h-matrix bem and fem ao solvers for large professional audio

The 21st International Congress on Sound and Vibration
13-17 July, 2014, Beijing/China
H-MATRIX BEM AND FEM AO SOLVERS FOR LARGE
PROFESSIONAL AUDIO SYSTEMS SIMULATION
Giuseppe Miccoli1 and Tommaso Nizzoli2
1
CNR-IMAMOTER, National Research Council of Italy, Ferrara, Italy
e-mail: [email protected]
2
Acoustics Vibration Consultant, Reggio Emilia, Italy
e-mail: [email protected]
The most innovative BEM & FEM analysis techniques and solvers implemented in a CAE
commercial code have proven to be an invaluable designer tool in a Professional Audio Application. The acoustic radiation on a quarter sphere of microphones of an Arrayable Loudspeaker’s Horn is calculated and compared to the measurements. This article documents the
progress we have done so far to check the capabilities of commercially available computational methodologies in relation to large professional audio systems simulations.
1.
Introduction
The possibility of designing components and/or whole systems in an efficient and costeffective way will call for reliable simulation methods to tackle real industrial cases.
Thoroughly investigation of the acoustic radiation of a Professional Audio’s component, midhigh frequency device, part of TTP5-A, true modular Point Source Loudspeaker designed by RCF
S.p.A is carried out by using state-of-the-art BEM and FEM solvers.
Siemens’s LMS Virtual.Lab R12 adds the H-Matrix boundary element method (H-Matrix
BEM) and the FEM Automatically Matched Layer Adaptive Order (FEM AO) to other advanced
computational techniques such as Fast Multipole BEM (FMBEM) and FEM Automatically
Matched Layers (FEM AML).
The acoustic radiation measured on a quarter sphere of microphones is compared to the simulation results and comparisons are carried out by means of acoustic balloons, horizontal and vertical
plots, beamwidth plots.
2.
Arrayable loudspeaker horn and experimental test setup
The RCF TTP5-A (Fig. 1) is a high power, two way active array module engineered to deliver
high fidelity output to be used in indoor and outdoor medium and large spaces. The system is designed to create horizontal or vertical arrays with a constant curvature. The system is planned to be
used from medium-small theatres to very large outdoor stadia and public spaces. The TTP5-A’s
high frequency point source audio waveguide and horn system covers an angle of 22.5° in the horizontal plane and 60° in the vertical one. The coverage angles are defined as the -6 dB decay from
the front axis acoustic response moving horizontal-wise or vertical-wise.
ICSV21, Beijing, China, 13-17 July 2014
1
21st International Congress on Sound and Vibration (ICSV21), Beijing, China, 13-17 July 2014
Figure 1. Compression driver and waveguide (left); Side view of the TTP5-A’s horn (right)
The RCF Test Lab Acoustic Balloon measurement setup (Fig. 2) rotates the DUT by two axes
around the centre of rotation (the horn’s throat) measuring the impulse response at each angle at a
10 m distance microphone flush to the ground (to avoid the first acoustic wave reflection).
Figure 2. Robot for the acoustic balloon measurement (left), Virtual microphone positions (right)
All the measurements (Figg. 3, 4, 5 and 6) are imported, processed and represented by AFMG
Speaker Lab Software. In this article test and simulation results are acquired and simulated on the
same Quarter Space π stearadians Acoustic Balloon of microphones (red dots of Fig. 2), centred on
the horn’s throat. The Full Space 4π Acoustic Balloon representation of results is processed automatically by the software. The virtual microphones are positioned in the same way on the quarter
space balloon with 10 m radius centred on the horn’s throat. The microphones have a one degree
spacing over the meridians and 5 degrees on the parallels. So the total number of nodes on each of
the 19 meridians is 181 and they have been numbered accordingly to define uniquely the angle position in space.
The 0 degree microphone is positioned in front, on axis at 10 m distance, while the 180 degree
microphone is at 10 m distance at the horn’s back. The horizontal meridian corresponds to 0°, the
vertical to 90°.
