Conference PaperPDF Available

Fast Methods applied to BEM Solvers for Acoustic Propagation Problems

Fast Methods applied to BEM Solvers
for Acoustic Propagation Problems
N. Balin
, G. Sylvand
, and J. Robert
Airbus Group Innovations, Toulouse, France
For the numerical simulation of wave propagation in acoustics, Airbus Group Innovations
relies on integral equations solved with the Boundary Elements Method (BEM), leading
to the need to solve dense linear systems. In this article, we intend to present two families
of fast solvers (Fast Multipole Method and H-Matrix method) that can be used on these
systems. We propose to underline their similarities, their connections and their differences,
to present their complementarity in future high performance solvers and to illustrate their
performances on industrial class applications.
I. Context
Airbus Group Innovations is the Airbus Group research center, dedicated to upstream research applied to
all Business Units (Airbus, Airbus Helicopters, Airbus Defence and Space ). The applied mathematics team
has developed over the years a software called ACTIPOLE 1destined to solve various acoustic propagation
problems using integral equations and boundary elements methods. This software suite is used in design
and research department to work on noise reduction.
The advantages of integral equations and BEM solver are well known: mainly accuracy, and simpler
(surfacic) mesh. The main algorithmic drawback is the need to cope with a dense matrix whose size can
be quite large for wave propagation problems, where the mesh step is governed by the wavelength of the
physical problem treated (in frequency domain).
For example, acoustic problems on a full size aircraft at 20 000 Hz (upper limit of audible frequencies)
can involve more than 108unknowns. Solving such linear systems with standard method is just impossible
(storage would require 80.000 terabytes of disk, factorization would take 100 years on all Airbus HPC
facilities).
Since the late 90’s, fast methods have been introduced to deal with these limitations. First, the Fast Mul-
tipole Method (FMM) allowed to compute fast matrix-vector products (in O(nlog2(n)) instead of O(n2) for
the standard algorithm), and hence to design fast solvers using iterative methods. Lately, H-Matrix methods
have gained wide acceptance by introducing fast direct solvers, allowing to solve systems in O(nlog2(n)) –
or less – without the hassle of using iterative solvers (unknown convergence rate and difficulty to find a good
preconditionner).
All these methods allow to compute the acoustic noise propagation for industrial configurations in pres-
ence of a uniform flow.
II. Boundary Element Method and Classical Resolution
A. Boundary Element Method
We are interested here into the modelling of the propagation of an acoustic field generated by an acoustic
source in an uniform flow (domain Ω). The acoustic source is supposed to be harmonic, defined by the
pulsation ω0. In the following we consider the exp(0t) time convention. The diffracted acoustic pressure
pverifies the convected Helmhotlz equation with associated boundary conditions on Γ = Ω.
Research Engineer, AGI
Expert, AGI/Inria
Research Engineer, AGI
1of13
American Institute of Aeronautics and Astronautics
Using the following Prandtl-Glauert transformation2,3
(x0=x+C(M0·x)M0
p(x) = p0(x0)eikM0·x0(1)
with C=1
M2
01
1M2
01and k=k0
1M2
0
,the problem is reduced to the classical Helmholtz equation
on the acoustic diffracted pressure
0p0+k2p0= 0 in Ω0
0p0·n0=−∇0pinc0·n0on Γ0
lim
|r0|→∞ |r0|p0·r0
|r0|ikp0= 0
(2)
with Ω0and Γ0the domain and boundary Ω and Γ after the transformation, n0the normal to Γ0oriented
outside Ω0and pinc0the transformed incident acoustic pressure.
For simplicity in the notations, we consider here only a rigid body and in the following, the prime has
been removed.
The knowledge of the acoustic pressure on Γ entirely solves the problem, since the following representation
theorem allows to compute the diffracted pressure pin any point x /Γ:4
p(x) = ZΓ∂G(x, y)
∂ny
p(y)dy, (3)
where G(x, y) is the Green’s function, solution of ∆u+k2u=δ0and given by G(x, y) = eik||xy||
4π||xy||. We
have the following results for xΓ:
∂p
∂n (x) = Dp(x),(4)
with:
Dp(x) = IΓ
2G(x, y)
∂nxny
p(y)dy. (5)
Using the rigid boundary conditions ptot
∂n = 0, we obtain the variationnal formulation:
find psuch that pt, we have:
IΓ×Γ
2G
∂nxny
p(y)pt(x)dydx =ZΓ
∂pinc (x)
∂n pt(x)dx. (6)
In order to discretize this system, we use a surfacic triangle mesh of the boundary Γ. The pressure trace
is discretized with P1 linear basis functions. We end-up with a complex, dense and symmetric system to
solve:
hDih pi=uinc
∂n .(7)
B. Classical Resolution
As we have seen in the previous section, the BEM formulation leads to the need to solve dense linear
systems, which is quite specific in the world of finite element methods. This has led to the development of
various solvers specifically designed for this purpose. In this section, we present a direct solver based on a
block-L.U or block-L.D.tLfactorization adapted to HPC platforms.5This is the most natural way to solve
a dense linear system, but its cost in terms of CPU time and storage grows fast with respect to the number
of unknowns. This has led to the introduction of advanced solver (FMM, H-mat) that we will present in the
next sections.
