Robust Truss Topology Design with Beam Elements via Mixed Integer Nonlinear Semidefinite Programming

Abstract

In this article, we propose a nonlinear semidefinite program (SDP) for the robust trusstopology design (TTD) problem with beam elements. Starting from the semidefinite formulation ofthe robust TTD problem we derive a stiffness matrix that can model rigid connections between beams.Since the stiffness matrix depends nonlinearly on the cross-sectional areas of the beams, this leads toa nonlinear SDP. We present numerical results using a sequential SDP approach and compare them toresults obtained via a general method for robust PDE-constrained optimization applied to the equationsof linear elasticity. Furthermore, we present two mixed integer semidefinite programs (MISDP), onefor the optimal choice of connecting elements, which is nonlinear, and one for the correspondingproblem with discrete cross-sectional areas.

Exact Optimal Designs of Experiments for Factorial Models via Mixed-Integer Semidefinite Programming

“…Due to the specificities of MISDP, and the advantages it offers in handling well-characterized design problems, it is currently getting increased attention (see Gally et al [48]). Several applications of MISDP have been addressed, including: the truss topology design [49]; the optimal placement of metering systems in distribution grids [50], where the authors solve the measurement placement problem by exploiting the M-optimality design criterion; and optimal sensor placement, using the A-and Doptimality criteria (see Schäfer [51]). Here, we consider another related application of MISDP: the construction of exact optimal designs of experiments.…”

Types of Uncertainty

“…In higher dimensions and to avoid using extreme values that lead to overestimation, the limiting intervals are typically replaced by ellipsoids for which the worst-case analysis becomes more complicated. For example and as shown in [11,[28][29][30]55], sophisticated optimisation techniques introducing uncertainty sets are necessary to master data uncertainty in this pessimistic setting, see also Sect. 6.1.…”

Strategies for Mastering Uncertainty

“…This can be extended to beam elements which can also represent bending. The new stiffness matrix, which depends nonlinearly on x, can be computed by using a finite element approach and inserted into (6.5), see [63]. The obtained non-convex SDP can be solved by a sequential SDP method based on [37], in which the nonlinear SDP constraint is linearised and iteratively solved by applying a suitable step length rule.…”