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 equation (along with the associated boundary conditions) numerically yields a discrete set of n linear equations
A p = F (2)
where A is an n × n matrix that contains a discrete rep- resentation of the continuous Helmholtz equation, p is a vector of unknown discrete acoustic pressures at the nodes of the discretized model, and F is an n × 1 vector containing information about boundary conditions and energy/load sources for the acoustics problem at hand. For completeness, we note that in the case of boundary element methods, one would start with the Helmholtz integral equation instead of the differential form given in Eq. 1. The details are omitted here for brevity.
High-Performance Computing for Acoustics
In high-performance computing (HPC) for acoustics, a challenge is to solve equations in the form of Eq. 2 when the number of unknowns (n) becomes too large for a single computer. Modern HPC platforms and the cor- responding software for domain decomposition and parallel communication are revolutionizing the numeri- cal solution of acoustic wave equations. This is enabling the solution of practical engineering problems in acous- tics such as airborne acoustic propagation (Hart et al., 2016), sonar applications, and aeroacoustic noise miti- gation that were not possible using previous generations of computers. The enabling HPC technology allows one to resolve acoustic wave propagation in ever-increasing domain sizes (or, equivalently, ever-increasing frequency ranges, e.g., megahertz) of interest for the wave propa- gation. The number of discrete equations to be solved increases with the frequency range and domain size. Eventually, the growing number of degrees of freedom and corresponding matrix storage requirements preclude the solution of the problem on one’s laptop or desktop as memory resources in the computer are exceeded.
Modern HPC platforms, based on either distributed cen- tral processing units (CPUs) and/or graphics processing units (GPUs), are built to optimize the use of memory resources on the largest problems in computational physics. In the case of acoustics, as the frequency range and/or domain size increases and the required in-core memory [aka random-access memory (RAM) for storing bits of information] resources correspondingly increase, an acoustics researcher can, in principle, simply employ
larger numbers of CPUs and/or GPUs on computing clusters to enable the numerical solution. Modern HPC clusters deployed by the US Department of Defense (DOD) and Department of Energy (DOE) laboratories have access to tens of thousands of CPUs/GPUs, each with substantial in-core memory resources. The prob- lem then becomes how to tailor the numerical method of interest so that it can be applied in these novel, distrib- uted memory and architectural computing environments.
Because a wide range of acoustics applications encoun- ter large-domain sizes and/or high frequencies of interest, the ability to numerically solve the acoustics equations in a scalable way is of broad interest across the field of acoustics engineering. Large-domain sizes present them- selves in underwater acoustics (Duda et al., 2019), waves in atmospheric propagation scenarios (Hart et al., 2016), aeroacoustics for airborne structures, architectural acoustics in large concert halls, and large-scale acoustic chambers for testing aerospace structures (Schultz et al., 2015), to name a few. In these applications, the large size of area where the acoustic solution is desired translates to large matrices for the numerical solution and hence the need for HPC. Equivalently, applications with high frequencies present precisely the same computational chal- lenges as large-domain sizes because in both cases the large number of wavelengths to be resolved requires more and more discrete elements and/or nodes to resolve the wave propagation. Ultrasound applications (Suslick, 2019) are an example where, due to the high frequencies and hence small wavelengths, numerical methods require large num- bers of degrees of freedom for the solution. HPC has the potential to enable the solution of these and other acous- tics problems across a variety of engineering disciplines.
In many acoustics applications, the FEM is an attractive numerical strategy. Some advantages of the method include • The ability to construct unstructured, body-fitted
meshes that capture curved interfaces between com-
plex fluid/structural domains;
• Sparse systems (i.e., matrices wherein most entries
are zeros) of algebraic equations that, when combined with a FEM of an elastic structure, render a coupled system of equations that is still sparse;
• The ability to solve either linear and/or nonlinear acoustic wave equations; and
• The ability to easily handle spatially varying material properties (e.g., capturing the speed of sound and
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