2of13
American Institute of Aeronautics and Astronautics
Direct vs. Iterative solver There are two main families of solvers for solving linear system : the direct
solvers apply a predetermined set of operations to compute the solution. The costs, in terms of storage and
floating point operations, are known in advance. For the linear system we are dealing with, L.U or L.D.tL
factorization are the most common approaches. The L.D.tLalgorithm is adapted to symmetric matrices,
and for a matrix of order Nit requires to store N2/2 scalar values and to computes 4N3/3 operations
(for complex scalar). The L.U is for non-symmetric matrices and requires twice the cost in storage and in
operation count.
On the other hand, iterative solvers repetitively apply a set of operations to compute an approximation of
the solution more and more precisely as the computation goes on. CG (Conjugate gradient,6) and GMRES
(Generalized Minimal Residual7) algorithms are the most widely used. For dense matrices, this kind of
approach is especially efficient when used in conjunction with a fast matrix-vector product such a the Fast
Multipole Method (see section III).
The SPIDO2 solver SPIDO2 is an in-house solver developed since the mid 90’s by Airbus Group Inno-
vations. It is a parallel direct solver, implementing both L.U and L.D.tLalgorithm, to solve dense linear
systems coming from boundary element method applied to wave propagation problems in electromagnetism
and acoustics. This solver features hybrid parallelism (with MPI and threads) and out-of-core execution
(allowing to store the matrix on disk during factorization, which is necessary since for large values of Nit
doesn’t fit in memory anymore).
Accuracy At the moment the SPIDO2 solver is our reference solver in terms of accuracy for solving
BEM problems. It has been tested during years against analytic results, measurements and other software
for electromagnetism and acoustic propagation problems.
Performance We will consider a problem with N= 1,3.106unknowns. This number of degrees of
freedom would allow to realize a computation for a full A321 aircraft as represented on figure 12 with an
acoustic frequency of 1500 Hz. For a nacelle computation as seen on figure 9, this size of problem would
allow to treat an acoustic frequency of 3100 Hz.
The test platform is Airbus Group’s current HPC system called HPC4. It a cluster based on Xeon E5-
2697 processors at 2.7 GHz, with a total of 68,000 cores, an infiniband interconnect, and an overall peak
performance of 1.5 Pflops (that is : 1.5.1015 floating point operations per second).
On this machine, using 80 nodes (or 1920 cores), a problem with N= 1,3.106unknowns is solved using
the block L.D.tLsolver in 27.5 hours. The factorization, the most expensive part of the computation,
takes 25.8 hours, with a performance above 65% of the machine peak performance. In this computation,
the number of right hand sides only impacts the solve time, which is very small when compared to the
factorization time : 27 minutes only for 362 RHS. It illustrates the fact that a direct solver is well suited for
problems with a large number of right hand sides. The matrix is stored distributed among the nodes and on
disk, representing approx. 160 Gbytes of data per nodes.
Nevertheless, on today’s machines, using this type of approach (with a storage growing like O(N2) and
a CPU time growing like O(N3)), it is difficult to consider doing larger computation than this. Hopefully,
several new approaches have emerged to solve this type of linear systems more efficiently.
III. Fast Multipole Methods
The initial FMM was introduced in the late 80’s for particle simulation.8Basically, the idea is to gather
the particle in clusters and to compute all the interactions not point-to-point, but cluster-to-cluster, using
approximations adapted to the considered kernel. A hierarchical approach for building the clusters leads to
a multi-level algorithm, which we refer to as “the” FMM. The introduction of FMM for Helmholtz kernel,910
paved the way to a very broad use of FMM in the field of wave propagation,11.12
Since then, new FMM formulations have been introduced. The directional FMM13 extends the black box
FMM to all oscillatory kernels, for instance for 2D applications. The advantage of this “black box” approach
is that it only relies on kernel evaluations, and not on an analytical decomposition of this kernel. In,14 the
authors deal with a numerical breakdown that prevents the classical FMM for Helmholtz from handling
low-frequency problems. Recently, this method has been improved and simplified,15 leading to a new FMM
scheme for Helmholtz stable at all frequencies. In our implementation of this method, we still rely on the
original approach available in the late 90’s.
3of13
American Institute of Aeronautics and Astronautics
The Fast Multipole Method (FMM) is a way to compute fast but approximate matrix-vector products.
In conjonction with any iterative solver, it is an efficient way to solve an integral equation problem such as
(7). The method is fast in the sense that CPU time is O(n. log2(n)) instead of O(n2) for standard matrix-
vector products, and approximate in the sense that there is a relative error between the “old” and the “new”
matrix-vector products ε103. Nevertheless, this error is not a problem since it is usually below the error
introduced by the iterative solver or by the approximation due to the mesh.
In practice, we want to compute the right hand side of (6) for all pt. In the mono-level FMM method,
we split the object into equally sized domain using for instance a cubic grid (as shown in Figure 1). The
degrees of freedom are then dispatched between these cubic cells. Interaction of basis functions located in
neighbouring cells (that is cells that share at least one vertex) are treated classicaly, without any multipole
acceleration.
