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INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, VOL.
39,909-922 (1996)
NUMERICAL ALGORITHMS FOR SOLUTIONS OF
LARGE EIGENVALUE PROBLEMS IN
PIEZOELECTRIC RESONATORS
YOOK-KONG YONG'
AND YOUNG CHO'
Department of Civil and Environmental Engineering, Rutgers University, P.O. Box 909, Piscataway,
N J 08855-0909, U.S.A.
SUMMARY
Two algorithms for eigenvalue problems in piezoelectric finite element analyses are introduced. The first
algorithm involves the use of Lanczos method with a new matrix storage scheme, while the second algorithm
uses a Rayleigh quotient iteration scheme. In both solution methods, schemes are implemented to reduce
storage requirements and solution time. Both solution methods also seek to preserve the sparsity structure of
the stiffness matrix to realize major savings in memory.
In the Lanczos method with the new storage scheme, the bandwidth of the stiffness matrix is optimized by
mixing the electrical degree of freedom with the mechanical degrees of freedom. The unique structural
pattern of the consistent mass matrix is exploited to reduce storage requirements. These major reductions in
memory requirements for both the stiffness and mass matrices also provided large savings in computational
time. In the Rayleigh quotient iteration method, an algorithm for generating good initial eigenpairs is
employed to improve its overall convergence rate, and its convergence stability in the regions of closely
spaced eigenvalues and repeated eigenvalues. The initial eigenvectors are obtained by interpolation from
a coarse mesh. In order for this multi-mesh iterative method to be effective, an eigenvector of interest in the
fine mesh must resemble an eigenvector in the coarse mesh. Hence, the method is effective for finding the set
of eigenpairs in the low-frequency range, while the Lanczos method with a mixed electromechanical matrix
can be used for any frequency range. Results of example problems are presented to show the savings in
solution time and storage requirements of the proposed algorithms when compared with the existing
algorithms in the literature.
KEY WORDS
piezoelectric resonators; numerical algorithms;large scale eigenvdue problems
1. INTRODUCTION
T h e analysis and design of piezoelectric resonator devices present a challenging set of problems to
the engineer: (1) The geometry a n d boundary conditions of the devices are complex, (2) the
resonating element is a composite, consisting of metal electrodes on a piezoelectric plate or
substrate, a n d in thin film devices, the piezoelectric layer is grown on a dielectric layer, a n d (3) the
devices usually operate at higher modes of vibration. Hence, the analysis of these devices demands
good analytical models, and detailed finite element solution of such analytical models. This is
especially relevant when an accurate prediction of the unwanted modes is required.
* Associate professor
Graduate research assistant
CCC 0029-5981/96/060909-14
0 1996 by John Wiley & Sons, Ltd.
Received 3 November I994
Revised 18 April 1995
910
Y.-K. YONG AND Y.CHO
The numerical models would generate a set of coupled piezoelectric eigenvalue equations of the
following form:
M,U
Kuuu K,g+ = F
(1)
+
+
K , ~ P+ K+,+ = Q
(2)
where the matrices Mu,,
K,, K , and K,, are the mechanical mass, mechanical stiffness,piezoelectric coupling, and dielectric matrices, respectively. The vectors u, 4, F and Q are, respectively, the
mechanical displacement, electric potential, mechanical force and electric charge vectors. A static
condensation' of the potential 4 in equations (1) and (2) renders a single equation written as
M,U
+ K*u = F*
(3)
where
K * = K,, - KUgK,&,lKg,
(4)
F* = F - K , + K i i Q
(5)
and
K* is the condensed electromechanical stiffness matrix and F* is the corresponding electromechanical forcing function. The components of the electric potential vectors are recovered
from equation (2) by
4 = K,-,'(Q - K&
(6)
A generalized eigenvalue problem can be obtained from equation (3) by assuming harmonic
solutions, and setting F* equal to zero to yield
(K* - W ~ M , , , ) =
U0
(7)
where o is the natural frequency.
