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Civil-Comp Proceedings
ISSN 1759-3433
CCP: 83
PROCEEDINGS OF THE EIGHTH INTERNATIONAL CONFERENCE ON COMPUTATIONAL STRUCTURES TECHNOLOGY
Edited by: B.H.V. Topping, G. Montero and R. Montenegro
Paper 211

Numerical Methods to Avoid Topological Singularities

V. Pomezanski

Computational Structural Mechanics Research Group, Hungarian Academy of Sciences and Budapest University of Technology and Economics, Hungary

Full Bibliographic Reference for this paper
V. Pomezanski, "Numerical Methods to Avoid Topological Singularities", in B.H.V. Topping, G. Montero, R. Montenegro, (Editors), "Proceedings of the Eighth International Conference on Computational Structures Technology", Civil-Comp Press, Stirlingshire, UK, Paper 211, 2006. doi:10.4203/ccp.83.211
Keywords: topology, optimization, corner node, corner contact, checkerboard patterns, diagonal chains, numerical method.

Summary
One of the most severe computational difficulties in finite element (FE) based topology optimization is caused by solid (or "black") ground elements connected only through a corner node. This configuration may appear in checkerboard patterns, diagonal element chains or as isolated hinges. Corner contacts in nominally optimal topologies are caused by discretization errors associated with simple (e.g. four-node) elements (e.g. Sigmund and Petersson [1]), which grossly overestimate the stiffness of corner regions with stress concentrations. In fact, it was shown by Gaspar et al. [2] that both checkerboard patterns and diagonal element chains may give an infinite compliance, if the latter is calculated by an exact analytical method. This makes them the worst possible solution, if an exact analysis is used in compliance minimization. Corner contacts may be suppressed by
(a)
a more accurate FE analysis of the ground elements, where the process may use several simple FEs per ground element (e.g. Rozvany and Zhou [3]), or higher order elements (Sigmund and Petersson [1]). Disadvantages of this approach are
  • greatly increased DOF for a given number of ground elements and
  • some diagonal chains remaining in the solution (Gaspar et al. [4]).
(b)
Modification of the original problem by using geometrical constraints or "diffused" sensitivities (filters), e.g. perimeter control (Haber and Bendsoe [5]) or filtering (e.g. Sigmund [6]) changes the original topology optimization problem and this usually results in a lower resolution, which may, in some cases, be somewhat nonoptimal in terms of the original problem [2].
(c)
Employing a constraint preventing corner contacts directly, here the "corner contact function" (CCF) with a high value for corner contacts and a low value for any other configuration around a corner node is an additional constraint (e.g. Poulsen [7]).
(d)
Correcting selectively the discretization errors by appropriately penalizing corner contacts, in this case the CCF is an additional term in the objective function representing penalty for corner contacts.

The approach (d) seems to be the most rational, because it rectifies the discretization errors, which lower incorrectly the value of the objective function (e.g. compliance). The proposed method is particularly effective in combination with the SIMP method, since the latter is a penalization method in its original form, and requires only a minor modification for corner contact control (CO-SIMP). An early corner contact function was suggested by Bendsoe et al. [8].

Defining and employing new CCFs, continuous functions which have a high value for corner contacts and a low value for any other configuration around a node, as an objective a new mathematical programming process CO-SIMP was developed [9]. The CO-SIMP method for the case of Michell's cantilever generates a similar result as the exact solution. The development of the CCFs properties, the numerical method, the modified SIMP algorithm and extensive numerical examples are included in the paper.

References
1
Sigmund, O.; Petersson, J. "Numerical instabilities in topology optimization: A survey on procedures dealing with checkerboards, mesh-dependencies and local minima", Struct. Optim., 16, 68-75, 1998. doi:10.1007/BF01214002
2
Rozvany, G.I.N.; Querin, O.M.; Gaspar, Z.; Pomezanski, V. "Weight increasing effect of topology simplification", Struct. Multidisc. Optim., 25, 459-465, 2003. doi:10.1007/s00158-003-0334-3
3
Rozvany, G.I.N.; Zhou, M. "Applications of the COC algorithm in layout optimization", In: Eschenauer, H Matteck, C.; Olhoff, N. (eds.) Engineering Optimization in Design Processes, Proc. Int. Conf. held in Karlsruhe, Germany, Sept. 1990), pp. 59-70, Springer-Verlag, Berlin, 1991.
4
Gaspar, Z.; Logo, J.; Rozvany, G.I.N. "Addenda and Corrigenda", Struct. Multidisc. Optim., 24, 338-342, 2002. doi:10.1007/s00158-002-0245-8
5
Haber, R.B.; Bendsoe, M.P. "A new approach to variable-topology shape design using constraint on the perimeter", Struct. Optim., 11, 1-12, 1996. doi:10.1007/BF01279647
6
Sigmund, O. "Design of material structures using topology optimization", Ph.D. Thesis, Dept. Solid Mech., TU Denmark, 1994.
7
Poulsen, T.A. "A simple scheme to prevent checkerboard patterns and one node connected hinges in topology optimization", Struct. Multidisc. Optim., 24, 396-399, 2002. doi:10.1007/s00158-002-0251-x
8
Bendsoe, M.P.; Diaz, A.; Kikuchi, N. "Topology and generalized layout optimization of elastic structures", In: Bendsoe, M.P.; Mota Soares, C.A. (Eds.) Topology design of Structures. (Proc. NATO ARW held in Sesimbra in 1992) Dordrecht: Kluwer, 1993.
9
Pomezanski, V.; Querin, O.M.; Rozvany, G.I.N., "CO-SIMP: extended SIMP algorithm with direct COrner COntact COntrol", Struct. Multidisc. Optim., 30, 164-168, 2004. doi:10.1007/s00158-005-0514-4

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