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The success of structural optimization in the s motivated the emergence of multidisciplinary design optimization MDO in the s.

Finite Element Multidisciplinary Analysis, Second Edition (AIAA Education Series)

Jaroslaw Sobieski championed decomposition methods specifically designed for MDO applications. The following synopsis focuses on optimization methods for MDO.

Finite element method - Gilbert Strang

First, the popular gradient-based methods used by the early structural optimization and MDO community are reviewed. Then those methods developed in the last dozen years are summarized. There were two schools of structural optimization practitioners using gradient -based methods during the s and s: optimality criteria and mathematical programming. The optimality criteria school derived recursive formulas based on the Karush—Kuhn—Tucker KKT necessary conditions for an optimal design.

The KKT conditions were applied to classes of structural problems such as minimum weight design with constraints on stresses, displacements, buckling, or frequencies [Rozvany, Berke, Venkayya, Khot, et al. The mathematical programming school employed classical gradient-based methods to structural optimization problems. Schittkowski et al. The gradient methods unique to the MDO community derive from the combination of optimality criteria with math programming, first recognized in the seminal work of Fleury and Schmit who constructed a framework of approximation concepts for structural optimization.

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They recognized that optimality criteria were so successful for stress and displacement constraints, because that approach amounted to solving the dual problem for Lagrange multipliers using linear Taylor series approximations in the reciprocal design space. In combination with other techniques to improve efficiency, such as constraint deletion, regionalization, and design variable linking, they succeeded in uniting the work of both schools.

Approximations for structural optimization were initiated by the reciprocal approximation Schmit and Miura for stress and displacement response functions. Other intermediate variables were employed for plates. Combining linear and reciprocal variables, Starnes and Haftka developed a conservative approximation to improve buckling approximations.

Fadel chose an appropriate intermediate design variable for each function based on a gradient matching condition for the previous point.


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Vanderplaats initiated a second generation of high quality approximations when he developed the force approximation as an intermediate response approximation to improve the approximation of stress constraints. Canfield developed a Rayleigh quotient approximation to improve the accuracy of eigenvalue approximations. Barthelemy and Haftka published a comprehensive review of approximations in In recent years, non-gradient-based evolutionary methods including genetic algorithms , simulated annealing , and ant colony algorithms came into existence.


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At present, many researchers are striving to arrive at a consensus regarding the best modes and methods for complex problems like impact damage, dynamic failure, and real-time analyses. For this purpose, researchers often employ multiobjective and multicriteria design methods. MDO practitioners have investigated optimization methods in several broad areas in the last dozen years. These include decomposition methods, approximation methods, evolutionary algorithms , memetic algorithms , response surface methodology , reliability-based optimization, and multi-objective optimization approaches.

Multidisciplinary design optimization - Wikipedia

The exploration of decomposition methods has continued in the last dozen years with the development and comparison of a number of approaches, classified variously as hierarchic and non hierarchic, or collaborative and non collaborative. Approximation methods spanned a diverse set of approaches, including the development of approximations based on surrogate models often referred to as metamodels , variable fidelity models, and trust region management strategies. The development of multipoint approximations blurred the distinction with response surface methods.

Some of the most popular methods include Kriging and the moving least squares method. Response surface methodology , developed extensively by the statistical community, received much attention in the MDO community in the last dozen years. A driving force for their use has been the development of massively parallel systems for high performance computing, which are naturally suited to distributing the function evaluations from multiple disciplines that are required for the construction of response surfaces.

Distributed processing is particularly suited to the design process of complex systems in which analysis of different disciplines may be accomplished naturally on different computing platforms and even by different teams. Evolutionary methods led the way in the exploration of non-gradient methods for MDO applications.

They also have benefited from the availability of massively parallel high performance computers, since they inherently require many more function evaluations than gradient-based methods. Their primary benefit lies in their ability to handle discrete design variables and the potential to find globally optimal solutions. Like response surface methods and evolutionary algorithms, RBO benefits from parallel computation, because the numeric integration to calculate the probability of failure requires many function evaluations.


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One of the first approaches employed approximation concepts to integrate the probability of failure. Professor Ramana Grandhi used appropriate normalized variables about the most probable point of failure, found by a two-point adaptive nonlinear approximation to improve the accuracy and efficiency. Southwest Research Institute has figured prominently in the development of RBO, implementing state-of-the-art reliability methods in commercial software.

This approach focuses on maximizing the joint probability of both the objective function exceeding some value and of all the constraints being satisfied. When there is no objective function, utility-based probability maximization reduces to a probability-maximization problem. When there are no uncertainties in the constraints, it reduces to a constrained utility-maximization problem. This second equivalence arises because the utility of a function can always be written as the probability of that function exceeding some random variable.

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