Abstract
When describing the mechanical behavior of some engineering materials, such as composites, grains, biological materials and cellular solids, the Cosserat continuum theory has more powerful capabilities compared with the classical Cauchy elasticity since an additional local rotation of point and its counterpart (couple stress) are considered in the Cosserat elasticity to represent the material microscale effects. In this paper, a parameterized level set topology optimization method is developed based on the Cosserat elasticity for the minimum compliance problem of the Cosserat solids. The influence of material characteristic length and Cosserat shear modulus on the optimized structure is investigated in detail. It can be found that the microstructural constants in the Cosserat elasticity have a significant impact on the optimized topology configurations. In addition, the minimum feature size and the geometric complexity of the optimized structure can be controlled implicitly by adjusting the parameters of the characteristic length and Cosserat shear modulus easily. Furthermore, the optimized structure obtained by the developed Cosserat elasticity based parameterized level set method will degenerate to the result by using the classical Cauchy elasticity based parameterized level set method when the Cosserat shear modulus approaches zero.
Graphic Abstract
A parameterized level set topology optimization method is developed based on the Cosserat elasticity for the optimization of minimum compliance of the structures with micropolar materials. The influence of characteristic length and Cosserat shear modulus of the micropolar material on the optimized structure has been investigated in detail. It can be found that the minimum feature size and the geometric complexity of the optimized structure can be controlled implicitly by adjusting the parameters of the characteristic length and Cosserat shear modulus easily.
Similar content being viewed by others
References
Anderson, W.B., Lakes, R.S.: Size effects due to Cosserat elasticity and surface damage in closed-cell polymethacrylimide foam. J. Mater. Sci. 29, 6413–6419 (1994)
Fleck, N.A., Muller, G.M., Ashby, M.F., et al.: Strain gradient plasticity: theory and experiment. Acta Metal Mater. 42, 475–487 (1994)
Cosserat, E., Cosserat, F.: Theorie des Corps Deformable. Herman Etfils, Paris. 3–12 (1909)
Toupin, R.A.: Elastic materials with couple-stresses. Arch. Ration. Mech. An. 11, 385–414 (1962)
Mindlin, R.D.: Micro-structure in linear elasticity. Arch. Ration. Mech. An. 16, 51–78 (1964)
Aifantis, E.: On the microstructural origin of certain inelastic models. ASME J. Eng. Mater. Technol. 106, 326–330 (1984)
Eringen, A.: On different equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J. Appl. Phys. 54, 4703–4710 (1983)
Tang, H.X., Guan, Y.H., Zhang, X., et al.: Low-order mixed finite element analysis of progressive failure in pressure-dependent materials within the framework of the Cosserat continuum. Eng. Comput. 34, 251–271 (2017)
Tang, H.X., Wei, W.C., Liu, F., et al.: Elastoplastic Cosserat continuum model considering strength anisotropy and its application to the analysis of slope stability. Copput. Geotech. 117, 103235 (2020)
Xiu, C.X., Chu, X.H., Wang, J., et al.: A micromechanics-based micromorphic model for granular materials and prediction on dispersion behaviors. Granul. Matter. 22, 74 (2020)
Sigmund, O.: A 99 line topology optimization code written in matlab. structural and multidisciplinary optimization. Struct. Mutidiscip. Optim. 21, 120–127 (2001)
Wang., M.Y., Wang., X., Guo., D.: A level set method for structural topology optimization. Comput. Method Appl. Mech. 192, 227 (2003)
Xie, Y.M., Steven, G.P.: A simple evolutionary procedure for structural optimization. Comput. Struct. 49, 885–896 (1993)
Guo, X., Zhang, W., Zhong, W.: Doing topology optimization explicitly and geometrically—a new moving morphable components based framework. J. Appl. Mech. 81, 081009 (2014)
