Level set-based topology optimization of micropolar solids under thermo-mechanical loading

TL;DR

The paper addresses designing micropolar solids under thermo-mechanical loading by integrating microstructure length-scales into a level-set topology optimization framework. It derives a coupled, interpolated strong/weak form that accounts for microrotation and couple stresses, and employs adjoint-based sensitivities within an augmented Lagrangian scheme implemented on Gridap.jl. Key findings show that increasing the micropolar coupling number $N$ and bending length scale $l_{ ext{b}}$ modifies optimal topologies from truss-like to frame-like, enhances stiffness, and mitigates thermal distortion, with size effects most pronounced when $H/l_{ ext{b}}$ is small. The method yields more efficient, thermally robust designs and provides an open-source, reproducible workflow for thermo-elastic micropolar topology optimization, with potential extensions to plasticity, transients, and multiphysics.

Abstract

We propose a novel level set-based topology optimization for micropolar solids subjected to thermo-mechanical loading. To capture the size effects, we have incorporated the microstructural length-scale information into the level set-based topology optimization method by adopting a micropolar theory. The proposed non-local topology optimization method can provide accurate topology optimization for size-dependent solids under thermo-mechanical loading. We have demonstrated the effectiveness of the proposed method through a few representative two-dimensional benchmark problems. The numerical results reveal the substantial influence of underlying micro-structures, incorporated in the model through micropolar parameters, and temperature on topology optimization, highlighting the necessity of the proposed thermo-mechanical micropolar formulation for materials with pronounced non-local effects. For the numerical implementation of the proposed model, we have used open-source finite element libraries, \texttt{Gridap.jl}, and \texttt{GridapTopOpt.jl}, available in Julia, to ensure transparency and reproducibility of the reported computational results.

Figures (27)

• Figure 1: Schematic representation of a material domain, $\mathcal{D}$, with a smooth boundary, $\partial \mathcal{D}$, subjected to specified mechanical and thermal boundary conditions. Dirichlet conditions are prescribed on $\partial \mathcal{D}_{(u,\theta, T)}$ (indicated in magenta), whereas the Neumann-type conditions—namely, the applied traction $\bar{\mathbf{t}}_\sigma = \boldsymbol{\sigma}\,\mathbf{n}$, $\bar{\mathbf{t}}_\sigma = \mathbf{m}\,\mathbf{n}$ and heat flux $\bar{q} = \mathbf{q} \cdot \mathbf{n}$—are enforced on $\partial \mathcal{D}_{(t_{\sigma},t_m)}$ and $\partial \mathcal{D}_{q}$ (shown in orange). The remainder of the boundary, $\partial \mathcal{D}^{0}_{(t_{\sigma},t_{m},q)}$ (highlighted in green), is subjected to homogeneous Neumann conditions, implying vanishing tractions and heat flux. These boundary subsets satisfy the partitioning relations $\partial \mathcal{D}_{(t_{\sigma},t_\mathrm{m},q)} = \partial \mathcal{D}_{(t_{\sigma},t_\mathrm{m})} \cup \partial \mathcal{D}_q \cup \partial \mathcal{D}^{0}_{(t_{\sigma},t_{m},q)}$ and $\partial \mathcal{D} = \partial \mathcal{D}_{(u,\theta,T)} \cup \partial \mathcal{D}_{(t_{\sigma},t_\mathrm{m},q)}$.
• Figure 2: Initial level set function $\phi_0(x_1,x_2)$ in a two-dimensional domain for $\zeta = 1$ and $b = 0.2$. (a) Initial level set function $\phi_0(x_1,x_2)$ with the reference plane at $\phi_0(x_1,x_2) = 0$, representing the design domain $\mathcal{D}$. (b) Intersection of $\phi_0$ with the reference plane at $\phi_0(x_1,x_2) = 0$. The blue region ($\phi_0(x_1,x_2) < 0$) denotes the material domain $\Omega$, the red line ($\phi_0(x_1,x_2) = 0$) indicates the evolving material boundary $\partial\Omega$, and the white region ($\phi_0(x_1,x_2) > 0$) corresponds to the void domain, $\mathcal{D}\setminus\Omega$.
• Figure 3: In (a), the design domain and boundary conditions are shown for a half-MBB (Messerschmitt–Bölkow–Blohm) beam in Example I under elastic loading. A symmetry condition is applied along the vertical axis to model half of the simply supported beam subjected to three-point bending. $L$ and $H$ denote the length and height of the half-MBB beam, respectively. The beam is subjected to a mechanical load $P$ at the top-left corner, and (b) shows the initial design.
• Figure 4: Topologically optimized designs of the half-MBB beam in Example I with dimensions $H=10\,\text{mm}, \,L=30\,\text{mm}$ for different bending length scale $l_{\text{b}}$ and micropolar coupling number $N$.
• Figure 5: Elastic strain energy $(\Psi_{\text{elas}})$, rotational strain energy $(\Psi_{\text{rot}})$, and coupled strain energy $(\Psi_{\text{coup}})$ are plotted against the change in $H/l_{\text{b}}$ ratio from $0.1$ to $100$ for micropolar coupling number $N = 0.5$ in Example I for the half-MBB beam with dimensions $H = 10$ mm, $L = 30$ mm.