We present a new framework for solving general topology optimization (TO) problems that find an optimal material distribution within a design space to maximize the performance of a structure while satisfying design constraints. These problems involve state variables that nonlinearly depend on the design variables, with objective functions that can be convex or non-convex, and may include multiple candidate materials. The framework is designed to greatly enhance computational efficiency, primarily by diminishing optimization iteration counts and thereby reducing the solving of associated state-equilibrium partial differential equations (PDEs). It maintains binary design variables and addresses the large-scale mixed integer nonlinear programming (MINLP) problem that arises from discretizing the design space and PDEs. The core of this framework is the integration of the generalized Benders' decomposition and adaptive trust regions. The trust-region radius adapts based on a merit function. To mitigate ill-conditioning due to extreme parameter values, we further introduce a parameter relaxation scheme where two parameters are relaxed in stages at different paces. Numerical tests validate the framework's superior performance, including minimum compliance and compliant mechanism problems in single-material and multi-material designs. We compare our results with those of other methods and demonstrate significant reductions in optimization iterations by about one order of magnitude, while maintaining comparable optimal objective function values. As the design variables and constraints increase, the framework maintains consistent solution quality and efficiency, underscoring its good scalability. We anticipate this framework will be especially advantageous for TO applications involving substantial design variables and constraints and requiring significant computational resources for PDE solving.

翻译：本文提出了一种用于求解一般拓扑优化问题的新框架，旨在设计空间内寻找最优材料分布，以在满足设计约束的同时最大化结构性能。此类问题涉及状态变量对设计变量的非线性依赖，目标函数可为凸或非凸，并可包含多种候选材料。该框架通过显著减少优化迭代次数，从而降低求解相关状态平衡偏微分方程的计算量，以大幅提升计算效率。该框架保持二元设计变量特性，并处理由设计空间和偏微分方程离散化产生的大规模混合整数非线性规划问题。其核心在于融合广义Benders分解与自适应信赖域方法，其中信赖域半径根据评价函数动态调整。为缓解极端参数值导致的病态问题，我们进一步引入参数松弛策略，通过不同步调分阶段松弛两个参数。数值实验验证了该框架的优越性能，包括单材料与多材料设计中的最小柔度问题和柔顺机构问题。通过与其他方法的对比，本框架在保持可比最优目标函数值的同时，将优化迭代次数降低约一个数量级。随着设计变量和约束条件的增加，该框架仍能保持一致的求解质量与效率，凸显其良好的可扩展性。我们预期该框架将在涉及大量设计变量与约束、且需消耗大量计算资源求解偏微分方程的拓扑优化应用中展现出显著优势。