Equation (7) is the piezoelectric eigenvalue problem based on Allik and Hughes' finite element
formulation, and has been widely used in most piezoelectric eigenvalue problems for over two
decades. It is found in some recent
The matrix K*, while still symmetric, does not retain
the sparsity structure of the original mechanical stiffness matrix K,,, and is now very densly
populated. Hence, direct solution of equation (7) requires extensive storage requirements and
consequently long solution time. In an effort to keep the sparsity structure of the stiffness matrix,
, ~ separates the
a perturbation method was proposed by Yong and Zhang' and Boucher et ~ l . which
mechanical eigenvalue problem from the piezoelectric eigenvalue problem. The piezoelectric effect
was then introduced into the mechanical eigenpairs as perturbation parameters. The accuracy of
the perturbation solution is predicated on the assumption that the perturbation parameters were at
least an order of magnitude smaller than the mechanical stiffness. Hence, the method does not
provide accurate eigenvalues for resonator materials such as lithium niobate, zinc oxide, and
aluminum nitride which have stronger piezoelectric coupling coefficients than, say, quartz. Guo et
ul.' proposed a scheme in which the piezoelectric eigenvalue matrix equations (1) and (2) are
solved directly using the Lanczos eigensolver. Their scheme retained the sparsity structures of the
finite element matrices, and provided savings in both memory and solution time when compared
with the direct solution of equation (7). However, it yields a very large half bandwidth in the
columns corresponding to the electrical degrees of freedom because of the large differences in
the degree of freedom numbers between the electrical degrees of freedom and mechanical degrees
of freedom in a node. For finite element matrices with large half bandwidths, such as those in
three-dimensional cubic problems, and circular plate problems, an iterative scheme which does
not require complete factorization and hence stores only the non-zero terms would provide
more savings in memory; and, for very large-scale problems, in computational time too.
PIEZOELECTRIC RESONATORS
91 1
2. ALGORITHMS FOR THE SOLUTION OF THE PIEZOELECTRIC
EIGENVALUE PROBLEM
In this paper, two sets of algorithms are presented to solve the piezoelectric eigenvalue problem
without static condensation of the electric potential. In the first set, the scheme proposed in
Reference 7 using a Lanczos eigensolver is enhanced by a new storage scheme for the stiffness and
consistent mass matrix. The stiffness matrix storage scheme uses an equation numbering system
that mixes the electrical and mechanical degrees of freedom in the assembled global matrix. The
equation number corresponding to the electrical degree of the freedom for any node is sequential
to equation numbers of the mechanical degrees of freedom for the same node. The storage scheme
for the consistent mass matrix follows that of Reference 9. This set of algorithms is denoted
subsequently as the Lanczos method with a mixed electromechanical matrix (LMEM). In the
second set of algorithms, an iterative scheme denoted as the multi-mesh Rayleigh quotient
iteration (MRQI) is presented. The scheme does not require complete factorization and exploits
the sparse matrix structure itself. Convergence of results is significantly improved by using initial
eigenpairs from a coarse mesh. Results from example problems using a three-dimensional brick
element are presented.