ICSV21, Beijing, China, 13-17 July 2014
2
21st International Congress on Sound and Vibration (ICSV21), Beijing, China, 13-17 July 2014
Figure 3. Horizontal and Vertical Beamwidths (-6dB angle) from measurements: the beamwidth frequency plot represents the angles by which the response decays -3dB, -6dB, -9dB from the 0° front measurement. Both the horizontal 0° meridian plot and the vertical 90° meridian plot beamwidths are shown
Figure 4. The in-plane representation of the acoustic pressure decay on the 0° meridian, sweeping
left-wise and right-wise (from the perspective of the horn’s mouth) from 0° front to 180° back. The plot on
the right is the vertical map which represents all the acoustic responses plotted in-plane sweeping in the 90°
meridian. Due to quarter space symmetry these plots are symmetric
Figure 5. Acoustic Balloon from measurements, (2.5 kHz and 4 kHz): the well-known Acoustic Balloon representation of the pressure amplitude (dB) is the relative pressure measured on all microphones (on
quarter space at 10 m and then duplicated to give the full space image) referenced to 0 dB on-axis in front of
the horn
ICSV21, Beijing, China, 13-17 July 2014
3
21st International Congress on Sound and Vibration (ICSV21), Beijing, China, 13-17 July 2014
Figure 6. Acoustic Balloon from measurements, 6.3 kHz and 10 kHz
3.
CAE analyses
3.1 H-Matrix BEM
The use of standard BEM techniques limits either the upper frequency or geometric size of the
analysis as six elements per wavelength are required in order to achieve good result accuracy. This
rapidly increases the size of the system with respect to frequency [1]. The H-Matrix Boundary Elements Method (H-Matrix BEM) computes acoustic radiation using a state-of-the-art Hierarchical
Matrix BEM solver. It uses recursive matrix storage and compression, based on the low rank approximation. H-Matrix BEM efficiently handles medium to large models, with key benefits:
•
•
•
Speed: faster computation as it uses matrix compression technology;
Efficiency: reduces the memory requirements as it uses hierarchical matrix storage and
compression;
Scalability: multi-load cases handled efficiently with direct solver approach.
The computational effort is reduced from O(n3) to O(n*log(n)) where n is the number of unknown variables as can be evinced from the graph below (Fig. 7). The Fast Multipole Boundary
Element Method is still a very competitive solver for very large problems [2] and as it proved to be
for this professional audio application [3].
Figure 7. Comparison between conventional BEM, H-Matrix BEM and Fast Multipole BEM Methods
Anyway H-Matrix is the optimal solver for this mid-large acoustic geometry and frequency
range of analysis. The model (Fig. 8) comprises 24k nodes and 48k shell elements and solves in 4h
24’ with a linear step sweep of 100 Hz in the 1400 Hz to 12000 Hz frequency range.
Results of the computation are reported in the following Figg. 9, 10, 11, 12.
ICSV21, Beijing, China, 13-17 July 2014
4
21st International Congress on Sound and Vibration (ICSV21), Beijing, China, 13-17 July 2014
Figure 8. Waveguide and Horn H-Matrix BEM Model
Figure 9. Horizontal and Vertical Beamwidths (-6dB angle) from H-Matrix BEM simulation
Figure 10. Horizontal and vertical coverage maps from H-Matrix BEM simulation
Figure 11. Acoustic Balloon from H-Matrix BEM simulation, 2.5 kHz and 4 kHz
ICSV21, Beijing, China, 13-17 July 2014
5
21st International Congress on Sound and Vibration (ICSV21), Beijing, China, 13-17 July 2014
Figure 12. Acoustic Balloon from H-Matrix BEM simulation, 6.3 kHz and 10 kHz
3.2 FEM AML Adaptive Order (FEM AO)
A truly breakthrough FEM solver technology is the Adaptive Order (FEM AO) Method implemented in LMS Virtual.Lab R12 [1].
FEM AO foresees the order of each element at each frequency to ensure accuracy. Higher order shape functions are used to represent the pressure inside each element. At order 10, an element
can span more than two acoustic wavelengths.
The solver increases element order with frequency and therefore the DOFs number, with key
benefits:
• Important savings on time and memory in lower frequencies;
• Development of smaller models on pre-processor which can be handled easier;
• Discretization only needs refinement in order to capture accurately the geometry and
boundary conditions.
