

2w,xyT

1 0 1 0

0 0 1

2

U,xxU,yy

2U,xy dxdy

+ sh

lx0

ly0

Uw dxdydz +

lx0

ly0

Pw dxdydz = 0

The wave equation is multiplied by a test function q(x,y,z) and integrated,lx0

ly0

lz0

P

1

c2P

q dxdydz = 0

Integrating the first term by parts gives

P

nq dA

lx

0 ly

0 lz

0 P q+

1

c2P q

dxdydz = 0 ,

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where the boundary integral can be written out as the sum over the side surfaces of the fluid pillar,

P

nqdA =

lx

0

ly

0P

zq

z=lz

dxdy lx

0

ly

0P

zq

z=0

dxdy

+

lx0

lz0

P

yq

y=ly

dxdz

lx0

lz0

P

yq

y=0

dxdz

+

ly0

lz0

P

xq

x=lx

dxdz

ly0

lz0

P

xq

x=0

dxdz .

or

P

nqdA =

lx0

ly0

P

zq

z=0

dxdy +

+

P

nqdA,

the first term on the left is the integral over the coupling surface and + is the union of the remainingsurfaces. If these surfaces are rigid then the integral over + vanishes.

Using the coupling condition (14) to replace the normal derivative on the coupling surface, thefluid variational equation is written as

1

f

lx0

ly0

lz0

P q dxdydz

+

P

ZnqdA

1

B

lx0

ly0

lz0

P q dxdydz

+

lx0

ly0

U q dxdy = 0

Collecting results, the variational formulation for the free undamped vibrations of a pressure-loaded plate finally assumes the form

K

lx0

ly0

w,xxw,yy

2w,xy

T 1 0

1 0

0 012

U,xxU,yy

2U,xy

dxdy + sh

lx0

ly0

Uwdxdy

+lx0

ly0

Pwdxdy = 0 (22)

1

f

lx0

ly0

lz0

P q dxdydz1

B

lx0

ly0

lz0

P q dxdydz

+

lx0

ly0

Uq dxdy = 0 . (23)

Damped Vibrations Vibrations in physical reality are always damped by various dissipa-tive mechanisms like material damping in the solid or boundary absorption in the fluid. Themathematical modeling of these effects is outlined below.

Slab (material damping, radiation damping): The effect of internal friction is de-scribed in the Kevin-Voigt hypothesis for the solid material. By this hypothesis, the totalstress is expressed as a sum of an elastic and a viscous stress,

= e + v = E + r ,

where r denotes a viscosity coefficient. This hypothesis affects only the material law; theequilibrium and kinematic conditions remain unchanged. The solid variational equation thencontains an additional term due to damping,

E

0

lU,x w,x dx + r

0

lU ,x w,x dx + s

0

lU wdx + P(0)w(0) = 0 .

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Assuming further that the radiation condition P = cfV is given at the far boundary ofthe fluid domain at x = l, the variational equation for the fluid is written as

l

0P,x q,x dx

fB

l

0P q d x + fU(0)q(0)

1

cP(l)q(l) = 0

This vibrational equation contains a damping term due to radiation.The coupled system of variational equations for the damped vibrations of the 1-D structure-

fluid continuum then finally assumes the form

E

0l

U,x w,x dx + r

0l

U ,x w,x dx + s

0l

U wdx + P(0)w(0) = 0 (24)

1

f

l0

P,x q,x dx 1

fcP(l)q(l)

1

B

l0

P q d x + U(0)q(0) = 0 (25)

Plate (boundary absorption): Assume that the boundary impedance Zn =

P /

Vn isgiven along the free edges, then the integral over + leads to the damping term+

P

nqdA = f

+

P

ZnqdA

Now the coupling condition (14) can be used to replace the normal derivative on the couplingsurface, to obtain

P

nqdA = f

lx0

ly0

Uzqz=0

dxdy f

+

P

ZnqdA,

and the fluid variational equation becomes

1

f

lx0

ly0

lz0P q dxdydz

+

P

ZnqdA

1

B

lx0

ly0

lz0

P q dxdydz

+

lx0

ly0

U q dxdy = 0

3 Finite Element Method

3.1 Finite Elements

The solid and fluid domains are partitioned into subdomains called finite elements. The

intersection points of the elements are called nodes. On this partition, the physical unknownsU and P are approximated by nodal functions Uh U, Ph P. The nodal functions can bewritten in the form

Uh(x, t) =j

Uj(t)j(x), Ph(x, t) =j

Pj(t)j(x), (26)

where Uj(t), Pj(t) are unknown nodal values, the functions j(x) are linear interpolants,and the summation extends over all nodes in the solid or fluid domain, respectively. Thenodal values are computed from a system of linear algebraic equations that is derived fromthe variational formulation of the coupled solid-fluid equations. The interpolants j are theco-called hat functions which are assembled from linear element shape functions Nk.

