Multipoint Approximations for Structural Optimization Problems with Noisy Response Functions

Fred van Keulen

Laboratory for Engineering Mechanics, Delft University of Technology,
Mekelweg 2, NL-2628 CD Delft, The Netherlands

Vassili V. Toropov

Department of Civil and Environmental Engineering
University of Bradford, Bradford, West Yorkshire, BD7 1DP, UK

Many 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 and (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) [6, 7, 14], for which the mesh density information can be influenced by the optimization history.

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

As an example of the noisy behaviour of response functions, consider a simple optimization problem of a square plate with a circular central hole [23]. The radius of the hole is treated as a single design variable. The normalised stress constraint (Fig. 1) and normalised stability constraint (Fig. 2) are shown for a coarse and a fine mesh (Fig. 3 and 4). In each case the mesh density was kept constant. It can be seen that the results, obtained with the coarse mesh, possess a significantly higher level of noise. Another obvious observation is that the numerical solution is also characterised by an offset from the exact solution. The magnitude of the offset depends on the specified mesh density and tends to zero with the increase of the mesh density.

Figure 1. Normalized stress constraint

Figure 2. Normalized stability constraint

Figure 3. Coarse mesh

Figure 4. Fine mesh

Another example (Polynkin et al. [13]) relates to the optimization of a conical shell made of a continuous fibre reinforced thermoplastic material (Fig. 5). The response analysis in this case involves two modelling steps, i.e. simulation of the thermoforming process and the FE analysis of the obtained product. The problem is formulated as maximization of the strain energy subject to the constant volume constraint. The design variables are the height of the cone and the base radius. Fig. 6 shows the noisy behaviour of the objective function.

Figure 5. CFRTP conical shell

Figure 6. Objective function for the CFRTP conical shell problem

Other examples of the noisy behaviour of functions in complex optimization problems are given by Free et al. [3], Haftka and co- workers [4, 11], Valorani and Dadone [29].

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

Rasmussen [15] investigated the influence of noise in the first order sensitivities on the performance of his ACAP technique, it was shown that even a 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 by Schoofs [17], Haftka with co-authors [4, 11, 16] and others, see reviews [1, 18]. These techniques are based on the least-squares surface fitting approach which was originally developed for dealing with problems where the function values are obtained as a result of physical measurements containing some level of noise. This approach allows to construct explicit approximations valid in the entire design variable space, but it is becoming computationally expensive as the number of design variables grows.

This study investigates the use of the Multipoint Approximation Method (MAM) (Toropov et al. [19, 20, 23, 24]) in the presence of numerical noise.

According to the MAM, the original optimization problem is replaced by a succession of simpler mathematical programming problems. The functions in each iteration present mid-range approximations of the corresponding original functions. These functions are noise-free or, at least, the level of noise does not cause problems with the 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. Each approximation function is defined as a function of design variables as well as a number of tuning parameters. The values of tuning parameters are determined by the weighted least squares surface fitting using the original function values (and its derivatives, when available) at several points of the design variable space considered as a plan of numerical experiments.

There are several important aspects in the basic methodology of the MAM which influence the convergence rate in the presence of numerical noise, namely: A simple but efficient choice of the structure of the approximations is an intrinsically linear (with respect to the tuning parameters) model, e.g. a multiplicative model. It has been successfully used for the variety of design optimization problems [6, 7, 9, 10, 12 - 14, 19, 20, 22, 24]. It should be noted that the requirements to the accuracy of the mid-range approximations can be more relaxed as compared to the case of the global ones but, obviously, the more accurate approximations would produce a faster convergence. Several new approaches are being investigated in the attempt to produce new high quality approximations valid for a larger range of design variables, e.g. Canterbury approximants [8], physically motivated (also called mechanistic [2]) models based on some prior knowledge about the behaviour of the system under consideration [21, 26], the use of simplified numerical models as approximations [21, 25, 26, 28], application of the Genetic Programming methodology to the choice of structure of the approximation function [27].

The weight coefficients characterize the relative contribution of the information about function values (and their derivatives, if available), their choice strongly influences the difference in the quality of the approximations in different regions of the design variable space. Since the optimum point usually belongs to the boundary of the feasible region, the approximation functions should be more accurate in that region. Thus, the information at the points located near the boundary of the feasible region is to be treated with higher weights. The weights will then depend on the values of the constraint functions, as these are indicators for the distance from a current point to the boundary of the feasible region, and also on the corresponding value of the objective function. The fact whether a point belongs to the feasible region or not is to be taken into account as well. 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 difference in the quality of the data, produced by the response analysis with different prescribed errors, can also be taken into account by means of the weight coefficients.

This can be efficiently used for the solution of problems where the response analysis is obtained using models of variable complexity (and accuracy). Van Keulen et al. [6, 7] showed examples where a good strategy for the choice of the prescribed error (i.e. complexity of the model) throughout the optimization process could significantly reduce the overall computational effort. The basic idea is not to keep the constant prescribed error (and model complexity) but to use a relatively rough model 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 improved in the process of optimization.

It is also significant how to choose the plan of numerical experiments as a set of points in a current subregion of the design variable space defined by the move limits. Better quality approximations can be obtained 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 (Fig. 7). Several schemes for updating the dimensions of this neighbourhood have been considered.

Figure 7. Search subregion and its neighbourhood

The move limit strategy shall reflects the progress of the optimization process. The applied reduction or enlargement of a next search subregion depends on (i) the quality of the approximation functions, (ii) whether the move limits at the current sub-optimum point are active or not, (iii) the angles between the previous move vectors (i.e. whether zigzagging occurs or not) and (iv) the quality of the present sub-optimum point in terms of the constraint violation and the value of the objective function as compared to the points examined during the last few optimization steps.

The applications of MAM to complex design optimization problems with noisy response analysis output include:
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