Figure 1: Use of a cubic grid to split the mesh
The interactions of unknowns located in non-neighbouring cells are accelerated with the FMM. The
base of this algorithm if the following addition theorem: given two points xand ylocated in two distant
(=non-neighbouring) boxes Cand C0centered in Mand M0, we have
G(|yx|) = ik
16π2lim
L+Zs∈S
eiks.xMTL
MM0(s)eiks.M0yds,(8)
where Sdenotes the unit sphere in R3, and TL
MM0is the transfer function defined on Sby
TL
MM0(s) = X
0lL
(2l+ 1)ilh(1)
l(k.|MM0|)Pl(cos(s,MM0)).(9)
with h(1)
lthe spherical Hankel function, Plthe Legendre polynomial, and the parameter Lis called
number of poles. It is chosen in accordance with the size of the box edge a, in order to have a good accuracy
in ((8)) and no divergence in ((9)): L=3ka satisfies these two conditions. Using (8) within (6), we can
replace an O(n2) computation by a sequence of three operations (traditionnaly called P2M, M2L and L2P
where P stands for particle, M for Multipole expansion and L for Local expansion). Each of these operations
uses a discretization of the unit sphere Sthat is chosen in accordance with the value of L. In practice, for
an optimal size of the grid, the complexity of the single level FMM is O(n3/2).
One can then introduce a more advanced variant of the FMM using a multilevel approach based on a
recursive subdivision of the diffracting object using an octree (see figure 2). The idea is to explore this tree
from the root (largest box) to the leaves, and at each level to use the single level multipole method to treat
interactions between non-neighbouring domains that have not yet been treated at an upper level. At the
finest level, we treat classicaly the interactions between neighbouring domains. The improvement adds two
new operations to the algorithm (M2M and L2L) to connect the different levels, but the operations used for
4of13
American Institute of Aeronautics and Astronautics
Figure 2: Subdivision of a plane through an octree
single level FMM remain unchanged. The optimal complexity of the multi level FMM is O(nlog2n). Hence,
for an iterative resolution of a linear system with NRHS right-hand sides requiring NIT E R iterations to reach
convergence, the complexity of the whole computation will be NRH S NIT E RO(nlog2n). The algorithm uses
a lot of functions and parameters, for a full description of the implementation and parallelisation, please
refer e.g. to.11 For examples of applications using FMM, see section B.
IV. H-Matrix
H-Matrix16 is a lossy, hierarchical storage scheme for matrices that, along with an associated arithmetic,
provides a rich enough set of approximate operations to perform the matrix addition, multiplication, factor-
ization (e.g. LU or LDLT) and inversion. It relies on two core ideas : (a) nested clustering of the degrees
of freedom (figure 3), and of their products; and (b) adaptive compression of these clusters. Several choices
exist in the literature for these two ingredients, the most common being Binary Space Partitioning for the
clustering and Adaptative Cross Approximation for the compression.
Figure 3: Example of geometry cluster tree
Each pair of cluster which are far enough from each other are said to be admissible (figure 4). They
yields to compressible blocks in the tree-like matrix structure (figure 5).
Once created, the structure is “filled” in a second step with low-rank approximations of the corresponding
matrix blocks, representing the interaction of two clusters. The algorithms then perform the operations on
this structure, using adaptive recompression to avoid inflating the matrix as the algorithm progresses.
The compression consists in replacing each block by a product of two low rank matrices (figure 6). During
the initial filling of the matrix, this is actived using the ACA+ algorithm. It has the big advantage of not
needing the full block but only the relevant lines and columns. This make the initial filling very fast compare
to uncompressed assembly. For each block the algorithm ensures that kBBcompressedk< ε kBkwhere εis
a user controlled parameter.
Together, they allow for the construction of a fast direct solver with complexity O(nlog2(n)) in some
5of13
American Institute of Aeronautics and Astronautics
(a) relative distance between cluster = 1 (b) relative distance between cluster = 3
Figure 4: Impact of different admissibility criteria (in blue, the admissible interactions with red group).
Figure 5: H-Matrix structure. Red blocks will not be compressed. White ones will.
' ×
Figure 6: Compression of a matrix block
6of13
American Institute of Aeronautics and Astronautics
cases,17 which is especially important for BEM applications as it gracefully handles a large number of Right-
Hand Sides (RHS). They also provide a kernel-independent fast solver, allowing one to use the method for
different physics.
Airbus Group Innovations has recently implemented the H-Matrix arithmetic and successfully applied it
to a wide range of industrial applications in electromagnetism and acoustics. A sequential version of this
implementation has been published as open source.18 These algorithms are hard to efficiently parallelize,
as the very scarce literature on the subject shows.19 We developed a parallel solver that goes beyond the
aforementioned reference, using innovative techniques on top of a state-of-the-art runtime system.20,21 This
enables the solving of very large problems, with a very good efficiency. In this presentation, we show some
results on the accuracy of this method on several challenging applications, and its fast solving time and
efficient use of resources.
V. Computation chain
In order to perform high frequency computations, an efficient and automated meshing methodology is
required. An in-house methodology has been developed.
