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INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING
Int. J. Numer. Meth. Engng. 46, 1997}2010 (1999)
EXTENDING INTEGRAL EQUATION TIME DOMAIN
ACOUSTIC SCATTERING ANALYSIS TO
LARGER PROBLEMS
V. SUCHIVORAPHANPONG, S. P. WALKER* AND M. J. BLUCK
Mechanical Engineering Department, Imperial College of Science ¹echnology and Medicine, Exhibition Road,
¸ondon S=7 2BX, ;.K.
SUMMARY
A method is presented to accelerate the execution of integral equation time domain analyses of exterior
acoustic scattering problems. Conventionally, these have costs which scale with the "fth power of the
frequency of the excitation, and practical limits to such computations are reached when bodies approach
perhaps &5}10 wavelengths long. The fast approach presented is based on exploiting the pulsed nature of
the illumination to omit much nugatory calculation. There is an associated slight accuracy loss; this is
investigated. The method has costs which can scale with frequency to the power as low as &3, such that, for
example, costs on a 18)5 wavelength body are reduced by a factor of about 28, with this factor itself increasing
with roughly the square of the body size. Associated with the reduction in operations is a reduction in the
scaling of storage required, from the third to the second power of frequency. Examples of analysis of large
scatterers are presented, extending to a &22 000 node &almond'. Copyright ( 1999 John Wiley & Sons, Ltd.
KEY WORDS: time domain integral equations; acoustic scattering; wave propagation; transient scattering; acoustic
cross-section
1. INTRODUCTION
Large acoustic scattering problems arise in a number of engineering applications where it is
desired to predict the acoustic "eld scattered by some body. For very large, smooth scatterers this
can be tackled using ray tracing and similar techniques, whilst for small scatterers (up to say
a handful of wavelengths long) full-"eld solutions are computationally practicable. However,
these full-"eld solutions have computational costs, both in operations and storage, which scale
very severely with problem size (a point to which we will return). There is a considerable range of
problem, from perhaps a few to a few tens of wavelengths long, or where signi"cant features are
small on an otherwise large body, where the "eld solutions are too expensive, but the ray-based
techniques can be insu$ciently accurate.
One approach for such analyses is the integral equation time domain technique [1}3]. In this
paper we will describe a modi"cation to the usual integral equation time domain technique which
* Correspondence to: S. P. Walker, Mechanical Engineering Department, Imperial College of Science Technology and
Medicine, Exhibition Road, London SW7 2BX, U.K. E-mail: s.p.walker@ic.ac.uk
CCC 0029-5981/99/361997}14$17.50
Copyright ( 1999 John Wiley & Sons, Ltd.
Received 17 August 1998
Revised 30 March 1999
1998
V. SUCHIVORAPHANPONG, S. P. WALKER AND M. J. BLUCK
reduces its cost and cost scaling sharply, and extends considerably the range of problem size
tractable. Recently, an approach was presented [4, 5] which was able to reduce cost scaling by
one power of frequency; the present one, based on work recently presented in electromagnetic
scattering [6] applications, improves on this, reducing the cost scaling by approaching two
powers.
In the remainder of this introduction we will discuss the origin of the high computational costs
of scattering calculations, "rstly by reference to the more widely employed integral equation
frequency domain and "nite di!erence time domain approaches.
In Section 2 we will outline the integral equation time domain method and its discretization,
and demonstrate the origin of the cost scaling of its conventional form, as a precursor to
discussing the fast method.
In Section 3 we present modi"cations we have developed to reduce costs, and describe their
practical incorporation into an integral equation time domain treatment. Section 4 will present
results from its use.
We seek to solve the scalar wave equation in the exterior region surrounding some hard
scatterer (i.e. a homogeneous Neumann boundary condition), where some incident wave impinges
upon the scatterer.
