Discrete Variable Topology Optimization Using Multi-Cut Formulation and Adaptive Trust Regions

Abstract

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.

Figures (14)

• Figure 1: Single-material minimum compliance: design domain and boundary conditions for two different problems.
• Figure 2: Single-material minimum compliance for the MBB design: Evolution of the adaptive trust-region radius and the objective function value during the optimization process, along with the resulting material configuration at different iteration steps. Here, the target volume fraction is $V_T = 0.4$; the discretization resolution is $240 \times 80$; and the initial trust-region radius is set as $d^0=0.3$. The minimum Young's moduli are $E_0 = 10^{-2}$ and $E_0 = 10^{-9}$, respectively, for the iteration steps before and after the gray dashed line.
• Figure 3: Single-material minimum compliance: The optimal topology obtained for the MBB design from different methods, with the discretization resolution of $360 \times 120$ and the target volume fraction of $V_T = 0.3$.
• Figure 4: Single-material minimum compliance for the MBB design: The final topological configuration with the target volume fraction of $V_T = 0.3$
• Figure 5: Single-material Compliant Mechanism: Design domain and boundary conditions.