2.1. Lanczos method with a mixed electromechanical matrix ( L M E M )for the piezoelectric eigenvalue problem
The piezoelectric eigenvalue matrix equations are obtained from equations (1) and (2) by
setting F and Q to zero, and assuming harmonic solution, which yields the matrix equation in the
form:
where
If the left-hand-side electromechanical matrix is stored in a skyline scheme as proposed in
Reference 7, the piezoelectric coupling K,, would have a very large half bandwidth equal to
(NMDOF-MDOF-I),where NMDOF is the total number of mechanical degrees of freedom in
the finite element model, and MDOF is the mechanical degrees of freedom per node. Depending
on the size of the finite element model, this half bandwidth will always be orders of magnitude
larger than the mean half bandwidth of the mechanical stiffness matrix K,,. Since the Lanczos
eigensolver requires factorization of the electromechanical matrix, all the zero values between the
non-zero values of K,, and K,, have to be stored. The half bandwidth of K,, can be reduced to
the same order of magnitude as the mean half bandwidth of K,, by rearranging the equation
numbers for the electrical degrees of freedom: For each node, the electrical degree of freedom
number is made sequential to the mechanical degrees of freedom number for the node, so that
K,,,K,, and K,, in equation (8) are all mixed together consisting of one large electromechanical
stiffness matrix with a mean half bandwidth of the same order of magnitude as the mean half
bandwidth of K,,. The consequence of this mixing of electrical and mechanical degrees of freedom
in equation (8) is great savings in memory and computational time when compared with the
previous scheme of Reference 7. The rearrangement results in zero diagonal values on the
right-hand mass matrix of equation (8) which correspond with the electrical degrees of freedom.
912
Y.-K. YONG AND Y.CHO
These zero 'mass' terms imply infinite eigenvalues for the electrical modes which are theorectically
correct since the piezoelectric eigenvalue equations assume infinite velocity for the electrical
waves. The eigenmodes which are of interest in the acoustic resonator correspond only to the
mechanical degrees of freedom, hence these infinite eigenvalues are neglected, and do not present
a problem in the algorithm. Pivotal problems resulting from the zero diagonal mass values can be
handled with a simple modification to the Cholesky's factorization of the mass matrix.
We use a piezoelectric eigenvalueequation with a mixed electromechanicalstiffnessmatrix [K]
instead of equation (8):
CK1 { X I = n c m { 4
(10)
is the vector of mechanical displacements and electric potential. The electric potential degree
of freedom is numbered sequential to the mechanical displacement degrees of freedom for a
node. For an eight-node bilinear brick element with three mechanical and one electrical degrees
of freedom per node, the element stiffness and mass matrices are formed by the following
equations:
K" =
sv
Me =
PI' CDl CBI dV
(12)
p[N]'[N]dV
J-V
where
0
0
CBI
N2.X2
0
=
N2.X3
0
0
0
N1
0
0
0
N1
0
0
O N 2
0 0 0
N I O O
9x32
0
Nz
0
0
0
0 0
N 2 O
...
(15)
The terms Ni, [C], [el and [ E ] are the shape functions, mechanical stiffness constants, piezoelectric constants, and dielectric permittivities, respectively. Since the [ N ] matrix contains zero
values in every fourth column corresponding to the electrical degree of freedom, the global mass
matrix [MIalso contains zero column values corresponding to every electrical degree of freedom.
913
PIEZOELECTRIC RESONATORS
Hence, [MI could be factorized using Cholesky's factorization without pivotal problems by
skipping the zero columns associated with the electrical degrees of freedom. An LDLT
factorization is performed on the mixed electromechanicalstiffness matrix [K].The skyline of the
matrices remain the same after factorization, and could be written in the form:
1
CLI CDl WIT{ X I
3,
=
c641 C641T{4.
(17)
Equation (17) could be further transformed into a standard eigenvalue matrix equation:
1
-
1
v = [A] v
(18)
where
v = [ 9 I T {x}
and
[A]
= [B]TIL]-TID]-l[L]-l
[64]
The application of the Lanczos algorithm' transforms equation (18) to the form
where V = [XI V * , [ T I is a tridiagonal matrix, and [XI is the Lanczos matrix.
In general, the zero values beneath the skyline of both the stiffness and mass matrices must be
stored since the factorization of these matrices would replace most of these zero values with
non-zero values. The consistent mass matrix however has a special sparse matrix form with
. ~ us investigate the structure of the consistent mass
a fill-in pattern which is known u p r i ~ r iLet
matrix of an eight-node brick element by reviewing its definition. The structure of the N matrix in
equation (15) causes the element mass matrix in equation (13) to be composed of subdiagonal
matrices in the form of:
M', = 0, if Ii - j l # lm
(22)
MFi=O, i f i = 4 p , p = 1 , 2 , . . . , 8
(23)
where M ; and M i are elements of Me, 1 is the difference in the node numbers associated with row
i and columnj, respectively, and m is equal to the number of degrees of freedom per node which is
four in this case. The structure of the mass matrix for an eight-node brick element as given by
equations (22) and (23) is shown in Figure 1. The assembled global mass matrix has the same
structure as the element mass matrix. Hence, equations (22) and (23) also apply to the global mass
matrix.
