Structural Design via Semidefinite Programming

We consider a general problem of Structural Design and demonstrate that under mild assumptions it can be posed as a Semidefinite program. The approach in question allows for unified treatment of single-load/multi-load Truss and Shape design with and without obstacles, same as for obstacle-free robust Truss and Shape design. We present three equivalent Semidefinite formulations of the Structural Design problem, discuss possibilities of their computational processing by interior point methods, same as the techniques for recovering optimal structures from the solutions to the corresponding SDP’s. 1 Structural Design: general setting The problem we are interested in is, in its general setting, as follows. Consider a mechanical construction S with finitely many degrees of freedom M ; thus, virtual displacements of the system are specified by vectors w ∈ RM . The potential energy capacitated by the construction under a displacement w is assumed to be a nonnegative quadratic function 12w TQw of the displacement, Q being a symmetric positive semidefinite matrix characterizing the construction; this matrix is assumed to depend linearly on the vector t of design parameters of the construction: Q = Q(t). We are given also a set W ⊂ RM of kinematically admissible displacements. The construction can be subject to an external load; mathematically, such a load is a vector f ∈ RM . The displacement caused by the load maximizes the function 2f w − wTQ(t)w over w ∈ W ; the corresponding optimal value compl(f ; t) = sup w∈W [ 2f w − wTQ(t)w ] is twice the compliance of the construction under load f ; the less is this compliance, the better are the rigidity properties of the construction with respect to the load. The problem of optimal structural design in its general setting is as follows: given a set F ⊂ RM of tentative loads and a set T of admissible values of the design vector t, we are 1) The research was partly supported by the Israel Science Foundation grant # 306/94-3, by the Fund for the Promotion of Research at Technion and by the G.I.F. contract No. I-0455-214.06/95