Development of MAM - Multipoint Approximation Method

Motivation for the development of MAM: Structural optimization and identification problems with computationally expensive and noisy response functions.

The majority of real-life structural optimization problems have the same common features: (i) the objective and constraint functions are evaluated as a result of expensive numerical computations, e.g. using FEM; (ii) function values and/or their derivatives may contain some level of noise. It means that function values, corresponding to two slightly different sets of design variables, may have a considerable random variation. Moreover, even the function values, corresponding to the same set of design variables, may be different depending on the optimization search history and/or iteration history of an iterative response analysis. As an example, we mention FEM in conjunction with Adaptive Mesh Refinement (AMR), for which the mesh density information might be influenced by the optimization history.

The level of noise can vary from negligible to quite significant (or even intolerable) depending on the formulation and the size of the optimization problem as well as the size and specific features of an response analysis model.

Examples of noisy behaviour of functions in complex optimization problems are given in [3, 4, 21, 23], among others. Other examples of the loss of accuracy of computations in a structural optimization problem may include the error accumulation in the semi-analytical sensitivity analysis [11].

During the last decade, several approximation techniques were proposed which take account of the negative effect of noise [1].

Rasmussen [14] investigated the influence of noise in first-order sensitivities on the performance of his ACAP technique, it was shown that even a quite considerable level of noise could be tolerated. The technique does not take into account the noise in the function values.

The performance of global approximation techniques was investigated in [4, 15, 25]. These techniques are based on the least-squares surface fitting approach which was originally developed for the situations where the function values are obtained as a result of physical measurements containing some level of noise. Such an approach allows to construct explicit approximations valid in the entire design space, but is restricted by relatively small optimization problems (up to ten design variables [25]).

Our research team at Bradford in collaboration with TU Delft (The Netherlands) and Nizhny Novgorod University (Russia) develops and investigates the use of Multipoint Approximation Method (MAM) (Toropov et al. [16, 17, 23]) in the presence of numerical noise.

According to MAM, the original structural optimization problem is replaced by a succession of simpler mathematical programming problems. The functions in each iteration present mid-range multipoint approximations of the corresponding original functions. These functions are noise-free or, at least, the level of noise does not cause problems with convergence of an individual optimization sub-problem. The solution of an individual sub-problem becomes a starting point for the next step, the move limits are changed and the optimization is repeated iteratively until the optimum is reached. Various optimization techniques can be used to solve the sub-problems because the approximation functions are chosen to be simple.

The efficiency of the optimization technique depends to a large extent on the accuracy of the approximate expressions. Employing the methods of regression analysis [2], each approximation is defined as a function of a number of tuning parameters. They are determined on the basis of the information about the original function (and, possibly, its derivatives) at several points of the design variable space. The use of the weighted least- squares method allows to determine the values of tuning parameters for each function.

The weight coefficients characterize the relative contribution of the information about function values (and their derivatives, if available). The proper choice of weight coefficients strongly influences the quality of the approximations. Since the optimum point usually belongs to a boundary of the feasible region, where at least one of the constraints is active, the approximation functions must be more accurate in such a point. Thus, the information on the points located near the boundary of the feasible region is to be treated with higher weights.

It is also significant how we choose the plan of numerical experiments as a set of points in a current subregion of the design variable space. Better quality approximations can be achieved by increasing the total number of points in the plan of experiments, but, on the other hand, it increases the number of calls for the response analysis. In order to reduce the total computational effort, the points from previous steps, belonging to a current subregion and its neighbourhood, can be used as well.

Other important aspects of any approximation method are the strategy of changing the move limits and the condition of termination.

In certain cases, e.g. when AMR techniques are applied to the shape optimization, it is possible to specify the error, requested from the numerical analysis procedure (error of the numerical experiment). The different tolerance of MAM to the data, produced by the response analysis with different prescribed errors, can be taken into account by means of the weight coefficients. In papers [5 - 7] it is shown that the proper choice of the prescribed error depending on the current status of the optimization process can significantly reduce the overall computational effort. The basic idea is not to keep the constant prescribed error (and mesh density) but to use a relatively coarse mesh in the beginning of the optimization process and to refine it as the process progresses. Correspondingly, the quality of approximations can be low in the beginning and must be increased in the process of optimization.

