# An Efficient and Robust Topology Optimization Method for Thermoelastically Damped Microresonators
## 1 Introduction
Among the non-Fourier heat conduction models, there is a specific branch dedicated to modeling non-Fourier phase-lag behavior. This branch attributes nonlinear non-Fourier heat conduction effects to the hysteresis effect between heat conduction and temperature gradients. One of the features of the non-Fourier phase-lag model is its ability to be degenerated by setting specific lag values [20]. According to the heat wave theory, the lag of heat conduction in solid structures usually affects the heat flux. In other words, the change of heat flux is driven by the changing temperature gradient [21]. Additionally, the single-phase lag and the dual-phase lag can be converted to each other [22]. For these reasons, we choose the dual-phase lag heat conduction model as the non-Fourier heat conduction law in this paper due to its universal nature. Figure 1: Schematic diagram of a structure considering thermoelastic damping. The thermoelastic structures with the dual-phase lag heat conduction model will cause a nonlinear thermoelastic eigenproblem, which is very different to handle. Often, the linearization technique based on the state-space method [23; 24] is used for solving the nonlinear eigenproblem. When the parametric partial differential equations of thermoelastic structures are discretized using commonly-used numerical techniques, the degrees of freedom of the governing equations will be very large. However, the dimension of the system matrix of the state-space method is much larger than the original dimension of the nonlinear thermoelastic eigenproblem, thereby resulting in a huge computational challenge. There has been extensive research on accelerating the solution of large-scale linear eigenproblems, which can, of course, be used for handling the state-space method of thermoelastic eigenproblems with the dual-phase lag heat conduction model. The reduced basis method, which finds the relationship between the design variable and the system information (such as the displacement field) to solve the problem, has been employed to reduce the dimensionality of the problem and computational cost [25]. Furthermore, the matrix operator method, when combined with other optimization algorithms, has been utilized to improve stability and accelerate the solution of partial differential equations [26], and has been widely applied to solve finite element eigenvalue problems in multi-field coupled dynamical systems [27]. Currently, there are many types of matrix operators, including diagonal matrix operators, reduced-basis matrix operators, and differential matrix operators, and these operators are often necessary for subspace solutions of partial differential equations [28; 29]. Among the types of matrix operators, diagonal and triangular matrix operators are known for their simplicity and fast convergence [30]. These efficient methods, although general, do not capture the essential features of thermoelastic structures. For the sake of further computational efficiency, [PERSON] et al. [31] constructed diagonal systemic matrices of the lumped mass and heat generation matrices. With these diagonal systemic matrices, an efficient Krylov subspace method is proposed for the thermoelastic eigenproblem. Note that the analysis of thermoelastic eigenproblems with the Fourier heat conduction model has been provided recently for commercial software COMSOL. However, these works mentioned above can only be used for thermoelastic eigenproblems with the Fourier heat conduction model, rather than the non-Fourier heat conduction model of interest. Furthermore, the repeated updating of topology optimization designs is computationally challenging, especially for large-scale linear eigenproblems of thermoelastic eigenproblems with the dual-phase lag heat conduction model. To the literature above, current system solutions of large-scale thermoelastic structures are difficult and time-consuming.