Performance of iterative solvers for acoustic problems. Part II.

Acceleration by ILU-type preconditioner

Stefan Schneider*, Steffen Marburg

Institut fur Festkorpermechanik, Technische Universitat, 01062 Dresden, Germany

Received 13 February 2002; revised 5 July 2002; accepted 14 July 2002

Abstract

We are presenting an incomplete factorization preconditioner for the boundary integral formulation of the time harmonic Helmholtz

equation in 3D. Out of the matrix of the near field interactions of a Fast Boundary Element Method, we are constructing the preconditioner

using an incomplete LU-factorization provided in the SPARSKIT V. 2 package by Saad. This preconditioner is tested and discussed for

Restarted GMRes and for CGNR. Application showed little influence on internal problems. Therefore, the paper is dedicated to external

problems only. These problems solved by the hypersingular formulation of Burton and Miller is strongly affected by this preconditioner.

Especially for non-smooth surfaces, the preconditioner reduces the number of iterations needed by GMRes or CGNR significantly. Examples

that are investigated are a cat’s eye, a tire noise problem and a piston compressor.

q 2003 Elsevier Ltd. All rights reserved.

Keywords: Iterative solvers; ILU-decomposition; Regular grid method

1. Introduction

In the second part of this article, we discuss possibilities

of accelerating the iterative solvers presented in the first

part. Namely, we will use the GMRes [1–4] and CGNR

[2–4] as they turned out as the most efficient and robust

iterative solvers in the first part of this paper. Therein, for

problems of moderate size (saying fewer then five thousand

unknowns) the full version of GMRes was always

applicable. In some cases, CGNR proved to converge

more efficiently. The Restarted Bi-Conjugate Gradient

Stabilized algorithm and the Transposed Free Quasi

Minimal Residual method did not supply satisfying

convergence. This became particularly evidence for exter-

nal problems.

In the following, we consider only problems where we

apply so-called Fast Boundary Element Methods. Namely,

the Regular Grid Method (RGM) [5] and the Multilevel Fast

Multiple Method [6–11] are used to overcome the OðN2Þ

memory requirement of the standard BEM. These methods

directly approximate the matrix–vector product Au by

Au ¼ ðI 2 Anear 2 AfarÞu ¼ ðI 2 AnearÞu þ vfarðuÞ ð1Þ

with a sparse matrix Anear: This matrix is calculated directly

and stored using standard sparse matrix techniques. The

vector vfar is evaluated directly without filling the dense

matrix Afar: It is approximated either by utilizing the Fast

Fourier Transformation or a Multipole Expansion.

It was already shown in the first part of this article that all

iterative solvers investigated there work very efficiently

with interior problems. This is caused by the fact that the

single and double layer operators are compact ones what is

well suited for iterative solvers like GMRes and CGNR (see

Refs. [12,13]). Moreover, the preconditioner that is

discussed in this paper was also tested for the three interior

problems of Part I. This clarified that they hardly give any

remarkable acceleration of solution.

Herein, we consider exterior problems only. To suppress

the spurious frequencies, we are using the Burton/Miller

approach [14] with a coupling parameter a ¼ i=k: The

hypersingular operator appearing in this approach is not any

more compact. This is one of the reasons why iterative

solvers do not converge well or even fail to converge at all.

This paper is organized as follows. In Section 2, we

define the problem we are interested to solve. Section 3

deals with the construction of a suitable preconditioner. This

preconditioner is then applied to three numerical examples.

0955-7997/03/$ - see front matter q 2003 Elsevier Ltd. All rights reserved.

doi:10.1016/S0955-7997(03)00016-X

Engineering Analysis with Boundary Elements 27 (2003) 751–757

www.elsevier.com/locate/enganabound

* Corresponding author.