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Solid-Fluid Interaction (3-D): The volumes are partitioned into hexaedral or tetraedralelements. The plot in Fig. 7 shows eight-node hexaedral solid and fluid elements with thenodal degrees of freedom (DOF) plotted for the first node.

Figure 7: Hexaedral solid and fluid element with nodal DOF.

Each solid node carries three DOF, while only one DOF is associated with a fluid node.The linear interpolation functions Nk(xloc) are explicitely given on a regular hexaedron inlocal element coordinates (the so-called master-element to which each physical element ismapped). The solid and fluid domains can be meshed independently; i.e., fluid and solid

nodes need not match on the wet surface.

Slab. The solid and fluid domains are partitioned into two-node line elements. Each node is asso-ciated with one DOF only, since only translations along the x-axis are computed in this example (Ingeneral, three translational DOF are associated with nodes on spatial truss elements.) Within eachelement, the displacement or the pressure are approximated by linear interpolants as shown in Fig 8.

Figure 8: Linear approximation on two-node elements.

Plate. The fluid domain is partitioned into hexaedral elements as shown in Fig. 7. The (mid-surface of the) plate is partitioned into rectangular elements. In order to represent the lateral bendingdeformation, each node is associated with three DOF, namely, the lateral deflection and two angles of

rotation. Only the deflection is coupled to the pressure in the fluid domain. 1 For each set of DOF,bilinear interpolation can be used within the plate element; see Fig. 9. If the Bernoulli hypothesis isassumed, the angles depend directly on the lateral deflection, and bicubic shape functions are used.

The plate and the fluid can be meshed independently.

3.2 Discrete equations

The nodes form a discrete subset of the continuous computational domain. Inserting thenodal approximations (26) into the continuous variational equations, these are transformed

1In general, six DOF are associated with nodes of spatial plate or shell elements, and the normal projection

of the translation components is coupled to the fluid pressure.

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The damping matrices are obtained from the corresponding terms of the variational equationsin the same way as the stiffness and the mass matrix. The symmetrised model of free dampedvibrations is written as

Ks 00 Kf

VP

+ Cs A

AT Cf V

P

+ Ms 0

0 Mf V

P

= 0

0

(32)

Time-harmonic vibrations: Assuming time-harmonic vibrations in the form U(t) =ueit etc., the stationary systems

Ks A0 Kf

2

Ms 0

AT Mf

up

=

00

(33)

or

Ks 0

0 Kf i 0 A

AT 0 2

Ms 0

0 Mf vp =

00 (34)

are obtained. The coupled problem (34) is a complex quadratic eigenvalue problem. Thestiffness and mass matrices are sparse and symmetric square matrices whos dimension ns ornf, respectively, are equal to the number of unconstrained DOF in all the solid or the fluidnodes, respectively. The coupling matrix is a rectangular matrix ns nf. It has, in general,non-zero entries in all nodes on the coupling surface.

Finally, the time-harmonic system of damped free vibrations can be written asKs 0

0 Kf

i

Cs A

AT Cf

2

Ms 0

0 Mf

vp

=

00

(35)

Remark: An alternative symmetric formulation of the undamped eigenvalue problem canbe obtained if one multiplies the first equation in (33) by . Then, setting u = u, one getsthe real quadratic eigenvalue problem

Ks 00 Kf

+

0 A

AT 0

2

Ms 0

0 Mf

up

=

00

(36)

Example 1: Slab A finite element partition of the coupled 1-D domain is shown in Fig. 10. Boththe solid and the fluid have been partitioned into two line element each, whereby the nodes 1,2,3 havebeen generated on each domain. The elements are of uniform length le = l/2.

Figure 10: FE partition of coupled solid-fluid domain.

The unknown functions U, P and the test functions w, q are now approximated by nodal func-tions, U Uh etc. On each element, the nodal functions can be written as products of unknownnodal values and element shape functions. For instance, on the second solid element, U(x, t) U2(t)N1(x) + U3(t)N2(x), and on the first fluid element, P(x, t) P1(t)N1(x) + P2(t)N2(x). Now the

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terms in the variational equations (18) and (19) can be calculated explicitely. The domain integralsare written as sums of element integrals viz.

