M-VCUT level set method for optimizing cellular structures. (English) Zbl 1442.74173
Summary: This paper presents a multiple variable cutting (M-VCUT) level set method for the topology optimization of cellular structures. This method first divides the reference design domain into cells, and then in each cell it uses multiple basic level set functions combined with their cutting functions to describe the microstructure. The basic level set functions are used to describe microstructure prototypes that are kept fixed during the course of optimization. Each basic level set function is cut by its own cutting function to achieve shape and topology variation of microstructure. After the cutting operations, multiple virtual microstructures are obtained in each cell, and they are further combined together through a union operation to generate an actual microstructure. Based on the multiple virtual microstructures, multiple combination options are available, thus resulting in an enlarged design space and performance improvement as compared to the VCUT level set method developed in our previous study. Moreover, as a generalization of the VCUT level set method, connections between microstructures in neighboring cells are naturally guaranteed without any extra constraints.
MSC:
| 74P15 | Topological methods for optimization problems in solid mechanics |
| 74S99 | Numerical and other methods in solid mechanics |
Keywords:
multiscale structure; multiscale optimization; level set method; VCUT; microstructure connectivity; topology optimizationReferences:
| [1] | Bendsøe, M. P.; Kikuchi, N., Generating optimal topologies in structural design using a homogenization method, Comput. Methods Appl. Mech. Engrg., 71, 197-224 (1988) · Zbl 0671.73065 |
| [2] | Takano, N.; Zako, M.; Ishizono, M., Multi-scale computational method for elastic bodies with global and local heterogeneity, J. Comput.-Aided Mater. Des., 7, 111-132 (2000) |
| [3] | Chellappa, S.; Diaz, A. R.; Bendsøe, M. P., Layout optimization of structures with finite-sized features using multiresolution analysis, Struct. Multidiscip. Optim., 26, 77-91 (2004) · Zbl 1243.74136 |
| [4] | Zhang, P.; Toman, J.; Yu, Y.; Biyikli, E.; Kirca, M.; Chmielus, M.; To, A. C., Efficient design-optimization of variable-density hexagonal cellular structure by additive manufacturing: Theory and validation, ASME J. Mech. Des., 137, Article 021004 pp. (2015) |
| [5] | Wang, Y. J.; Xu, H.; Pasini, D., Multiscale isogeometric topology optimization for lattice materials, Comput. Methods Appl. Mech. Engrg., 316, 568-585 (2017) · Zbl 1439.74301 |
| [6] | Wang, Y. J.; Arabnejad, S.; Tanzer, M.; Pasini, D., Hip implant optimization design with three-dimensional porous material of graded density, AMSE J. Mech. Des., 140, Article 111406 pp. (2018) |
| [7] | Wu, Z. J.; Xia, L.; Wang, S. T.; Shi, T. L., Topology optimization of hierarchical lattice structures with substructuring, Comput. Methods Appl. Mech. Engrg., 345, 602-617 (2019) · Zbl 1440.74321 |
| [8] | Rodrigues, H.; Guedes, J. M.; Bendsøe, M., Hierarchical optimization of material and structure, Struct. Multidiscip. Optim., 24, 1-10 (2002) |
| [9] | Coelho, P. G.; Fernandes, P. R.; Guedes, J. M.; Rodrigues, H. C., A hierarchical model for concurrent material and topology optimisation of three-dimensional structures, Struct. Multidiscip. Optim., 35, 107-115 (2008) |
| [10] | Liu, L.; Yan, J.; Cheng, G. D., Optimum structure with homogeneous optimum truss-like material, Comput. Struct., 86, 1417-1425 (2008) |
| [11] | Xia, L.; Breitkopf, P., Concurrent topology optimization design of material and structure within \(\textsc{FE}{}^2\) nonlinear multiscale analysis framework, Comput. Methods Appl. Mech. Engrg., 278, 524-542 (2014) · Zbl 1423.74770 |
| [12] | Li, H.; Luo, Z.; Zhang, N.; Gao, L.; Brown, T., Integrated design of cellular composites using a level-set topology optimization method, Comput. Methods Appl. Mech. Engrg., 309, 453-475 (2016) · Zbl 1439.74286 |
| [13] | Wang, Y. J.; Xu, H.; Pasini, D., Multiscale isogeometric topology optimization for lattice materials, Comput. Methods Appl. Mech. Engrg., 316, 568-585 (2017) · Zbl 1439.74301 |
| [14] | Long, K.; Han, D.; Gu, X. G., Concurrent topology optimization of composite macrostructure and microstructure constructed by constituent phases of distinct Poisson’s ratios for maximum frequency, Comput. Mater. Sci., 129, 194-201 (2017) |
