## 1 Introduction  
The repeated updating of parametric designs is computationally challenging, especially for large-scale multi-physics models. Thermoelastic structures are involved in many engineering fields. For example, micro-electronic mechanical systems with thermodynamic behavior now dominate the billion-level market, including aviation, space, and daily life [1]. Thermoelastically coupled microresonators are usually used as the core component of micro-electronic mechanical systems and have two main characteristics: the resonant frequency and dissipative ratio [2; 3; 4]. A higher resonant frequency leads to a higher reaction speed and sensitivity, while a lower dissipative ratio means better accuracy. There are dense analyses of the characteristics of thermoelastically coupled microresonators, including analytical, numerical, and experimental analyses of the resonant frequency and dissipative sources [5; 6; 7]. [PERSON] and [PERSON] investigated thermoelasticity within the Laplace transform domain and gave analytical solutions for thermoelasticity in spherical and cylindrical systems [8; 9; 10; 11]. On the other hand, with the maturity of microresonator fabrication techniques [12; 13], rational design methods for microresonators with complex substructures are more needed than ever. The structural topology optimization method is one of the rational gradient-based optimization methods which can be used to design the frequencies and dissipative ratios of microstructures [14]. [PERSON] et al. established a generative algorithm for steady state heat conduction, and the optimal designs obtained by the proposed method were compared with commercial finite element software [15]. [PERSON] et al. presented an efficient level set-based topology optimization method for weakly coupled thermoelastic systems, and an augmented Lagrangian method was developed to explore Pareto topologies [16]. [PERSON] et al. proposed a temperature-constrained topology method for thermoelastically coupled problems, and the numerical results showed that the proposed method significantly reduced the system temperature while maintaining the system's stiffness [17]. However, for large-scale thermoelastically coupled systems, the computational cost of topology optimization is extremely high due to the asymmetry of the system matrices. Furthermore, the repeated iterative processes could even prolong the unbearable computation time. Although there is intense research on the topology optimization methods of multi-physical coupled systems [15; 18; 19], most of them focus on weakly coupled and small-scale systems, and there are few studies on improving the computational efficiency of optimization methods. The most important feature of gradient-based optimization methods is the sensitivity numbers of elements in the design region, while the time-consuming steps in the process of obtaining the sensitivity of design variables focus on the eigenvalue solution and global sensitivity analysis. Since the material distribution of the structure alters during the iterative process of topology optimization, the eigenvalue solution has to be performed at each iteration. Regardless of the objective of the design's character, designing microresonators with complex geometry generally means structural modifications [14; 20]. Structural modifications are encountered frequently in practical designs to improve or enhance the performance of a structural or mechanical system by changing its design parameters. One of these structural modification problems is to predict the eigenfrequencies and mode shapes of the modes of a modified structural system. The modal modification problem usually needs to be repeated several times, which may arise in different subjects, including conceptual design problems, structural optimization, damage analysis, probabilistic analysis, structural control, and smart or adaptive structures.