Now, we investigate the fill-in pattern in the Cholesky factorization procedure:
~
1
=Mi','
1
(
u..
lI =
M.. -
(24)
i- 1
1
uii)'I2
i = 2,3,. . . ,n, i
+ 4p,
p = 1,2,. . .
(25)
k=I,k#4p
i- 1
k=l.k#4p
where uij are the elements of the factorized upper triangular matrix U,and n is the order of the
matrix.
914
Y.-K. YONG AND Y.CHO
*
SYMMETRIC
I
Non-zero terms
000
00
0
0Zero terms
32x32
Figure 1. Structure of the mass matrix for an eight-node brick element
Examining equations (22) and (26), the matrix U is found to be a sparse matrix and have the
same structure as M ,namely,
uij=O, ifli-jl#lm
forjai
uii=O, i f i = 4 p , p = l , 2 , . . .
(27)
(28)
From equations (27) and (28), we observe that the fill-in's would not happen at the locations with
li - j I # lm and every fourth column and row which contain zero values can be discarded without
affecting other values. Hence, we could store, inside the envelope, only the terms with I i - j I = lm
for the mass matrix. With this storage scheme, the percentage savings of memory over the skyline
storage is about ((m- l)/m) x 100per cent when the problem size is large. The savings in memory,
for example, is about 75 per cent for a problem which has 4 degrees of freedom per node. There is
also a decrease in computational time due to a smaller number of floating point operations.
2.2. Multi-mesh, Rayleigh Quotient Iteration Method (MRQI) for the piezoelectric eigenvalue
problem
A multi-mesh, Rayleigh Quotient Iterative Method (MRQI) is proposed to gain further savings
in memory and computational time for large-scale piezoelectric eigenvalue problems. This
method is advantageous if only a small set of eigenpairs is desired, and the eigenvectors could be
represented on a coarser mesh of the final finite element mesh. Hence, the method is useful for the
extreme eigenpairs in the low-frequency range.
Rayleigh quotient iteration. The Rayleigh quotient iteration" is a revision of the symmetric
QR algorithm. It is useful when a few eigenvalues and eigenvectors are desired. The Rayleigh
quotient is
1 = r(u) =
u*K*u
u Mu
915
PIEZOELECTRIC RESONATORS
2 x 2 x 2 Coarsest Mesh
4 x 4 x 4 Coarser Mesh
8 x 8 x 8 Original Mesh
Figure 2. Hierarchy of the cubic mesh applied in the iterative solution
which minimizes (1 (K* - 1M)u 11. The scalar r(u) is called the Rayleigh quotient of u. If u is an
approximate eigenvector, then r(u) is a reasonable choice for the corresponding eigenvalue. On
the other hand, if 1 is an approximate eigenvalue, then the inverse iteration theory gives the idea
that the solution to (K* - 1M)u = b will also be a good approximate eigenvector. This idea
constitutes the basic algorithm of the Rayleigh quotient iteration. By starting with an arbitrary
non-zero vector uo, a desired accuracy level of eigenpair solutions are obtained after a moderate
number of iterations. However, the number of iterations required for convergence depends on the
size of the problem, which results in excessively long solution time for large size problems. We
found that by starting with an initial vector which resembles the final eigenvector, rather than an
arbitrary vector uo, the required number of iterations for convergence is significantly reduced.
The initial eigenpairs could be obtained from a coarse mesh of the problem, and the eigenvectors
could then be represented in the final mesh by interpolation. The coarse mesh problem could be
solved cheaply by the Lanczos method proposed in the previous section, and would not involve
much memory since the memory allocated for the final mesh could be utilized temporarily.