(a) Tesselated geometry (b) Half-automated with
commercial software
(c) Amibe (d) Amibe with Afront
Figure 7: Comparison of different mesher on a 1250Hz mesh
The geometry is first transformed to a clean and sewed tesselated surface (figure 7a). Then a mesh is
created for each frequency using a batch mesher named Amibe.22 Amibe is Delaunay mesher so it is robust
but not good at creating high quality triangles. Robustness is what allows to do batch meshing, as no human
must be involved in fixing the mesh. To get better triangles we use Afront.23 Afront is not robust but, as
it is a frontal mesher, it creates very good quality triangles. Then hybridizing Amibe with Afront gives a
robust and high quality mesher (as shown on figure 8) with more than 90% of the edges length at the target
±5%. In our case, the number of degrees of freedom has been reduced by 30% with this method.
Having good quality triangles help reducing the number of degrees of freedom, increasing the effectiveness
of our FMM algorithm and then reducing the computation time.
Frequency Number of triangles Time
3150Hz 9,347,590 52min
4000Hz 15,065,208 1h 7min
5000Hz 23,525,739 2h 18min
6300Hz 36,263,028 3h 50min
Table 1: Time for the generation of the mesh
VI. Numerical results
Two examples of application are presented here to highlight the possibilities of the Fast Multipole Method
and the H-Matrix solvers. They have been run on a machine with 2 intel Xeon(R) “Ivy Bridge EP” processors
with 12 cores running at 2.7GHz and 192GB RAM per node, and and infiniband QDR network to connect
the nodes.
7of13
American Institute of Aeronautics and Astronautics
Figure 8: Repartition of the edge length on a 1250Hz mesh (representative of the mesh quality)
A. Nacelle treatment characterisation
We are interested here in characterising nacelle acoustic treatments effects. The figure 9shows a nacelle
geometry mesh provided by Airbus. The computation configurations are: uniform flows (with different flow
conditions depending on the pre-defined flight phase analysed), a modal source and several frequencies, which
are related to the engine regime (through the fan blade count and rotation speed).
Figure 9: Example of mesh for a nacelle simulation
We study the takeoff case for different frequencies (cf. table 2). The Mach number of the carrying flow
is around 0.6 and all the propagative modes have an energy of 100dB. Due to the number of propagative
modes (Right Hand-Side (RHS) terms), the Fast Multipole Method is not effective. So the H-Matrix solver
is only compared to the direct solver. In the following computations we used ε= 103.
1500Hz 2400Hz
Elements 201 780 877 305
Total unknowns 268 416 783 648
Number of propagative modes 466 1 690
Table 2: Nacelle computations characteristics
For the modes coefficients, the relative errors between the H-Matrix and the classical direct solvers are
illustrated in table 3.
Another quantity of interest is the broadband noise obtained by equi-repartition of the energy on the
8of13
American Institute of Aeronautics and Astronautics
Type Relative Error
Min 5.887 104
Average 1.839 104
Max 4.144 103
Table 3: Nacelle computations validation
azimuthal modes
RMSdB(θ) = 10 log10 1
MX
m
1
Nm
Nm
X
n=1
Pmn(θ)2!10 log10 P2
ref(10)
with θthe radiation direction, Pref the reference pressure (2.105Pa), Nmthe number of radial modes for
azimuthal mode mand Pmn the radiated pressure of the mode m, n.
The figures 10 and 11 illustrate the broadband noise obtained at 2400Hz on an arc of radius of 46m
centered on the inlet and in the vertical plane for H-Matrix and standard (SPIDO2) methods and the
relative error between these two methods. The error is very low and more important for the lowest levels of
noise.
Figure 10: Nacelle computation - broadband noise on an arc at 2400Hz
The computation time at 2400Hz is presented in the table 4. With the new H-Matrix solver, a speed-up
of 60 for the CPU time is observed and the maximum memory is reduced by 15 compared to the direct
classical solver.
Total Assembly Solver Max mem.
(s) (s) (s) (Go)
direct
(480 cores) 85406 32874 34448 107 * 20
H-Matrix
ε= 10329195 14019 10833 143
(24 cores)
Table 4: Nacelle computations time
9of13
American Institute of Aeronautics and Astronautics
Figure 11: Nacelle computation - error on the broadband noise at 2400Hz
B. Ramp Noise
The other application presented here is the prediction of the installation effects for ramp noise sources. Here,
we consider an A321 rigid aircraft and we are interesting in computing the noise generated by Environmental
Control System (ECS) or Auxilliary Power Unit (APU) sources at servicing points in a frequency range
between 500 Hz and 10 kHz. The ground plane will be taken into account in the Boundary Element Method
by changing the Green kernel. The sources considered are modal sources.
The high frequency imposes the use of the FMM algorithm with an iterative resolution. The figure 12
shows the mesh adapted to the frequency of 1250 Hz and the table 5the computation characteristics.
Figure 12: Mesh at 1250Hz
Figures 13 and 14 illustrate the broadband noise obtained using equation (10) at 1250Hz on the receivers
for the H-Matrix and FMM methods and the relative error between these two methods. A very good
agreement is observed between the two methods in this case.
The FMM methods shows to be performant and accurate for these kind of computations allowing to
compute Ramp Noise on the whole aircraft for high frequencies (5kHz). At the moment the H-Matrix
solver is not able to run on the 2500Hz and 5000Hz frequencies.
