Singular, Large-Scale Solutions in Local Stress-Constrained Topology Optimization

In this paper, the union of sequential approximate optimization (SAO) and the ‘simultaneous analysis and design’ (SAND) formulation of the local stress-constrained topology optimization problem is delineated. The method is a standard structural optimization approach based on strictly convex and separable approximate subproblems, except for a straight-forward extension to include nonlinear equality constraints. The reason is, unlike conventional ‘nested analysis and design’ (NAND) methods, the finite element equilibrium equations are retained in the optimization problem as a set of nonlinear equality constraints. This implies that the state variables—i.e., the displacements—are primal optimization variables alongside the material-density variables. Because all optimization variables are independent, the computational cost and the dense coupling which is often consequential to nested formulations and calculation of the associated sensitivity derivatives, reduces to the manipulation of simple and sparse partial derivatives. Moreover, the equilibrium constraints may be violated up to the point of convergence, and, because the global stiffness matrix is not inverted per se, the material-density variables may take on a value of zero (0) exactly on the lower bound. The decoupling of the design and state variables on the one hand, and the exact representation of void material on the other, permits convergence to the singular optima which are typically not available in nested formulations without resorting to constraint relaxation techniques. Finally, numerical experiments on a simple benchmark problem indicates that high levels of computational efficiency may be achieved.