2. Problem

The main goal of our work is to use finite element surface

meshes of a structure and the nodal results of an FEM-

simulation directly for the calculation of radiated or

scattered sound field of such an object using the Boundary

Element Method in the frequency domain. Hence, we like to

solve the boundary integral representation of the Helmholtz

equation

pðyÞ þðG

kðx; yÞpðxÞdG ¼ f ðyÞ ð2Þ

or, equivalently

ðI2AÞp ¼ f ð3Þ

with a given right hand side f representing surface loads. To

suppress the spurious frequencies, we are using the

Burton/Miller approach. The appearing hypersingular

operator is a non-compact operator. Thus, the eigenvalues

of A are not clustered. This results in the fact that the

eigenvalues of the matrix representing the discretization of

the integral operator will not lie in a small circle around a

center l [15]. The eigenvalue distribution of a matrix can be

used as convergence indicators for iterative solvers [16,17]

like GMRes and CGNR.

One drawback of the application of the Fast Boundary

Element Methods is that the approximation of Au seems to

modify eigenvalues which influence the convergence rate of

the iterative solver. Even if we have only a small error in the

approximated solution in comparison with the standard

BEM, the number of iterations needed by the iterative solver

differs significantly.

Further, we recall the result of Part I of this work. There

we stated that the iterative solvers are sensitive to non-

smooth surface. But when using finite element surface

meshes, we have lots of edges and vertices on the surface.

What may cause slow convergence of iterative solvers.

The aim of this paper is to find a way to overcome these

problems arising with the Burton/Miller approach, the

application of the Fast Boundary Element Methods, and the

fact that the surface G is not smooth. It is the idea to utilize a

suitable preconditioner.

3. Construction of the preconditioner

In the following, we want to construct a preconditioner

for the (non-symmetric) dense linear system

ðI 2 AÞp ¼ b ð4Þ

which arise from the discretization of a hypersingular

integral operator. Thus we will solve the preconditioned

system

ðI 2 AÞM21y ¼ b ð5Þ

and

p ¼ M21y ð6Þ

To make GMRes and CGNR perform more efficiently, we

aim on the matrix ðI 2 AÞM21 to have clustered eigen-

values. Or with other words, we want the operator ðI2

AÞM21 to be compact. Further, a suitable preconditioner M

should possess the following properties:

† only the entries of Anear are needed for its construction

† sparse representation of M21 (OðNÞ entries)

† ðI 2 AÞM21 has clustered eigenvalues.

The first two restrictions follow directly from the fact that

Fast Boundary Element Methods are applied.

There exist various ways of preconditioning for dense

linear systems arising from the Boundary Element Method.

We mention only Sparse Approximate Inverse, Diagonal

Block Approximate Inverse and Operator Splitting Pre-

conditioner (OSP). For a more detailed description we

referred to Ref. [15,18]. The ideal method matching our

requirements among them is the OSP. Its construction is

based on the fact that the product of a compact operator with

a bounded operator is compact. Thus the operator

ðI2AÞM21 ¼ ðI2Anear 2AfarÞM21 ð7Þ

is compact (plus an identity) if we chose

M ¼ I2Anear ð8Þ

as Afar is already compact because it is an integral operator

with an continuous kernel.

The choice of the preconditioner satisfies all the criteria

of suitable precondition for our problems if we use a sparse

representation of the inverse of M: For this task, some kind

of incomplete LU-decomposition seems to be well suited. In

detail, we apply the routines provided in the SPARSKIT V.

2 PACKAGE by Saad, namely, we are using the ilutðt; pÞ

routines [19]. With the parameter t; the minimal relative

size of an entry in the factors L and U is prescribed. The

second parameter p controls the maximum number of fill-in

elements per row. Hence, we can prescribe the memory

requirement with the parameter p whereas the threshold

parameter t can influence the time for the calculation of the

decomposition and its application in the iterative solution.

We like to emphasize at this point that the time for

evaluating a matrix–vector product by the Fast Boundary

Element Methods is dominated by calculating vfarðuÞ: The

time for the evaluation of Anearu and for solving LUu ¼ z is

negligible since these are sparse matrix operations.

