Keywords: reduced-basis, design optimisation, heat conduction.

In the last decade, the development and use of optimisation techniques have been applied to more realistic applications, associated with practical engineering problems. As a consequence, approximation based optimisation [1] has received more attention, as in general, these techniques produce very fast computations for the multiple functions/derivatives evaluations required by the optimisers. In the above mentioned context, the reduced basis method (RBM) [2,3,4] is appropriate. The RBM is a Galerkin projection onto low order approximation spaces comprising solutions of the problem of interest at selected points in the parameter/design space. The purpose of such a scheme is therefore to obtain high fidelity model information without high computational expense.

In this work a standard structural sizing optimisation (SSO) [3] algorithm is developed which encompass several aspects such as: geometric definition, mesh generation, heat transfer analysis, sensitivities analysis and mathematical programming algorithm. The RBM will be implemented in the thermal analysis and sensitivity analysis modules to obtain fast optimal designs. Comparisons will be conducted with the traditional (costly) finite element (FE) based SSO approach. The main focus of RBM is to construct an approximation for the solution fields (here temperature) and consequently for any solution output satisfying efficiency and accuracy requirements. The approach is to calculate the solution at several points on the manifold (samplings) corresponding to several parameter values and then for any new parameter the new solution is approximated by some linear combination of the known solutions.

The computation of the reduced basis method is conducted in the so-called reference domain [2,3,4]. As will be shown, to compute the solutions of a real domain, affine mapping transformations between the real and the reference domain will be implicit into the governing equations. The use of an affine decomposition together with the separability concept [2,3] to the conductivity "stiffness" matrix and load terms of the problem is the requirement to perform low cost calculations.

One important question addressed in the RBM is how to select the number of sampling points to achieve a desired accuracy with maximum computational efficiency. A posteriori error estimator is taken into account in this method to address such issues.

Using RBM, the equations to obtain the output and its derivative and conduct error estimation involves two classes of terms respectively, parameter/no parameter dependent. Therefore an off-line/on-line strategy is used for the computational implementation of the method.

The procedures described in this work were applied for the analysis and optimisation of a thermal fin problem extracted from reference [4]. Firstly, some accuracy studies were conducted for different number of sampling of points. The results were compared with the FE method and the reported solution of literature. As the number of sampling point increases, the approximated results monotonically converges to the FE solution but also the observed error was quite small even for small values of the number of samplings.

To illustrate the application of the RBM into the SSO procedure the structure presented was optimised. Twenty five (25) samplings were used. Both conventional and RBM approaches were used to obtain the optimum design. Two design variables were associated with the problem. They determined the optimum thickness and the optimum heat transfer coefficient of the problem. The objective function was a combined function [4] involving the total volume and the heat transfer coefficient of the structure. Apart from the imposed side constraints, the average temperature on the root was also constrained. Both conventional FE an approximated RBM techniques converged to the same optimum design, validating our implementation. Faster solutions were obtained using the RBM. In the final design, a 36% improvement was obtained compared to the initial design. Also, the results computed were in good agreement to those presented in the literature [4].

1

F. van Keulen, R. Haftka, "Special issue on approximations in optimization", Structural and Multidisciplinary Optimization Journal, 27(5), pp310-302, 2004. doi:10.1007/s00158-004-0443-7

2

C. Prud'homme, D. Rovas, K. Veroy, L. Machiels, Y. Maday, A.T. Patera, "Reliable Real-time Solution of Parametrized Partial Differential Equations: Reduced-basis Output Bound Method", Journal of Fluids Engineering, 124,70-79, 2003. doi:10.1115/1.1448332

3

T.M.M. Albuquerque, S.M.B Afonso, "The use of Reduced-basis Method for Optimization of 2D Elasticity Problems", in CILAMCE 2004, Recife, Brazil, 2004.

4

I.B. Oliveira, A.T. Patera, "Reduced-Basis Techniques for rapid Optimization of Systems Described by Parametric Partial Differential Equations", Optimization and Engineering (submitted for consideration), Kluwer, 2003. doi:10.1007/s11081-007-9002-6

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