Abstract
We suggest an algorithm to generate the topology of load-bearing structures with help of a phase field model. The objective function homogenizes equivalent stress within the isotropic elastic material. However, local inhomogeneities in the stress field, e.g., at concentrated loads, do not distract the convergence of the algorithm. Beside a certain threshold in the equivalent stress field, the desired filling level of the design space is the main parameter of our objective function. The phase field parameter describes the density and stiffness of the substance in a closed interval. An Allen-Cahn equation regulates the phase transition, which is not conserving the mass of the system. The model evolves continuous regions of voids or dense material, whereas voids retain an infinitesimal residual stiffness, which is a million times smaller than the stiffness of the dense material. The evolution of structures is discussed by numerical examples.
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References
G. Allaire, F. Jouve, A. Toader, Structural optimization using sensitivity analysis and a level-set method. J. Comput. Phys. 194, 363–393 (2004)
M. Bendsøe, Optimal shape design as a material distribution problem. Struct. Optim. 1(4), 193–202 (1989)
M. Bendsøe, N. Kikuchi, Generating optimal topologies in structural design using a homogenization method. Comput. Methods Appl. Mech. Eng. 71, 197–224 (1988)
J. Cahn, S. Allen, A microscopic theory for domain wall motion and its experimental verification in fe-al alloy domain growth kinetics. J. de Physique Colloques 38(C7), 51–54 (1977)
J. Deaton, R. Grandhi, A survey of structural and multidisciplinary continuum topology optimization: post 2000. Struct. Multidiscip. Optim. 49, 1–38 (2014)
C. Le, J. Norato, T. Bruns, C. Ha, D. Tortorelli, Stress-based topology optimization for continua. Struct. Multidiscip. Optim. 41(4), 605–620 (2010)
I. Münch, Ch. Gierden, W. Wagner, A phase field model for stress based evolution of load-bearing structures. Int. J. Numer. Methods Eng., 1–21 (2018). https://doi.org/10.1002/nme.5909
D. Munk, G. Vio, G. Steven, Topology and shape optimization methods using evolutionary algorithms: a review. Struct. Multidiscip. Optim. 52, 613–631 (2015)
G. Rozvany, On design-dependent constraints and singular topologies. Struct. Multidiscip. Optim. 21(2), 164–172 (2001)
O. Sigmund, K. Maute, Topology optimization approaches. Struct. Multidiscip. Optim. 48, 1031–1055 (2013)
N. van Dijk, K. Maute, M. Langelaar, F. van Keulen, Level-set methods for structural topology optimization: a review. Struct. Multidiscip. Optim. 48(3), 437–472 (2013)
M. Wang, X. Wang, D. Guo, A level set method for structural topology optimization. Comput. Methods Appl. Mech. Eng. 192, 227–246 (2003)
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Muench, I. (2019). Evolution of Load-Bearing Structures with Phase Field Modeling. In: Radu, F., Kumar, K., Berre, I., Nordbotten, J., Pop, I. (eds) Numerical Mathematics and Advanced Applications ENUMATH 2017. ENUMATH 2017. Lecture Notes in Computational Science and Engineering, vol 126. Springer, Cham. https://doi.org/10.1007/978-3-319-96415-7_29
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DOI: https://doi.org/10.1007/978-3-319-96415-7_29
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