In what follows, we only investigate the influence of the

parameters p to the efficiency of the preconditioner. For the

threshold parameter t; we use 1025 throughout this paper.

Hence, the number of entries in the factors L and U is only

prescribed by the parameter p: There are some papers on the

influence of ordering of the unknowns to the efficiency of

ILU-type preconditioners [20,21]. We have found that for

S. Schneider, S. Marburg / Engineering Analysis with Boundary Elements 27 (2003) 751–757752

our problems (even for larger values for t), an ordering with

the Reverse Cuthill–McKee algorithm leads to a negligible

improvement only and does not pay off the extra costs.

4. Numerical examples

We apply the presented preconditioner to both iterative

solvers, GMRes and CGNR. Hence, we will solve

ðI 2 AÞM21p ¼ b ð9Þ

with GMRes. CGNR requires hermitian preconditioning for

convergence. Therefore, the hermitian system

M2HðI 2 AÞHðI 2 AÞM21p ¼ M2HðI 2 AHÞb ð10Þ

is solved. We define the residual at the n-th iteration step as

kðI 2 AÞM21pn 2 bkkbk

¼ 1n ð11Þ

The iteration process is terminated if 1n , 1026 is satisfied.

Note that CGNR needs two matrix–vector products per

iteration while GMRes requires only one.

While the first example considers a cat’s eye with smooth

surfaces and few edges or vertices, the remaining two

numerical examples investigate if the presented precondi-

tioner will also work for more practical problems.

4.1. Scattering by cat’s eye

As the first example, we investigate the cat’s eye from

the first part. Three boundary element models with 7680,

10,830, and 18,750 linear discontinuous elements are used.

These numbers correspond to systems of equations with

30,720, 43,320, and 75,000 unknowns. Additionally,

the finest mesh of Part I with 108,000 constant elements is

briefly reconsidered.

We investigate the behavior of CGNR and GMRes.

Throughout these three models, the CGNR solver performs

best. GMRes does only converge within an acceptable

number of iteration when only very few restarts occur.

Otherwise, the solver virtually tends to stagnate. Such a

behavior was already reported by Ref. [22].

Also, we have applied the strategies mentioned in Refs.

[17,22,23]. They consist in keeping the new search

directions orthogonal to a subspace of the previously

calculated Krylov Space or adding approximated eigenvec-

tors of A of the Krylov Space. Storage of eigenvectors

requires extra memory in addition to the memory that is

used for storing the Krylov basis. These strategies are not

successful if the total amount of memory in use is kept

constant compared to a preconditioned restarted GMRes.

The typical convergence behavior of the GMRes solver

in shown in Fig. 1. After a certain number of iterations, we

have exponential convergence. But convergence rate

decreases with every restart.

If preconditioning is applied with CGNR, the required

memory for the solver is of acceptable size. The precondi-

tioned CGNR needs only 158 matrix–vector products at

1600 Hz (see Fig. 2), but even the unpreconditioned version

requires only 289 matrix – vector products while

GMRes(600) needs 1135 iterations. Here, we can state

that the GMRes solver is not suitable for this problem at the

frequency range of interest. CGNR shows completely

different behavior.

It was shown in Part I for iterative solution without

precondition that the number of required matrix–vector

products for convergence decreases with increasing fre-

quency. We observe the opposite behavior if precondition-

ing is in use. At higher frequencies and with a small fill-in

parameter p for the ilutðt; pÞ; the preconditioned version

converges even slower (see Fig. 3) than the unprecondi-

tioned one. With a sufficiently large fill-in parameter, the

number of iterations needed is significantly reduced. This is

shown in Fig. 4.

Fig. 1. Model of cat’s eye with 128 elements along the perimeter (total 7680

elements, 30,720 unknowns) at 1600 Hz, comparison of convergence with

and without application of a preconditioner for iterative solver

GMRes(600).

