Microresonators have recently emerged as ubiquitous devices in various fields. Frequency and quality factor play, respectively, a very important role in the sensitivity and resolution of a microresonator and have, however, a competitive relation. A thermodynamic design methodology is proposed herein for optimizing the overall performance (frequency–quality product) of microresonators in the framework of thermodynamics. A definition of optimization problem is formulated for achieving ultra-high frequency–quality product of microresonators. A three-dimensional 8-node thermoelastic hexahedral element is developed to predict the frequency–quality product of microresonators with complex geometric substructures. By using the three-dimensional finite element method, we formulate the thermoelastic eigenproblem of microresonators whose eigenvalues are complex. The dominated frequency (working frequency in a microresonator) can be achieved from the real part of the first-order eigenvalue, and the quality factor can be evaluated by using the ratio of the real part of the first-order eigenvalue to its imaginary part. With the help of the numerical solution technique, a topology optimization method is developed to solve the optimization problem by tailoring the substructures of microresonators. By using the developed thermodynamic design methodology, some numerical examples with different boundary conditions are discussed. Numeric examples demonstrate that a very good improvement of the frequency–quality product can be achieved for beam- and plate-type microresonators. Therefore, the proposed thermodynamic design methodology has the capability to enlarge the overall performance of microresonators by tailoring their substructures, regardless of the competitive relation between frequency and quality factor.

微谐振器最近已成为各个领域中无处不在的设备。频率和品质因数分别在微谐振器的灵敏度和分辨率中起着非常重要的作用，并且具有竞争关系。本文提出了一种热力学设计方法，用于在热力学框架内优化微谐振器的整体性能（频率质量乘积）。为实现微谐振器的超高频质量产品，制定了优化问题的定义。开发了一个三维 8 节点热弹性六面体单元来预测具有复杂几何子结构的微谐振器的频率质量积。通过使用三维有限元方法，我们制定了本征值为复数的微谐振器的热弹性本征问题。主频（微谐振器中的工作频率）可以从一阶特征值的实部获得，质量因子可以通过一阶特征值的实部与其虚部的比值来评估。借助数值求解技术，开发了一种拓扑优化方法，通过定制微谐振器的子结构来解决优化问题。通过使用开发的热力学设计方法，讨论了一些具有不同边界条件的数值例子。数值例子表明，对于梁式和板式微谐振器，可以实现频率质量乘积的非常好的改进。所以，