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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 212
A New Algorithm for ThreeDimensional Topology Optimisation in Structural Mechanics Using a LevelSet Method H. Andrä^{1}, S. Amstutz^{2}, I. Matei^{1} and E. Teichmann^{1}
^{1}Fraunhofer Institut für Techno und Wirtschaftsmathematik, ITWM, Kaiserslautern, Germany
H. Andrä, S. Amstutz, I. Matei, E. Teichmann, "A New Algorithm for ThreeDimensional Topology Optimisation in Structural Mechanics Using a LevelSet Method", 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 212, 2006. doi:10.4203/ccp.83.212
Keywords: topology optimisation, levelset, topological sensitivity.
Summary
In this paper, we present a new algorithm for 3D topology
optimisation in structural mechanics which combines the concept of
topological gradient and the representation of the domain by a
levelset function. Unlike other algorithms which have been
developed in this area, we use the levelset function only as a
boundary description tool and not for propagating the boundary by
a HamiltonJacobi equation as in the classical levelset method.
The way we use the levelset function makes topology changes of
all kind, including nucleation of holes, very easy and natural.
After the presentation of a model optimisation problem having the equations of linear isotropic elasticity as constraints, a short review of the topological gradient and the levelset function approach is given. The main part consists of a detailed description of the algorithm and the new features it provides. Two examples from the context of structural mechanics show the performance of the algorithm. The iterative scheme presented is based on two main components: The boundary of the structure is represented by a levelset function , which has originally been devised by Osher and Sethian [1] to keep track of the movement of the boundary. We choose as the signed distance function. The criterion used to determine how the topology has to be changed is the topological sensitivity, which has been first developed by Schumacher [2]. The complete optimisation loop is outlined as follows: After generation of a tetrahedral finite element mesh of the domain, a structural analysis is performed. From the resulting stresses and strains, the topological gradient is computed. This quantity is used to update the levelset function describing the original structure, i.e., to modify the shape and, or the topology such that the objective function is decreased. Thereafter, a marching cube algorithm produces a surface triangulation of the updated levelset function, which is the basis for generation of a new tetrahedral finite element mesh. Transferring the boundary conditions to this new mesh closes the loop. As the topology optimisation problem is wellknown to be illposed, some regularisation has to be applied. The levelset function and, or the topological gradient are smoothed by convolution with a nonlinear filter kernel. The quality of the surface triangulation is improved by application of a LaplaceBeltrami type operator. The crucial point in this optimisation loop is the update of the levelset function , which is equivalent to a modification of the structure's topology. In the domain , has to be increased in points where the topological gradient in order to create holes in these points. The actual value of in a point x is of interest only in a small surrounding of the structure's boundary (narrow band) because these points are used to reconstruct the exact position of the boundary. These considerations lead to a new way of computing the update of : If the topological gradient indicates that a hole of radius is to be created in a point , it is sufficient to set and to adapt the values of in a neighbourhood of the boundary of the hole such that the new levelset function remains a correct signed distance function. This procedure is very simple and thanks to the narrow band technique very inexpensive. In addition to the efficient levelset update, the algorithm provides several other desirable features:
The last item is of special interest to the authors since consideration of the effects of the casting process such as residual stresses and eigenstrains on the optimisation is part of the project "MIDPAG  Innovative Methods for Integrated Dimensioning and Process Design for Cast Parts". This is to be presented in a forthcoming paper. References
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