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The predominant methods for solving linear systems with the large dimension n beyond the reach of direct solv- ers are preconditioned iterative approaches (Smith et al.,
1996; Dohrmann et al., 2010). These methods solve the linear system in an iterative fashion rather than using a direct factorization. A common preconditioned iterative approach is based on the concept of a divide-and-con- quer strategy where the physical domain is divided into disjoint partitions and each partition is handled by a separate CPU (Smith et al., 1996; Dohrmann and Wid- lund, 2010). Other preconditioners based on multigrid are popular in other applications, wherein each partition is further subdivided into another level of partitions.
Here, we present examples of applications where HPC has been utilized to solve acoustics problems of the form in Eq. 2, with a large dimension n that would not be possible with smaller scale computing platforms. Solutions of problems with over 2.2 billion degrees of freedom are presented.
Underwater Acoustics for Ship Shock
One application of underwater acoustics is ship-shock testing. Vessels in the Navy fleet must undergo ship-shock tests before they are certified for service. These tests involve setting off large underwater explosives near the vessel of interest, typically around 75% of the expected failure load. The purpose is not to sink the vessel but to find out what breaks when the ship is exposed to nearby explosions (e.g., electronics, chairs, pipes). Such at-sea tests are extremely expensive and take a vessel out of the fleet for many months or even years. The more a ship is damaged in such a test, the longer it takes to return to the fleet. Computational modeling of a ship-shock event is one strategy to design components that will survive a ship-shock test. In the far field, the large pressures generated by explosives can be modeled as acoustic pressure waves impinging on the ship. These underwater acoustics applications exhibit large simu- lation domains and frequency ranges of interest that result in many wavelengths in the domain. As such, they lend themselves well to solution via HPC (Moyer et al., 2016).
Figure 2 shows slices of the acoustic pressure field reflected from a stiffened cylinder in a transient-coupled structural-acoustics simulation. Acoustic loading is due to an underwater explosion away from the submerged, air-filled structure. Rings on the surface of the cylinder indicate the mechanical response. The near-field fluid
domain consists of an ellipsoidal region composed of tetrahedral elements with an acoustic material formula- tion. The far-field, semi-infinite domain is approximated by infinite elements, which are not shown. Figure 2, inset, shows a typical gauge time history predicted by the analysis.
Simulation of Ground-Based Acoustic Tests
Qualification tests of aerospace structures and flight vehicles require that the structures be subjected to acoustic loads that are representative of the environ- ments that will be experienced in-flight. One way to achieve this, of course, is to conduct a full-scale flight test on the structure. The associated accelerometer and/ or pressure sensor measurements can then be used to assess the acoustic environment, and the resulting structural response.
However, flight tests tend to be very expensive, and due to instrumentation and telemetry limitations, only limited accelerometer data are typically available from such tests. As a result, ground-based acoustic testing is a common alternative wherein the structure is sub- jected to representative acoustic fields in an acoustic test chamber. Typically, high-powered speakers and other acoustic sources are used to generate the acoustic fields. The advantages of ground-based testing are that the cost is typically only a small fraction of that of a flight test
  Figure 2. The simulated response of a stiffened cylinder subject to underwater explosion is a demonstration of a typical Navy use of the FEM. The incident pressure wave excites the structure as it is reflected off the surface. Inset: a typical gauge time history. These results can be used to design structures to withstand explosive detonations in the surrounding medium (water).
26 Acoustics Today • Summer 2020

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