0l

U,x w,x dx =

le

l

P,x q,x dx +

0le

P,x q,x dx ,

l

0

P,x q,x dx =

le

0

U,x w,x dx +

l

le

U,x w,x dx ,

and similarly for the inertia terms.Now the nodal approximations are introduced. Consider, for example, the integrals over the first

solid element,

E

le0

U,x w,x dx =

11

(U, ,x ) (w, ,x )dx

dd

=2E

le

w1w2

T 11

N1, N1, N1, N2,N2, N1, N2, N2,

d

U1U2

=

w1w2

TEle

1 1

1 1

U1U2

.

The result can be written in matrix notation as wTK(e)U, thereby introducing the element stiffnessmatrix K(e). Similarly, the inertia term leads to the element mass matrix M(e) via

s

le0

U w dx =

11

U wdx

dd

=sle

2

w1w2

T 11

N1N1 N1N2N2N1 N2N2

d

U1U2

= w1

w2T sle

6 2 1

1 2 U1

U2

Likewise, the evaluation of the fluid variational equation on the first fluid element leads to

1

f

le0

P,x q,x dx =

q1q2

T1

fle

1 1

1 1

P1P2

1

B

le0

P q dx =

q1q2

Tsle

6

2 11 2

P1P2

,

defining the element stiffness and mass matrices of the fluid. The fluid and solid domains are coupledin the conincident nodes 3 (solid) and 1 (fluid). The coupling terms thus are written as P(0)w(0) =P1w3, U(0)xq(0) = U3q1. Computing all integrals and summing up over the elements, a linear system

of discrete variational equations is obtained (all non-written matrix entries are zero). Note that themass matrix entries have been lumped, and the equation for the constrained solid node 1 has beeneliminated from the system of unconstrained equations.

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w2

w3q1q2q3

T

2Ele Ele

EleEle

1

1lef

1lef

1lef 21

lef 1lef

1lef1

lef

U2

U3P1P2P3

+

w2w3q1q2q3

T

1fc

U2U3P1P2P3

+

w2w3q1q2q3

T

sle

sle

1 leB

leB

leB

U2U3P1P2P3

= 0

Dividing by the nodal test vector [w2, w3, q1, q2, q3], one obtains the system of discrete finiteelement equations for the damped vibrations of a 1-D solid-fluid slab

Ks A

0 Kf

UP

+

Cs A0 Cf

UP

+

Ms 0AT Mf

UP

=

00

(37)

with

Ks =E

le

2 1

1 1

, Cs =

0 00 0

, Ms = sle

1 00 1

,

A =

1 0 00 0 0

,

Kf =1

lef

1 1 01 1 1

0 1 1

, Cf =

1

fc

0 0 00 0 0

0 0 1

, Mf =

1

B

1 0 00 1 0

0 0 1

.

Example 2: Plate The coupled system of variational equations (23,23) for the undamped vibra-tions of a fluid-loaded plate can be written short as

K

A

(dw)TC(dU) dA + textrmsh

A

U w dA +

A

P w dA = 0 , (38)

1

textrmf

V

P q dV 1

B

V

PqdV +

A

U q dA = 0 , (39)

where C denotes the elasticity matrix and d =

xx ,

yy , 2

xy ,

. The FE-model is now constructed

from these equations by the same procedure as for the slab:

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Figure 11: FE meshes on the plate: structural mesh (left); faces of solid elements on couplingsurface (right).

1. FE Partition (Mesh). The domains are meshed independently. The meshes do not have tomatch on the coupling surface. A sample configuration is shown in Fig. 11.

2. Partition of domain integrals. The integrals in the above equations are decomposed intosums of element integrals.

3. FE interpolation. The functions in the element integrals are interpolated by the FE shapefunctions.

4. Numerical evaluation of element integrals. The element integrals are evaluated by ap-propriate methods of numerical integration.

5. Matrix assembly. The components of the system matrices are obtained by summation ofnodal entries from the element integrals.

The coupled system of discrete equations has the formKs A

0 Kf

UP

+

Ms 0AT Mf

U

P

=

00

,

Figure 12: Coupling of structural and fluid nodes.

The coupling of structural and fluid nodes is illustrated in Fig. 12. A 4-node fluid element is

highlighted in the plot on the right. Denoting its surface area by Ae, the coupling integral fromequation (39) becomes

Ui

Ae

Nsi Nfj dA

qj

where the summation index i includes all nodes that belong to the structural elements in the shadedarea of the left plot while the index j runs through the four nodes of the fluid element. Similarly,the coupling integrals for all elements in the shaded structural area include the fluid shape functionsNfj from the marked fluid element. In the result, the four nodes of the fluid element on the right arecoupled to all structural nodes from the highlighted area on the left. Special care has to be taken inthe selection of Gauss points in the coupled elements in order to obtain the same numerical entries tothe coupling matrices.

