Michell Structures within L-Shaped Domains
By recalling the main mathematical results concerning the theory of Michell structures, the present paper puts forward an interpretation of the selected numerical methods for constructing their approximants, that is, trusses with a large number of nodes. The efficiency of one of these methods: the ground structure method in its adaptive version is shown in the context of the L-shaped design domain problem. A large family of highly accurate truss approximants corresponding to the point loads acting at selected vertices is constructed and discussed.
KeywordsMichell structures, Structural optimization, trusses,
References1. K. Bołbotowski, T. Sokół, New method of generating Strut and Tie models using truss topology optimization, [in:] M. Kleiber, T. Burczynski, K. Wilde, J. Górski, K. Winkelmann, Ł. Smakosz [Eds], Advances in mechanics: Theoretical, computational and interdisciplinary Issues, pp. 97–100, CRC Press, 2016.
2. G. Bouchitté, I. Fragalà, Optimality conditions for mass design problems and applications to thin plates, Archive for Rational Mechanics and Analysis, 184(2): 257–284, 2007.
3. G. Bouchitté, W. Gangbo, P. Seppecher, Michell trusses and lines of principal action, Mathematical Models and Methods in Applied Sciences, 18(9): 1571–1603, 2008.
4. M. Gilbert, A. Tyas, Layout optimization of large-scale pinjointed frames, Engineering Computations, 20(8): 1044–1064, 2003.
5. W.S. Hemp, Optimum structures, Oxford, Clarendon Press, 1973.
6. T. Lewinski, G.I.N. Rozvany, Exact analytical solutions for some popular benchmark problems in topology optimization III: L-shaped domains, Structural and Multidisciplinary Optimization, 35(2): 165–174, 2008, doi: 10.1007/s00158-007-0157-8.
7. T. Lewinski, G.I.N. Rozvany. T. Sokół, K. Bołbotowski, Exact analytical solutions for some popular benchmark problems in topology optimization III: L-shaped domains revisited, Structural and Multidisciplinary Optimization, 47(6): 937–942, 2013, doi: 10.1007/s00158-012-0865-6.
8. T. Lewinski, T. Sokół, C. Graczykowski, Michell Structures, Springer International Publishing AG, Cham, Switzerland, 2019, doi: 10.1007/978-3-319-95180-5.
9. T. Sokół, Multi-load truss topology optimization using the adaptive ground structure approach, [in:] T. Łodygowski, J. Rakowski, P. Litewka [Eds], Recent Advances in Computational Mechanics, pp. 9–16, CRC Press, London, 2014.
10. T. Sokół, A new adaptive ground structure method for multi-load spatial Michell structures, [in:] M. Kleiber, T. Burczynski, K. Wilde, J. Górski, K. Winkelmann, Ł. Smakosz [Eds], Advances in mechanics: Theoretical, computational and interdisciplinary issues, pp. 525–528, CRC Press, 2016.
11. G. Strang, R.V. Kohn, Hencky-Prandtl nets and constrained Michell trusses, Computer Methods in Applied Mechanics and Engineering, 36(2): 207–222, 1983.