Improving stability of the transient boundary element method
using the CHIEF-Method
Michael STÜTZa , Martin OCHMANNb and Michael MÖSERa
a Institut
für Strömungsmechanik und Technische Akustik,
Einsteinufer 25 10587 Berlin, Germany, e-mail: [email protected]
b Beuth Hochschule für Technik Berlin,
Luxemburger Straße 10 13353 Berlin, Germany, e-mail: [email protected]
ABSTRACT
In the Frequency Domain BEM (FD-BEM) internal resonances of the structure may corrupt
the solution for radiation problems. The combined Helmholtz integral equation formulation
(CHIEF) is a popular technique in the FD-BEM that can overcome the so called non-uniqueness
problem. In Time Domain BEM (TD-BEM) one encounters a similar problem. The internal
resonances of the structure affect the exterior sound field, which may lead to inaccurate results
or even unstable behavior. The use of the CHIEF method in the time domain does not show
the desired results if the solution is computed stepwise in time. But applying the method to
the non-iterative one-step solution shows a significant reduction of the internal resonances and
their influence on the exterior sound field. This approach makes it necessary to solve a huge
overdetermined matrix. The solution process is not an easy task because of the immense size of
the matrices involved.
1. INTRODUCTION
The boundary element method (BEM) is a widely used numerical tool. Calculations may be
performed in the frequency domain (FD) or in the time domain (TD). The fundamentals of the
TD-BEM were developed in 1960 from Friedman and Shaw [1] and later in 1968 from Cruse
and Rizzo [2]. The FD-BEM is limited to the simulation of stationary processes. Instationary processes like moving sources, engine acceleration, impulses etc. must be simulated in the
time domain. Mansur [3] developed one of the first boundary element formulations in the time
domain for the scalar wave equation and for elastodynamics with zero initial conditions. The
extension of this formulation to non-zero initial conditions was presented by Antes [5]. Later
Jäger [6] and Antes and Baaran [7] extended the time domain formulations to analyze 3D noise
radiation caused by moving sources. Since then this method is subject to continuous enhancement, whereby however the frequency range algorithms exhibit a clear lead in development.
The main reason that the TD-BEM-Method is not being widely used till now is the instable
behavior that may appear. Ergin et al [8], Chappell et al [9] and Stütz and Ochmann [10]
showed numerical evidence that the instable behavior is caused by the eigenfrequencies of the
structure. Ergin and Chappell used the Burton-Miller-Method [11] to prevent instabilities. In
this paper a different approach, the use of the so-called CHIEF Method, will be presented.
2. NUMERICAL MODEL
The TD-BEM-Method is based on the Kirchhoff integral. A detailed derivation can be found
e.g. in Meise [4].
Z q ∂r p
1 ∂p
−4πd(~x0 ) p(~x0 ,t0 ) =
+
dΓ
(1)
+
r ∂ n r2 cr ∂t
ret
Γ
p denotes the sound pressure, r the distance between the observation point and source point and
c the speed of sound. Γ is the boundary surface, n the surface normal vector pointing into the
domain and q is the flux, following the relationship derived from Eulers equation
q = −ρ0
∂ vn
∂t
(2)
The index ret in Eq 1 states that all time dependant variables are taken with respect to the
retarded time tret = t − cr . Discretizing time itself in equidistant time steps ti = i∆t with (i =
1, 2, ...) we get tri = i∆t − cr for the retarded time. The boundary surface Γ is divided into N
planar elements with a uniform spatial pressure and flux distribution. For q a constant approach
and for p a linear approach in time are chosen. A more detailed description can be found in [10]
and [13]. With these approaches one can derive the following discretized integral equation
i
− 2π pi (~x0 ) =
N
∑∑
m=1 n=1
Z
1 n
qm Ψ(tri )dΓn
Γn r
∂r 1 n
n
+
(i
−
m
+
1)p
−
(i
−
m)p
Ψ(tri )dΓ
m
m−1
2
Γn ∂ n r
Z
(3)
with
(
1 ;tri ∈ [tm−1 ,tm )
Ψ(tri ) =
0 ; else
(4)
Using the Collocation method the following corresponding linear system of equations can be
derived
µmax
µmax
−2π p̂i = ∑ Gµ q̂i−µ+1 + ∑ Hµ µ p̂i−µ+1 − (µ − 1) p̂i−µ .
(5)
µ=1
µ=1
Gµ and Hµ are square matrices describing the system at µ = i − m + 1.
µ= 1
µ= 2
µ= 3
µ= 4
Figure 1. Division of sphere surface in regions of relativ time influence µ for one collocation point and
the resulting sparse matrices for all collocation points.
Compared to the FD-BEM the resulting matrices are very sparse, because for each matrix with
index µ the integration must only be performed within the integration limits of an inner radius
(µ − 1)c∆t and an outer radius µc∆t (see Fig. (1)). A difficulty that arises from that condition
is that one must not integrate over the whole element but over the part that lays within the integration limits. The sparsity of the matrices is explained due to the fact that most elements
are not within the integration limits and therefore have a zero entry. Eq. 5 can be written as




