Keywords: structural shape optimization, adaptive remeshing, sensitivity analysis, evolutionary algorithms, mesh sensitivity, differential evolution.

It is well accepted that evolutionary methods are a very powerful and robust tool for the solution of general optimization problems with the ability of not getting trapped in local minima, as in the case of deterministic methods. Nevertheless, the use of these methods requires the analysis of an important number of different designs. In structural shape optimization problems, the computational cost and the quality of the solutions are very much dependent on the quality of the finite element meshes used for the analysis. One important ingredient of the numerical analysis is the strategy for the generation of a proper mesh for each design. Here we can see two types of strategies:

1. To adapt a single existing mesh to the geometries of all different designs. Some existing strategies [1] permit adapting an existing mesh for very great modifications of the boundary shape preventing the elements from being too much distorted. Nevertheless, despite the fact that this type of strategy provides a valid mesh for each design, there is no control of the discretization error contained in the results of each analysis.
2. To perform a classical adaptive remeshing procedure for the analysis of each different design. Of course, this procedure ensures good quality results in the numerical analysis of each design, but the total computational cost grows significantly because each design is computed more than once.

This work presents a new strategy that allows generating an adapted mesh for each design without the necessity of performing a full adaptive remeshing procedure. It is based on the use of sensitivity analysis of all magnitudes related with adaptive remeshing (location of nodes, error estimation) with respect to the design variables. This sensitivity analysis is performed only once using a reference geometry and it is used to project the results of the corresponding analysis to all other designs to be analyzed. The projected informationpermits the generation of an appropriate adapted mesh for each new design usually in one shot, greatly reducing the computational cost compared with the second described strategy. This method was previously developed and used in the context of the solution of shape optimization problems using deterministic methods [2].

Two optimization problems have been used to test the behavior of the proposed methodology. The first one, with known analytical solution, consists of finding the best shape for the external boundary of a pipe subjected to a uniform pressure applied over the internal circular boundary. The numerical analysis clearly shows that the results converge to the analytical solution. The second problem consist of finding the optimum shape of a fly-wheel subjected to centrifugal and tangential loads. In this case the proposed methodology has been used to optimize a preliminary layout optimization obtained by topology optimization. The results of this problem have been shown to be similar to previous results obtained by a classical deterministic shape optimization strategy based on shape sensitivity analysis [3].

The examples show that the integration of the adaptive remeshing strategy into the evolutionary algorithm does not affect the convergence of the optimization process and ensures a good evaluation of the objective function and the constraints for each different design.

The proposed strategy provides a control on the quality of the analysis of each design in the least expensive way because only one single analysis is performed for each different individual.

1

G. Chiandussi, G. Bugeda, E. Oñate, "A simple method for automatic update of finite element meshes", Communications in Numerical Methods in Engineering, 16, 1-19, 2000. doi:10.1002/(SICI)1099-0887(200001)16:1<1::AID-CNM310>3.0.CO;2-A

2

G. Bugeda, J. Oliver, "A general methodology for structural shape optimization problems using automatic adaptive remeshing", International Journal for Numerical Methods in Engineering, 18, 3161-3185, 1993. doi:10.1002/nme.1620361807

3

J. Sienz, E. Hinton, G. Bugeda, S. Bulman, "Some studies on Integrating Topology and shape optimization" Innovative Computational Methods for Structural Mechanics. Saxe-coburg Publications. Edt. M Papadrakakis and B.H.V. Topping. 223-256, 1999. doi:10.4203/csets.1.11

purchase the full-text of this paper (price £20)

go to the previous paper
go to the next paper
return to the table of contents
return to the book description
purchase this book (price £105 +P&P)