10 of 13
American Institute of Aeronautics and Astronautics
Frequency # dof # RHS # cores time
1250Hz 0.9M 3 24 0.9h
2500Hz 3.6M 9 96 3.5h
5000Hz 14.3M 32 88 86h
Table 5: Computation characteristics on A321 Ramp Noise
Figure 13: Ramp noise computation - broadband noise at 1250Hz
Figure 14: Ramp noise computation - error on the broadband noise at 1250Hz
11 of 13
American Institute of Aeronautics and Astronautics
VII. Conclusion
High performance solvers have been implemented into ACTIPOLE software and allow to run large scale
industrial applications. A very good agreement has been obtained between the three solvers tested here in
terms of accuracy.
The three solvers are complementary: SPIDO remains the reference solver in term of accuracy but is very
expensive and can not address large problems, H-Matrix solver has to be preferred for medium problems
especially with a large number of RHS and at the moment, FMM solver remains the reference solver for
huge problems. For instance, FMM solver is able to compute the ramp noise of an airbus A321 at 5kHz.
An automated meshing chain has also been developed in order to manage huge problems with minimal
human interaction.
All these features associated to the current computing power allow to address large industrial problems
during the design phases as well as optimization, uncertainty and sensibility analysis and then to propose
a strategy based on acoustic numerical modelling. Work is on-going for the extension of the modelling to
non-uniform flows.2
Acknowledgements
This work has been funded by the French Government in the frame of Hi-BOX and ABRICOT pro jects.
The authors would like to thank D. Lizarazu from the AIRBUS Acoustics team for her contribution to this
work.
References
1Delnevo, A., Le Saint, S., Sylvand, G., and Terrasse, I., “Numerical methods: Fast multipole method for shielding effects,”
AIAA Paper, Vol. 2971, 2005, pp. 2005.
2Balin, N., Casenave, F., Dubois, F., Duceau, E., Duprey, S., and Terrasse, I., “Boundary element and finite element
coupling for aeroacoustics simulations,” Journal of Computational Physics, Vol. 294, 2015, pp. 274–296.
3Dubois, F., Duceau, E., Mar´echal, F., and Terrasse, I., “Lorentz Transform and Staggered Finite Differences for Advective
Acoustics,” Tech. rep., EADS and arXiv:1105.1458, 2002.
4ed´elec, J., Acoustic and Electromagnetic Equations: Integral Representations for Harmonic Problems, Vol. 144 of
Applied Math. Sciences , Springer, 2001.
5Liz´e, B., Solveur Direct Haute Performance, Master’s thesis, Ecole Centrale Paris, 2009.
6Hestenes, M. R. and Stiefel, E., Methods of conjugate gradients for solving linear systems , Vol. 49, NBS, 1952.
7Saad, Y. and Schultz, M. H., “GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear
systems,” SIAM Journal on scientific and statistical computing , Vol. 7, No. 3, 1986, pp. 856–869.
8Greengard, L. and Rokhlin, V., “A fast algorithm for particle simulations,” Journal of computational physics, Vol. 73,
No. 2, 1987, pp. 325–348.
9R. Coifman, V. R. and Wandzura, S., “The fast multipole method for the wave equation: a pedestrian prescription,”
IEEE Antennas and Propagation Magazine , Vol. 35, No. 3, 1993, pp. 7–12.
10Darve, E., “The fast multipole method I: error analysis and asymptotic complexity,” SIAM Journal on Numerical
Analysis, Vol. 38, No. 1, 2000, pp. 98–128.
11Sylvand, G., La m´ethode multipˆole rapide en ´electromagn´etisme. Performances, parall´elisation, applications, Ph.D.
thesis, Ecole des Ponts ParisTech, 2002.
12Sylvand, G., “Performance of a parallel implementation of the FMM for electromagnetics applications,” Int. J. Numer.
Meth. Fluids, Vol. 43, 2003, pp. 865–879.
13Messner, M., Schanz, M., and Darve, E., “Fast directional multilevel summation for oscillatory kernels based on Chebyshev
interpolation,” Journal of Computational Physics, Vol. 231, No. 4, 2012, pp. 1175–1196.
14Darve, E. and Hav´e, P., “Efficient fast multipole method for low-frequency scattering,” Journal of Computational Physics,
Vol. 197, No. 1, 2004, pp. 341–363.
15Collino, F., “Analyse th´eorique d’une m´ethodes multipˆoles stable `a toutes ´echelles pour le noyau d’Hemholtz,” Tech. rep.,
CERFACS, 2013.
16Hackbusch, W., “A sparse matrix arithmetic based on H-matrices. part i: Introduction to H-matrices,” Computing,
Vol. 62, No. 2, 1999, pp. 89–108.
17Grasedyck, L. and Hackbusch, W., “Construction and arithmetics of H-matrices,” Computing, Vol. 70, No. 4, 2003,
pp. 295–334.
18“hmat-oss,” https://github.com/jeromerobert/hmat- oss, Accessed: 2016-04-20.
19Kriemann, R., “Parallel-Matrix Arithmetics on Shared Memory Systems,” Computing , Vol. 74, No. 3, 2005, pp. 273–297.
