Compliance-based topology optimization of structural components subjected to thermo-mechanical loading

Abstract

Apart from mechanical actions, structural components in the construction industry may be subjected to a thermal gradient, causing (internally) restrained thermal expansion. These thermal loads can alter the mechanical response of components in a structural topology optimization procedure. Therefore, the influence of thermal loading should be considered in the sensitivity analysis to efficiently update the structural layout of material. In this paper, a density-based topology optimization procedure is developed for compliance minimization of structural components subjected to thermo-mechanical loads considering steady-state heat conduction and weak thermo-mechanical coupling. The adjoint method is employed to obtain the analytical sensitivities, taking into account the influence of the design-dependent temperature field and thermal properties. The proposed topology optimization procedure is demonstrated on the MBB problem, extended with thermal loading, to investigate the influence of the proposed sensitivities on the optimized results. Furthermore, the thermo-mechanical load ratio is quantitatively defined and its effect on the resulting topologies is studied. The results show that the thermo-mechanical load ratio significantly changes the topology of the optimized results. Finally, the topology optimization procedure is presented in an efficient 138-line MATLAB code and provided as supplementary material, serving as a basis for further research.

Appendixes

1.1 Appendix 1: Verification sensitivity analysis

The verification of the analytical sensitivities obtained with the adjoint sensitivity analysis (Sect. 2.2.3) is performed based on the example of the MBB beam in Sect. 4. The analytical sensitivities of several arbitrary elements are compared with their corresponding numerical equivalent values, which are calculated with a central finite difference scheme (Cho and Choi 2005), as expressed in Eq. (53).

$$\frac{\Delta c}{2\Delta {x}_{i}}=\frac{c\left({x}_{i}+\Delta {x}_{i}\right)-c\left({x}_{i}-\Delta {x}_{i}\right)}{2\Delta {x}_{i}}$$

(53)

with $c$ the structural compliance, $\Delta {x}_{i}$ a small perturbation of the element density of $\mathcal{O}\left({10}^{-8}\right)$ , and ${x}_{i}$ the relative density of element $i$.

The relative (mean) difference between the analytical ${\text{d}}c/{\text{d}}{x}_{i}$ and numerical sensitivities $\Delta c/2\Delta {x}_{i}$, denoted $\overline{\delta c }$ here, is determined with Eq. (54), adapted from Tang et al. (2019).

$$\overline{\delta c } =\frac{\left|\frac{\Delta c}{2\Delta {x}_{i}}-\frac{{\text{d}}c}{{\text{d}}{x}_{i}}\right|}{\left|\overline{\frac{{\text{d}}c}{{\text{d}}{\varvec{x}}}}\right|}\times 100\%$$

(54)

with $\left|{\overline{\text{d}c/{\text{d}}{\varvec{x}}}}\right|$ the mean value of the analytical sensitivities of all elements.

A comparison of the sensitivities is made on a coarser mesh of $10\times 30$ elements for the sake of clarity. The sensitivities and corresponding differences of an arbitrary selection of 6 elements (indicated in Fig. 20) is provided in Table 5. The average iteration time considering the numerical sensitivities is 1.465 s, which is more than 200 times slower compared to 0.0064 s in case of the analytical sensitivities and clearly shows the advantage of the adjoint method for a high element count.

Fig. 20

Mesh and element set for verification

The results in Table 5 show insignificant differences between the analytical and numerical sensitivities, which confirms the accuracy of the sensitivities and justifies the use of the adjoint method in the proposed TO procedure. In addition, the verification results are visually presented in Fig. 21.

Table 5 Sensitivity analysis verification results

Fig. 21

Sensitivity analysis verification

Note that using the mean value $\left|\overline{{\text{d}c/{\text{d}}{\varvec{x}}}}\right|$ ensures a valid comparison for the large range of values, as in other formulations (Tang et al. 2019), the relative errors would become much larger in case of very small (but relatively very different) sensitivity values, e.g., elements 8, 168, 240 in Table 5.

1.2 Appendix 2: A 138-line MATLAB code for topology optimization with steady thermo-mechanical loads