One approach is to employ a frequency domain integral equation treatment. The di!erential
equation is transformed to an integral equation on the surface of the body [7]:
P)
2n/(r)"4n/*/#(r)#
L
/(r@)
A
B
R ) n@ 1
#jk e+kR ds@
R2 R
(1)
Here the "eld at a point on the surface is given as a function of the (unknown) "elds at all
other points on the surface. The surface must be discretized. Nodal spacing would depend
on the highest frequency f of interest, with spacing typically &1/10 of the shortest relevant
wavelength employed. With only the surface discretized, the number of nodes scales
with f 2. In discretized form (1) becomes a dense matrix equation of size nodes by nodes, and
thus size proportional to f 2. Direct solution then has a cost scaling with f 6. Storage of this
dense matrix is also di$cult. As an example, a 10 wavelength diameter sphere with &10
nodes/wavelength, would have about 30 000 nodes, and an associated matrix occupying approaching 10 Gb. Recent work, especially in the computationally similar area of electromagnetic scattering, has applied &fast multipole' techniques to this matrix, with impressive results [8].
Alternatively, the di!erential equation can be tackled directly in the time domain (with
either harmonic on pulsed time domain illumination, with frequency domain results extracted via
Fourier transformation, if required). Using a "nite di!erence approach, we would discretize
a region of the space surrounding the scatterer (truncating the region some way from the
scatterer, and applying some approximate absorbing boundary condition). The volume discretization now results in the total number of nodes scaling with f 3. With the number of timesteps for
which we must analyse similarly proportional to f we obtain a nominal cost scaling of f 4.
In practice, as problems become bigger, more nodes per wavelength are required to maintain
phase accuracy as disturbances propagate through a larger mesh [9]; with three spatial
and one temporal dimension a!ected this raises the cost scaling to rather more than the fourth
power of frequency. Storage is again also a di$culty, but less so than for the integral equation
treatment.
Copyright ( 1999 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng. 46, 1997}2010 (1999)
1999
TIME DOMAIN SCATTERING ANALYSIS TO LARGER PROBLEMS
2. THE INTEGRAL EQUATION TIME DOMAIN APPROACH
The time domain equivalent of (1) is
2np(r, t)"4np
*/#
P ) R3 C p(r@, t*)# c Lt@ (r@, t*)D ds@
R)n
(r, t)#
R Lp
(2)
L
2.1. Numerical treatment
A fuller description of this has been presented elsewhere [2, 10]; we will only summarize here.
The surface of the scatterer is represented via curvilinear quadratic surface patches, of (say) eight
nodes each, over which the geometry is approximated using polynomial functions
S (m, g), a"1, . . . , 8
(3)
a
where m, g are parameterized intrinsic co-ordinates. The position vector to a point on such a patch
is then
8
rm(m, g)" + S (m, g)r
(4)
a
j(m,a)
a/1
where j"j(m, a) are the node numbers of the local nodes on element m, and the position vector of
each of these nodes is r
. The pressure variation is represented similarly
j(m,a)
8
(5)
p (m, g; t)" + S (m, g)pm (t)
b
m
a
a/1
and the temporal variation of the "eld is modelled using one-dimensional quadratic Lagrangian
basis functions with a nodal separation *t.
Inserting these representations into (2) allows us to write
2np(r , k*t)"4np (r , k*t)
i
*/# i
M
n@ (m, g) ) R
1 8
3
m
m
#+
+ S (m, g) + ¹ (q(R , k))pml
ab
a
b
m
R2
R
m
m a/1
m/1 m,g
b/1
PP G
M
#+
m/1
C
C
PPm,g G m
DH
DH
n@ (m, g) ) R 1 8
3
m
+ S (m, g) + ¹Q (q(R , k))pml
ab
a
b
m
R2
c
m
a/1
b/1
DJ(m, g)Ddm dg
DJ(m, g)Ddm dg
(6)
Integrals are evaluated using Gaussian quadrature. Some elements involve singularities, and are
partitioned to integrate these accurately [11]. Doing this, we obtain
2np(r , k*t)"4np (r , k*t)
i
*/# i
8 3
M N1 N' N' n@ (m , g ) ) R
m q r
m + + S (m , g )¹ (q)pml DJ(m , g )Du u
#+ + + +
ab
a q r b
q r q r
R3
m
a/1 b/1
m/1 p/1 q/1 r/1
M N1 N' N' n@ (m , g ) ) R 1 8 3
m q r
m
+ + S (m , g )¹Q (q)pml DJ(m , g )Du u
#+ + + +
a q r b
ab
q r q r
R2
c
m
a/1 b/1
m/1 p/1 q/1 r/1
(7)
G
G
Copyright ( 1999 John Wiley & Sons, Ltd.