20Liz´e, B., esolution directe rapide pour les ´el´ements finis de fronti`ere en ´electromagn´etisme et acoustique: H-Matrices.
Parall´elisme et applications industrielles , Ph.D. thesis, Paris 13, 2014.
12 of 13
American Institute of Aeronautics and Astronautics
21Augonnet, C., Thibault, S., Namyst, R., and Wacrenier, P.-A., “StarPU: a unified platform for task scheduling on
heterogeneous multicore architectures,” Concurrency and Computation: Practice and Experience , Vol. 23, No. 2, 2011, pp. 187–
198.
22“Amibe,” http://jcae.sourceforge.net/amibe.html, Accessed: 2016-04-20.
23Schreiner, J., Scheidegger, C., Fleishman, S., and Silva, C., “Direct (Re)Meshing for Efficient Surface Processing,”
Computer Graphics Forum (Proceedings of Eurographics 2006), Vol. 25, No. 3, 2006, pp. 527–536.
13 of 13
American Institute of Aeronautics and Astronautics
... The numerical acoustic computations in this study are performed with the Airbus code ACTIPOLE (see Ref. [8]) based on the Boundary Element Method (BEM), which solves the Helmholtz equation and computes the propagated and scattered acoustic field for a variety of simplified acoustic sources (monopoles, dipoles, quadrupoles, modal surfaces, etc.). The meshed boundaries can be rigid or treated and the effects of a uniform flow can be included, although it is not the case here. ...
... This approach is theoretically equivalent to modelling a single microphone and 171 monopole sources (as it was done in Ref. [3]), but enables faster computations. The computations were run using the H-matrix solver of ACTIPOLE (see Ref. [8]) and a frequency resolution of 4 points per third octave band between 50Hz and 5000Hz. Plate installation effects (i.e the modeling of a step with respect to the ground) are studied by varying the thickness of the reflecting plates with respect to the surrounding ground. ...
Conference Paper
Full-text available
View Video Presentation: https://doi.org/10.2514/6.2021-2158.vid This paper introduces a new design of reflecting plate for aircraft noise measurements on the ground, which exhibits a low sensitiveness to installation effects and that allows to obtain accurate noise measurements even in the case of degraded installation conditions. Its improved acoustic performance is shown by comparison with an existing state-of-the art plate design, using both numerical predictions and flight test data. The effects of varying the ground impedance and the plate step with respect to the ground are investigated through numerical simulations, for both plate designs. Finally, some recommendations are proposed regarding ground microphone set-ups for aircraft noise measurements.
... The effect of aircraft noise on the environment is a main concern for the aviation industry and research institutes. There is an impelling need of fast numerical methods to predict aircraft noise sources and the corresponding installation effects during design cycles (Balin et al., 2016). Particularly for aeronautical applications, the problem involves sound with short wave length propagating for long distances and through non-uniform flows. ...
... Boundary element methods (BEM) coupled with fast algorithms, such as the fast multipole method (Gumerov and Duraiswami, 2004) or the H-Matrix algorithm (Hackbusch, 1999), can solve large-scale short-wavelength noise propagation around aircraft (Balin et al., 2016). On the other hand, a boundary element formulation which solves either the linearised Euler equations or the full potential linearised wave equation is yet to be provided-BEM needs a boundary integral solution and the corresponding Green's function. ...
... To this end, there is an impelling need to devise robust numerical methods to predict these effects during design cycles. 1 In this case, a large-scale short-wavelength sound propagation problem with a non-uniform flow must be solved. ...
... On the other hand, boundary element methods (BEM) in combination with fast algorithms, such as the fast multipole method 6 or the H-Matrix approach, 7 can efficiently solve large-scale short-wavelength noise propagation around aircraft. 1 Nevertheless, boundary element methods can only solve approximate physical models including non-uniform mean flow effects on wave propagation such as the Taylor wave equation 8 and the small mean flow gradient formulation proposed by Tinetti and Dunn. 9 While non-uniform mean flow effects are conventionally included using variable transformations, in order to reduce an approximate operator with mean flow to the Helmholtz operator with quiescent media, [9][10][11][12] solutions in the physical space, i.e. without transformation, have been proposed. ...
Conference Paper
Full-text available
A boundary element formulation is used in order to solve acoustic wave propagation in non-uniform potential subsonic mean flows. An approach based on a Taylor--Lorentz variable transformation is proposed for low Mach number mean flows. The variable transformation reduces an approximate formulation of the linearised potential wave equation to the standard wave equation with quiescent media. In the transformed domain, a standard boundary element method can then be used. This paper presents a strategy to exploit existing boundary element solvers for wave propagation with quiescent media to compute mean flow effects. The solution is proposed in the frequency domain for prescribed velocity boundary conditions. Numerical experiments are performed to test the proposed variable transformation for three-dimensional applications. A comparison between the results based on either the Taylor--Lorentz or a more conventional Lorentz transform assuming a uniform flow in the entire domain is provided. In the near field, the solutions based on the Taylor--Lorentz transform slightly improve on boundary element solutions based on the Lorentz transform. However, they provide similar results as soon as a region of uniform mean flow is reached.
