Robust Truss Topology Design via Semidefinite Programming

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Robust Truss Topology Design via Semidenite Programming










Robust Truss Topology Design via Semidenite
Programming
A. Ben-Tal
�
A. Nemirovski
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November 20, 2004
Abstract
We present and motivate a new model of the Truss Topology Design
problem, where the rigidity of the resulting truss with respect bothto given loading scenarios and small occasional loads is optimized.It is shown that the resulting optimization problem is a SemideniteProgram. We derive and analyze several equivalent reformulations ofthe problem and present illustrative numerical examples.
Keywords: structural optimization, truss topology design, robustness, semidef-inite programming, interior point methodsAMS Subject Classication 19C25, 19C30,19C50, 73K40
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Introduction
Truss Topology Design (TTD) deals with the selection of optimal congu-ration for structural systems (mechanical, civil engineering, aerospace) andconstitutes one of the newest and most rapidly growing elds of StructuralDesign (see the excellent survey paper by Rozvany, Bends�e and Kirsch[12]). The TTD problem was studied extensively, both mathematically andalgorithmically, in [1, 2, 3, 4, 5].

Faculty of Industrial Engineering and Management at Technion the Israel Institute
of Technology, Technion City, Haifa 32000, Israel (morbt@ie.technion.ac.il)

Faculty of Industrial Engineering and Management at Technion the Israel Institute
of Technology, Technion City, Haifa 32000, Israel (nemirovs@ie.technion.ac.il)
�
This research was partly supported by the Israel Science Foundation grant # 306/94-3
and by The Fund for Promotion of Research at the Technion
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ROBUST TRUSS DESIGN VIA SDP
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In this paper we bring forth the issue of the robustness of the truss; here
we say that a truss is robust, if it is reasonable rigid with respect both tothe given set of loading scenarios and to all small uncertain (in size anddirection) loads which may act at any of the active nodes of the truss, i.e.,those which are linked at least by one bar. In the engineering literaturerigidity is modeled by considering dierent loading scenarios on the structure(the multi-load TTD problem) or by imposing upper and lower bounds onnodal displacements. The rst approach depends on the engineers abilityto guess right the relevant scenarios, while the second approach leadsto a mathematical problem which is not tractable computationally. Herewe suggest a new modeling approach which circumvents both of the abovementioned diculties.
The paper is organized as follows. Section 2 describes the modeling ap-
proach in question. The preliminary Section 2.1 presents the basic notionsrelated to the TTD problem and the traditional formulations of the problem.We demonstrate by simple example (Section 2.2) that robustness restrictions(which are basically ignored in the traditional formulations) are critical toobtain reasonable designs; this observation motivates our modeling approachpresented in Section 2.3. Its computational tractability is demonstr