Marcus Redhe, Tomas Jansson and Larsgunnar Nilsson Division of Solid Mechanics
Link¨oping University, Link¨oping, Sweden e–mail: marre@ikp.liu.se
ABSTRACT
Summary The aim of this work is to determine if space mapping technique using surrogate models together with response surfaces can be used for structural optimization of crashworthiness problems.
Further, the efficiency of optimization using space mapping will be compared to traditional structural optimization using the Response Surface Methodology.
Structural optimization often uses gradients for the objective and constraints in order to find the optimum solution. For nonlinear transient problems traditional structural optimization methods, e.g. Svanberg [1], do not work due to noise in the response.
To be able to determine the optimum the Response Surface Methodology (RSM) is often used.
Surfaces based on polynomial expressions are created for the objective and constraint functions.
These surfaces smoothen out the noisy response and the gradients can easily be calculated from the surface approximation. For further reading about RSM, see Myers and Montgomery [2].
If the number of design variables is not too large, RSM only needs a few number of evaluations to calculate the surface approximation. Still the simulation time for each design can be rather long. There is therefor a need for methods where simplified models can be used for the functional evaluation instead of using the full simulation models. The simplified models can be developed using less number of elements, other solvers, simple approximations with analytic solutions etc.
One method which makes this possible is called space mapping where a surrogate model is used together with the full model. The surrogate model (coarse model) determines in what direction the optimization will continue and the full model (fine model) will determine the design point for the next iteration. The use of the coarse model makes it possible to reduce the total simulation time and the fine model is used to get an accurate solution.
The first space mapping paper was written by Bandler et al. [3] where the basic theory was stated.
In Bandler et al. [4] the problem was formulated for using Broydens method for non-linear equa-tions and Bakr et al. [5] introduced a trust-region methodology. A mathematical viewpoint of space mapping can be found in Madsen and Sondergaard [6] .
The space mapping method has until now mainly been used in electromagnetics and circuit op-timization applications, see e.g. Bandler et al. [3] and Bakr et al. [7]. Leary et al. [8] used it in structural optimization on a simple cantilever beam.
The aim of this work is to study if the space mapping technique can be used on nonlinear transient dynamic problems, e.g. crashworthiness problems.
Table 1: Results from optimization of the beam for RSM and space mapping optimization.1
250,2
0:55, cost in minutes (SM space mapping).
x
f
iter fine coar cost RSM-L 21.83;25.0 0.8510 250.4;0.482 4 20 0 40
SM 21.63;25.0 0.8435 250.9;0.487 5 5 45 10
Optimization methods
The response surface methodology is a method for constructing global approximations of the objective and constraint functions based on functional evaluations at various points in the design space. The strength of the method is in applications where gradient based methods fails, i.e. when design sensitivities are difficult or impossible to evaluate. The selection of approximation functions to represent the actual behavior is essential. These functions can be polynomials of any order or be the sum of different basis functions, e.g. sine and cosine functions. To determine the unknown coefficients a least square approach is used. This approximation is applicable for all types of objectives and constraints in the optimization problem.
The idea of space mapping is to use two models for optimization. One coarse model, the surrogate model, that is fast to solve but not accurate enough, and one fine model that takes long time to solve but is more accurate. With these two models the optimization algorithm using space mapping takes advantage of the coarse model short solution time and the accuracy of the fine model. Therefore the vast amount of function evaluations are performed on the coarse model and corrections are made with the fine model. The theory for space mapping can be found in Leary et al. [8].
Examples and results
Three examples are used to study the algorithm where all examples are constrained structural optimization problems with one or two design variables. The problems are: one optimization of a beam and two crashworthiness problems. The crashworthiness problems are one square tube and one symmetric model of a vehicle front from Saab Automobile AB which both impacting onto a stone wall. All models are Finite Elements (FE) models.
One coarse model is developed for each system, used as surrogate model in the optimization using space mapping. For the beam problem the analytic solution to the problem is used as a coarse model and the fine model is solved with the nonlinear FE program LS-DYNA Hallquist [9] with a nonlinear material model. For the square tube both the fine and coarse models are solved in LS-DYNA, but the mass of the coarse model is increased in order to lower the total CPU-time.
Finally, in the coarse model of the vehicle model only the most important parts are included in the model. The two vehicle FE models are shown in Figure 1.
The optimum solution and the total CPU time used for the optimization using space mapping and RSM with linear (RMS-L) and quadratic (RMS-Q) surface approximations are given below for all examples. The vehicle model is only optimized using space mapping due to the excessive solution time.
Table 2: Results of square tube for RSM and space mapping optimization.1110, cost in hours (SM space mapping).
x
f
iter fine coar cost
RSM-Q 1.347;55 0.7771 110.0 3 28 0 32.6
RSM-L 1.349;55 0.7781 109.7 3 16 0 18.7
SM 1.395;55 0.775 110.4 4 4 36 8.87
Table 3: Design variable, objective and constraint history for the vehicle reanalysis using space mapping,420001.
iter 0 1 2 3 4 5
x
k 1.650 1.568 1.490 1.475 1.436 1.450
f
k —- 2.995 2.846 2.819 2.742 2.751
k —- 44508 43425 43578 42318 42702
Conclusions
The algorithm converged to the optimum solution for all problems. The total CPU time for conver-gence was reduced with up to 53% using space mapping compared to RSM. The conclusions are that optimization using space mapping can be used for optimization of crashworthiness problems with a significant reduction in CPU time. The drawback of the algorithm using space mapping is that it seems to be more unstable compared to RSM, e.g. if a bad starting point is chosen space mapping might not converge due to a too large derivation between the models. The starting point must be ’intelligently’ chosen based on knowledge of how the space mapping technique works and how the model behaves for parameter changes.
This work is founded by Saab Automobile AB, Volvo Car Corporation, the research programmes:
Figure 1: The fine and coarse model for the vehicle reanalysis
ENDREA and PROPER through the Swedish Foundation for Strategic Research. Saab Automobile AB has provided geometry and material data for the vehicle model application. We acknowledge Nielen Stander at LSTC for allowing us to use the optimization package LS-OPT.
REFERENCES
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