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3.3 Modal analysis of the forced response

In many applications one is interested in the dynamic response to harmonic excitations ofthe form F(t) = Feit. The discrete model is then given by an inhomogeneous system oflinear algebraic equations. This system can be solved by the method of modal superposition

as follows:

1. Modal Analysis: Find - separately - sets of uncoupled eigenmodes of the solid andthe fluid from the equations

[Ks Ms] x = 0 , (40)

[Kf Mf] x = 0 . (41)

For each of the problems, the number of eigenvectors is equal to the respective matrixdimensions. The full sets of all eigenvectors are bases of the linear function spaces inwhich the FE nodal functions are defined. However in most practical applications, only asmall number of eigenvectors is computed. The remaining (not computed) eigenvectorsare called residual vectors. The complete sets of all vectors can thus be represented asunions of computed and residual vectors,

{xsj}nsj=1 = {xsj}

Nsj=1 {xsj}

nsj=Ns+1

, (42)

{xfj}nfj=1 = {xfj}

Nfj=1 {xfj}

nfj=Nf+1

. , (43)

where Ns, Nf denote the numbers of computed eigenvectors and ns, nf are the matrixdimensions.

2. Modal projection: Assume that the sets of computed eigenvectors are assembled inrectangular matrices Xs and Xf, respectively. Then seek the solutions of the coupled

system of forced response as modal superpositions

U(t) = XsYs(t) (44)

P(t) = XfYf(t) (45)

with unknown complex-valued modal coefficients Ys, Yf. The modal superpositionsproject the forced response onto the subspaces that are spanned by the computed solidand fluid eigenvectors, respectively.

3. Modal transform: Insert the modal superpositions (44, 45) into the equations for theforced response, and multiply these equations from the left by XTs and X

Tf , respectively,

XTsXTf

T Ks A

0 Kf

XsYsXfYf

+

XTsXTf

T Ms 0

AT Mf

XsYsXfYf

=

XTsXTf

T FsFf

or, after multiplication,Ks A

0 Kf

YsYf

+

Ms 0

AT Mf

YsYf

=

FsFf

. (46)

The submatrices Ks = XTs KsXs, Ms = X

Ts MsXs and Kf = X

Tf KfXf, Mf = X

Tf MfXf

are called the modal transforms of the stiffness and mass matrices, respectively. Bythe properties of symmetric eigenvalue problems, these modal transforms are squarediagonal matrices of dimensions Ns and Nf, respectively.

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The modal transform A = XTs AXf of the coupling matrix is a fully populated rectan-gular matrix with the size Ns Nf.

It is convenient to introduce damping effects in the structure and the fluid by meansof square diagonal damping matrices Cs and Cf. The system of equations for the

computation of the damped forced response then finally assumes the formKs A

0 Kf

YsYf

+

Cs 0

0 Cf

YsYf

+

Ms 0

AT Mf

YsYf

=

FsFf

(47)

The modal damping matrices can be superposed with the modal transforms of materialor other damping matrices that were introduced prior to modal transform.

The corresponding symmetrised system can be written asKs 0

0 Kf

YsYf

+

Cs A

AT Cf

YsYf

+

Ms 0

0 Mf

YsYf

=

Fs

Ff

(48)

where the modal coefficients Ys correspond to the solid velocity, andFs results fromthe time derivative of the solid excitation vector.

4. Solution of the modal equations and back-transform. The modal equations canbe solved by a direct complex solver to obtain the modal coefficients Ys and Yf. Thephysical solutions are then found from equations (44) and (45).

4 Statistical Energy Analysis

The finite element method is well suited for the vibrational analysis of structures or structure-fluid interaction. Using this method, the frequency response of dynamical systems is eval-

uated in specified spatial points (the FE nodes) and for specified frequencies. The FEM isthus a deterministic method, both in spatial variables and in the time (frequency) variable.With increasing frequency, the spacing of the nodes has to be very fine in order to resolvethe decreasing wavelength. The Statistical Energy Analysis (SEA) is an alternative methodthat is particularly well suited for computations at large frequencies. The vibrational sys-tem is partitioned into relatively large subsystems whose average vibrational energy can berepresented by a single statistical parameter, the acoustic temperature. The computationaims at determining the temperature distribution in the dynamical system when the influx ofexternal vibrational or acoustic energy is given. This distribution is computed from a systemof balance equations that describe the power flow between the subsystems.

