Validating topology optimization for acoustics

Topology optimization is a mathematical method for tuning a system’s shape or mass distribution to achieve a specific function or behavior. Used for more than two decades in structural mechanics—for such purposes as maximizing stiffness or minimizing cost—it has found application in numerous other areas, including acoustics. But regardless of how optimized the calculated designs are in theory, they must still satisfy such real-world constraints as manufacturability. The methodology must also be validated through real-world confirmation. Rasmus Christiansen, Ole Sigmund, and Efren Fernandez-Grande of the Technical University of Denmark now provide that validation in an acoustics setting: the topological optimization of an acoustic cavity. The design goal was to minimize, in two dimensions, the acoustic pressure in a 2 cm × 2 cm region (blue in the figure) near the lower right corner of an 18 cm × 9 cm rectangular cavity by allowing the shape of the upper wall to vary. The walls are perfectly reflecting, except for a speaker near the lower left corner that outputs a single frequency. The figure shows the calculated optimized design of the upper wall and the resulting simulated sound pressure in decibels. The researchers fabricated the contoured surface using a 3D printer and then, while precisely controlling the cavity humidity and temperature, used a microphone to measure the sound at a regularly spaced array of points in the green region. The experimental positions of the pressure maxima and nodal lines agreed well with optimization results, especially when factoring in microphone positioning errors and background noise. But the measurements highlighted the need for the calculations to account for damping, frequency and environmental variations, and manufacturing tolerance. (R. E. Christiansen, O. Sigmund, E. Fernandez-Grande, J. Acoust. Soc. Am. 138, 3470, 2015, doi:10.1121/1.4936905 .)