Fig. 2. Model of cat’s eye with 128 elements along the perimeter (total 7680

elements, 30,720 unknowns), comparison of convergence with and without

application of a preconditioner for iterative solver CGNR.

S. Schneider, S. Marburg / Engineering Analysis with Boundary Elements 27 (2003) 751–757 753

Finally, we consider the large model of 108,000 constant

elements that has already been discussed in Part I. As given

there, solution requires 320 and 312 matrix–vector product

evaluations for solution at 3400 ðkR ¼ 20pÞ and 4000 Hz,

respectively. Solution with application of ilutðt; pÞ with p ¼

50 requires 290 and 326 matrix–vector product evaluations,

respectively. While the number of iterations decreases with

increasing frequency for the unpreconditioned solver, the

iteration count also increases for the preconditioned version.

The crossover point is apparently found between 3400 and

4000 Hz. It shall be mentioned that a higher value p does not

affect performance of the solver. We assume that constant

elements are better suited than linear discontinuous

elements for solving the linear system of equation arising

from RGM without preconditioning.

For this example, we can conclude that CGNR,

especially if used together with ilutðt; pÞ preconditioning

is a very efficient solver for exterior problems at high

frequencies.

4.2. Tire noise excitation of a car

The second example being the model of a car standing on

a rigid ground has been described in Part I of this article.

There we found that the number of iterations needed by the

GMRes grows rapidly with increasing frequency. Now, we

aim on reducing the costly part of the solution process, i.e.

the iterative solution, by applying the ilutðt; pÞ precondi-

tioner. CGNR is not successfully applied in this example,

even with preconditioning.

We use a fill-in parameter of p ¼ 50: With this choice,

the cost of the preconditioner in terms of memory

requirement is approximately 15% of the total memory at

highest frequency (<235 Mb) in use. Again, we distinguish

the two meshes available (Part I). In Fig. 5, the tremendous

Fig. 3. Model of cat’s eye with 152 elements along the perimeter (total

10,830 elements, 43,320 unknowns), comparison of convergence for

different values of ilutðt; pÞ fill-in parameter p for iterative solver CGNR.

Fig. 4. Model of cat’s eye with 200 elements along the perimeter (total

18,750 elements, 75,000 unknowns), comparison of convergence in terms

of frequency with and without preconditioning for iterative solver CGNR.

Fig. 5. Sedan tire noise analysis, comparison of convergence in terms of

frequency with and without preconditioning for iterative solver GMRes.

Fig. 6. Radiation of piston compressor: finite element mesh of structure and

boundary element mesh of fluid coincide.

S. Schneider, S. Marburg / Engineering Analysis with Boundary Elements 27 (2003) 751–757754

reduction of number of iterations needed by the GMRes

solver is shown. The number of iterations is more then

halved over the entire frequency range. The expenditure of

preconditioner calculation and its application in the iterative

solution process is negligible. So, the total solution time is

also halved.

In the unpreconditioned version, the larger model of

continuous topology (39,502 elements) requires about 50–

100 iterations more than the smaller model of discontinuous

topology (25,810 elements). If preconditioning is applied,

we observe hardly any difference in convergence between

both discretizations. GMRes converges even better for the

larger continuous model. Concerning this tire noise

example, we can summarize that performance of GMRes

is remarkably affected by the ilutðt; pÞ preconditioner.

Speedups between two and five compared to unprecondi-

tioned solution are reported.

4.3. Radiation of piston compressor

In our last numerical example, we will calculate

the radiated sound field of a piston compressor of

KNORR–Brake Munich. For a detailed description of the

compressor, we refer to Ref. [24].

With the given finite element discretization, we calculate

the eigenfrequencies in the range of 100–2000 Hz. Out of

these, we select eigenfrequencies with characteristic mode

shapes, for example, first, second and fourth plate bending

modes of the plane parts of the housing. The normal

displacements of the nodes are used as normal velocities for

the acoustic simulation. These values are adjusted to the

values measured at the Institute fur Festkorpermechanik

with a laser scanning vibrometer. We measured a normal

surface velocity of approximately 7 mm/s for the eigen-

frequency at 1431 Hz and approximately 1 mm/s for the

eigenfrequencies at 1115 and 1940 Hz, respectively.

