Department of Mechanical Engineering
University of Houston
4800 Calhoun Road
Houston, TX 77204 - 4792
U.S.A
(713) 743-4535 
(713) 743-4503 Research
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Abstract :
In this paper, we study the rich class of formulations that arise in the limit analysis and design of elastic/plastic structures in the presence of contact constraints. It is well known that in the absence of contacts, both the limit analysis and limit design problems can be written as linear programs. However, when contact constraints are present, the structure effectively exhibits both softening and stiffening behavior under monotonic loading. The resulting limit analysis and limit design problems are nonconvex and are difficult to solve due to complementary type of equality constraints. We show that using a mixed form of the minimum principle, we can restate the limit analysis and limit design problems as two and three level formulations, respectively. Further, under a strong assumption on the problem and solution data, we can take advantage of the underlying convexity to reduce both these multilevel formulations to equivalent linear programs. While it may not be possible to always verify this assumption in practice, we show that a two-step iterative procedure is effective in reaching a solution to the limit design problem.
postscript version of the complete paper
Abstract :
When a design problem is viewed not individually but in the context of an overall system time-line, a multi-player game naturally arises. These models contrast with single-player, multi-objective models commonly used in design optimization. In this paper, simple two-player formulations are developed for the design of automotive suspensions where several game protocols are set up between a braking performance player, and a suspension designer. After the models are derived, we obtain and interpret Nash, Pareto and Stackelberg solutions by exploiting the monotonicities in the two models.
postscript version of the complete paper
Abstract :
This paper presents several equivalent formulations for a structural design problem where the load carrying capacity is maximized for a prescribed volume subject to bounds on complementary energy and stresses. This limit design model covers the full range of strictly elastic, elastic/plastic and strictly plastic designs and is based on the unified analysis model of Ben-Tal and Taylor(1988). While this design model is convex, it is nonlinearly constrained and is of a very high dimension for topology design problems. Application of duality principles leads to several simpler but nonsmooth equivalent models. In particular, for the case when the design variables do not have explicit bounds, the dual models reduce to a minimization, subject to a single linear constraint, of a pointwise maximum of a finite number of convex functions. More importantly, these simpler design models are of greatly reduced size, since they contain only nodal variables. Further, the two cases of strictly plastic as well as strictly elastic limit design models can be reduced to linear programs both of which, unexpectedly, are shown to be equivalent to the more widely studied model for minimum compliance topology design of elastic trusses. Several numerical examples illustrate the usefulness of these new dual formulations.
postscript version of the complete paper
Abstract :
Concurrent modeling, as an emerging theme in engineering design research, also offers interesting new challenges in applied optimization. However, it is not always clear how best to construct mathematical models that represent such a concurrent decision-making process. In this paper, Stackelberg game models are investigated as a new class of formulations to study concurrent design. Such game models can provide unexpected and useful insights even in conventional design problems - for example, a standard class of structural optimization problems is shown to be a two-player Stackelberg game with Synthesis as the leader and Analysis as the follower. Situations arising from a reversal of roles of these two players are studied and a Pareto model of a Design versus Analysis game is also investigated. Stackelberg games are then viewed in the broader context of concurrent design with Design and Manufacturing as two prototypical decision-making players. The potential benefits of such an approach are illustrated in the design of a stiffened plate structure where a two-player model is formulated in which one objective is to maximize the collapse load and the other objective is to minimize the welding cost. This application turns out to have multiple Stackelberg solutions one of which is also then identified as a cooperative or Pareto solution.
postscript version of the complete paper
Abstract :
Bilevel problems are general models that arise in game-theoretic and decomposition related applications in engineering design. The principal difficulty in solving the bilevel models is that the lower level model is parametrically deformed and the resulting deformed solution is implicitly used by the upper level model. The effects of this parametric deformation are not always obvious. In this paper, this parametric model deformation is studied in the context of (i) two-level monotonicity analysis for constraint activity deduction, and (ii) the computation of sensitivities to data perturbations. The results and the examples show that while a general bilevel model can exhibit complex singularities, such a model is much easier to solve if the monotonicity information is available explicitly.