C
C
DH
DH
Int. J. Numer. Meth. Engng. 46, 1997}2010 (1999)
2000
V. SUCHIVORAPHANPONG, S. P. WALKER AND M. J. BLUCK
where N is the number of partitions (1 usually; 2 or 4 for singular integrals), N is the order of the
1
'
Gauss quadrature, u is the qth Gauss weight and s is the parametric retarded time t*.
q
Some (most) quantities p on the right-hand side are known; the associated retarded time was
longer ago than one timestep. Such terms, matrix B below in (8), are multiplied out to form
a vector say McN. Coe$cients multiplying unknown (nearby and present) values are gathered, and
together with the free term form a sparse matrix equation for the "eld at time k:
[A]Mp N"[B]Mp
N"McN
(8)
k
j,j:k
The number of non-zero terms on any row of A is independent of f; for large problems the vast
majority of the surface lies further away than c*t. The cost of repeated solution of this matrix
equation is small. As discussed further below, the main work of the method is associated with the
formation of the coe$cients in B (and A), and the multiplications by B to form McN.
2.2. Cost scaling in the conventional IE¹D
Equation (2) and its discretized counterpart (7) show the "eld at any point on the surface, at
some time, to be an integral of (some geometrical function of ) "eld values elsewhere on the surface
at the associated retarded times. Again discretized with some frequency-dependant re"nement we
have the number of surface nodes and patches proportional to f 2, such that the cost of evaluation
of the "eld at any point and at any timestep is proportional to f 2. Doing it at each node thus
generates a cost at each timestep proportional to f 4, and with again the number of timesteps
proportional to f the overall cost scaling is seen to be with f 5. This cost scaling is implicit in the
summations of equation (7).
Storage needs scale as, but are rather greater than, those of the frequency domain integral
equation approach. The coe$cients of B in (8) represent a dense matrix, of size a few times nodes
by nodes. As in the frequency domain, for multi-wavelength problems this is far too large to store.
The approach of necessity adopted is the recalculation of the matrix coe$cients as required. Cost
scalings are unchanged, but a considerable cost penalty is naturally incurred. In exchange storage
needs are vastly reduced. The next largest requirement is to store one transit's worth of the "eld
history. With nodes scaling with f 2, and the timesteps in a transit with f, this scales with f 3. For
the example sphere mentioned earlier, it comprises about 30 Mb.
3. THE FAST FORM OF THE IETD
3.1. Fast approximation
The reduction in cost involves introduction of an approximation based upon the behaviour of
high-frequency transient "elds. For a large body, illuminated with a short transient pulse, the
pulse will tend to propagate over the body as a fairly narrow excited region, with perhaps
a modest &wake', and with propagation of other similarly narrow active regions spawned by
re#ections from geometrical discontinuities. There are two respects in which this can be exploited
to save costs.
(i) Over most of surface locations, most of the time, the "eld will be (close to) zero. If we can
identify these locations and times we need not evaluate at them. If the excited region has
Copyright ( 1999 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng. 46, 1997}2010 (1999)
TIME DOMAIN SCATTERING ANALYSIS TO LARGER PROBLEMS
2001
a dimension related to the pulse width (&1/f ), the fraction of the surface which is excited at
any instant should decline with f. If so, this provides a reduction in cost scaling by one
power of frequency.
