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.

翻译：功能梯度材料的热导率可通过拓扑优化进行高效设计，从而获得能够主动调控热流的热超结构。与传统解析设计方法相比，拓扑优化能够处理任意几何形状、边界条件和设计要求，并为非唯一性问题提供多种设计方案。此外，就超结构设计而言，拓扑优化无需依赖基于直觉的坐标变换或控制方程的形式不变性（如变换热学方法所需）。本研究探索了连续体系下的等几何密度拓扑优化方法，该方法与功能梯度材料具有天然的契合性。在此框架中，密度场、几何构型及控制方程的解均通过非均匀有理B样条实体进行参数化表征。相应地，热传导问题采用等几何分析方法求解。我们通过设计不同场景下的二维与三维热超结构，展示了该方法的有效性与普适性。此外，基于架构多孔材料（功能梯度材料的一个特殊子类）及其通过数值均匀化计算得到的经验材料本构关系，我们也设计了相应的热超结构。