According to the experimental data, these are the dominant

surface velocities.

The compressor is located 1 m above the ground (the axis

of the cylinders are parallel to the ground) which is assumed

to be rigid. This corresponds to the measurements

performed at KNORR–Brake Munich to evaluate the

radiated sound pressure level of such a compressor.

The mesh of the housing of the compressor is shown

in Fig. 6. This mesh consists of 18,503 triangular and

Fig. 7. Radiation of piston compressor: mode shapes for eigenfrequencies at 1431 and 1940 Hz.

Fig. 8. Radiation of piston compressor: dependency of the number of

iterations on frequency. Test of different fill-in parameters for

GMRes(200).

Fig. 9. Radiation of piston compressor: efficiency of preconditioning for

GMRes(200) as number of iterations and as memory requirements in terms

of ilu fill-in parameter.

S. Schneider, S. Marburg / Engineering Analysis with Boundary Elements 27 (2003) 751–757 755

rectangular elements (Fig. 8). Element size is limited to a

maximum of 2.8 cm. This guarantees at least six

elements per wavelength up to a frequency of 2000 Hz.

Using constant acoustic boundary elements, we have the

same number of unknowns for the sound pressure at the

mid points of the elements. The problem is solved using

the two Fast Boundary Element Methods that were

explained above.

Two selected structural mode shapes for the eigenfre-

quencies at 1431 and 1940 Hz are shown in Fig. 7.

With this example, we study the influence of the fill-in

parameter. Obviously, we expect the preconditioner to be

more efficient if we use a large fill-in parameter. Thus, a

larger fill-in parameter will lead to a smaller number of

iterations (see Fig. 8)—because of a more accurate

ILU-decomposition—but memory costs increase as well

and the preconditioner becomes less efficient. This situation

is shown in Fig. 9.

A GMRes(200), i.e. restart after 200 iterations, is applied

in this example. It preconditioning is used restart is seldom

required. If preconditioning is omitted, we found no

convergence within 600 iterations.

The effect of the preconditioner is remarkable. With

only two additional off diagonal entries (ilu2) GMRes

reaches the required residual with 188 iterations at 1940 Hz

and only 95 iterations at 1115 Hz. Up to a fill-in of about

10, we get a significant reduction of the number of

iterations needed. A further increase of this parameter

reduces the number of iterations only slightly. But much

more memory is needed as the memory requirement of the

ILU-decomposition grows linearly with p: Out of this

example, we conclude that for this specific case, the choice

of p ¼ 10–20 is nearly optimal.

The situation changes if we reduce the restart parameter.

Due to the stagnation occurring after a restart of the

GMRes, the choice of the fill-in parameter determines

whether the solver converges within an acceptable number

of iterations or not. The comparison of the performance of

GMRes(50) with GMRes(200) is shown in Table 1. An

integer value in the columns 2–12 gives the number of

iterations needed to reach the required residual. A floating

point value gives the reached residual after 630 iterations.

If the ILU is accurate enough, no or only one restart will

occur. Otherwise, GMRes stagnates before reaching the

required residual.

The CGNR is also tested with this model. Without

preconditioning, this iterative solver does not coverage

within 400 iterations. The success of preconditioning

strongly depends on the size of the fill-in parameter.

Table 2 shows the behavior of CGNR with a maximum

number of iterations of 400. An integer value in the column

2–5 gives the number of iterations needed to reach the

required residual. A floating point value gives the reached

residual after the maximum number of iterations. To

achieve convergence at every frequency, a value of

p ¼ 200 will be necessary for the ILU-decomposition. In

terms of memory requirement, this corresponds to a

GMRes(400). A GMRes(400) without preconditioning

does not converge within 400 iterations. Again we find

that the number of iterations increases with increasing

frequency when using the CGNR. This may be caused by

the RGM as the size of the matrix. Anear is decreasing with

increasing frequency what may result in a worse perform-

ance of the preconditioner.

