Topological design problems arise in many important manufacturing and scientific applications, such as additive manufacturing and the optimization of structures built from trusses. Topology design optimizes the material layout within a design space to satisfy given global load constraints and boundary conditions, represented by partial differential equations (PDEs). Traditionally, the PDEs have been discretized using the finite-element method with additional continuous variables that model the density of the material within each finite element, giving a (discretized) layout of the material.
In contrast, we model the material layout using binary variables to represent the presence or absence of material in each finite element. This approach results in a massive mixed-integer nonlinear optimization (MINLO) problem, where the nonlinearities arise from the discretization of the governing PDEs. At first sight, such an approach seems impractical, because it requires a massive number of binary variables (one per finite element): the number of binary variables is driven by the accuracy requirements of the finite-element discretization. We show empirically that traditional MINLO algorithms that search a branch-and-bound tree cannot solve these topology optimization problems with even a coarse finite-element discretization.
Recently, a new class of methods have been proposed for solving MINLOs that arise from the discretization of differential equations. These methods solve a single continuous relaxation of the MINLO, which relaxes the integrality requirement on the binary variables. Given the optimal solution of this relaxation, we apply a rounding technique that relies on space-filling curves. These rounding techniques have a remarkable convergence property, and can be shown to asymptotically converge to the global optimum as we refine the finite-element mesh. We illustrate these solution techniques with examples from topology optimization arising in the design of electro-magnetic cloaking devices.
