# An Efficient and Robust Topology Optimization Method for Thermoelastically Damped Microresonators
## 1 Introduction  
Micro-electromechanical system (MEMS) resonators are currently dominating the billion-dollar market, particularly in the fields of biomedicine, aerospace, and others [1, 2]. High-frequency resonators are widely used as the key components of MEMS. Two critical dynamic characteristics for microresonators are known as resonance frequency and energy dissipation (inverse quality factor). Frequency affects the sensitivity of the response of resonators, while the energy dissipation rate affects their accuracy and noise. Enlarging the quality factor of microresonators can, therefore, reduce their signal loss and improve stability [3]. The attainment of both high sensitivity and high accuracy is a vital requirement for most MEMS resonators; unfortunately, resonance frequency and energy dissipation are generally mutually exclusive. Although the quest continues for higher frequency resonators, these have little-to-no use as MEMS resonators without appropriate accuracy (qualify factor). One of the major damping sources of microresonators is thermoelastic damping [4], which cannot be reduced by external measurement-and-control methods. Thermoelastic damping determines the upper limit of the performance of high-frequency MEMS resonators [5, 6], although theirresonance frequency is generally high [7]. For these reasons, the development of high-frequency yet low-dissipation microresonators is an optimization problem for an exercise in the compromise between frequency and quality. As depicted in Figure 1, a thermoelastic structure undergoes vibration, resulting in the emergence of compressed and stretched regions within the structure. Since temperature increases in the compressed region but decreases in the stretched region, the temperature gradient-derived heat conduction will lead to irreversible thermoelastic damping. It has been examined by some works [8; 9] that microstructural features can affect the thermoelastic damping of microresonators, and structural topology design methods can be used to enlarge the quality factor of microresonators. On the other hand, advancements in lithography manufacturing processes and 3D printing technology have gradually loosened the constraints of manufacturing processes on the design of thermoelastic structures. As a result, it has become possible to fabricate microresonators with intricate internal configurations [10]. The fundamental theory of the structural topology optimization method is defined by combining homogenization theory with numerical optimization methods. This method achieves optimal structural performance by modifying the micro-structural features within the design domain [11]. There have been extensive studies in this field, including the classic solid isotropic material with the penalization (SIMP)-based optimization criteria method [12], the evolutionary and bi-directional evolutionary structural optimization [13], the Smooth-edged material distribution for optimizing topology algorithm [14], and the floating projection topology optimization [15]. The heat transfer mechanism plays a significant role in the analysis and design of thermoelastic structures. It has been demonstrated through a large number of experiments that the assumption of infinite heat conduction velocity, which is required for linear Fourier heat conduction, is no longer valid for devices at the micron or even nanoscale. Instead, nonlinear non-Fourier heat conduction gradually dominates as the size of thermoelastic structures is down from the micrometer to the nanometer [16]. For instance, extreme temperatures in a micro-cantilever not only affect its Young's modulus but also change the tension in the cantilever due to differential thermal expansion, resulting in the variation of both frequency and dissipation [17]. To better characterize these nonlinear heat conduction effects, scholars have extensively studied the phenomenon since the classical Maxwell-Cattaneo-Vernotte equation [18; 19].