A global optimization method for designing resonant shapes in two-dimensional fluid domains with plane-wave...

Acoustic resonance has found applications in ultrasonic cleaning, sonochemistry, and particle manipulation. However, conventional approaches optimize resonant structures by performing finite element analysis while iteratively updating the shape, making them computationally expensive and prone to local optima. To address these problems, this study presents a method that first determines the optimal internal acoustic field and then extracts the boundary shape. The acoustic field is represented as a superposition of plane standing waves. Since the resonator is driven by a vibrating boundary, not all such superpositions are physically realizable; we identify a subspace of fields satisfying the boundary conditions and optimize within it. Global optimization using Lagrange multipliers avoids local optima. Since the particle velocity is tangent to its streamlines, rigid boundaries placed along them satisfy the zero-normal-velocity condition without disturbing the optimized field; the resulting shapes form the resonator structure. To validate our method, we derived resonator structures for two design targets: concentrating acoustic energy on a prescribed target line, and producing a standing wave tilted with respect to a rigid piston boundary. In both cases, the optimization reproduced the intended acoustic fields. Comparison with finite element analysis showed that the relative acoustic field distributions agreed, with a maximum difference of 0.028 when each field is normalized by its own maximum value, confirming the validity of the approach. This method enables rapid global optimization for designing resonant shapes while maintaining a broad search space and allowing quick assessment of the physical feasibility of acoustic fields.