Multiscale concurrent topology optimization for thermoelastic structures under design-dependent varying temperature field

Abstract

In this paper, a new multiscale concurrent topology optimization method for thermoelastic structures considering the iterative variation of temperature field is proposed for the first time, which breaks the limitation that previous multiscale concurrent topology optimization studies being compliable merely to uniform temperature field. In this method, the iterative variations of macroscopic structural heat transfer, structural temperature, structural force transfer, structural displacement, design-dependent thermal stress load, microscopic effective thermal conductivity, effective elasticity and effective thermal expansion coefficient are all taken into consideration. In order to establish a compact hierarchical thermoelastic coupling equation on the above iterative factors, firstly, a thermoelastic coupling matrix with a distinct physical meaning is proposed to address the issues on accuracy of thermal stress loads and solution of adjoint sensitivity multipliers caused by design-dependent varying temperature field, and this matrix can be used as a new manner to solve homogenized effective thermal stress coefficient. Secondly, the compact coupling equation is derived using multiscale adjoint sensitivity analysis and its effectiveness is illustrated by comparative cases. Finally, the generality and stability of proposed method are illustrated through diverse scenarios involving compliance optimization, multimaterial concurrent design, maximum displacement control, multicellular structure design, asymmetric boundary conditions and three-dimensional structures. It is obvious that this pioneering approach has a broad potential in advanced integrated structures and materials design of thermoelastic structures.

Appendices

Appendix A: Detailed deviration procedure for Eq. (39)

Based on energy-based homogenization method (Xia and Breitkopf 2015), the perturbed strain fields in microstructures is equivalent to the averaged strain fields. Thus, Eq. (9) can be expressed as the form in terms of element mutual energies (Sigmund 1994), i.e.:

$$\begin{aligned} E_{ijkl}^H = \frac{1}{{|{\textrm{Y}}|}}\int _{\textrm{Y}} {{E_{pqrs}}\varepsilon _{pq}^{A(ij)}\varepsilon _{rs}^{A(kl)}} {\textrm{dY}}, \end{aligned}$$

(A1)

where ${\mathbf{\varepsilon }}_{pq}^{A(ij)}$ are the strain fields solutions in energy-based homogenization method corresponding to the unit test strain fields ${{\mathbf{\varepsilon }}^{0(ij)}}$. Similarly, Eq. (16) can be rewritten as:

$$\begin{aligned} {{\beta }}_{ij}^H = \frac{1}{{|{\textrm{Y}}|}}\int _{\textrm{Y}} {{{{E}}_{pqrs}}{{\varepsilon }}_{pq}^{A\alpha }{{\varepsilon }}_{rs}^{A(ij)}} {\textrm{dY}}, \end{aligned}$$

(A2)

where ${{\varepsilon }}_{pq}^{A\alpha }$ are the thermal strain fields solutions in energy-based homogenization method corresponding to a unit thermal load and can be further expressed as ${{\varepsilon }}_{pq}^{A\alpha } = {{{\alpha }}_{pq}}{{{\bar{\gamma }} }}$. Thus, Eq. (A2) can be further expressed as:

$$\begin{aligned} {{\beta }}_{ij}^H = \frac{1}{{|{\textrm{Y}}|}}\int _{\textrm{Y}} {{{{E}}_{pqrs}}{{{\alpha }}_{pq}}{{{\bar{\gamma }} }}{{\varepsilon }}_{rs}^{A(ij)}} {\textrm{dY}}. \end{aligned}$$

(A3)

With FEA, Eq. (A3) can be expressed as:

$$\begin{aligned} {\varvec{\upbeta }}_{ij}^{\textrm{H}} = \frac{1}{{|{\textrm{Y}}|}}\sum \limits _{{\breve{e}} = 1}^{\breve{N}} {{{({{\varvec{\upchi }}_{{\breve{e}} }}^{A(ij)})}^T}\int _{{{\textrm{Y}}^{\breve{e}}}} {{{\left( {{\textbf{LN}}} \right) }^T}\alpha {\textbf{D}}{\varvec{\upvarphi }}{{\textbf{n}}_{\textrm{t}}}{{{{\bar{\varvec{\upgamma }} }}}_{{\breve{e}} }}{\textrm{d}}{{\textrm{Y}}^{\breve{e}}}} }, \end{aligned}$$

(A4)

where combining with Eq. (38), we have:

$$\begin{aligned} {\varvec{\upbeta }}_{ij}^{\textrm{H}} = \frac{1}{{|{\textrm{Y}}|}}\sum \limits _{{\breve{e}} = 1}^{\breve{N}} {{{({{\varvec{\upchi }}_{{\breve{e}} }}^{A(ij)})}^T}{{\textbf{H}}^{\breve{e}}}{{{{\bar{\varvec{\upgamma }} }}}_{\breve{e}}}}. \end{aligned}$$

(A5)

In energy-based homogenization method, Eq. (A5) can be used directly. In asymptotic homogenization method, based on the equivalence relation between energy-based homogenization method and asymptotic homogenization method (Sigmund 1994), Eq. (A5) should be modified to:

$$\begin{aligned} {\varvec{\upbeta }}_{ij}^{\textrm{H}} = \frac{1}{{|{\textrm{Y}}|}}\sum \limits _{{\breve{e}} = 1}^{\breve{N}} {{{{({\varvec{\upchi }}{{_{\breve{e}}^0}^{(ij)}} - {{\varvec{\upchi }}_{\breve{e}}}^{(ij)})}^T}} {{\textbf{H}}^{\breve{e}}}{{{{\bar{\varvec{\upgamma }} }}}_{\breve{e}}}}. \end{aligned}$$

(A6)

Equation (A6) is just the Eq. (39) in the main text, which ends the proof.

Appendix B: The convergence history curves

Fig. 13

The convergence history of the proposed method for the square initial design

Fig. 14

The convergence history of the proposed method for the two objectives of a compliance and b displacement

Fig. 15

The convergence history of the proposed method for the multiphase materials concurrent design

Fig. 16

The convergence history of the proposed method for the multicellular structure design

Fig. 17

The convergence history of the proposed method for the optimization under asymmetric boundary conditions

Fig. 18

The convergence history of the proposed method for the 3D case

From Figs. 13, 14, 15, 16, 17 and 18, it can be concluded that the proposed method based on SIMP within three field scheme exhibits strong stability and good convergence except for the first few steps where objective values jump because of the sharp changes in microvolume fraction.