Figure 2 shows, for example, the use of a 2 x 2 x 2 cube mesh as a coarsest mesh and 4 x 4 x 4 cube
8
mesh problem. One could also use
mesh as a coarser mesh for the original 8 x 8 ~ cube
a 2 x 2 x 2 mesh as a coarser mesh for 8 x 8 x 8 mesh problem by interpolating twice. However,
this will result in poor initial conditions leading to more iterations and longer solution time.
Table I shows the algorithm for solving a piezoelectric eigenvalue problem using the Rayleigh
quotient iteration. In this study, the convergence criterion is set as
where 11 11 indicates the Euclidean norm of a vector and e4 is the convergence tolerance for the
required eigensolutions.
Preconditioned conjugate gradient iteration. In Table I, which shows the Rayleigh quotient
iteration algorithm, the step involving the solution of the linear matrix equation Bzk+ = Muk
provides for the next iterate Z k + 1. The matrix B is usually not positive definite. Hence, a conjugate
gradient iterative methodlo for obtaining zk+ will not normally perform well. However,
and quite fortunately, the Rayleigh quotient iteration does not need accurate solutions of z k +
to yield satisfactory convergence patterns in the final eigenpairs, and a conjugate gradient
iterative method could yield reasonable estimates of z k + even when B is not positive definite.
A preconditioned conjugate gradient iterationlo is adapted for the solution of this linear
equation.
916
Y.-K. Y O N G A N D Y. CHO
Table I. Algorithm for rayleigh quotient iteration
Initial vector uo is obtained from the eigenvectors of a coarse mesh
For k = 0,1, . . . ,n, repeat
Calculate #& = r(uk)
exit the loop
The convergence speed of the conjugate gradient iteration method can be accelerated by
applying an appropriate preconditioning matrix. Theoretically the ideal preconditioning matrix
is a completely factorized matrix of B itself. A complete factorization is however not the right
choice because it defeats the primary purpose of this algorithm, namely, to use minimal amounts
in the
of memory storage. We note that the contribution of the piezoelectric term K,,&$K&
matrix B is relatively small, and is not formed explicitly. Hence, one choice for the preconditioning matrix is the incompletely factorized matrix (K,,,,- l k M ) where the piezoelectric term is
neglected. The LDLT decomposition is used in the incomplete factorization. The convergence
criterion for the solution of this linear equation is set as
where is the convergence tolerance. Table I1 shows the algorithm for the preconditioned
conjugate gradient iteration.
3. NUMERICAL EXAMPLES AND RESULTS
Numerical examples and their results are presented in this section to demonstrate the efficiency of
the proposed algorithms. A cubic solid and a prismatic bar are used to show the performances of
each algorithm. The two models are discretized with different fineness of meshes to generate
different problem sizes. The results of the proposed LMEM and MRQI schemes are compared
with (a) the Lanczos method with a condensed electromechanical matrix (LCEM) from
Reference 1 and equation (7), and (b) the Lanczos method with a separated electromechanical
matrix (LSEM)from Reference 7 and equation (8). All the schemes except the MRQI scheme have
an efficient memory storage method described in equations (22) and (23) for the consistent mass
matrix. The stiffness matrix in the LCEM scheme is a dense matrix, and hence is stored in a full
matrix. The stiffnessmatrix in the LSEM and LMEM schemes is stored in a skyline format. In the
MRQI scheme, only the non-zero terms of both the mass and stiffness matrices are stored. The
examples use a three-dimensional, eight-node bilinear element with three mechanical, and one
electrical degrees of freedom per node. The material type of the examples is PZT4 and the
917
PIEZOELECTRIC RESONATORS
Table 11. Algorithm for preconditioned conjugate gradient iteration
Solve for Z k + in Bzk+ = Muk where B = (K,
+ K.,K;tK&
- L,M)
Set initial vector zP+ = 0
Set ro = Muk
For j = 1,. . .,n,repeat
Solve P y j - = rj- for y j - where P: preconditioner
Calculate flj = yJ-lrj-l/yJ-2rj-2 If j = 1, fll = 0
Calculate Pi = y j - + PjPj- If j = 1, P 1 = ro
Compute aj = y:- Irj- ,/[Pj'BPj]
Obtain updated vector z:+~ = z { i t + a,Pj
Compute rj = rj- - ajBP,
Check for convergence ;
coefficients of the material are obtained from Reference 6. The convergence criteria for the MRQI
scheme are set as lo-' for &+ in the Rayleigh quotient iteration (Table I), and lo-' for el in the
preconditioning conjugate gradient iteration (Table 11). The relatively large parameter of 10- for
is justified by the fact that the marginal return for setting smaller parameters is very low,
namely, the Rayleigh quotient iteration needs only a reasonable estimate of Z k + i for the good
convergence rates, and the convergence rate of the preconditioned conjugate gradient iteration is
decelerated near the exact solution.
