RPG® is presently using a powerful new surface optimization capability — which optimally shapes surfaces to provide the desired scattering — to develop a new generation of optimized surfaces. This new technology is a feature of the CHAOS collaboration with acousticians worldwide to design custom surfaces for specific applications. To help explain the power of this new approach, we will describe a case study prompted by a request from an acoustical consulting firm.

Because of existing site restrictions and aesthetics, the architect requested a concave surface in the rear of a recital and rehearsal hall. Concave surfaces have plagued acousticians for centuries and so it seemed like an ideal challenge for the new technique. To minimize the focusing effects associated with a concave surface, we selected the amplitude modulation feature of the Shape Optimizer™. This feature is useful when optimization is desired, while maintaining the general features of the requested surface.

Optimization Parameters
To optimize this concave arc profile, we required the desired width and depth, scattering coverage, depth modulation percent (we used 30%), number of harmonics, and source and receiver locations. The program generates a surface shape from the specified set of sinusoidal harmonics which are then used to amplitude modulate the concave arc. The optimized surface may be constrained to only include concave sections whose focal point is less than the closest receiver positions. However, ray tracing and Boundary Element Method (BEM) analysis indicate that this is a minor issue.

Figure 1
In Figure 1 we illustrate the original concave surface and the amplitude modulated optimized profile. In this example, sources and receivers were placed between two and six meters from the surface and scattered between ±80°. The concave arc has a radius of about 2.3 meters, so sources at that location focus straight back onto themselves. Before and after optimization we must evaluate the performance using a full BEM analysis. To describe the concave surface before optimization, we place a source at 2.3 meters and evaluate the scattered pressure on a receiver arc of radius 2.3 meters. In Figures 2 and 3, the focusing from the concave arc is apparent by the concentrated energy at 90°.

Results
After the optimization has converged to an acceptable solution, the optimized surface is evaluated as before using a BEM analysis.

Figure 2

Comparison between the 500Hz angular response of the original concave surface and the optimized profile.

Figure 3

Comparison between the 2KHz angular response of the original concave surface and the optimized profile.

In Figures 2 and 3, we compare the pressures on a 2.4 meter arc from a source at 2.3 meters for 500 Hz and 2 KHz, respectively. It can be seen in the red lines that the focusing present with the concave surface has been minimized. In this case, ideal diffusion would be a flat horizontal line. It can be seen that the optimized profile provides excellent uniform coverage.