Evaluation and Maximization of Robustness of Trusses by using Semidefinite Programming (original) (raw)
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Robustness Analysis of Trusses with Separable Load and Structural Uncertainties
International Journal of Solids and Structures
This paper discusses evaluation techniques of the robustness function of trusses, which is regarded as one of measures of structural robustness, under the uncertainties of member stiffnesses and external forces. By using quadratic embedding of the uncertainty and the S-procedure, we formulate a quasiconvex optimization problem which provides lower bounds of the robustness functions. A bisection method is proposed, where we solve a finite number of semidefinite programming problems in order to obtain a global optimal solution to the proposed quasiconvex optimization problem. The lower bounds of the robustness functions are computed for various trusses under several uncertainty circumstances.
A robust truss optimization scheme, as well as an optimization algorithm, is presented based on the robustness function. Under the uncertainties of external forces based on the info-gap model, the maximization problem of robustness function is formulated as the optimization problem with infinitely many constraint conditions. By using a semidefinite relaxation technique, we reformulate the present problem to a nonlinear semidefinite programming problem. A sequential semidefinite programming method is proposed which has the global convergent property. It is shown, in numerical examples, that optimum designs of various trusses can be found without any di#culty. Keywords: robust optimization; info-gap model; semidefinite program; structural optimization; successive linearization method + Department of Urban and Environmental Engineering, Kyoto University e-mail: kanno@archi.kyoto-u.ac.jp # Department of Urban and Environmental Engineering, Kyoto University e-mail: takewaki@archi.kyoto-u...
Optimization under uncertainty with applications to design of truss structures
Structural and Multidisciplinary Optimization, 2008
Many real-world engineering design problems are naturally cast in the form of optimization programs with uncertainty-contaminated data. In this context, a reliable design must be able to cope in some way with the presence of uncertainty. In this paper, we consider two standard philosophies for finding optimal solutions for uncertain convex optimization problems. In the first approach, classical in the stochastic optimization literature, the optimal design should minimize the expected value of the objective function with respect to uncertainty (average approach), while in the second one it should minimize the worst-case objective (worst-case or min-max approach). Both approaches are briefly reviewed in this paper and are shown to lead to exact and numerically efficient solution schemes when the uncertainty enters the data in simple form. For general uncertainty dependence however, these problems are numerically hard. In this paper, we present two techniques based on uncertainty randomization that permit to solve efficiently some suitable probabilistic relaxation of the indicated problems, with full generality with respect to the way in which the uncertainty enters the problem data. In the specific context of truss topology design, uncertainty in the problem arises, for instance, from imprecise knowledge of material characteristics and/or loading configurations. In this paper, we show how reliable structural design can be obtained using the proposed techniques based on the interplay of convex optimization and randomization.
Semidefinite programming for robustness analysis of structures under large uncertainties
A numerically tractable algorithm based on the semidefinite program (SDP) is proposed for computing a lower bound of the robustness function of structures under non-probabilistic load and structural uncertainties. By using the robustness function, we define a robust structural optimization problem which attempts to maximize the robustness function. A sequential SDP method is proposed to solve the robust structural optimization problem under large uncertainties.
Sequential Semidefinite Program for Maximum Robustness Design of Structures under Load Uncertainty
Journal of Optimization Theory and Applications, 2006
A robust structural optimization scheme as well as an optimization algorithm are presented based on the robustness function. Under the uncertainties of external forces based on the info-gap model, the maximization problem of the robustness function is formulated as an optimization problem with infinitely many constraints. By using the quadratic embedding technique of uncertainty and the S-procedure, we reformulate the problem presented into a nonlinear semidefinite programming problem. A sequential semidefinite programming method is proposed which has the global convergent property. It is shown through numerical examples that optimum designs of various linear elastic structures can be found without any difficulty.
Global optimization of robust truss topology via mixed integer semidefinite programming
Optimization and Engineering, 2010
This paper discusses a global optimization method of robust truss topology under the load uncertainties and lower bound constraints of the member cross-sectional areas. We consider a non-stochastic uncertainty model of external loads, and attempt to minimize the maximum compliance corresponding to the most critical load. A design-dependent uncertainty model of external loads is proposed in order to consider the variation of truss topology rigorously. It is shown that this optimization problem can be formulated as a 0-1 mixed integer semidefinite programming (0-1MISDP) problem. We propose a branch-and-bound method for computing the global optimal solution of the 0-1MISDP. Numerical examples illustrate that the topology of robust optimal truss depends on the magnitude of uncertainty.
Robust Truss Topology Design via Semidefinite Programming
SIAM Journal on Optimization, 1997
We present and motivate a new model of the truss topology design problem, where the rigidity of the resulting truss with respect both to given loading scenarios and small "occasional" loads is optimized. It is shown that the resulting optimization problem is a semidefinite program. We derive and analyze several equivalent reformulations of the problem and present illustrative numerical examples.
Robustness Analysis of Interactive Structures against Load and Structural Uncertainties
2005
In this paper, we discuss evaluation techniques of the robustness function of trusses, which is regarded as one of measures of structural robustness, under the uncertainties of member stiffnesses and external forces. By using quadratic embedding of the uncertainty and the S-procedure, we formulate a quasiconvex optimization problem which provides the lower bounds of the robustness functions. A bisection method is proposed, where we solve a finite number of semidefinite programming problems in order to obtain a global optimal solution to the proposed quasiconvex optimization problem. The lower bounds of the robustness functions are computed for various trusses under several uncertain circumstances.
Direct Evaluation of Robustness Functions of Trusses associated with Stress Constraints
2004
A direct computational method is presented for the evaluation of the robustness function of linear elastic trusses associated with stress constraints. Under the uncertainties of external forces based on the info-gap model, the robustness function is formulated as the optimal objective value of an optimization problem with infinitely many constraint conditions. By using the strong duality theory of the second-order cone programming problem, we reformulate the present problem to a numerically tractable form without any variable. The robustness functions are computed for various trusses under several uncertain loading circumstances.
A direct computational method is presented for the evaluation of the robustness function of linear elastic trusses associated with stress constraints. Under the uncertainties of external forces based on the info-gap model, the robustness function is formulated as the optimal objective value of an optimization problem with infinitely many constraint conditions. By using the strong duality theory of the second-order cone programming problem, we reformulate the present problem to a numerically tractable form without any variable. The robustness functions are computed for various trusses under several uncertain loading circumstances. Keywords: robustness; info-gap model; second-order cone program; duality; data uncertainty; reliability + Department of Urban and Environmental Engineering, Kyoto University e-mail: kanno@archi.kyoto-u.ac.jp # Department of Urban and Environmental Engineering, Kyoto University e-mail: takewaki@archi.kyoto-u.ac.jp 1.