Design of a 3D phononic-fluidic sensor using shape optimization

A phononic-fluidic sensor consists of a fluidic cavity resonator with 3D phononic crystal (PnC) layers around it. The phononic structures around the cavity effectively improve the boundary conditions of the cavity resonator, significantly increasing quality factor and resolution. Thereby, such combination allows to measure volumetric properties of liquids, for example, density and speed of sound, especially in small volumes. This sensor concept was realized in one- and two-dimensional arrangements [1, 2]. Additionally, this concept was implemented in three-dimensional arrangements [3, 4]. The main motivation of this work is to suggest a new sensor design using shape optimization of air inclusions of phononic structures, which provides acoustic resonance peaks with higher quality factors. In this work we describe the first application of shape optimization to improve our 3D phononic-fluidic structures.
For optimization we used the computational setup shown in Fig. 1a. The model is meshed to cover the minimum wavelength encountered in a study: the maximum element size is set to a/12, where a is the lattice constant of the PnC, see Fig. 1b. Since a shape optimization process is computationally demanding, we decided to use a semi-infinite model using periodic boundary conditions (PBC). Contact with emitter and receiver to excite and detect transmitted waves is modeled as a low-reflection impedance boundary condition with the effective transducer surface impedance. Furthermore, we excited the emitter with a constant time-harmonic velocity amplitude. As fluid domain, we used two arbitrary liquids with different speeds of sound. The optimization problem is formulated as increasing the quality factor Q of the acoustic resonance peak. This is a challenging multifrequency optimization problem, which requires solving at least three frequencies. Additionally, the optimization problem was constrained with a maximal displacement from the initial shape of 0.04a, in order to avoid intersection of PnC air inclusions and cavity walls. As optimizer we used the method of moving asymptotes (MMA).

To demonstrate the capability of optimization we started with a 2D model. In Fig. 2a we displayed the deformed shape after 200 iterations and in Fig. 2b we illustrated the transmission spectra of initial and optimized designs. As a result, the optimized design shows a Q-factor of 268, an increase by 2.6 times. In Fig. 3a we demonstrated the deformed shape for a 3D case, starting from the initial guess of our cubic cell with spherical void [4]. For the 3D case the Q-factor is increased by 65%. (Fig. 3b). Moreover, a secondary effect of optimization for the 3D case is that the spurious, undesired bump near the first resonance peak in the
optimized design is at 235 kHz, i.e. further than for the initial one at 222 kHz. This effect helps us to get a less disturbed signal around the resonance. Finally, the semi-infinite optimized mesh geometry is exported, expanded into a finite structure including bottom and side walls around the cavity, and fabricated using additive fabrication methods (Fig. 4). Transmission measurements are carried out to compare the Q-factor of initial and optimized design to validate simulation results.

Moreover, we are planning to test different initial shapes of PnC air inclusions and to perform the optimization of cavity geometry simultaneously with PnC layers to enhance the benefits of optimization. In addition, a further step of our study is to apply geometrically-dependent constraints, in order to provide more freedom for the optimizer that may also increase sensor performance.