Zong, H., Liu, H., Ma, Q., et al.: VCUT level set method for topology optimization of functionally graded cellular structures. Comput. Methods Appl. Mech. Eng. 354, 487–505 (2019)
Liu, H., Zong, H., Shi, T., et al.: M-VCUT level set method for optimizing cellular structures. Comput. Methods Appl. Mech. Eng. 367, 113154 (2020)
Xia, Q., Zong, H., Shi, T., et al.: Optimizing cellular structures through the M-VCUT level set method with microstructure mapping and high order cutting. Compos. Struct. (2020), in press. https://doi.org/10.1016/j.compstruct.2020.113298
Gei, M., Rovati, M., Veber, D.: Effect of internal length scale on optimal topologies for Cosserat continua. In: IUTAM Symposium on Topological Design Optimization of Structures, Machines and Materials. pp. 157–166 (2006)
Rovati, M., Veber, D.: Optimal topologies for Cosserat solids. Struct. Mutidiscip. Optim. 33, 47–59 (2007)
Optimal topologies in structural design of Cosserat materials: Arimitsu, Y., Karasu, K., Wu., Z.Q. Procedia Eng. 10, 1633–1638 (2011)
Liu, S.T., Su, W.Z.: Topology optimization of couple-stress material structures. Struct. Mutidiscip. Optim. 40, 319 (2010)
Li, L., Zhang, G., Khandelwal, K.: Topology optimization of structures with gradient elastic material. Struct. Mutidiscip. Optim. 56, 1–20 (2017)
Li, L., Khandelwal, K.: Topology optimization of structures with length-scale effects using elasticity with microstructure theory. Comput. Struct. 157, 165–177 (2015)
Bruggi, M., Taliercio, A.: Maximization of the fundamental eigenfrequency of Cosserat solids through topology optimization. Struct. Mutidiscip. Optim. 46, 549–560 (2012)
Su, W.Z., Liu, S.T.: Topology design for maximization of fundamental frequency of couple-stress continuum. Struct. Mutidiscip. Optim. 53, 395–408 (2016)
Su, W.Z., Liu, S.T.: Size-dependent microstructure design for maximal fundamental frequencies of structures. Struct. Mutidiscip. Optim. 62, 543–557 (2020)
Veber, D., Taliercio, A.: Topology optimization of three-dimensional non-centrosymmetric Cosserat bodies. Struct. Mutidiscip. Optim. 45, 575–587 (2012)
Wang, S.Y., Wang, M.Y.: Radial basis functions and level set method for structural topology optimization. Int. J. Numer. Methods Eng. 65, 2060–2090 (2006)
Wei., P., Li., Z., Li., X., et al.: An 88-line MATLAB code for the parameterized level set method based topology optimization using radial basis functions. Struct. Mutidiscip. Optim. 58, 831 (2018)
Liu, H., Zong, H., Tian, Y., et al.: A novel subdomain level set method for structural topology optimization and its application in graded cellular structure design. Struct. Mutidiscip. Optim. 60, 2221–2247 (2019)
Xiu, C.X., Chu, X.H., Wang, J.: Prediction on dispersion in elastoplastic unsaturated granular media. Theor. Appl. Mech. Lett. 10, 74–78 (2020)
Chang, J., Chu, X., Xu, Y.: Finite-element analysis of failure in transversely isotropic geomaterials. Int. J. Geomech. 15, 04014096 (2014)
Xia, Q., Shi, T.L.: Constraints of distance from boundary to skeleton: For the control of length scale in level set based structural topology optimization. Comput. Method. Appl. Mech. 295, 525–545 (2015)
Guo, X., Zhang, W.S., Zhong, W.L.: Explicit feature control in structural topology optimization via level set method. Comput. Methods Appl. Mech. Eng. 272, 254–378 (2014)
Acknowledgements
This work was supported by the National Natural Science Foundation of China (Grants 12072242, 11772237, and 11472196), the Hubei Provincial Natural Science Foundation (Grant 2020CFB816), and the Fundamental Research Funds for the Central Universities (Grant 2042018kf0016).
Author information
Authors and Affiliations
Corresponding author
Additional information
Executive editor: Xu Guo
Rights and permissions
About this article
Cite this article
Chen, L., Wan, J., Chu, X. et al. Parameterized level set method for structural topology optimization based on the Cosserat elasticity. Acta Mech. Sin. 37, 620–630 (2021). https://doi.org/10.1007/s10409-020-01045-z
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10409-020-01045-z