... Instead we develop a general parametric geometry as well as automatic meshing rules to provide numerical models that are tractable using advanced linear algebra such as the H-matrix (HMAT) and the Fast Multipole Method (FMM) solvers, all implemented in the in-house Airbus ACTIPOLE software (Ref. Balin et al. [2016]) used throughout this paper. This parametric model allows us to control and achieve high accuracy solutions and seems to be robust to several plate shapes. ...
Conference Paper
Full-text available
View Video Presentation: https://doi.org/10.2514/6.2021-2141.vid A new numerical acoustic wave-absorbing geometry is proposed to evaluate the acoustic response of reflecting plates for ground microphones, used to measure the exterior noise of aircraft in flight. It is defined as a revolution surface, by rotation of a modified Euler spiral curve around the vertical axis. This allows us to control the effects of truncation of the infinite plane by varying the size and curvature of this domain. Optimized mesh rules are derived from a posteriori error estimates in order to control the error on the observable with the minimum number of unknowns. The models are validated against the infinite hardwall model. Simulations are done with the ACTIPOLE software which is based on the Boundary Element Method and makes use of fast linear solvers. It is shown that the acoustic pressure is precisely predicted for a large band of acoustic impact angles and frequencies.
... It should, however, be mentioned that several other well validated and documented codes for acoustic shielding calculations are around. Prominent examples, of which the author is aware of, are the Fast Scattering Code (FCS) developed at NASA Langley Research Center [48,50,49,18] and the ACTIPOLE code from Airbus Group Innovations [17,8,9]. Two important topics which will not be discussed in the present paper are (1) sound absorbing boundary conditions, which are important for the assessment of passive noise control measures like, e. g., acoustic liners, and (2) the treatment of broadband noise problems with Boundary Element Methods. ...
Article
Acoustic installation effects are considered as scattering problem. Two methods to investigate them are presented. First, a fast multipole method boundary element method (FMM-BEM) which can solve the Helmholtz wave equation for full scale aircraft configurations at frequencies of some kHz. There, low Mach number potential mean flow fields can be taken into account by a so-called Taylor transformation. Second, a discontinuous Galerkin method (DGM) which solves acoustic perturbation equations (APE) for realistic mean flow fields is presented. DGM calculations are very expensive and can be performed for full scale aircrafts only at low frequencies. Its main purpose so far is to assess the accuracy of the FMM for selected cases. The Taylor-transformed Helmholtz equation is derived and the fundamentals of the FMM are introduced. Some details of the DLR FMM code FMCAS are given. The basic DGM equations are derived for the APE and some implementation details of the DLR DGM code DISCO++ are discussed. For generic geometries results of the FMM and DGM are compared and the limits of the Taylor transformation are shown. Finally, scattering results for a 1 kHz Monopole at a full scale aircraft geometry will be presented.
... Galilean transformation [4] is the direct transformation, but it combines the time components in the space, and is not preferred for the discretization. Lorentz transformation (Prandtl-Glauert transformation) [5][6][7][8]9] is the most popular transformation in the uniform flow Helmholtz euqation. Recently the transformation that can contain the weak non-uniform background flow has attract the attentions. ...
Article
The simulation of the acoustic fields propagating in moving flows and the interaction of such fields with the mean flow itself is of importance in aviation acoustic prediction. Volume methods are preferred in this area. Boundary integral method is also concerned because of its own advantages. The transformation space is usually adopted in solving the convective Helmholtz Equation, physical space is also preferred for the convenient boundary condition implementation and combination with other methods. In this paper, the Convective Helmholtz Equation is solved by the Single layer Regularized Meshless Method (SRMM) in the physical space. The SRMM has been proposed by the author for the Laplace equation and Helmholtz Equation. SRMM has similar idea with the Equivalent Source Method (ESM), which represents the fields by the monopoles. But the source points overlap physical points. It overcomes the difficulty of the source points location choice, but the singularity arises. The subtraction and adding-back technique is adopted to get the amplitude of the singularity. The Dual-Surface Method is adopted to avoid irregular frequencies.
Article
Full-text available
We consider the scattering of acoustic perturbations in a presence of a flow. We suppose that the space can be split into a zone where the flow is uniform and a zone where the flow is potential. In the first zone, we apply a Prandtl-Glauert transformation to recover the Helmholtz equation. The well-known setting of boundary element method for the Helmholtz equation is available. In the second zone, the flow quantities are space dependent, we have to consider a local resolution, namely the finite element method. Herein, we carry out the coupling of these two methods and present various applications and validation test cases. The source term is given through the decomposition of an incident acoustic field on a section of the computational domain's boundary.
Article
Full-text available
We propose a novel surface remeshing algorithm. While many remeshing algorithms are based on global parametrization or local mesh optimization, our algorithm is closely related to surface reconstruction techniques and it requires no explicit parameterization. Our approach is based on the advancing-front paradigm, and it can be used to both incrementally remesh the complete surface, or simply to remesh a portion of it with a high-quality mesh. It is accurate, fast, robust, and suitable for use with interactive mesh processing applications that require local remeshing. We show a number of applications, including matching the resolution of meshes when doing Boolean operations such as unions and intersections. We also show how to adapt the algorithm to blend and merge mixed-mode objects — for example, to compute the union of a point-set surface and a triangle mesh.