'
3. I . Cubic solid example
In this example, the cubic solid model is discretized using meshes of increasing fineness starting
from 3 x 3 x 3 to 16 x 16 x 16 elements. Figure 3 shows the comparison of storage requirements
for the four schemes versus the number of equations (number of degrees of freedom). Some
savings are gained from the LSEM scheme which uses a skyline storage of the separated
electromechanical matrix when compared with the LCEM scheme which requires full matrix
storage. The LMEM scheme which uses a skyline storage of the mixed electromechanical matrix
(equation (10)) shows big savings in the storage requirements when compared with the two
previous schemes. Since the MRQI scheme needs to store only the non-zero values of the stiffness
and mass matrices, it has the smallest storage requirements. For the 16 x 16 x 16 element mesh
problem with approximately 20 OOO degrees of freedom, the LSEM scheme uses about 43 per cent
of the storage requirement of the LCEM scheme; while the LMEM and MRQI schemes use only
15 and 1.8 per cent, respectively.
Figure 4 shows the solution-time comparisons of the four schemes when two eigenpairs are
calculated. The solution time for the MRQI scheme includes time for generating initial eigenpairs
from a coarse mesh. For the MRQI scheme, the 6 x 6 x 6 mesh problem used initial eigenpairs
obtained by the Lanczos method from a 3 x 3 x 3 coarse mesh; and the 16 x 16 x 16 mesh problem
used eigenpairs obtained by the MRQI scheme from a 8 x 8 x 8 coarse mesh, which in turn used
eigenpairs solved by the Lanczos method from a 4 x 4 x 4 mesh problem. The graph indicates that
the LMEM scheme shows the best performance for problem sizes smaller than the 10 x 10 x 10
918
Y.-K. YONG AND Y. CHO
3
-
,
,k lo3 ; --)5
Y
u)
c
C
:
:
lo-’
LCEM scheme [I]
--A--LSEMschemeV]
MRQl scheme (proposed)
LMEM scheme (PrOpOSed)
I
I
1o3
104
1
No. of Equations
Storage requirement vs. number of equations in the cube example
Figure 3. Storage requirements vs. number of equations in the cube example
Solution time vs. number of equations in the cube example (2 eigensolutions)
Figure 4. Solution time vs. number of equations in the cube example (2 eigensolutions)
919
PIEZOELECTRIC RESONATORS
Table 111. Total number of iterations required for
2 eigenpairs
Type of iteration
Rayleigh quot.
Problem size
iter.
4X4X4
6 x 6 ~ 6
8X8X8
10 x 10 x 10
12 x 12 x 12
1 6 x 1 6 x 16
6
5
4
4
4
4
Prec. conj.
grad. iter.