Other interests of our research team, closely collaborating with researchers of TU Delft, include the development of new types of mid-range approximations, e.g. using simplified numerical models as approximations [24] and also the use of genetic programming (GP) methodology for the selection of the structure of approximations.

The variety of applications of MAM to complex structural optimization problems with computationally expensive and noisy response analysis output include:

References

  1. Barthelemy, J.-F.M.; Haftka R.T. (1993) Approximation concepts for optimum structural design - a review. Structural Optimization, 5, 129-144.
  2. Box, G.E.P.; Draper, N.R. Empirical model-building and response surfaces. New York: John Wiley & Sons, 1987
  3. Free, J.W.; Parkinson, A.R.; Bryce, G.R.; Balling, R.J.; Approximations of computationally expensive and noisy functions for constrained nonlinear optimization. J. of Mech. Trans. Auto. Des. 109, 1987, pp. 528- 532
  4. Giunta, A.A.; Dudley, J.M.; Narducci, A.; Grossman, B.; Haftka, R.T.; Mason, W.H.; Watson, L.T. Noisy aerodynamic response and smooth approximations in HSCT design. AIAA-94-4376-CP, Proc. 5th AIAA/USAF/NASA/ISSMO Symp. on Multidisciplinary and Structural Optimization, Panama City, Florida, 1994, pp. 1117-1128
  5. van Keulen, F.; Toropov, V.V.; Polynkine, A.A. Optimization of geometrically nonlinear shell structures using multi-meshing and adaptivity, AIAA-94-4361-CP, Proc. 5th AIAA/USAF/NASA/ISSMO Symp. on Multidisciplinary and Structural Optimization, Panama City, Florida, 1994, pp. 995-1005
  6. van Keulen, F., Toropov, V.V.; Polynkine, A.A.: Shape optimization strategies using the multi-point approximation method and adaptive mesh refinement. N. Olhoff, G.I. Rozvany (eds.). Proceedings of The First World Congress of Structural and Multidisciplinary Optimization, Goslar, Germany, May 1995, pp. 67 - 74, Pergamon, 1995.
  7. van Keulen, F.; Toropov, V.V.; Markine, V.L.: Recent refinements in the multi-point approximation method in conjunction with adaptive mesh refinement. In: McCarthy, J.M. (ed.), Proc. ASME Design Engineering Technical Conferences and Computers in Engineering Conference, August 18-22 1996, Irvine CA, 96- DETC/DAC-1451, 12 p.
  8. Markine, V.L.; Meijers, P.; Toropov, V.V.; Meijaard, J.P.: Multilevel optimization of the dynamic behaviour of mechanical systems. Proceedings of Int. Forum on Aeroelasticity and Structural Dynamics 1995, Manchester, June 1995, pp. 31.1-31.8, Royal Aeronautical Soc., 1995.
  9. Markine, V.L.; Meijers, P.; Meijaard, J.P.; Toropov, V.V.: Optimization of the dynamic response of linear mechanical systems using a multipoint approximation technique. In: D. Bestle, W. Schiehlen (eds.), Proceedings of IUTAM Symposium on Optimization of Mechanical Systems, Stuttgart, March 1995, pp. 189-196, Kluwer, 1996.
  10. Markine, V.L.; Meijers, P.; Meijaard, J.P.; Toropov, V.V.: Multilevel optimization of dynamic behaviour of a linear mechanical system with multipoint approximation. Eng. Optimization, vol. 25, 1996, pp.295-307.
  11. Pedersen, P.; Cheng, G.; Rasmussen, J. On accuracy problems for semi-analytical sensitivity analysis. Mech. Struct. Mach. 17, 1989, pp. 373-384
  12. Polynkine, A.A.; van Keulen, F.; Toropov, V.V.: Optimization of geometrically nonlinear thin-walled structures using the multipoint approximation method. Struct. Optimiz., 9, 105-116, 1995.