−2π 



p̂1
p̂2
..
.
..
.
p̂i


G1
  G2
 
 
 =  G3
 
  .
  ..
0
0
G1
···
0
G2
G1
···
···
0
0
..
.
..
···
Gµmax
.
···








G1
q̂1
q̂2
..
.
..
.
q̂i


H1
2H2
 
 
 
 +  3H3 − H2
 
 
..
 
.
0
0
H1
···
0
2H2
H1
···
···
0
0
..
.
..
···
−(µmax − 1)Hµmax
.
···
H1








p̂1
p̂2
..
.
..
.
p̂i








(6)
and in matrix notation one has to solve
(2πE + Hg ) p̂g = −φ̂g
(7)
to get the time response at once (with E being the identity matrix). This results in a BlockToeplitz-Matrices Hg that is not easy to handle because of the relatively big size (square of
(number of elements x time steps)). For example a small model consisting of 1000 elements
with a calculation time of only 500 time steps result in a matrix with 250 billion entries. Thats
why the time response is solved time step wise, starting with solving the first row of Eq. 6
(−2π p̂1 = G1 q̂1 + H1 p̂1 ) and proceeding till the last row. This approach is numerically much
easier to handle.
3. CHIEF METHOD
A common problem of the TD-BEM reported in the literature is the occurrence of spurious instabilities. In case of instability the results start to oscillate at high frequency with exponentially
rising amplitude. Using a sphere to simulate a monopole source one gets stable results as shown
in Figure 2. But apparently the sound radiated by the sphere is influenced by the eigenfrequencies of the structure. This connection however is still mathematically unproven. The magnitude
of influence of the eigenfrequencies depends on β , which describes the ratio of time step size
∆t and element size h.
β = c∆t/h
(8)
95
p [dB]
analytic
∆t=0.25ms, β=0.5
∆t=0.5ms, β=1
∆t=0.75ms, β=1.5
90
dsphere=n λ/2
85
85
170
255
340
f [Hz]
6 elements/λ
425
510
595
Figure 2. Sound radiation of sphere (radius r=1m, 1300 triangular elements) influenced by the eigenfrequencies marked by the vertical dotted lines
The problem of the natural frequencies is well known in frequency domain BEM calculation,
also called the nonuniqueness problem. Various methods have been derived to find a modified
equation with a valid solution for all frequencies. Well known approaches are the combined
field integral equation by Burton and Miller [11] and the combined Helmholtz integral equation
formulation (CHIEF) by Schenk [12]. For time domain calculations first Ergin [8] in 1999 and
later Chappell et al [9] in 2006 used a Burton-Miller approach to avoid the above described
problem.
As shown in section 2 a linear system of equations can be derived using the collocation method.
The collocation points are located on the boundary surface. Placing some additional points,
the so called CHIEF Points, inside of the structure we can derive a second linear system of
equations.
i < µs
4π p̂ci = 0
i = µs
4π p̂ci = µs Hµcs p̂1 + φ̂1c
i = µs + 1
4π p̂ci = µs Hµcs p̂2 + ((µs + 1)Hµcs+1 − (µs − 1)Hµcs ) p̂1 + φ̂2c
i = µs + 2
..
.
4π p̂ci = µs Hµcs p̂3 + ((µs + 1)Hµcs+1 − (µs − 1)Hµcs ) p̂2 + · · ·
..
.
(9)
Below µs we get a zero on the right hand sight of Eq. (9) due to the fact that the radiated sound
needs µs time steps to travel from the boundary to the CHIEF points. The principle of the
CHIEF method is to set the pressure in the CHIEF points to zero to force the inner sound field
to zero. In matrix notation we get
0 = Hgc p̂g + φ̂gc
(10)
with




