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On the (non-)optimality of Michell structures

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Abstract

Optimal analytical Michell frame structures have been extensively used as benchmark examples in topology optimization, including truss, frame, homogenization, density and level-set based approaches. However, as we will point out, partly the interpretation of Michell’s structural continua as discrete frame structures is not accurate and partly, it turns out that limiting structural topology to frame-like structures is a rather severe design restriction and results in structures that are quite far from being stiffness optimal. The paper discusses the interpretation of Michell’s theory in the context of numerical topology optimization and compares various topology optimization results obtained with the frame restriction to cases with no design restrictions. For all examples considered, the true stiffness optimal structures are composed of sheets (2D) or closed-walled shell structures (3D) with variable thickness. For optimization problems with one load case, numerical results in two and three dimensions indicate that stiffness can be increased by up to 80 % when dropping the frame restriction. For simple loading situations, studies based on optimal microstructures reveal theoretical gains of +200 %. It is also demonstrated how too coarse design discretizations in 3D can result in unintended restrictions on the design freedom and achievable compliance.

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Notes

  1. 1 We note that the term “optimal”is grossly overused in the discussion of numerical topology optimization results in the literature, however, the variable thickness sheet problem is indeed a convex problem and hence convergence to global optima can in this case be guaranteed.

  2. 2 Note that the analytical Michell solution is pin-supported at the lower corners but this would cause stress singularities when modeled using a continuum formulation. We have found that reducing the line support to the lower three rightmost and leftmost elements, mimicking a pinned support does not change the conclusions of this study.

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Acknowledgments

The authors dedicate this paper to George Rozvany (1930-2015): partly due to his life-long and great efforts in providing and promoting benchmarks to the structural optimization society; partly for valuable email exchanges concerning the benchmark examples used in this paper; and partly for his general and significant scientific and personal impact on the field of structural and multidisciplinary optimization and on the first author’s scientific career in particular.

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Authors and Affiliations

  1. Department of Mechanical Engineering, Solid Mechanics, Technical University of Denmark, 2800 Kgs., Lyngby, Denmark

    Ole Sigmund, Niels Aage & Erik Andreassen

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  1. Ole Sigmund
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  2. Niels Aage
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  3. Erik Andreassen
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Corresponding author

Correspondence to Ole Sigmund.

Additional information

This paper is dedicated to George Rozvany (1930-2015) - see more in the acknowledgement

This work was funded by the Villum Foundation through the NextTop project.

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Sigmund, O., Aage, N. & Andreassen, E. On the (non-)optimality of Michell structures. Struct Multidisc Optim 54, 361–373 (2016). https://doi.org/10.1007/s00158-016-1420-7

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