Circuit theoretical method for efficient finite element analysis of acoustical problems

the structure. C is usually defined as

C = a M + @ K ( 2 )

where o and are constants.

Once the vector Q is found, the variations of the displacements. stresses and strains in the elements can he approximated.

C m cmr

SIMULATION

circuits are defined hy the equation

In modified node analysis ( M N A ) formulation linear

C j.= -G z

+

bu(t) (3) where C is the nlatrix of capacitances and inductors. C, is the admit,tance matrix. u ( t ) is the voltage or current excitation at thc nodes defined by vector b. a d .c is the vect,or of nodal voltages. inductor currents aud independent voltage supply currents.

The dynanlic quation of motion (1) is of order 2 ; hut it can bc converted into an equation of order 1 [7]: s o that it, can hc solved using circuit, simulation tech- niques.

:\lolrir.nt : l l u t t : l l i y ~ , nnd Oliler Reductton

; U vlrr.,lit siunhf.ion in order to reduce the execution

Til<, I I ~ ~ ~ I I I ~ I ~ ~ - I I ~ ~ ~ ~ . , : ~ ~ ~ I I ~ 1,echniques are widely used f i l w [ G ] . [E]. [9j 111 t h e x t,echniques by approximating 1 1 w ~1,nrIillallt Im l w o f the circuit, wit,ll a lower order 1 1 ~ , ~ 1 ~ 1 . t h Iwll;Lvioln of the circuit is obtained.

~ ~ ~ ~ ~ I I ~ ~ ~ I I I - I ~ I ~ ~ , ~ . I I ~ I I ~ uscs the coefficients (moments) ot t,h(, i,xpallsiou of thR system transfer function, H ( s ) ,

: t r ( ~ ~ n l d a rmillt i l l t h e complex s-plane. The Taylor

srlics cwpausioll o f

H(,,)

around so is given as:

~ ( s ) = ,,lii

+

(.\ - .so)rnl

-

( . g -

+ .

. .

(4)

Aftkr thri mulnmlts are generated. they are matched (,U a ratio of two polynomials [S], [X], [lo] or to a low- order set of poles and residues [g] by using Pad6 ap- proximatio~l.

The transfer function for a circuit which consists of lumped elements can be approximated as

i%(.s) = ( I

+

n A

+

u2A2

+

. . .

+

u"A")r ( 5 ) where

S = s o +L7

r = ( G + s , C ) - ' b A = - ( G +s,C)-'C

As a result the moments are found using m, = A"r. During the generation of moments the LU decompo- sition of the circuit matrix is calculated once for the

first moment. Ot,her moments are obtained using for- ward and backward substitutions.

transfer function is approximated with a rational func- After the moments are generated, the entries of the

In AWE, to approximate the behaviour of the cir- cuit, the Taylor expansion around so = 0 is evaluated. Since the information carried by the mnments is ac- curate at low frequenc.y region, the AWE t,echnique is efficient in extracting the low frequency poles of the circuit. To improve the accuracy of AWE a t rel- atively higher frequencies multi-point Pad6 approxi- mation techniques were proposed [G]: 191. [lo]. In the work by Chiprout et. al. [Y] conlplrx frequency hop- ping (CFH) technique is introduced. MAWE circuit simulation tool uses the method developed in t,he work of Celik et. al. [lo]. MAWE provides good results also in high frequency regions as it uses the shifted mo- ments, which provide the necessary information about the frequency range of interest. However, this ap- proach requires the solution of the circuit matrix at several frequency points.

EQULVALENT CIItCUIT E X T l l A C T l O N

t,ems of structures and circuits. we rnay solve the There are many similarities in analyzing t h e sys- acoustic field analysis problems using a circuit sim- ulator.

tial equation (1) into a first order differcntial cquation. The first step is to change thc scc.ond order differen-

We choose the _unknowns as follows: displacements ( Q ( t ) ) i branch currents

velocities

( Q ( t ) )

+

potential difference between the nodes. is constructed. The algorirhm is of order

N2.

and

By looking at the cnt,ries of the system the circuit the time consumptior, is very small compared 10 the simulation time.

The circuit is realized using currcut through the in- ductors (L) the displacement. and thc voltage differ- ence between the induct.or nodes as velocity. Currents

through the capacitors are time derivatives of veloc- ity. The other entries are simulated using current and voltag? controlled current sources (CCCS and VCCS). ditional equations are needed, but to couple the cur- A!l the circuit elements come in parallel so that no ad- rents one needs to add 0 volt voltage supplies (VS). In the circuit there are no negative valued capacitors and inductors, and no voltage source-inductor loops are introduced.

For each displacement-velocity pair, there are 6 rows in the circuit matrix, so its size is 6 times the original matrix sizes ( M , C and K ) , but it is sparser. If the sparsity of the original matrices are a then the circuit matrix sparsity will be about

6.

EXAMPLES

In this section we will give some examples to demonstrate the speed of the method compared to the FEM solvers. The simulations are performed cn a SPARCstation 20 with 5OMHz clock frequency by using FEM simulation program ANSYS (version 5 . 2 ) , electrical circuit simulation program HSPICE tool MAWE.