p̂g = 


p̂1
p̂2
..
.
..
.
p̂i











φ̂gc = 


φ̂1c
φ̂2c
..
.
..
.
φ̂ic







and



Hgc = 

µs Hµcs
0
c
((µs + 1)Hµs+1 − (µs − 1)Hµcs ) µs Hµcs
..
.
0
Rearranging Eq.10 we get
···
···
0
···
···
0
0
...
c
−(µmax − 1)Hmax
···
Hgc p̂g = −φ̂gc
µs Hµcs



 . (11)

(12)
Combining Eq. (10) with Eq. (7) we than get the combined Helmholtz integral equation (CHIEF)
2πE + Hg
φ̂g
p̂g = −
.
(13)
Hgc
φ̂gc
To the authors knowledge the use of the CHIEF method in time domain BEM calculations was
never shown before. This may have different reasons. One reason is that the CHIEF method
does not work using the iterativ solution process. Applying the method to the non-iterativ onestep solution makes it necessary to solve the huge overdetermined matrix. The solution process
is not an easy task because of the immense size of the matrices involved. To deal with this
problem, the special symmetry of the matrices must be used. Using an iterativ solver that can
handle rectangular matrices, like the LSQR method [14], only one line of the matrix has to be
build at a time. This makes it possible to solve Eq. (13), but the numerical costs are quite high,
depending on the number of iterations.
4. RESULTS
The results in Figure 3 show that the CHIEF method successfully regularizes the solution at the
first 2 eigenfrequencies of the structure. The influence of the 3rd one at 510 Hz could only be
reduced. With increasing frequency the probability increases that a CHIEF point is located in a
knot of the eigenmode. In this case the CHIEF point will have no effect.
Also a fine spatial and temporal resolution has a positive effect on preventing the occurrence
of internal eigenfrequencies as can be seen in Figure 4. This seems to be an evidence that the
instabilities of internal eigenfrequencies are caused by numerical errors due to discretization.
The TD-BEM can be easily extended to calculate scattering problems by adding a source term
on the right hand sight of Eq. (1). The results for scattering of a monopol sound field by a sphere
with spatial discretization of 384 / 2400 quad elements and a constant time step of ∆t = 0.2ms
can be seen in Figure 4. Even at the quite coarse discretization of 384 elements no occurrence
of internal eigenfrequencies can be observed. With a finer spatial discretization the accuracy of
the results can be extended from 340 Hz to 550 Hz. Since the errors at higher frequencies are
caused by low discretization the CHIEF-Method will not improve the results.
95
p [dB]
∆t=0.25ms, β=0.5
∆t=0.5ms, β=1
∆t=0.75ms, β=1.5
analytic
90
dsphere=n λ
85
170
6 elements/λ
340
f [Hz]
510
Figure 3. Sound radiation of sphere (radius r=1m, 1300 triangular elements) calculated with (full lines)
and without (dotted lines) CHIEF points. The use of CHIEF points prevents the influence of
first two eigenfrequencies.
95
p [dB]
β=1.4, 384 quad elements
β=1.4, 1296 triangular elements
β=1.4, 2400 quad elements
analytic solution
90
85
85
170
255
f [Hz]
340
425
Figure 4. Sound radiation of sphere with spatial discretization of 384 and 2400 elements. With a fine
spatial discretization the influence of the eigenfrequencies does not occur.
60
0 π analytic
1 π analytic
0 π , 384 quad
1 π ,384 quad
0 π ,2400 quad
1 π ,2400 quad
p [dB]
50
40
30
20
10
85
f=254 Hz
170
90
255
340
510
90
60
25
f=450 Hz
30
180
210
270
60
25
30
0
210
analytic solution
384 quad elements
2400 quad elements
765
50
180
330
300
680
120
150
0
240
595
50
120
150
425
f [Hz]
330
240
300
270
Figure 5. Scattering of monopol sound field by a sphere with spatial discretization of 384 / 2400 quad
elements and a constant time step of ∆t = 0.2ms. Top figure shows scattered sound pressure in
