  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.