| [15] | Vogiatzis, P.; Ma, M.; Chen, S.; Gu, X. D., Computational design and additive manufacturing of periodic conformal metasurfaces by synthesizing topology optimization with conformal mapping, Comput. Methods Appl. Mech. Engrg., 328, 477-497 (2018) · Zbl 1439.74298 |
| [16] | Wang, Y. G.; Kang, Z., Concurrent two-scale topological design of multiple unit cells and structure using combined velocity field level set and density model, Comput. Methods Appl. Mech. Engrg., 347, 340-364 (2019) · Zbl 1440.74318 |
| [17] | Zong, H. M.; Liu, H.; Ma, Q. P.; Tian, Y.; Zhou, M. D.; Wang, M. Y., VCUT level set method for topology optimization of functionally graded cellular structures, Comput. Methods Appl. Mech. Engrg., 354, 487-505 (2019) · Zbl 1441.74184 |
| [18] | Zhou, S. W.; Li, Q., Design of graded two-phase microstructures for tailored elasticity gradients, J. Mater. Sci., 43, 5157-5167 (2008) |
| [19] | Zhou, S. W.; Li, Q., Microstructural design of connective base cells for functionally graded materials, Mater. Lett., 62, 4022-4024 (2008) |
| [20] | Radman, A.; Huang, X.; Xie, Y. M., Topology optimization of functionally graded cellular materials, J. Mater. Sci., 48, 1503-1510 (2013) |
| [21] | Wang, Y. Q.; Chen, F. F.; Wang, M. Y., Concurrent design with connectable graded microstructures, Comput. Methods Appl. Mech. Engrg., 317, 84-101 (2017) · Zbl 1439.74239 |
| [22] | Cramer, A. D.; Challis, V. J.; Roberts, A. P., Microstructure interpolation for macroscopic design, Struct. Multidiscip. Optim., 53, 489-500 (2016) |
| [23] | Wu, J.; Clausen, A.; Sigmund, O., Minimum compliance topology optimization of shell-infill composites for additive manufacturing, Comput. Methods Appl. Mech. Engrg., 326, 358-375 (2017) · Zbl 1439.74303 |
| [24] | Wu, J.; Aage, N.; Westermann, R.; Sigmund, O., Infill optimization for additive manufacturing-approaching bone-like porous structures, IEEE Trans. Vis. Comput. Graphics, 24, 1127-1140 (2018) |
| [25] | Liu, C.; Du, Z. L.; Zhang, W. S.; Zhu, Y. C.; Guo, X., Additive manufacturing-oriented design of graded lattice structures through explicit topology optimization, J. Appl. Mech., 84, Article 081008 pp. (2017) |
| [26] | Deng, J. D.; Pedersen, C. B.W.; Chen, W., Connected morphable components-based multiscale topology optimization, Front. Mech. Eng., 14, 129-140 (2019) |
| [27] | Pantz, O.; Trabelsi, K., A post-treatment of the homogenization method for shape optimization, SIAM J. Control Optim., 47, 1380-1398 (2008) · Zbl 1161.49042 |
| [28] | Groen, J. P.; Sigmund, O., Homogenization-based topology optimization for high-resolution manufacturable microstructures, Internat. J. Numer. Methods Engrg., 113, 8, 1148-1163 (2018) · Zbl 07874747 |
| [29] | Allaire, G.; Geoffroy-Donders, P.; Pantz, O., Topology optimization of modulated and oriented periodic microstructures by the homogenization method, Comput. Math. Appl., 78, 2197-2229 (2019) · Zbl 1443.74246 |
| [30] | Zhang, W. H.; Zhao, L. Y.; Cai, S. Y., Shape optimization of Dirichlet boundaries based on weighted B-spline finite cell method and level-set function, Comput. Methods Appl. Mech. Engrg., 294, 359-383 (2015) · Zbl 1423.74730 |
| [31] | Sethian, J. A.; Wiegmann, A., Structural boundary design via level set and immersed interface methods, J. Comput. Phys., 163, 489-528 (2000) · Zbl 0994.74082 |
| [32] | Osher, S.; Santosa, F., Level-set methods for optimization problems involving geometry and constraints: Frequencies of a two-density inhomogeneous drum, J. Comput. Phys., 171, 272-288 (2001) · Zbl 1056.74061 |
| [33] | Allaire, G.; Jouve, F.; Toader, A. M., A level-set method for shape optimization, C. R. Acad. Sci. Paris I, 334, 1-6 (2002) |
| [34] | Allaire, G.; Jouve, F.; Toader, A. M., Structural optimization using sensitivity analysis and a level-set method, J. Comput. Phys., 194, 363-393 (2004) · Zbl 1136.74368 |
| [35] | Wang, M. Y.; Wang, X. M.; Guo, D. M., A level set method for structural topology optimization, Comput. Methods Appl. Mech. Engrg., 192, 227-246 (2003) · Zbl 1083.74573 |
| [36] | Wang, M. Y.; Wang, X. M., Color level sets: A multi-phase method for structural topology optimization with multiple materials, Comput. Methods Appl. Mech. Engrg., 193, 469-496 (2004) · Zbl 1060.74585 |
| [37] | Wang, M. Y.; Wang, X. M., A level-set based variational method for design and optimization of heterogeneous objects, Comput. Aided Des., 37, 321-337 (2005) |
