Most topology optimization methods seek optimal topologies that satisfy a minimum compliance with a pre-specified volume constraint in the design domain. However, practical designs often include various functional constraints and the optimal solid volume ratios are unknown a priori, which implies a gap between topology optimization methods and practical designs in industries. This paper studies the performance-based topology optimization (PTO) problem that searches for the optimal topology with minimum compliance to satisfy the pre-specified functional constraints without a pre-specified volume constraint. A novel element-based evolutionary switching method (ESM), which can automatically adjust solid volume ratio and material distribution, is developed and implemented using the commercial finite element software ABAQUS. The effects of displacement constraints on the optimal topologies are investigated, and the differences between PTO problems and the topology optimization problem which has a volume constraint are discussed. Numerical examples demonstrate that the optimal topologies are mainly determined by the load pattern and locally changed with respect to the location of the active displacement constraints. In addition, the displacement constraints to a large extent control the solid volume ratio of optimal topologies according to the allowable displacements in PTO problems. Finally, the proposed ESM could provide conservative solutions to the topology optimization with multiple displacement constraints problems.

Functional constraints and bi-functional constraints are an important constraint class in Constraint Programming (CP) systems, in particular for Constraint Logic Programming (CLP) systems. CP systems with finite domain constraints usually employ Constraint Satisfaction Problem(s)-based solvers which use local consistency, for example, arc consistency. We introduce a new approach which is based instead on variable substitution. We obtain efficient algorithms for reducing systems involving functional and bi-functional constraints together with other nonfunctional constraints. It also solves globally any CSP where there exists a variable such that any other variable is reachable from it through a sequence of functional constraints. Our experiments on random problems show that variable elimination can significantly improve the efficiency of solving problems with functional constraints.