RPG has developed a proprietary new Shape Optimizer™ software that allows custom shape design that simultaneously satisfies the architectural and acoustical requirements. RPG is presently using this new technology to develop a new generation of optimized surfaces, as well as collaborating with acousticians worldwide to design custom surfaces for specific application. To help explain the power of this new approach, we will describe a case study prompted by a request from an acoustical consulting firm. The architect requested a concave surface in the rear of a recital/rehearsal hall, because of existing site restrictions and aesthetics. 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.

Figure 1. Comparison between the original concave surface (blue) and the amplitude modulated optimized profile (red)

Optimization Parameters
To optimize this concave arc profile, we require the desired width and depth, scattering coverage, depth modulation percent (we used 30%), number of harmonics, and source and receiver locations. 2D surface optimization will be the subject of a future discussion. 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. In Figure 1 we illustrate the original concave surface and the amplitude modulated optimized profile. In this example, sources and receivers were placed between 2 and 6 m from the surface and scattered between +/- 800. The concave arc has a radius of about 2.3 m 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 m and evaluate the scattered pressure on a receiver arc of radius 2.3 m.

Figure 2. Comparison between the 500 Hz angular response of the original concave surfaced and the optimized profile.

Figure 3. Comparison between the 2 kHz angular response of the original concave surface and the optimized profile.

Results
In figures 2 and 3, the focusing from the concave arc is apparent by the concentrated energy at 900. (blue lines). After the optimization has converged to an acceptable solution, the optimized surface is evaluated as before using a BEM analysis. In Figures 2 and 3, compare the pressures on a 2.3 m arc from a source at 2.3 m for 500 Hz and 2 kHz, respectively. The red lines indicate how the focusing present with the concave surface (blue lines) has been minimized. Ideal diffusion in this case would be a flat horizontal line. it can be seen that the optimized profile provides excellent uniform coverage. In Figure 4 we illustrate two photos of the actual installation at Edwina Palmer Hall (1) and (2), along with a rendering of the surface from the Shape Optimizer™ program (3). The hall was very well received and there was no evidence of focusing effects.

Figure 4. Comparison of the rendered image from the Shape Optimizer™ and an actual photo of the completed installation

Home: Research & Development: Research Topics: Concave Surface Optimization