An Application of High Performance Computing to Improve Linear Acoustic Simulation Fouad Butt Abdolreza Abhari Jahan Tavakkoli Department of Computer Science Department of Computer Science Department of Physics Ryerson University Ryerson University Ryerson University Toronto, Canada Toronto, Canada Toronto, Canada f2butt@ryerson.ca aabhari@scs.ryerson.ca jtavakkoli@ryerson.ca Abstract A model describing the acoustic field resulting from an acous- tic source vibrating in a rigid planar baffle is found to be com- putationally intensive if the field values are computed sequen- tially. The temporal complexity of the model is firstly due to the large number of computations required to integrate over the surface area of an arbitrarily-shaped source and secondly, due to volume of the acoustic field itself. Thus, the model is assessed and it’s workload characterization derives directly from the data-level parallelism inherent in the computation of the acoustic field. Two high performance computing ap- proaches are developed and lead to improvements in both the precision and efficiency of the model with computation speedups that are beyond theoretical expectations. A further reduction in temporal complexity is introduced as a result of the axial-symmetric properties of the acoustic fields. The result is a particularly useful tool for high perfor- mance simulation of 3-dimensional ultrasound fields gener- ated by realistic sources in various fluid media. 1. INTRODUCTION A particular phenomenon, known as diffraction, is ob- served when a wave continues to propagate through an aper- ture or encounters a change in media. The formation of the diffraction pattern is described by the Rayleigh diffraction integral, though others have also expressed different solu- tions (see [1] for descriptions of the Schoch and King in- tegrals). Sommerfeld applied a Green’s function approach to the Rayleigh diffraction integral, which results in the Rayleigh-Sommerfeld integral [1]. Ocheltree and Frizzell begin with the Rayleigh- Sommerfeld integral to arrive at an expression for the pressure amplitude at a point p 0 [2]. High performance computing (HPC) solutions are sought to improve the preci- sion and efficiency of simulating an acoustic field based on their expression. A primary assessment of the model results in the identification of inherent data-level parallelism. The integration of a point in the acoustic field is independent of that same integration process at another point. The acoustic field is then decomposed to enhance the performance of the simulation by exploiting data-level parallelism and executing the integrations in parallel. An approach relying on multiple threads in a symmetric multi-processing (SMP) environment is implemented along with a secondary hierarchical and distributed approach that builds on multiple threads by supplementing them with mul- tiple processes. The simulations are performed on a single CPU for the multi-threaded approach and on multiple CPUs in a cluster for the multi-processed or cluster approach. The reflective symmetry of the values in the acoustic field is ex- ploited to reduce computations to only one-quarter of the acoustic field. The following outlines research related to scalar diffraction in linear acoustics and parallel computing. Subsequently, a detailed description of the linear acoustic simulation model is presented and the multi-threaded and cluster approaches are explained. An elaboration on a technique to reduce computa- tions is next and results of the performance enhancements are then discussed with conclusive remarks and possibilities for future work. 2. SCALAR DIFFRACTION IN LINEAR ACOUSTICS The Laplace transform of the velocity potential at a point P situated in a homogeneous, isotropic, non-dissipative medium for a piston source with a uniform velocity distribution vibrat- ing in an infinitely rigid baffle is described as [1]: Φ(r , s)= Z S V n (r 0 , s)e −sR/c 2πR dS (1) where the integration is performed over the surface area S of the transducer, V n is the particle velocity normal to the transducer surface, R is the distance from a spatial point P to the transducer in the plane z = 0 and c is the speed of sound in the medium. r and r 0 describe the distance from a point P to the origin and the distance from an infinitesimal element dS to the origin, respectively. A geometric illustration of this type of acoustic source model is presented in Fig. 1 and adapted from [1]: