Keywords: topology optimisation, level-set, topological sensitivity.

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 level-set function. Unlike other algorithms which have been developed in this area, we use the level-set function only as a boundary description tool and not for propagating the boundary by a Hamilton-Jacobi equation as in the classical level-set method. The way we use the level-set 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 level-set 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 level-set 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 level-set 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 level-set 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 well-known to be ill-posed, some regularisation has to be applied. The level-set 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 Laplace-Beltrami type operator.

The crucial point in this optimisation loop is the update of the level-set 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 level-set 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 level-set update, the algorithm provides several other desirable features:

• No a priori knowledge on the optimal topology is required.
• Since the level-set function is defined on an independent Cartesian grid, boundary representation and FE computation are completely decoupled. This allows for high flexibility in choosing accuracy and resolution.
• Due to remeshing, the results obtained are highly accurate.
• In each iteration, there is a sharp boundary and complete finite element model of the structure at hand. This makes possible not only a subsequent structural analysis without any manual pre-processing (e.g. assignment of boundary conditions), but also the inclusion of multi-physics simulations using existing independent codes.

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.

1

S. Osher and J.A. Sethian, "Front propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations", J. Comp. Phys. 78, pp. 12-49, 1988. doi:10.1016/0021-9991(88)90002-2

2

A. Schumacher, "Topologieoptimierung von Bauteilstrukturen unter Verwendung von Lochpositionierungskriterien", Universität-Gesamthochschule-Siegen, 1995.

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