4.1 Energy content of vibrational subsystemsThe acoustic temperature of a subsystem is defined as

T =Etot

N, (49)

where Etot is the total energy (sum of the time-averaged kinetic energies over all modes withinthe frequency band), and N is the number of modes in the frequency band over which thestatistical average is taken. The acoustic temperature of a subsystem is thus its space- andtime-averaged (or frequency-averaged) energy content in a given frequency band (usually,thirds or octaves). It is not important whether a subsystem is a single resonator, (part of)

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Spatial averaging. The averaged velocity v is defined by

v2 =1

A

A|v|2dA

Assume that a modal decomposition of the velocity is given in the form v(x, y) = n

vnn(x, y),where the modes are orthonormalized with respect to the inner product (, ) =

A dA.

Then

v2 =1

A

A|v|2dA =

n

|vn|2 , (55)

The modal coefficients depend on the distributed load p(x, y) by the well-known relation

vn =i

[2n(1 + i) 2] m

A

pndA ,

where m denotes the mass of the plate. If the plate is loaded by a point force F(xo, yo) then

vn

(xo

, yo

) =iFn(xo, yo)

[2n(1 + i) 2] m.

Space-averaging (55) over the positions of the point forces now leads to

v2 =1

A

A|v(xo, yo)|

2dxodyo =F2

m2

n

2

(2 2n)2 + 24n

. (56)

Frequency averaging. Considering a frequency interval = [1, 2], the mean-squarefrequency averages of the velocity and the forces are written as

v2 =1

21

v2d , F 2 =1

21

F2d .

Applying this averaging to equation (56), one arrives after some simplifications at the ap-proximate relation

v2 F2m2

1

N2n=N1

2n,

where the sum is taken over the eigenfrequencies in the interval .

Modal density. This relation is further simplified if the eigenfrequencies are approximatedby the center frequency of the band, setting n ,

v2 F2

m2

2

N

, (57)

where N is the number of modes in the frequency band . The last term of (57) is thus ameasure of modal density. According to Cremer and Heckel[1], the above approximations arevalid under the condition N 5, assuming that all modes have the same energy contenton the average.

Dissipated internal energy. The left-hand side of the balance equation (54) can now bewritten (in the absence of coupling loss) as

E = mv22

=F2m

4

N

. (58)

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Applying integration by parts on the left we see that the volume integralsfGp dV

fpG dV cancel. The left-hand side can thus be written in terms of boundary integrals,

S+

G

np

p

nG dA =

f

pdV ,

where S+ denotes a large sphere with radius R . It can be shown (cf. [2, 1.1.3]) that theboundary integral over S+ vanishes if p satisfies the Sommerfeld radiation condition. Theintegral on the right can be evaluated as p(x) where

= (x) =

0 x R \ f12 x 1 x f

The Helmholtz boundary integral equation is now finally obtained as

p(x) +

G(x, y)

ny p(y) dA(y) = vn(y)G(x, y) dA(y) (62)

The left-hand side of this equation can be interpreted as a linear operator A : p(x) Ap(x)acting on the unknown function p. The right-hand side is a computable function f(x) whichdepends on the velocity input v. The linear integral equation can thus be written alternativelyas a linear operator equation

Ap = f .

5.2 Boundary element method

The radiating boundary is subdivided into elements, similarly to the FEM. These elements

are needed for the approximate evaluation of the boundary integral

G(x,y)

ny p(y) dA(y).Consider, for instance, the partition of the 1-D boundary shown in Fig. 13. Assuming thatthe function p is piecewise constant on the elements, the boundary integral can be writtenapproximately as

Ap(x) = p(x) +j

hjp(yj)G(x, yj)

where hj denotes the length of element j (distance between the nodes) and yj denotes thelocation of the element midpoints.

Figure 13: Boundary partition.

The operator equation Ap(x) = f(x) can be solved by collocation. Letting x = xj, j =1, . . . , N one obtains a set of linear algebraic equations

Ap(xj) = f(xj), j = 1, 2, . . . N

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or, if written as a matrix-vector equation,

BP = F

where P is the vector of pressures in the collocation points of the method. This system

matrix B is, in general, fully populated and frequency-dependent (since the Green functiondepends on the frequency). Also special care must be given to the evaluation of the elementintegral when the collocation point lies inside the element, since |G| if x y.

References

[1] L. Cremer and L. Heckl, Structure-Borne Sound, Springer-Verlag Berlin 1973

[2] F. Ihlenburg, Finite Element Analysis of Acoustic Scattering, Springer Verlag New York 1998

[3] P. Bremner, R. Langley, A hybrid method for the vibration analysis of complex structural-acousticsystems, J. Acoust. Soc. Am. 105(3), 1999, 16571670

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81177621 Computational Methods for Vibroacoustic Simulations

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