(h97.2), and in-house-developed circuit simulation

Esample I

First example is the harmonic analysis of a simple bar (Fig. 1). We have divided the bar into N ele- ments, apply the force at the mid-point and observe the displacement at l j l 0 0 of the length. The system bas N - 1 UY unknowns (displacement in Y direc- tion) and N

+

1 ROTZ unknowns (rotation around the z-axis), which gives a total of 2N unknowns. The summary of extracted circuit is given in Table I. In Table SI. times consumed (in seconds) during the har- monic analysis are shown. In AWE analysis 6th order Pad4 approximation is used, evaluation point is S = 0. As the number of unknowns increase, the ratio of the zxecution times gets larger. It can be seen in Fig. 2

Fig. 2 . Harmonic analysis results for Example I. T.kBLE I1

T I M I N G RESULTS FOR EXAMPLE 1

1

N

I

ANSYS

I

HSPICE

1

AWE

I

4.83 6.63 4918 13.80 34.78

that the harmonic response outputs are indistinguish- able.

Ezample I I

In this example, the transient behaviour o f the mid point of the same bar, when a step functiou with fi- nite rise time is applied to the same point(Fig. 1). is simulated. Approximation order is 6; and the matrix inversion is done only for S = 0. In Table III the time consumptions [in seconds) are shown. It can be seen in Fig. 3 that the asymptotic waveform evalu- ation technique gives very good results in transient analysis.

T.4BLE 111

T I M I N G RESClL'rS FOR E X ~ M P L E 11

N

I

ANSYS

I

HSPICE

I

AWE

1

76.55 18.25

CONCLUSION

In this paper, we propose a method to convert the finite element formulations of the structural problems

TABLE 1

C l R C L i T SUMMARY FOR E X A M P L E 1

FE Problem Electrical Circuit

Unknowns

I

matrix size

Ij

L

I

C

1

cCCS

I

vs

1

Circuit nodes

1

circuit matrix size 200

I

200

/I

200

1

200

I

1788

I

400

I

601 1201 800 800 ₈₀₀

1

₈₀₀

I

x

-

B 0 3 - P

ii

/

l

l J

into circuit, analysis problen~s so that they can be solved with a general purpose circuit simulator. program which takes the total mass and stiffness ma-

Equivalent circuit extraction is done by a computer

7138

I

1600

I

2401 4801

a better sparse matrix sparse matrix solver, the ex- pected speed-up rat,io is almost equal to the number of frequency points divided by the number of expan- sion points.

REFERENCES

[l] Gabriel Kron, 'LEqnivalent &cults of the elastic field,"

Journal of Applied Mcchanecs; YOI 11, pp. A149~4161,

(21 Janne V i i n i n e n , "Circuit theoretical approach to couple September 1944.

two-dimensional finite d e m e n t models with external circuit equatmns," I E E E T h n s o c f t o n s on Muqnetics, v01 32, no.

[3] Thomas E. McDermott,, r i n g Zhou and John Gllmare.

2 , pp. 400-410, March 1996.

ated from finite element solutions," I E E E Tmnsnctzans on

"Electromechanical system simulation with models gener-

[4] Jia Tzer Hsu and L W V u ~ Q ~ m c . '..4 rational formulatmrl

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Circuits and Sp,~temr- 1: F m d a m c t n l Theory and A p p k

[S] META-SOFTWARE: Campbell, CA, HSPICE Use7 ',S

cations. vol. 13, no Cl, pp. 721L732, Sept,ember 15%

Mnnuol, 1996.

[G] M. Sungur, A S . Eliin~l and 4 . Atalar, ".4n eficicnt, pro-

gram for analysis of interconnec.t circuits," Intt:motmnal

system of equations at each frequency point moment- evaluation for timing analysis." I E E E Rananctwns on

matching techniques can be used. The main reason Computer Aided D e s i g n of IntqrntPd Crrcuits and S y s t e m s ,

. .

behind the efficiency of this method is the decrease in vol. 9, pp. 352-.366, April 1990. the number of LU decompositions. Multi-point AWE

techuique requires one LU decomposition per expan- networks using complex frequency hopping(CFH):" Tmnsocttans on Computer Aided Ueszgn of Integrated I E E E Car-

[g] E. Chiprout and hl. S Kaklrla. "iinalysis Of interconnect

sion point. Generally, the number of expansion points and Systems. vol. 14, no. 2 , PP. 1 8 6 ~ 2 0 0 . Fcbrllary

is less than the number of frequency points to get a n

[lo] M.

cdik,

0 .

odi,

h l . ~ Tan altd . A Atalar, "Pole-

1995.

accurate solution. colnputatiorl i n microwave circuits using m u h point

Several examples have been studied using the prc- posed method and an accurate match with the finite

pade approximation," I E E E Ransactiorla O n czlcvits and

systems-l- Fundamental Theory and .Applzcotaons, VU1 42.

element method results has been obtained. Without pp. 6-13, 1995 a significaxt loss of accuracy, the simulation speed

is improved using AWE. With our in house devel- oped sparse matrix solver, which is not a particularly fast one. this potential improvement with respect to HSPICE program cannot be seen in the figures. With