| [38] | Guo, X.; Zhang, W. S.; Zhong, W. L., Stress-related topology optimization of continuum structures involving multi-phase materials, Comput. Methods Appl. Mech. Engrg., 268, 632-655 (2014) · Zbl 1295.74081 |
| [39] | Wang, Y. Q.; Luo, Z.; Kang, Z.; Zhang, N., A multi-material level set-based topology and shape optimization method, Comput. Methods Appl. Mech. Engrg., 283, 1570-1586 (2015) · Zbl 1423.74769 |
| [40] | Xia, Q.; Wang, M. Y.; Shi, T. L., A level set method for shape and topology optimization of both structure and support of continuum structures, Comput. Methods Appl. Mech. Engrg., 272, 340-353 (2014) · Zbl 1296.74084 |
| [41] | Xia, Q.; Wang, M. Y.; Shi, T., Topology optimization with pressure load through a level set method, Comput. Methods Appl. Mech. Engrg., 283, 177-195 (2015) · Zbl 1423.74774 |
| [42] | Xia, Q.; Shi, T., Topology optimization of compliant mechanism and its support through a level set method, Comput. Methods Appl. Mech. Engrg., 305, 359-375 (2016) · Zbl 1425.74381 |
| [43] | Xia, Q.; Shi, T., Optimization of structures with thin-layer functional device on its surface through a level set based multiple-type boundary method, Comput. Methods Appl. Mech. Engrg., 311, 56-70 (2016) · Zbl 1439.74305 |
| [44] | Sigmund, O., Design of Material Structures using Topology Optimization (1994), Technical University of Denmark, (Ph.D. thesis) · Zbl 1116.74407 |
| [45] | Sigmund, O., Tailoring materials with prescribed elastic properties, Mech. Mater., 20, 351-368 (1995) |
| [46] | Cadman, J. E.; Zhou, S.; Chen, Y.; Li, Q., On design of multi-functional microstructural materials, J. Math. Sci., 48, 1, 51-66 (2013) |
| [47] | Vogiatzis, P.; Chen, S. K.; Wang, X.; Li, T.; Wang, L., Topology optimization of multi-material negative Poisson’s ratio metamaterials using a reconciled level set method, Comput. Aided Des., 83, 15-32 (2017) |
| [48] | Wang, Z. P.; Poh, L. H., Optimal form and size characterization of planar isotropic petal-shaped auxetics with tunable effective properties using IGA, Compos. Struct., 201, 486-502 (2018) |
| [49] | Dunning, P. D., Design parameterization for topology optimization by intersection of an implicit function, Comput. Methods Appl. Mech. Engrg., 317, 993-1011 (2017) · Zbl 1439.74266 |
| [50] | Svanberg, K., The method of moving asymptotes - a new method for structural optimization, Internat. J. Numer. Methods Engrg., 24, 359-373 (1987) · Zbl 0602.73091 |
| [51] | Svanberg, K., A class of globally convergent optimization methods based on conservative convex separable approximations, SIAM J. Optim., 12, 555-573 (2002) · Zbl 1035.90088 |
| [52] | Wang, Z. P.; Abdalla, M.; Turteltaub, S., Normalization approaches for the descent search direction in isogeometric shape optimization, Comput. Aided Des., 82, 68-78 (2017) |
| [53] | Liu, Y.; Li, Z. Y.; Wei, P.; Wang, W. W., Parameterized level-set based topology optimization method considering symmetry and pattern repetition constraints, Comput. Methods Appl. Mech. Engrg., 340, 1079-1101 (2018) · Zbl 1440.74304 |
| [54] | H. Liu, H.M. Zong, Y. Tian, Q.P. Ma, M.Y. Wang, A novel subdomain level set method for structural topology optimization and its application in graded cellular structure design, Struct. Multidiscip. Optim. http://dx.doi.org/10.1007/s00158-019-02318-3. |
| [55] | Wang, M. Y.; Zong, H. M.; Ma, Q. P.; Tian, Y.; Zhou, M. D., Cellular level set in B-splines (CLIBS): a method for modeling and topology optimization of cellular structures, Comput. Methods Appl. Mech. Engrg., 349, 378-404 (2019) · Zbl 1441.74172 |
| [56] | Zhang, W. H.; Sun, S. P., Scale-related topology optimization of cellular materials and structures, Internat. J. Numer. Methods Engrg., 68, 9, 993-1011 (2006) · Zbl 1127.74035 |
| [57] | Zhang, H. W.; Liu, H.; Wu, J. K., A uniform multiscale method for 2D static and dynamic analyses of heterogeneous materials, Internat. J. Numer. Methods Engrg., 93, 714-746 (2013) · Zbl 1352.74459 |
| [58] | Liu, H.; Wang, Y.; Zong, H. M.; Wang, M. Y., Efficient structure topology optimization by using the multiscale finite element method, Struct. Multidiscip. Optim., 58, 1411-1430 (2018) |
| [59] | Zhang, W. S.; Li, D. D.; Kang, P.; Guo, X.; Youn, S., Explicit topology optimization using iga-based moving morphable void (MMV) approach, Comput. Methods Appl. Mech. Engrg., 360, Article 112685 pp. (2020) · Zbl 1441.74182 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
