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INNOVATION IN ENGINEERING COMPUTATIONAL TECHNOLOGY
Edited by: B.H.V. Topping, G. Montero, R. Montenegro
Design Optimisation with Randomness: A Taylor Expansion Approach
I. Doltsinis* and Z. Kang+
*Faculty of Aerospace Engineering and Geodesy, University of Stuttgart, Germany
I. Doltsinis, Z. Kang, "Design Optimisation with Randomness: A Taylor Expansion Approach", in B.H.V. Topping, G. Montero, R. Montenegro, (Editors), "Innovation in Engineering Computational Technology", Saxe-Coburg Publications, Stirlingshire, UK, Chapter 8, pp 145-168, 2006. doi:10.4203/csets.15.8
Keywords: stochastic optimisation, robust design, structures, deformation processes, Taylor series approximation.
Synthetic Monte Carlo sampling and analytic Taylor series expansion offer two different techniques for the treatment of random input scatter . This paper expounds on the Taylor series approximation as applied to the stochastic analysis and design optimisation of structures and deforming solids, including robustness against uncertainties. A unified approach is presented starting from linear elastic structures , extending to nonlinear and path dependent response  and progressing to deformation processes of inelastic solids . The methodology refers to finite element systems, and assumes that the response varies continuously as a function of the input; representation of the probability distribution is restricted to mean and variance. The approach is applicable to input scatter of practical relevance and is computationally efficient; its analytic nature allows utilisation of optimisation algorithms.
The randomness inherent to input parameters induces fluctuations of the response and of the performance. Characterization of the performance by mean and variance demands the determination of these two quantities for the response variables. An expansion by Taylor series about the mean input is used for the approximation of the mean response to the second order, and of the variance to the first order. The method requires continuity of the functional dependence between input and output variables at least up to the order of the approximation, and represents the probability distribution by the two first moments. The computational effort is determined by the number of the random input parameters. Monte Carlo computations, on the other hand, depend on the size of the statistical sample to be investigated and are seen to be more expensive; the probability distribution can be exposed in more detail, however.
The random variability of the performance entails considerations on design optimisation different from deterministic. If only the mean is targeted, the statement of the optimisation problem and its algorithmic treatment are as in the deterministic counterpart, but the participating quantities result from stochastic analysis. Apart from best performance in the mean, least variability is frequently requested as well. Then the design variables are to be specified such that the system is not sensitive to the scatter of the input, be it external actions or internal parameters. This is the issue of robust optimum design which demands minimization of both the mean value and the variance of the performance as defined by the objective function. The usually conflicting requirements are compromised by a weighted combination which constitutes the desirability function.
The analysis formalism is developed for the finite element representation of deformation problems. The case of linear elastic structures exemplifies the determination of mean response and variance using the Taylor series approximation. In addition, the computation of sensitivity expressions with respect to the design variables is pursued, which allows employment of gradient based optimisation algorithms. Advancing the theory, nonlinear structures with path dependent response as in elastoplasticity are treated by incrementation. The stochastic analysis is executed consistently, as is the formulation of the design sensitivity expressions. Deformation of inelastic solids is considered next, which requires attention to the propagation of fluctuations by the process. The stochastic problem is treated for viscous solids under stationary conditions as well as for unsteady deformation. The computational technique complies with a solution of the primary deterministic problem by the secant matrix of the system in which case a tangent operator is not available.
The theoretical exposition is complemented by numerical examples elucidating design optimisation for robustness. Apart from the instructive case of a planar cantilever truss, selected applications include a nonlinear space truss structure and preform design for net shape forming. A software package based on sequential quadratic programming  is used to solve the respective optimisation problem in conjunction with the input supplied by the stochastic finite element analysis.
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