525
135
136
178
218
316
element mesh (5323 degrees of freedom), while the MRQI scheme is clearly superior for larger
problems. The reason for the poor performance of the MRQI scheme in small size problems is
that the small problem size forces the scheme to use meshes which are too coarse for good initial
eigenpairs, which in turn leads to large increases in the number of iterations required for
convergence of the final eigenpairs. Overall, the LCEM scheme is the worst performer. For
example, in the 12 x 12 x 12 element mesh problem (8787 degrees of freedom), the solution times
for the MRQI, LMEM, and LSEM schemes are respectively, about 0.6,1, and 6 per cent that of
the LCEM scheme.
Table 111 shows the total number of iterations required in the MRQI scheme using the given
convergence criteria. The table shows that the total number of iterations in the Rayleigh quotient
iteration is unchanged by the problem size (except the first two small size problems). Other
iteration schemes, such as, the Gauss-Seidel method, and the conjugate gradient iteration, require
more iteration steps for larger problem size. This is because the finer mesh problem uses initial
conditions which are more accurate, i.e. for example, the 16 x 16 x 16 element mesh problem uses
initial conditions from a 8 x 8 x 8 element mesh, while the 8 x 8 x 8 element mesh problem uses
initial conditions from a 4 x 4 x 4 element mesh. The preconditioned conjugate gradient iteration,
however, shows a consistent increase in number of iterations with the problem size, with the
exception of the 4 x 4 x 4 element mesh problem. The poor showing in the 4 x 4 x 4 mesh problem
is due to poor initial conditions from the 2 x 2 x 2 element coarse mesh.
3.2. Prismatic bar example
A prismatic bar example is also conducted to compare the storage requirements and solution
times for the four schemes. The following meshes of cubic shaped elements are used in Figures 5
and 6: 2 x 2 x 8,4 x 4 x 16,6 x 6 x 24,8 x 8 x 32, and 10 x 10 x 40 element meshes. Figure 5 shows
the storage requirements versus the number of equations (degrees of freedom) for the four
schemes. The performances in terms of memory requirements of the schemes are similar to those
discussed in the previous cubic solid example. The MRQI scheme uses the least amount of
memory. For the 10 x 1 0 x 4 0 element mesh, the MRQI, LMEM and LSEM schemes use,
respectively, 2, 6, and 24 per cent of the memory required for the LCEM scheme.
Figure 6 shows the solution times of the four schemes versus the number of equations when two
eigenpairs are calculated. We observe that the LMEM and LSEM schemes exhibit an improved
performance over the cubic solid problem because the mesh for the prismatic bar yields a smaller
920
Y.-K. YONG AND Y.CHO
/
LCEM scheme [I1
LSEMscheme
MRQl scheme (proposed)
LMEM scheme (proposed)
lo-'1
I
1o3
I
I
10'
No. of Equation
Storage requirements vs. number of equations in the bar example
Figure 5. Storage requirements vs. number of equations in the bar example
5
:
8
v)
Y
@105
E
--+-LCEM scheme [I]
- -A-- LSEM scheme [7]
MRQl scheme (proposed)
LMEM scheme (proposed)
F
-
.-s
:lo4
v)
lo3
lo2
10' 7
loo I
I
1o3
I
lo4
1
No. of Equations
Solution time vs. number of equations in the bar example (2 eigensolutions)
Figure 6. Solution time vs. number of equations in the bar example (2 eigensolutions)
PIEZOELECTRIC RESONATORS
921
half bandwidth in the stiffness matrix, which leads to smaller storage requirements and faster
solution time for these two schemes. The figure shows that the LMEM scheme is the best
performer for this prismatic bar example. For the 8 x 8 x 32 element mesh, the LMEM, MRQI,
and LSEM scheme require approximately only 0-6,2,and 5 per cent of the solution time needed
for the LCEM scheme. For practical, very large-scale finite element models, we expect that the
MRQI scheme would prove to be quite competitive with the LMEM scheme. The LMEM
scheme is expected to be best performer for mesh problems with narrow half bandwidths. Overall,
the LCEM scheme is the worst performer.
