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CivilComp Proceedings
ISSN 17593433 CCP: 83
PROCEEDINGS OF THE EIGHTH INTERNATIONAL CONFERENCE ON COMPUTATIONAL STRUCTURES TECHNOLOGY Edited by: B.H.V. Topping, G. Montero and R. Montenegro
Paper 204
BiasSpecified Robust Design Optimization: An Alternative Approach G. Steenackers and P. Guillaume
Acoustics & Vibration Research Group, Department of Mechanical Engineering, VUB University Brussels, Belgium G. Steenackers, P. Guillaume, "BiasSpecified Robust Design Optimization: An Alternative Approach", in B.H.V. Topping, G. Montero, R. Montenegro, (Editors), "Proceedings of the Eighth International Conference on Computational Structures Technology", CivilComp Press, Stirlingshire, UK, Paper 204, 2006. doi:10.4203/ccp.83.204
Keywords: optimization, robust design, regression techniques.
Summary
This paper discusses an existing robust optimization approach,
suggested and applied in the work of the authors Shin et al.
[1], and presents an alternative robust optimization
approach. Shin et al. suggest an optimization model, using
Lagrange multipliers to combine both constraints and slack
variables to change the inequality constraints to equality
constraints. The cost function used, proposed by the authors,
consists of minimizing the variance () while the extra
inequality constraint of
is
considered and taken into account by using Lagrange multipliers.
The optimization strategy is to minimize the variance while the
difference between process mean and target value should be
situated within (allowable) userspecified interval boundaries.
Our proposed algorithm, based on a generalized robust optimization
approach, will consist out of a standard GaussNewton optimization
algorithm [4] in combination with a mean squared error
function as the cost function to be minimized. An alternative robust
design optimization approach consists of using a different cost
function by minimizing the mean squared error, which taken into
account both the variance () and the difference
() between process mean and target value. Lagrange
multipliers are again used in order to take into account the extra
inequality constraint.
Let and represent the fitted response functions for the mean and standard deviation of the design parameter vector X . The secondorder response functions fitted for the mean and standard deviation are obtained as: with and regression constants, a and b ()vectors with constant values and A and B ()matrices with constant element values. The optimization problem is solved by using a numerical regression technique. For solving the generalized least squares problem, one needs to specify three initial design parameter value pairs where the correct value of the updating parameter is situated in between [4]. The least squares problem is solved with the boundary values used as starting values. A regression algorithm is used to calculate the regression coefficients of the interpolation polynomials. The difference in this alternative optimization approach is the fact that the mean squared error is minimized by using a cost function consisting of both the process variance and the difference between process mean and process target. In the optimization approach described in [1] only the variance was minimized, taking into account the extra inequality constraint that the process mean must be situated in a region around the process target, predefined by the value. As an example, both optimization approaches are illustrated with a numerical test case. The results are explained and discussed. The generalized robust optimization approach, minimizing the mean squared error of both process mean deviation and variance, will have a mean value closer to the process target value for almost the same variance value, reached with the original robust optimization approach. The reason for this better optimization result is the fact that the generalized optimization approach minimizes the mean square error of () and instead of only minimizing . Ignoring rounding errors, both robust optimization algorithms give the same results when using the same cost function expression. Also, when using a cost function only containing , the process mean is within the predefined target region. Thus, it is not necessary to use Lagrange multipliers in order to take into account the extra inequality constraint. The optimized process mean is situated in the allowable userdefined target region and the inequality constraint is automatically satisfied without the usage of Lagrange multipliers. When considering a cost function, consisting only out of the difference between process mean and bias (), the process bias reaches 0 which is the target when applying optimal design instead of robust design. References
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