(ii) Where the "eld is to be found, it is found by integrating contributions from all the rest of the
surface evaluated at the appropriate retarded times. By the same arguments as above, most
of these integrations will be over regions in which the historical "elds were similarly
(almost) zero. Similarly, the fraction of the surface exhibiting non-zero historical "elds
should decline with frequency, providing a reduction in cost scaling by one further power of
frequency.
These savings will of course be an upper bound; the relevant regions are unlikely to be
identi"ed perfectly, and there will be some &housekeeping' cost associated with their identi"cation.
Note that one implication of the above is that interactions between most surface-point pairs are
never actually calculated; only a fraction of the coe$cients of the full matrix is ever evaluated
during the entire course of the analysis. The tiny fraction needed at each timestep is calculated as
needed.
Surface "eld history is again a major storage requirement, but this fast approach reduces this,
and its scaling, very sharply. At any one location only those periods of history which are excited
need of course be stored; a modest, frequency-independent number of values. Total history
storage should then scale with the number of nodes; frequency squared. This is the same scaling
indeed as merely the mesh geometry itself, and in practice mesh-related storage actually is the
dominant requirement.
3.2. Computational implementation
There are two aspects to achieve the cost saving; identifying where to calculate the "eld, and
when doing so, identifying over which portion of the surface to integrate.
We have investigated employing (rather complicated) searching algorithms to identify where to
integrate. However, in practice a simple brute force approach is actually very economical, and
accurate. For each location at which we are calculating the "eld, we simply test the retarded "eld
magnitude in every element. This is both a very quick test, and is done for only the centre node of
an element, so the total time spent on this testing is a tiny (O(1 per cent)) fraction of the total time.
For an explicit treatment, a dynamically adaptive selection of locations at which to calculate
would be simplest to implement: if at some time the "eld is found to be signi"cant at a point, it is
calculated at neighbouring points till a halo of insigni"cant "eld values is discovered. In the
implicit treatment it is not possible to "nd a "eld value in isolation from its neighbours, so this
cannot be done.
It is essentially always the case that the surface is active in the region where the incident wave
impinges, and this is used as one criterion for the selection of where to calculate. Additionally, the
progress of "elds as they move over the surface of the body is tracked. They propagate at the wave
speed, and so are captured by including all locations within a distance &c *t' of those locations
found to be active at the last step. These two approaches between them accommodate all convex
geometries. They do not cope with propagation across cavities or between multiple scatterers;
these are treated by identifying a small number of &watch nodes', where the "eld is calculated
always, with these watch nodes becoming nuclei for the growth of calculation regions whenever
a "eld is calculated there.
Copyright ( 1999 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng. 46, 1997}2010 (1999)
2002
V. SUCHIVORAPHANPONG, S. P. WALKER AND M. J. BLUCK
Even using these fast methods, multi-wavelength scattering calculations are computationally
large, and parallel computers need to be used. The fast algorithm is rather more complex to
parallelize than conventional IETD. The nodes at which calculations are performed di!er at each
timestep, and the elements over which integration must be performed varies from node to node,
and from timestep to timestep. This makes load balancing rather awkward, but good balancing
can still be achieved. (In part, by the deliberate random renumbering of nodes and elements, in
contrast to "nite element treatments!). Most of the larger problems reported in this present paper
were analysed on parallel machines; either a Cray T3D (512 processors) or a Fujitsu AP3000 (64
processors).
4. EXAMPLES OF PERFORMANCES
4.1. Accuracy
There is naturally some loss of accuracy associated with neglect of calculation at and
integration over some locations. Obviously, the savings achieved and the accuracy loss are linked,
Figure 1. Surface pressure on front-centre of sphere, for indicated thresholding levels, with zoomed inset (diameter 1,
pulse half-width 0)4, 1538 node mesh)
Copyright ( 1999 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng. 46, 1997}2010 (1999)
TIME DOMAIN SCATTERING ANALYSIS TO LARGER PROBLEMS
2003
and what accuracy loss is tolerable depends on the application, but as the following examples will
demonstrate, it is generally small at levels of approximation which still provide marked cost
reductions.