front of sphere facing the monopol source (φ = 0π) and behind the sphere (φ = 1π) Bottom
figures show the directivity of the scattered sound pressure at selected frequencies 254 Hz and
450 Hz.
5. CONCLUSIONS
Numerical test cases show that the TD-BEM-Method can be influenced by the inner resonant
sound field of the structure. The temporal and spatial discretization plays a fundamental role,
because numerical errors seem to stimulate the influence of the eigenfrequencies. In order to
ensure accurate simulation results, it was shown that the CHIEF method can be used to regularize the TD-BEM-Method. Also a fine spatial discretization combined with an appropriate time
step can prevent this problem. Both approaches increase the numerical cost. It still has to be
investigated if there is a parameter range of temporal and spatial resolution that ensures stable
and accurate results. The factor β , which describes the ratio of time step size ∆t and element
size h (Eq. 8), is not a sufficient descriptor. More research regarding the effects of spatial and
temporal discretization has to be done.
References
[1] M. Friedmann, R. Shaw: Diffraction of pulses by cylindical Obstacles of Arbitrary cross
Section, J. Appl. Mech.,Vol. 29 (1962), S.40-46
[2] T.A. Cruse, F.J. Rizzo: A direct formulation and numerical solution of the general transient
elastodynamic problem I, J. Math. Analysis and Applic., Vol.22 (1968), S.244-259
[3] Mansur, W. J.: A Time-Stepping technique to solve wave propagation problems using the
boundary Element Method, PhD thesis, University of Southampton, 1983
[4] Meise T.: BEM calculation of scalar wave propagation in 3-D frequency and time domain
(in German). PhD Thesis, Technical Reports Nr.90-6, Fakultät für Bauingenieurwesen,
Ruhr-Universität, Bochum, Germany; 1990
[5] H. Antes, A Boundary Element Procedure for Transient Wave Propagations in Twodimensional Isotropic Elastic Media, Finite Elements in Analysis and Design 1 (1985)
313-322
[6] Jäger M. Entwicklung eines effizienten Randelementverfahrens für bewegte Schallquellen.
PhD Thesis. Fachbereich für Bauingenieur- und Vermessungswesen der TU CaroloWilhelmina zu Braunschweig, Germany; 1993
[7] Antes H, Baaran J. Noise radiation from moving surfaces. Engineering Analysis with
Boundary Elements; Volume 25, Issue 9, October 2001, Pages 725-740 Combustion
[8] A.A.Ergin, B. Shanker, E. Michielssen: Analysis of transient wave scattering from rigid
bodies using a Burton-Miller approach, JASA, 106, 1999, pp. 2396-2404
[9] D.J.Chappell, PJ. Harris, D. Henwood, R. Chakrabarti: A stable boundary element method
for modeling transient acoustic radiation,JASA,120, 2006, pp. 74-80
[10] M.Stütz, M.Ochmann, Stability behaviour and results of a transient boundary element
method for exterior radiation problems, Acoustics’08-Paris, 2008
[11] A. J. Burton, G. F. Miller: The Application of Integral Equation Methods to the Numerical
Solution of Some Exterior Boundary-Value Problems, Proceedings of the Royal Society of
London. Series A, Mathematical and Physical Sciences, Vol. 323, No. 1553, A Discussion
on Numerical Analysis of Partial Differential Equations (Jun. 8, 1971), pp. 201-210
[12] Harry A. Schenck: Improved Integral Formulation for Acoustic Radiation Problems,
JASA , July 1968, Vol 44, Issue 1, pp. 41-58
[13] M.Stütz, M.Ochmann : Calculation of the acoustic radiation of an open turbulent flame
with a transient boundary element method, International Congress on Acoustics in Madrid
(ICA), 2007
[14] Paige, C. C. and M. A. Saunders, "LSQR: An Algorithm for Sparse Linear Equations And
Sparse Least Squares," ACM Trans. Math. Soft., Vol.8, 1982, pp.43-71