Article
Full-text available
A class of matrices (\(\Cal H\)-matrices) is introduced which have the following properties. (i) They are sparse in the sense that only few data are needed for their representation. (ii) The matrix-vector multiplication is of almost linear complexity. (iii) In general, sums and products of these matrices are no longer in the same set, but their truncations to the \(\Cal H\)-matrix format are again of almost linear complexity. (iv) The same statement holds for the inverse of an \(\Cal H\)-matrix. This paper is the first of a series and is devoted to the first introduction of the \(\Cal H\)-matrix concept. Two concret formats are described. The first one is the simplest possible. Nevertheless, it allows the exact inversion of tridiagonal matrices. The second one is able to approximate discrete integral operators.
Article
Full-text available
In previous papers hierarchical matrices were introduced which are data-sparse and allow an approx- imate matrix arithmetic of nearly optimal complexity. In this paper we analyse the complexity (storage, addition, multiplication and inversion) of the hierarchical matrix arithmetics. Two criteria, the sparsity and idempotency, are sufficient to give the desired bounds. For standard finite element and boundary element applications we present a construction of the hierarchical matrix format for which we can give explicit bounds for the sparsity and idempotency.
Article
This paper describes an algorithm for the rapid evaluation of the potential and force fields in systems involving large numbers of particles whose interactions are described by Coulomb's law. Unlike previously published schemes, the algorithm of this paper has an asymptotic CPU time estimate of O(N), where N is the number of particles in the simulation, and does not depend on the statistics of the distribution for its efficient performance. The numerical examples the authors present indicate that it should be an algorithm of choice in many situations of practical interest.
Article
This paper describes the parallel fast multipole method implemented in EADS integral equations code. We will focus on the electromagnetics applications such as CEM and RCS computation. We solve Maxwell equations in the frequency domain by a finite boundary-element method. The complex dense system of equations obtained cannot be solved using classical methods when the number of unknowns exceeds approximately 105. The use of iterative solvers (such as GMRES) and fast methods (such as the fast multipole method (FMM)) to speed up the matrix–vector product allows us to break this limit. We present the parallel out-of-core implementation of this method developed at CERMICS/INRIA and integrated in EADS industrial software. We were able to solve unprecedented industrial applications containing up to 25 million unknowns. Copyright © 2003 John Wiley & Sons, Ltd.
Article
An algorithm is presented for the rapid evaluation of the potential and force fields in systems involving large numbers of particles whose interactions are Coulombic or gravitational in nature. For a system of N particles, an amount of work of the order O(N2) has traditionally been required to evaluate all pairwise interactions, unless some approximation or truncation method is used. The algorithm of the present paper requires an amount of work proportional to N to evaluate all interactions to within roundoff error, making it considerably more practical for large-scale problems encountered in plasma physics, fluid dynamics, molecular dynamics, and celestial mechanics.
Article
The solution of the Helmholtz and Maxwell equations using integral formulations requires to solve large complex linear systems. A direct solution of those problems using a Gauss elimination is practical only for very small systems with few unknowns. The use of an iterative method such as GMRES can reduce the computational expense. Most of the expense is then computing large complex matrix vector products. The cost can be further reduced by using the fast multipole method which accelerates the matrix vector product. For a linear system of size N, the use of an iterative method combined with the fast multipole method reduces the total expense of the computation to NlogN. There exist two versions of the fast multipole method: one which is based on a multipole expansion of the interaction kernel expιkr/r and which was first proposed by V. Rokhlin and another based on a plane wave expansion of the kernel, first proposed by W.C. Chew. In this paper, we propose a third approach, the stable plane wave expansion (SPW-FMM), which has a lower computational expense than the multipole expansion and does not have the accuracy and stability problems of the plane wave expansion. The computational complexity is NlogN as with the other methods.
Article
Many applications lead to large systems of linear equations with dense matrices. Direct matrix–vector products become prohibitive, since the computational cost increases quadratically with the size of the problem. By exploiting specific kernel properties fast algorithms can be constructed.A directional multilevel algorithm for translation-invariant oscillatory kernels of the type K(x,y)=G(x−y)eık∣x−y∣, with G(x−y) being any smooth kernel, will be presented. We will first present a general approach to build fast multipole methods (FMMs) based on Chebyshev interpolation and the adaptive cross approximation (ACA) for smooth kernels. The Chebyshev interpolation is used to transfer information up and down the levels of the FMM. The scheme is further accelerated by compressing the information stored at Chebyshev interpolation points using ACA and QR decompositions. This leads to a nearly optimal computational cost with a small pre-processing time due to the low computational cost of ACA. This approach is in particular faster than performing singular value decompositions.This does not address the difficulties associated with the oscillatory nature of K. For that purpose, we consider the following modification of the kernel Ku=K(x,y)e−ıku·(x−y), where u is a unit vector (see Brandt [1]). We proved that the kernel Ku can be interpolated efficiently when x−y lies in a cone of direction u. This result is used to construct an FMM for the kernel K.Theoretical error bounds will be presented to control the error in the computation as well as the computational cost of the method. The paper ends with the presentation of 2D and 3D numerical convergence studies, and computational cost benchmarks.