In Figure 1 we show the time dependant surface pressure on a unit diameter sphere, comprising
1538 nodes, illuminated with a Gaussian pulse of half-width 0)4. (In this, as in all cases, results are
normalized: body dimensions and pulse widths are expressed in arbitrary units, pressures are
expressed as a multiple of peak incident pressure, itself unity, and the wave speed is unity). This is
done with a range of values of the threshold pressure below which calculation and integration are
neglected, with results compared also to the analytical solution. As is seen, there is a steady
decline in accuracy with threshold, but even with a threshold of 5 per cent of the incident wave
magnitude the maximum error is &3 per cent of the peak "eld.
A frequency domain surface pressure can be extracted from this time domain result, and this is
used to obtain the bistatic acoustic cross section shown in Figure 2. The frequency extracted
corresponds to the sphere being 1 wavelength in diameter. As before, little error is introduced by
2
the thresholding.
Figures 3 and 4 show similar tests on an almond-shaped target [12]. There is no analytical
result for comparison, but the modest degradation with threshold is again apparent. The (greatly)
Figure 2. Acoustic cross-section of sphere, for indicated thresholding levels (wavelength 0)4 diameters, 1538 node mesh)
Copyright ( 1999 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng. 46, 1997}2010 (1999)
2004
V. SUCHIVORAPHANPONG, S. P. WALKER AND M. J. BLUCK
Figure 3. Surface pressure on-axis, rear surface of almond, for conventional (0 per cent) and indicated thresholding
levels, with zoomed inset (plane wave incident along axis on tip, pulse half-width 0)674, almond length 10 2450
node mesh)
zoomed inset in Figure 3 shows decaying oscillatory behaviour after the unit magnitude incident
pulse has passed; having a peak amplitude of only &0)003, this oscillation is lost in the
thresholded analysis.
One of very few large published time domain computations was recently presented by Ergin
[3], showing the surface "eld history on a 10 wavelength long almond. The case studied is
reproduced here as Figure 5. We have access only to the printed graph, but a few points measured
from the graph are included in Figure 5. Whilst this is only a comparison between codes, rather
than with experiment or analytical solution, the reasonable agreement does provide some
comfort.
4.2. Cost savings and scaling
We now turn to evaluate the cost saved in exchange for the introduction of errors such as those
discussed.
Copyright ( 1999 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng. 46, 1997}2010 (1999)
TIME DOMAIN SCATTERING ANALYSIS TO LARGER PROBLEMS
2005
Figure 4. Near "eld for the almond (location 0, 3)5, 0 relative to almond de"nition co-ordinates) for conventional
(0 per cent) and indicated thresholds (plane wave incident along x-axis on tip, pulse half-width 0)674, almond length xx,
2450 node mesh)
Figure 6 shows the computational cost of analysing propagation over a series of spheres,
ranging in diameter up to nine wavelengths, and comprising up to 23 554 nodes. Timesteps, nodal
separations, pulse widths and frequencies extracted are changed in concert between spheres,
to maintain similar &nodes per wavelength' and so on. Details are summarized in Table I.
Thresholding of 5 per cent is employed.
Computational cost comprises overwhelmingly ('95 per cent) activities directly proportional
to the number of surface patch integrations performed, allowing this to be used as an accurate
measure over the wide range of machine types necessarily employed. The graph shows clearly the
large savings being achieved, with the cost for the largest sphere reduced by a factor of about 25,
and with that factor increasing sharply with body size. (Note that the larger spheres cannot be
analysed using the conventional code, but accurate cost ratios are available merely by noting the
fraction of the total work (integrations) which the fast method actually does). The logarithmic
inset allows the scaling of cost with frequency to be inferred; depending on how it is extracted, it
ranges between about 2)8 and 3)2; more or less the third power discussed above.