  13. Polynkine, A.A.; van Keulen, F.; Toropov, V.V.: Optimization of geometrically non-linear structures based on a multi-point approximation method and adaptivity. Engineering Computations. Int. J. for Computer-Aided Engineering & Software, vol. 13, 2/3/4, 1996, pp.76-97.
  14. Rasmussen, J. Accumulated approximations - a new method for structural optimization by iterative improvements. Proc. 3rd USAF/NASA Symp. on Recent Advances in Multidisciplinary analysis & Optimization, pp. 253-258
  15. Schoofs, A.J.G. Experimental design and structural optimization. TU Eindhoven: Ph.D. dissertation, 1987
  16. Toropov V.V. Simulation approach to structural optimization. - Struct. Optimiz., 1, No 1, 1989, pp. 37-46.
  17. Toropov, V.V.; Filatov, A.A.; Polynkin, A.A. (1993) Multiparameter structural optimization using FEM and multipoint explicit approximations. Struct. Optimiz., 6, 7-14.
  18. Toropov, V.V.; van der Giessen, E. Parameter identification for nonlinear constitutive models: Finite element simulation - optimization - nontrivial experiments. In: Optimal design with advanced materials P.Pedersen (ed.), Proc. of IUTAM Symp. 1992, Elsevier Sci. Publ. 1993, pp. 113-130
  19. Toropov, V.V.; Carlsen, H.: Optimization of Stirling engine performance based on multipoint approximation technique. In: Gilmore, B.J. et al (eds.), Advances in Design Automation 1994, Vol. 2, Robust Design Applications, Decomposition and Design Optimization, Optimization Tools and Applications. 20th Design Automation Conference, Minneapolis, September 11-14, 1994, pp.531-536, 1994.
  20. Toropov, V.V.; Markin, V.L.; Carlsen, H.: Discrete structural optimization based on multipopint explicit approximations. In: W. Gutkowski, J. Bauer (eds.), Discrete Structural Optimization, Proc. of IUTAM Symposium on Discrete Structural Optimization, Zakopane, Poland, August 31-September 3, 1993, pp.98- 107, Springer-Verlag, 1994.
  21. Toropov, V.V.: Multipoint approximation method for structural optimization problems with noisy function values. In: K. Marti, P. Kall (eds.), Stochastic Programming. Numerical Techniques and Engineering Applications. Lecture Notes in Economics and Mathematical Systems 423. Proc. of 2nd GAMM/IFIP Workshop on Stochastic Optimization: Numerical Methods and Technical Applications, Munich, June 1993, pp. 109-122, Springer-Verlag, 1995.
  22. Toropov, V.V.; van der Giessen, E.; Yoshida, F.: Material parameter identification based on nontrivial experiments, numerical simulation and optimization. In: M.I. Friswell, J.E. Mottershead (eds.). Proceedings of Int. Conf. on Identification in Engineering Systems, Swansea, March 1996, pp. 328-337. The Cromwell Press Ltd.
  23. Toropov, V.V.; van Keulen, F.; Markine, V.L.; de Boer, H.: Refinements in the multi-point approximation method to reduce the effects of noisy responses. 6th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, Bellevue WA, September 4-6 1996, Part 2, pp. 941-951
  24. Toropov, V.V.; Markine, V.L.: The use of simplified numerical models as mid-range approximations. 6th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, Bellevue WA, September 4-6 1996.
  25. Vanderplaats G.N. Effective use of numerical optimization in structural design. Finite Elements in Analysis and Design 6, 1989, pp. 97-112
  26. Yoshida, F,; Urabe, M; Toropov, V.V.: Identification of material parameters in constitutive model for sheet metals from cyclic bending tests. In: Abe, T., Tsuta, T. (eds.), Proc. of Asia-Pacific Symp. on Advances in Engineering Plasticity and its Applications, 21-24 August 1996, pp. 271-276, Pergamon, 1996

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