4. CONCLUSION
Two sets of algorithms for solving piezoelectric eigenvalue problems were proposed. The
performances of the Lanczos method with a Mixed Electromechanical Matrix (LMEM) and the
Multi-mesh, Rayleigh Quotient Iteration Scheme (MRQI) were compared with the conventional
Lanczos method using a condensed electromechanical matrix’ (LCEM), and the Lanczos method
using a separated electromechanical matrix’ (LSEM).
The proposed MRQI scheme shows the best performance in terms of storage requirements. In
the examples conducted, significant savings in storage requirements were obtained with this
method in large-size problems. In terms of solution times, the MRQI scheme is advantageous for
problems with a stiffness matrix of large half bandwidth, such as that in the cubic solid example.
The MRQI scheme, however, has a restriction that the desired eigenmode must be reasonably
well-represented on a coarser mesh. Hence, the scheme is only well-suited for finding eigenpairs in
the low-frequency spectrum of very large-scale problems.
The LMEM scheme, while requiring more memory than the MRQI scheme, is much less
restrictive, and could be used for finding eigenpairs in any bandwidth of frequencies. The LMEM
scheme shows big savings in solution times over the existing LCEM and LSEM schemes,
especially for problems with a stiffness matrix possessing a compact half bandwidth, such as that
in the prismatic bar example.
The LCEM scheme shows the worst performance. The LSEM scheme in which the skyline
storage method is maintained through the Lanczos algorithm shows an improvement in performance over the LCEM scheme. Overall, these two methods are not competitive with the
proposed LMEM scheme, or the MRQI scheme.
ACKNOWLEDGEMENT
Support by the F A A Center for Computational Modeling of Aircraft Structures (CMAS) at
Rutgers, The State University of New Jersey, is gratefully acknowledged.
REFERENCES
1. H. Allik and T. J. R. Hughes, ‘Finite element method for piezoelectric vibrations’, Int. j . numer methods eng., 2,
151-157 (1970).
2. H. S. Tzou and C. I. Tseng, ‘Distributedmodel identification and vibration control of continua: piezoelectric finite
element formulation and analysis’,J . Dyn. Systems, Measurement, Control, Tran. ASME, 113, 500-505 (1991).
3. J. A. Hossack and G. Hayward, ‘Finiteelement analysis of 1-3 composite transducers’, IEEE Trans. Ultrason.
Ferroelec. Freq. Controls, 38, 618-629 (1991).
4. S. K. Ha, C. Keilers and F.-K. Chang, ‘Finite element analysis of composite structures containing distributed
piezoceramic sensors and actuators’,A I A A J., 30, 772-780 (1992).
5. Y.-K. Yong and Z. Zhang, ‘A perturbation method for finite element modeling of piezoelectricvibrations in quartz
plate resonators’,IEEE Trans. Ultrason. Ferroelec. Freq. Control, 40, 551-562 (1993).
922
Y.-K. YONG AND Y. CHO
6. D. Boucher, M. Lagier and C. Maerfeld, ‘Computation of the vibrational modes for piezoelectric array transducers
using a mixed finite element-Perturbation method’, IEEE Trans. Sonics Ultrason., su-28, 318-330 (1981).
7. N. Guo, P. Cawley and D. Hitchings, ‘The finite element analysis of the vibration characteristics of piezoelectric discs’,
J . Sound Vibration, 159, 115-138 (1992).
8. T. J. R. Hughes, The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, Prentice-Hall,
Englewood Cliffs, N.J., 1987.
9. Y.-K.
Yong and Z. Zhang, ‘Numerical algorithms and results for sc-cut quartz plates vibrating at the third harmonic
overtone of thickness shear’, IEEE Trans. Ultrason. Ferroelec. Freq. Control, 41, 6855693 (1994).
10. G. H. Golub and C. F. Van Loan, Matrix Computations, 2nd edn., John Hopkins Univ. Press, Baltimore, MD,
1989.
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