Essentially, higher orders are used at high frequencies and/or for large elements and low orders will be employed at low frequencies and/or for small elements.
With 85725 nodes and 444036 solid elements only (Fig. 13) one mesh is necessary to span an
acoustic frequency range of a decade from 1 kHz to 10 kHz.
Figure 13. Waveguide and Horn FEM AO model, AML surface highlighted
Moreover, a FEM Automatically Matched Layer surface boundary condition is applied on the
horn aperture (Fig. 13). Whereas with the PML method the (theoretically unbounded) acoustical
domain is meshed with standard finite elements up to a certain distance from the radiating or scattering structure, in the AML implementation the generation of the PML is automated (thus the name
ICSV21, Beijing, China, 13-17 July 2014
6
21st International Congress on Sound and Vibration (ICSV21), Beijing, China, 13-17 July 2014
“Automatically Matched Layer”) and all the user has to do is to build the standard FE mesh of the
near-field radiating region.
The FE mesh extends to 5 cm from the horn’s mouth. Acoustic complex pressure results are
solved exactly as for the H-Matrix BEM model at a distance of 10 m on the quarter space balloon of
microphones. Fig. 14 shows memory required and time for each step for a computation run.
Figure 14. Charts showing Memory (MB) required to complete and Time for each step
All the analyses to be referred to the two simulation methods here tested have been carried out
using a 4 core Intel Xeon CPU @ 3,60 GHz DELL PC, 1 processor (4 cores) and 16 GB RAM.
Results of the computation are reported in the following Figg. 15, 16, 17 (figure showing horizontal and vertical coverage maps not included here for a matter of space).
Figure 15. Horizontal and Vertical Beamwidths (-6dB angle) from FEM AO, AML simulation
Figure 16. Acoustic Balloon from FEM AO, AML simulation, 2.5 kHz and 4 kHz
ICSV21, Beijing, China, 13-17 July 2014
7
21st International Congress on Sound and Vibration (ICSV21), Beijing, China, 13-17 July 2014
Figure 17. Acoustic Balloon from FEM AO, AML simulation, 6.3 kHz and 10 kHz
4.
Conclusions
This paper would like to shed light on the H-Matrix BEM and FEM AO with AML advanced
simulation techniques in a Professional Audio Transducer Application. The competitive solver
times and the creation of the GLL file (Generic Loudspeaker Library) of the virtual source uprights
virtual prototyping from early design stage through sound reinforcement applications.
In the following Table 1 the measurement and simulation times are compared. The measurements have to be referred to a very fine step in the 20 Hz – 20 kHz frequency range. The simulations are solved in the 1400 Hz – 12 kHz frequency range, 100 Hz linear step sweep for H-Matrix
BEM and in the 1400 Hz – 10.5 kHz, 100 Hz linear step sweep for FEM AO. The H-Matrix BEM
model has been discretized taking into account the 6 elements per wavelength rule with a 12 kHz
maximum analysis frequency and a 4.7 mm maximum element length.
Table 1. Simulation Models Characteristics & Computation Performance Comparison
Category
Quarter Balloon
Measurements
H-Matrix BEM
FEM AO with AML
Measurement
&
Simulation
Time (Hrs)
3
4
2
Model
Nodes
Model
Elements
Multi CPU
processing
RAM
usage
24 k
85725
48 k
444036
8 CPUs
1 CPU
12 GB
variable
The authors wish to thank RCF S.p.A. for the use of products, software and instruments of its
R&D Test Laboratory.
REFERENCES
1
2
3
LMS Virtual.Lab R12 User’s Manual, (2014)
R. Hallez, K. De Langhe, Solving large industrial acoustic model with the fast multipole method, Proceedings of
the 16th International Congress on Sound and Vibration, ICSV16, Krakòw, (2009)
G. Miccoli, T. Nizzoli, Arrayable Loudspeaker Horn BEM/FMBEM & FEM/AML modeling and analysis, Proceedings of the 20th International Congress on Sound and Vibration, ICSV20, Bangkok, (2013)
ICSV21, Beijing, China, 13-17 July 2014
8