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Abstract :
In this paper, the objective is to study game-theoretic approaches for concurrent decision-making in engineering systems when well-defined, quantitative models are available to represent the different stages in the product life-cycle. The design and operation of a single-point metal cutting tool is used as an illustrative case-study and explicit models are developed for the design and manufacturing stages of the life-cycle of the tool. Monotonicities in these two models have been evaluated symbolically and numerically, leading to the deduction of active constraints and the convenient computation of the rational reaction sets of the two players. Cooperative (Pareto), non-cooperative (Nash) and leader-follower (Stackelberg) solutions are obtained which show that the unique Pareto solution coincides with the Stackelberg strategy when Design is the leader but this is not necessarily the case when Manufacturing is the leader.
postscript version of the complete paper
Abstract :
In this article we examine the behavior of optimal design models when the parameters are progressively unfixed in a process referred to as design aggregation. While this is a common modeling practice, its consequences on solution properties are not always predictable. Using no special assumptions on problem data, we develop an upper bound result for the aggregation process and highlight the difficulties which arise when even a single parameter is unfixed. When such an aggregated model is in turn parametrically decomposed, we recover the result where sufficiency at the lower level is required for necessary optimality conditions at the upper level. We also present a generalization of the commonly studied ``portal frame'' problem as a design case study. While aggregation has only benign effects on the solution of this model, interesting new results are obtained when model perturbations more severe than simple aggregation are imposed on this problem.
postscript version of the complete paper
Abstract :
One way to model concurrent decision-making is to formulate multi-player mathematical games. The purpose of this paper is to illustrate a variety of concurrent decision-making models that arise in two simple applications where explicit models can be obtained to represent the typical stages of design and manufacturing in product development. The resulting models are bilevel formulations and have the interpretation of a Stackelberg game between two players. The solutions in these models show that while a rich class of concurrent designs are obtained, such bilevel models also lead to different types of singularities in their solution sets.
postscript version of the complete paper
Abstract :
Mathematical programming methods have traditionally been used extensively in the analysis and design of elastic/plastic structures. However, some of the recent results from parametric nonlinear programming have not been fully exploited in mechanical applications, particularly those relating to the causes of singularities due to parametric variations. In this paper, we reexamine the phenomenon of nonunique displacements in elastic/perfectly-plastic structures under proportional loading. Once the analysis model as derived from a minimum energy principle is formulated as a quadratic program, it turns out that the response can be characterized as being unique or nonunique depending on the satisfaction of the linear independence or the Mangasarian-Fromovitz constraint qualification. A completely analogous result is then also derived for elastic/locking structures, thus indicating the usefulness of constraint qualifications in analyzing the nonunique behavior in other applications as well.
postscript version of the complete paper
Abstract :
In the broad context of modeling for system design, it is normally assumed that all decision-makers cooperate fully and thus avoid conflict. However, this is not always possible, in which case the design process is best modeled and studied as a multi-player game. The objective in this paper is to present and illustrate such a general game-theoretic framework for design modeling. In particular, explicit solutions are computed and interpreted for conservative or minmax, Pareto, Nash and Stackelberg games in a simple pressure vessel design example in which two players interact strategically, one of whom wants to minimize the weight and other wishes to maximize the volume. In another example in which a rotating disk is to be optimized for two objectives related to its design and manufacture, the Pareto solutions show how a novel manufacturing cost function can improve, under assumptions of a cooperative game, the optimal shape that is obtained under a single objective formulation related to design alone.
postscript version of the complete paper
Abstract :
Design models may be subject to a variety of perturbations or uncertainties, such as in variables, parameters and the constraint right-hand sides. In this paper, we study the specific cases of deterministic variations in the variables and the parameters. The goal is to obtain designs that not only reduce cost but which are also insensitive to such perturbations. We formulate bilevel models for this purpose and illustrate with examples the nature of insensitive designs, as well as the singularities that occur in such bilevel models.
postscript version of the complete paper
Abstract :
In this paper, we study a design problem in which the macro structural elements can be freely relocated around obstacles in a given design domain. The purpose is to examine this system as an illustrative example of a general feature-based design methodology where the design variables are both the location and the sizes of macro functional blocks. In this problem, the four frame elements of a structure are optimally located around an obstacle to minimize the compliance. The resulting mathematical program has a multiply connected feasible region, and has multiple local minima. A simulated annealing algorithm was used to obtain the different solutions to this problem. It was found that vertical frame members were usually able to jump over the obstacles in the optimization process, as desired. The results also indicate that the performace of the algorithm in this constrained problem depends critically on the step size adjustment parameters.
postscript version of the complete paper
Abstract :
Bilevel models have attracted increasing attention in recent optimization literature for two reasons - on one hand, they pose many new and difficult problems in numerical computation due to the underlying nonsmoothness of a two-level problem and on the other hand, they also offer a very rich set of interpretations via game-theoretic modeling in general system design. The goal of this paper is to illustrate these properties of multilevel modeling in two applications. In the first example, a three-level model is posed for the limit design of elastic/plastic structures, which under a strong assumption reduces to just a one level linear program. In the second example, various game models, including Stackelberg games, are formulated for the design of a metal-cutting tool. The different optimal strategies offer insights into how bilevel models capture real-life trade-offs in concurrent design.
postscript version of the complete paper
Dr.J.R.Jagannatha Rao
Department of Mechanical Engineering
University of Houston
4800 Calhoun Road
Houston, TX 77204 - 4792
U.S.A
(713) 743-4535
(713) 743-4503