Topology optimization for transient thermoelastic structures under time-dependent loads

Abstract

Most of the previous topology optimization methods for thermoelastic structures use steady-state heat transfer and static equations, which are not applicable to time-dependent loads. In this article, a generic topology optimization method considering transient thermoelastic coupling is proposed based on transient heat transfer and dynamic equations. First, a thermoelastic coupling matrix is proposed to address the issues on calculation error of thermal stress loads and solution difficulty of adjoint multipliers under transient conditions. Second, a compact transient thermoelastic sensitivity equation with a distinct physical implication is derived based on adjoint sensitivity analysis. Finally, various comparative case studies of structural dynamics, transient heat conduction, and transient thermoelastic optimizations (including multi-material and 3D structures) are carried out to demonstrate the effectiveness, stability, and versatility of the proposed method. Additionally, this study discovers that the transient thermoelastic optimizations present strong transient effect, and the corresponding mechanism is also revealed. The proposed method can be widely used in structural optimization considering transient thermoelasticity under time-dependent loads.

Appendix: Impact of optimization parameters and mesh density

As is known, the volume fraction constraint, the density filtering radius, and the meshing density have influence on the topology optimization results. In this part, all these factors will be discussed in detail.

First, the effect of the filter radius on the optimization is investigated. Taking the load case of $q=40$ kW/$\textrm{m}^2$ and t=1s in Table 5 as an example, the filter radius ${R_{\min }}$ varies from 2.5 to 5 mm and 10 mm, and other conditions remain the same as the case in Sect. 5.3.1.

Table 8 Effect of filter radius on optimization

The optimization results are shown in Table 8. Comparing Table 8 with Table 5, the topological conformation results differ with different filter radius ${R_{\min }}$. Since ${R_{\min }}$ has the effect of minimum size control, the small size holes and structures in the configurations under ${R_{\min }}$ = 2.5 and 5 mm disappear when ${R_{\min }}$ becomes 10 mm. However, too large ${R_{\min }}$ tends to lead to more gray elements within the topological configurations and consequently causes worse structural thermal conduction and an increase in temperature.

For the volume fraction constraint, taking the load case of $q=40$ kW/$\textrm{m}^2$ and t=10s in Table 5 as an example, ${V^*}$ is constrained to 0.4 and 0.6, respectively, and other conditions remain the same as the cases in Sect. 5.3.1.

Table 9 Effect of volume fraction constraint on optimization

The optimization results are shown in Table 9. Along with the results in Table 5 , evidently the topological configurations vary with different volume fraction constraints. And a larger volume fraction constraint results in more material distribution to reduce the temperature of the system.

Finally, it should be noted that the mesh density has little influence on optimization due to the utilization of the filtering technique. Compared with the cases in Sect. 5.3.1, the mesh of the design domain is refined to 120 $\times$ 120, and other conditions remain the same.

Fig. 10

Topology configurations with the refined meshes 120 $\times$ 120 under the load cases in Table 5. a $q=40$ kW/$\textrm{m}^2$, t = 1 s; b $q=40$ kW/$\textrm{m}^2$, t = 10 s; c $q=40$ kW/$\textrm{m}^2$, t = 20 s; and d $q=60$ kW/$\textrm{m}^2$, t = 20 s

With the refined meshing, all the cases in Table 5 are recalculated in the same order and the corresponding optimization results are shown in Fig. 10a–d. Comparing the obtained configurations between Table 5 and Fig. 10, obviously there is basically no difference between the corresponding configurations with the refined mesh density and the original mesh grid of 60 $\times$ 60. This is due to the introduction of density filtering, which can effectively eliminate the mesh dependence. Therefore, the mesh density can be reduced appropriately in optimizations to reduce the computational cost. However, it is important to note that the mesh density should not be too coarse, because too coarse mesh will lead to the inaccuracy of the finite element solution, and thus affect the optimization results.