Structural Optimization 14, 256-263 (~) Springer-Verlag 1997 Topology optimization with displacement constraints: a bilevel programming approach M. KoSvara Institute of Applied Mathematics, University of Erlangen, Martensstr. 3, D-91058 Erlangen, Germany e-mail kocvara@am.uni-erlangen.de Abstract We consider the minimum-complianceformulation of the truss topology problem with additional linear constraints on the displacements: the so-called displacement constraints. We propose a new bilevel programming approach to this problem. Our primal goal (upper-level) is to satisfy the displacement constraint as well as possible - we minimize the gap between the actual and prescribed displacement. Our second goal (lower level) is to minimize the compliance - we still want to find the stiffest structure satisfying the displacement constraints. On the lower level we solve a standard truss topology problem and hence we can solve it in the formulation suitable for the fast interior point alogrithms. The overall bilevel problem is solved by means of the so-called implicit programming approach. This approach leads to a nonsmooth optimization problem which is finally solved by a nonsmooth solver. 1 Introduction We consider the minimum-compliance formulation of the truss topology problem. To answer also the question of the optimal number of bars and the optimal position of nodes, we consider the so-called ground structure approach: we start the optimization with a dense mesh of nodes and potential bars and allow the bar volumes to be zero, such that only a few of the potential bars are presented in the optimal struc- ture. It is well-known that standard optimization software fails to solve the problem in the standard formulation even for a moderate number of nodes and bars. In recent years the problem has been reformulated by means of convex analysis; very large problems can now he solved by modern interior- point methods (Bendsr et al. 1994; Jarre et al. 1996). In this paper we add further linear constraints on the dis- placements to the original minimum-compliance formulation. These are usually called displacement constraints (not to be confused with unilateral contact conditions - here we have no obstacles). The need for such kinds of constraints often comes from the technological background of the problem: we want certain nodes of the deformed truss to lie within given boxes; or the displacements of certain nodes that are not un- der load should be minimized; or a boundary of the deformed truss should take a prescribed shape (circle, parabola). Un- fortunately, the above-mentioned problem reformulation does not work with these additional constraints. We propose a new approach which still benefits from the reformulation of the unconstrained problem. We introduce a bilevel optimization problem. Our primal goal (upper level) is to satisfy the displacement constraint as well as possible; we minimize the gap between the actual and prescribed dis- placement. If the constraint is infeasible for the given truss problem, we find a structure, the displacements of which are closest to the prescribed values. Our second goal (lower-level) is to minimize the compliance - we still want to find the stiffest structure satisfying the displacement constraints. On the lower level we solve a standard truss topology problem and hence we can solve it in the formulation suitable for the interior-point method. The overall bilevel problem is solved by means of the so-called implicit programming approach (cf. e.g. Ko~vara and Outrata 1995). This approach leads to a nonsmooth optimization problem which is finally solved by the nonsmooth solver BT (Schramm and Zowe 1992). The paper is organized as follows. In the next section we introduce the problem with displacement constraints, in Section 3 the bilevel programming approach, Section 4 con- tains computations needed in this bilevel approach, Section 5 implementation issues and results of two model examples. Section 6 is devoted to the discussion on drawbacks of our approach as well as on general difficulties met while solv- ing displacement constrained problems. In the Appendix we show the relation of our problem to problems with unilateral contact constraints. 2 Problem formulation We want to design a pin-jointed framework (so-cMled truss) which is as stiff as possible under a given load f. The prob- lem is modelled by a mesh of N tentative nodal points in ~dim, where dim is 2 for planar and 3 for spatial trusses. The nodal points are connected by m slender bars of con- stant mechanical properties characterized by their Young's moduli Ei, i = 1, 2,..., m. The given load f is a vector of forces fj C ~dim acting at nodes j. As usual, the vector of nodal displacements is denoted by u. Let p be the num- ber of fixed nodal coordinates, i.e. the number of components with prescribed homogeneous boundary condition. We omit these fixed components from the problem formulation reduc- ing thus the dimension of u to n = dim.N-p. Analogously, the external load f is considered as an element from ~u. Let tl,t2,...,tm be the volumes of the bars, our design variables by which the designer can control the displacement vector u. We consider the truss topology design problem writ-