基于多相材料的稳态热传导结构轻量化设计

在多相材料的结构拓扑优化问题中,通常给定各相材料体积约束或材料总重量约束作为材料的控制用量.在结构轻量化设计的实际工程背景下,以结构总重量最小化为目标的优化模型具有明确的工程意义.针对含多相材料的稳态传热结构拓扑优化问题,提出了以结构总重量最小化为目标和给定热柔顺度为约束的多工况连续体结构拓扑优化建模方法.遵循独立连续映射建模方式,采用两类独立拓扑变量分别表征单元热传导矩阵和单元重量状态.推导了热柔顺度和总重量对设计变量的敏度,基于一阶和二阶泰勒展开得到各自的近似表达式.通过求解偏微分方程,实现了约束函数一次项过滤,消除了棋盘格现象和网格依赖性问题,并保证了约束方程在过滤后严格成立.建立的近似优化模型具有二次函数形式的目标函数和一次函数形式的约束函数.基于对偶序列二次规划方法对优化模型进行求解直至收敛.通过四个三维结构数值算例分析对比了热柔顺度约束限值、不同材料混合及多工况、多约束条件对优化结果的影响.数值算例结果表明,本文提出的优化方法在基于多相材料的多工况稳态热传导结构轻量化设计中具有可行性和有效性.

STRUCTURAL LIGHT DESIGN FOR STEADY HEAT CONDUCTION USING MULTI-MATERIAL

In topology optimization problems of structures containing multiphase materials, it is common practice to set the volume constraint of each constituent phase or total mass of entire constituent phase constraint to control the final material usage. On the practical engineering background for lightweight design, it is of significance that the minimized weight is taken as the objective in optimal model from the engineering point of view. To solve the topology optimization problem of steady heat conductive with the multiple candidate materials, a new modeling method of weight minimization with the given thermal compliance constraint under multiple load cases is proposed. Following the modeling manner of independent continuous mapping method, two sets of independent topological variables are employed to identify ele-mental thermal conductive matrix and elemental weight, respectively. The sensitivities of thermal compliance and global weight with respect to the design variable are derived, and their approximate expressions are calculated based on the first-order and second-order Taylor expansion. To eliminate checkerboard patterns and mesh-dependence, the first term of the constraint function is filtered as a solution of the partial differential equation, which also ensures the constraint equa-tion is consistent. The approximate optimal model with the objective and constraint in the form of quadratic and linear function is established. The topological optimization model is solved by dual sequential quadratic programming. Various effects such as the constraint value of thermal compliance, the selection of multiple materials, and the multiple constraints in multiple load cases on the optimal result are discussed in four 3D numerical examples. The results demonstrate the feasibility and effectiveness of the proposed optimization approach regarding structural light design using multi-material in steady heat conduction.