Design of thermal meta-structures made of functionally graded materials using isogeometric density-based topology optimization

TL;DR

This work introduces a fully isogeometric density-based topology optimization framework for designing thermal meta-structures made of functionally graded materials (FGMs) and architected cellular materials (ACMs). By parameterizing density, geometry, and solution with NURBS and solving the heat conduction problem via Isogeometric Analysis, the method supports arbitrary geometries, boundary conditions, and design constraints, while addressing checkerboarding and sensitivity calculations through adjoint methods. It validates the approach through 2D and 3D cloaks, concentrators, rotators, and sensors across six material models, including ACM-based homogenization, and demonstrates reconstruction of complex ACMs from optimized densities. The results show high flexibility, robustness to non-uniqueness, and potential for manufacturing FGMs/ACMs using additive processes, offering a powerful design tool beyond traditional transformation thermotics. The framework achieves accurate cloaking and concentration performance with smooth density distributions and integrated design-analysis-optimization workflows, highlighting its practical impact for thermal management and stealth applications."

Abstract

The thermal conductivity of Functionally Graded Materials (FGMs) can be efficiently designed through topology optimization to obtain thermal meta-structures that actively steer the heat flow. Compared to conventional analytical design methods, topology optimization allows handling arbitrary geometries, boundary conditions and design requirements; and producing alternate designs for non-unique problems. Additionally, as far as the design of meta-structures is concerned, topology optimization does not need intuition-based coordinate transformation or the form invariance of governing equations, as in the case of transformation thermotics. We explore isogeometric density-based topology optimization in the continuous setting, which perfectly aligns with FGMs. In this formulation, the density field, geometry and solution of the governing equations are parameterized using non-uniform rational basis spline entities. Accordingly, the heat conduction problem is solved using Isogeometric Analysis. We design various 2D & 3D thermal meta-structures under different design scenarios to showcase the effectiveness and versatility of our approach. We also design thermal meta-structures based on architected cellular materials, a special class of FGMs, using their empirical material laws calculated via numerical homogenization.

Figures (34)

• Figure 1: Domain description of the boundary value problem. $\Omega_{\rm design}$ represents a design region where thermal meta-structure is optimized, $\Omega_{\rm in}~\&~\Omega_{\rm out}$ are the inside and outside regions with respect to $\Omega_{\rm design}$, respectively. $\Omega$=$\Omega_{\rm in}\cup\Omega_{\rm out}\cup\Omega_{\rm design}$. $\Gamma=\partial \Omega=\Gamma_D\cup\Gamma_N \cup\Gamma_R$. The solid black line shows an explicitly defined interfaces $\Gamma_{I_{\rm in}}$ and $\Gamma_{I_{\rm out}}$, $\Gamma_{I}=\Gamma_{I_{\rm in}}\cup\Gamma_{I_{\rm out}}$. The square is shown in detail highlighting the matching of control points of connecting patches at the interface $\Gamma_I$.
• Figure 1: The solution mesh refinement strategy using the knot insertion procedure. At each stage, new knots are added at midpoints of existing knot spans in each parametric direction.
• Figure 2: Parametrization of a point from the parametric domain to a point in the physical domain using NURBS basis functions.
• Figure 2: Relative error in volume fraction $v$ value over $\Omega_{\rm design}$ with respect to the number of degrees of freedom of solution mesh for (a) 2D thermal cloak problem and (b) 2D thermal concentrator problem. The last refinement solution is considered as the reference solution to calculate the relative error. We consider $2\%$ relative error as an acceptable error, which is represented by the black horizontal line.
• Figure 3: Effective thermal conductivity $\kappa_{\rm eff}$ and its derivative with respect to the relative density $v$ for all models shown in Table \ref{['table: effective thermal conductivity models']}.