# Aalto subproject 3

# Enhancing finite and boundary element methods for efficient high-performance computing

## Demand

The capability of accurately and reliably predicting acoustic wave propagation with numerical simulations plays an important role in many industrial research and development tasks. Simulations based on accurate numerical approximations of physical phenomena can provide a considerable alternative for tedious prototyping and measurements. In acoustics, a linear wave equation is an acceptable model for sound propagation in many fluids, such as air and water. If the acoustic field is periodic, time-harmonic, the wave equation reduces to the Helmholtz equation. This essentially simplifies the solution process, but on the other hand, requires separate solutions for each time-harmonic frequency.

Boundary element method (BEM) is an attractive technique for solving acoustic wave problems involving unbounded domains. The method is based on the equivalence principle, where the solutions of Helmholtz equation are expressed in terms of the boundary values of an acoustic pressure. The great benefit of BEM is that it requires only surface meshes and unknowns. This property significantly simplifies the mesh generation procedure and leads to much reduced data structures. On the other hand, the main drawback of BEM is its high computational cost. The time and memory required to solve the associated linear system scale very badly as the number of degrees of freedom is increased. To solve large-scale problems with BEM, therefore special fast solvers that significantly reduce the high computational load of the traditional BEM are of paramount importance.

## Solution

The main goal of the SIMPRO subproject Aalto 3 was to enhance the existing fast acoustic boundary element solver developed in the Aalto University. This solver, called
*CompA*, utilises the
*multilevel fast multipole algorithm* (MLFMA) to speed up the computations. It can be used to predict sound pressure levels for time harmonic acoustic scattering and emission from structures modelled with an acoustic surface impedance. The solver is able to consider large-scale acoustic problems with millions degrees of freedom on a broad range of frequencies from a couple of Hz up to tens of kHz. The method is particularly well-suited for exterior acoustic problems where high accuracy is needed, e.g., in audio acoustics.

During the SIMPRO project, several technical improvements were made on the CompA solver. These improvements have further enhanced the efficiency, stability and robustness of the solver. From a scientific point of view, the most significant result was the development of a novel combination of the local and global interpolators that opens the door for high-performance distributed memory parallelisation. From the practical numerical simulation point of view, the most important result was the utilisation of MLFMA to speed up the computation of the BEM source term in the case of large vibrating surfaces and in the evaluation of the sound pressure levels in the case of high number of field points. These improvements remove the two major practical bottlenecks of the previous version of CompA solver.

## Benefits and use cases

The improved CompA solver enables faster acoustic analyses of large and complex problems and can be utilised for e.g. speaker and microphone design and simulation of room and cabin acoustics. By combining the solver with a mechanical simulation tool allows accurate estimation of noise levels of vibrating structures and machines. With the high-performance CompA solver it is possible to optimise a design so that the sound pressure levels are as desired or vibration of the structure and noise emitted to the environment is minimised.

The CompA solver has been integrated with Kuava's
*Waveller Cloud* audio and noise simulation technology. In Figure 1, a head-related transfer function is simulated for a human torso using standardized acoustic mannequins and individual head models. As a second example, the surface velocity of a gear box is computed using a structural dynamics FEM software and the resulting velocity profile is then used as an input in acoustics solver to find noise emission. Figure 2 shows the computed surface pressure.

*Figure 1. A head-related transfer function is simulated for a human torso using standardized acoustic mannequins and individual head models. (Image courtesy: Kuava Oy.)*

*Figure 2. An example of computed surface pressure. (Image courtesy: Kuava Oy.)*

## Subproject deliverables and results

Deliverables:

- Ylä-Oijala, P.
*et al.***Comforming boundary elements in acoustics**

Engineering Analysis with Boundary Elements, vol. 50, January 2015, Pages 447–458

DOI: 10.1016/j.enganabound.2014.10.002

Link: http://www.sciencedirect.com/science/article/pii/S0955799714002446 - Järvenpää, S. & Ylä-Oijala, P.
**Multilevel fast multipole algorithm with global and local interpolators***Antennas and Propagation*, IEEE Transactions on , vol.62, no.9, pp.4716–4725, Sept. 2014

DOI: 10.1109/TAP.2014.2333056

Link: http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=6846325 - Järvenpää, S. & Ylä-Oijala, P.
**A global interpolator with low sample rate for multilevel fast multipole algorithm***Antennas and Propagation*, IEEE Transactions on , vol.61, no.3, pp.1291-1300, March 2013

DOI: 10.1109/TAP.2012.2231927

Link: http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=6374227 - Järvenpää, S. & Ylä-Oijala, P.
**MLFMA with local and global interpolators***Radio Science Meeting (Joint with AP-S Symposium), 2015 USNC-URSI*, vol., no., pp.84–84, 19–24 July 2015

DOI: 10.1109/USNC-URSI.2015.7303368

Link: http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=7303368 - Järvenpää, S. & Wallen, H.
**Improvements to the numerical computation of the evanescemt part in broadband MLFMA***Radio Science Meeting (Joint with AP-S Symposium), 2015 USNC-URSI*, vol., no., pp.85–85, 19–24 July 2015

DOI: 10.1109/USNC-URSI.2015.7303369

Link: http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=7303369

Other project results:

- Conforming boundary elements in acoustics
- Calderon preconditioners for acoustic surface integral equations
- Combined global and local interpolators for the multilevel fast multipole algorithm
- Pressure evaluation speed up with the multilevel fast multipole algorithm
- New algorithms computing the translators of the broadband multilevel fast multipole algorithm
- SIMD (Single Instruction, Multiple data) acceleration of the calculation of system matrix elements and sound pressure levels

## Contacts

- Dr. Doc. Pasi Ylä-Oijala, Aalto University, pasi.yla-oijala@aalto.fi
- Dr. Seppo Järvenpää, Aalto University, seppo.jarvenpaa@aalto.fi