The memory requirements of these same cases are shown in Figure 7. The elimination of the
history storage scaling with frequency cubed has a marked e!ect, with the factor by which storage
Copyright ( 1999 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng. 46, 1997}2010 (1999)
2006
V. SUCHIVORAPHANPONG, S. P. WALKER AND M. J. BLUCK
Figure 5. Comparison of the results of Ergin [3] (black circles) with a 1 per cent threshold case. Almond, 3)75 pulsewidths
(&10 wavelengths) long
is reduced being about 2 for the 9 wavelength case, with this factor increasing roughly linearly
with frequency.
Figure 8 shows a comparison of the conventional and fast approaches for a series of almonds,
ranging in size up to 21 552 nodes, and 18)5 wavelengths. (Details are summarized in Table II).
Similar behaviour to the spheres is seen, with costs reduced greatly; here by a maximum factor of
&28 in the 18)5 wavelength case. Cost scaling is again to the power 2)9}3)2 of frequency. A "gure
is not shown, but savings in storage are also similar to those of the spheres; for the 18)5
wavelength case storage needs are reduced by a factor of about 2.
5. DISCUSSION AND CONCLUSIONS
The modi"cation to the usual IETD has been shown to reduce costs greatly, and to reduce the
scaling with frequency of these costs by about two powers of frequency. On the biggest (18)5
wavelength) example shown here, the factor by which costs are reduced is &28. These reductions
Copyright ( 1999 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng. 46, 1997}2010 (1999)
2007
TIME DOMAIN SCATTERING ANALYSIS TO LARGER PROBLEMS
Figure 6. Computational cost (expressed in terms of integrations required) versus body size for a series of spheres,
conventional and 5 per cent thresholding
Table I. Unit radius sphere mesh and illumination information. The pulsewidth parameter g gives the full-width at half-maximum of the incident Gaussian plane wave pulse
Number of
nodes
2402
4706
6146
9602
16 226
23 554
Bodysize
(wavelengths)
Timestep
(and biggest nodal
separation)
Ave. nodal
separation
g
2)765
3)870
4)423
5)529
7)187
8)659
0)0879
0)0657
0)0546
0)0459
0)0385
0)0364
0)0723
0)0517
0)0452
0)0362
0)0278
0)0231
0)3516
0)2628
0)2190
0)1800
0)1529
0)1442
in operations are accompanied by reductions in the scaling of memory needed from the third to
the second power of frequency. There is an associated slight loss in accuracy, but costs savings of
the size mentioned are typically associated with maximum errors of only a few per cent of peak
"eld magnitudes.
Copyright ( 1999 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng. 46, 1997}2010 (1999)
2008
V. SUCHIVORAPHANPONG, S. P. WALKER AND M. J. BLUCK
Figure 7. Storage requirement versus body size for a series of spheres, conventional and 5 per cent thresholding
Table II. NASA almond (9)936 units long) mesh and illumination information
(timestep size"1)323]average nodal separation)
Number of
nodes
1202
2962
5538
10 866
21 552
Bodysize
(wavelengths)
4)380
6)868
9)400
13)150
18)510
Copyright ( 1999 John Wiley & Sons, Ltd.
Timestep size
Biggest
nodal
separation
Ave. nodal
separation
g
0)3000
0)1914
0)1399
0)1000
0)0710
0)3657
0)1814
0)1639
0)1019
0)0895
0)2270
0)1447
0)1057
0)0755
0)0537
1)2000
0)7655
0)5595
0)4000
0)2840
Int. J. Numer. Meth. Engng. 46, 1997}2010 (1999)
TIME DOMAIN SCATTERING ANALYSIS TO LARGER PROBLEMS
2009
Figure 8. Computational cost (expressed in terms of integrations required) versus body size for a series of almonds,
conventional and 1 per cent thresholding. Inset: logarithmic plot
ACKNOWLEDGEMENTS
It is a pleasure to acknowledge helpful discussions with Dr. S. J. Dodson, previously of this
Department.
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Copyright ( 1999 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng. 46, 1997}2010 (1999)
2010
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Int. J. Numer. Meth. Engng. 46, 1997}2010 (1999)
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