Finally, we conclude that, in this example, GCNR is not

competitive to GMRes. Moreover, we conclude that the best

strategy to solve problems as described above is to share the

memory between the GMRes and the preconditioner in such

a way that we have no restart in the GMRes solver. One fill-

in element per row in the factorization costs the same as an

increase of the restart parameter by two. In other words, if

we are using p ¼ 10 instead of p ¼ 20; we can use a

GMRes(m þ 20) instead of GMRes(m). In general, this

choice will perform much better.

Table 1

Radiation of piston compressor: comparison of GMRes(50) and GMRes(200) as number of matrix–vector product evaluations being necessary for

convergence (floating point values indicate residuals if convergence is not reached after 630 matrix–vector product evaluations)

Frequency (Hz) ilu2 ilu5 ilu10 ilu30 ilu50 ilu100

50 200 50 200 50 200 50 200 50 200 50 200

1115 181 99 90 65 57 52 39 38 36 35 31 31

1431 613 126 194 86 102 68 80 55 52 51 49 49

1940 0.3 £ 100 188 0.7 £ 1021 133 0.1 £ 1021 110 0.7 £ 1023 89 0.1 £ 1023 84 0.2 £ 1024 79

Table 2

Radiation of piston compressor: effect of size of fill-in parameter on CGNR

as number of matrix–vector product evaluations being necessary for

convergence (floating point values indicate residuals if convergence is not

reached after 800 matrix–vector product evaluations)

Frequency (Hz) No ilu ilu30 ilu100 ilu200

1115 3.0 0.8 £ 1023 350 198

1431 1.7 0.1 £ 1021 498 282

1940 1.5 0.7 £ 1021 0.1 £ 1024 520

S. Schneider, S. Marburg / Engineering Analysis with Boundary Elements 27 (2003) 751–757756

5. Conclusion

The Incomplete LU-decomposition of the matrix

representing the near field interactions of a Fast Boundary

Element Method is well suited as a preconditioner for

exterior acoustic problems. It significantly reduces the

number of iterations needed by the iterative solvers

investigated. The extra time for calculation and appli-

cation of the preconditioner is negligible since the time

for evaluation of Au is dominated by the evaluation of

vfarðuÞ:

Especially for problems with highly non-smooth sur-

faces, the usage of a preconditioner is essential as Restarted

GMRes and CGNR do not converge at all in the

unpreconditioned cases. Full GMRes converges slowly if

no preconditioning is applied.

The preconditioned GMRes performed the best in the low-

to mid-frequency range as long as no or only a few restarts

occur. Thus, a good balance of the memory distribution

between the preconditioner and the basis for the Krylov

Space must be found. In general, a value of p ¼ 10–20for the

fill-in parameter leads to the most efficient results.

CGNR performed excellent in the high frequency range

for models with a closed and smooth surface even without

preconditioning. If the surface has many edges and vertices,

CGNR fails to converge. Application of the ilutðt; pÞ

preconditioner with large fill-in parameter reduces influence

of the surface and the iterative solver performs reasonable.

Obviously, the presented preconditioner requires

additional memory. However, a single iteration step itself

is very costly. Therefore, a reduction of the number of

iterations is often more interesting than the gain of some

computer memory.

It is assumed that similar effect of preconditioning can be

observed in the case of interior problems that involve

hypersingular operators.

Acknowledgements

The authors wish to thank KNORR–Brake Munich for

providing us with the CAD model of the piston compressor.

Furthermore, it is acknowledged that the computation was

run on the SGI Origin 2000 at the Zentrum fur Hochleis-

tungsrechnen of the Technishe Universitat Dresden.

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