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Original Articles

Level-set topology optimization for multimaterial and multifunctional mechanical metamaterials

, , , &
Pages 22-42 | Received 19 Apr 2015, Accepted 02 Mar 2016, Published online: 06 Apr 2016

References

  • Allaire, G. 2002. Shape Optimization by the Homogenization Method. New York: Springer.
  • Allaire, G., C. Dapogny, G. Delgado, and G. Michailidis. 2014. “Multi-phase Structural Optimization via a Level Set Method”. ESAIM - Control Optimisation and Calculus of Variations, EDP Sciences 20: 576–611. doi: 10.1051/cocv/2013076
  • Allaire, G., F. Jouve, and A.-M. Toader. 2004. “Structural Optimization Using Sensitivity Analysis and a Level-Set Method”. Journal of Computational Physics 194 (1): 363–393. doi:10.1016/j.jcp.2003.09.032.
  • Belytschko, T., S. P. Xiao, and C. Parimi. 2003. “Topology Optimization with Implicit Functions and Regularization”. International Journal for Numerical Methods in Engineering 57 (8): 1177–1196. doi:10.1002/nme.824.
  • Bendsøe, M. P., and N. Kikuchi. 1988. “Generating Optimal Topologies in Structural Design Using a Homogenization Method”. Computer Methods in Applied Mechanics and Engineering 71 (2): 197–224. doi: 10.1016/0045-7825(88)90086-2.
  • Bendsøe, M. P., and O. Sigmund. 1999. “Material Interpolation Schemes in Topology Optimization”. Archive of Applied Mechanics 69 (9–10): 635–654. doi:10.1007/s004190050248.
  • Bendsøe, M. P., and O. Sigmund. 2003. Topology Optimization: Theory, Methods and Applications. Berlin: Springer.
  • Buehler, M., B. Bettig, and G. G. Parker. 2004. “Topology Optimization of Smart Structures Using a Homogenization Approach”. Journal of Intelligent Material Systems and Structures 15 (8): 655–667. doi: 10.1177/1045389X04043944.
  • Chen, H., and C. Chan. 2007. “Acoustic Cloaking in Three Dimensions Using Acoustic Metamaterials”. Applied Physics Letters 91: 183518. doi:10.1063/1.2803315.
  • Cherkaev, A. 2000. Variational Methods for Structural Optimization. New York: Springer.
  • Choi, K. K., and N. H. Kim. 2005. Structural Sensitivity Analysis and Optimization I: Linear Systems. New York: Springer.
  • Dijk, N. P., K. Maute, M. Langelaar, and F. Keulen. 2013. “Level-Set Methods for Structural Topology Optimization: A Review”. Structural and Multidisciplinary Optimization 48 (3): 437–472. doi:10.1007/s00158-013-0912-y.
  • Evans, K. E., and A. Alderson. 2000. “Auxetic Materials: Functional Materials and Structures from Lateral Thinking”. Advanced Materials 12 (9): 617–628. doi: 10.1002/(SICI)1521-4095(200005)12:9<617::AID-ADMA617>3.0.CO;2-3.
  • Gao, T., and W. Zhang. 2011. “A Mass Constraint Formulation for Structural Topology Optimization with Multiphase Materials”. International Journal for Numerical Methods in Engineering 88 (8): 774–796. doi:10.1002/nme.3197.
  • Gibiansky, L. V., and O. Sigmund. 2000. “Multiphase Composites with Extremal Bulk Modulus”. Journal of the Mechanics and Physics of Solids 48 (3): 461–498. doi:10.1016/S0022-5096(99)00043-5.
  • Grima, J. N., E. Chetcuti, E. Manicaro, D. Attard, M. Camilleri, R. Gatt, and K. E. Evans. 2012. “On the Auxetic Properties of Generic Rotating Rigid Triangles”. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science 468 (2319): 810–830. doi:10.1098/rspa.2011.0273.
  • Haber, E. 2004. “A Multilevel, Level-Set Method for Optimizing Eigenvalues in Shape Design Problems”. Journal of Computational Physics 198 (2): 518–534. doi:10.1016/j.jcp.2004.01.031.
  • Hashin, Z., and S. Shtrikman. 1963. “A Variational Approach to the Theory of the Elastic Behaviour of Multiphase Materials”. Journal of the Mechanics and Physics of Solids 11 (2): 127–140. doi:10.1016/0022-5096(63)90060-7.
  • Kang, Z., and Y. Wang. 2011. “Structural Topology Optimization Based on Non-local Shepard Interpolation of Density Field”. Computer Methods in Applied Mechanics and Engineering 200 (49–52): 3515–3525. doi:10.1016/j.cma.2011.09.001.
  • Lakes, R. 1987. “Foam Structures with a Negative Poisson’s Ratio”. Science 235 (4792): 1038–1040. doi: 10.1126/science.235.4792.1038.
  • Lakes, R. 1996. “Cellular Solid Structures with Unbounded Thermal Expansion”. Journal of Materials Science Letters 15 (6): 475–477.
  • Lu, L., T. Yamamoto, M. Otomori, T. Yamada, K. Izui, and S. Nishiwaki. 2013. “Topology Optimization of an Acoustic Metamaterial with Negative Bulk Modulus Using Local Resonance”. Finite Elements in Analysis and Design 72: 1–12. doi:10.1016/j.finel.2013.04.005.
  • Luo, Z., W. Gao, and C. Song. 2010. “Design of Multi-phase Piezoelectric Actuators”. Journal of Intelligent Material Systems and Structures 21 (18): 1851–1865. doi:10.1177/1045389X10389345.
  • Luo, Z., L. Tong, M. Y. Wang, and S. Wang. 2007. “Shape and Topology Optimization of Compliant Mechanisms Using a Parameterization Level Set Method”. Journal of Computational Physics 227 (1): 680–705. doi:10.1016/j.jcp.2007.08.011.
  • Luo, Z., L. Tong, P. Wei, and M. Y. Wang. 2009. “Design of Piezoelectric Actuators Using a Multiphase Level Set Method of Piecewise Constants”. Journal of Computational Physics 228 (7): 2643–2659. doi:10.1016/j.jcp.2008.12.019.
  • Luo, Z., M. Y. Wang, S. Wang, and P. Wei. 2008. “A Level Set-Based Parameterization Method for Structural Shape and Topology Optimization”. International Journal for Numerical Methods in Engineering 76 (1): 1–26. doi:10.1002/nme.2092.
  • Luo, Z., N. Zhang, W. Gao, and H. Ma. 2012. “Structural Shape and Topology Optimization Using a Meshless Galerkin Level Set Method”. International Journal for Numerical Methods in Engineering 90 (3): 369–389. doi:10.1002/nme.3325.
  • Luo, Z., N. Zhang, Y. Wang, and W. Gao. 2013. “Topology Optimization of Structures Using Meshless Density Variable Approximants”. International Journal for Numerical Methods in Engineering 93 (4): 443–464. doi:10.1002/nme.4394.
  • Makhija, D., and K. Maute. 2014. “Numerical Instabilities in Level Set Topology Optimization with the Extended Finite Element Method”. Structural and Multidisciplinary Optimization 49 (2): 185–197. doi:10.1007/s00158-013-0982-x.
  • Mei, Y., and X. Wang. 2004. “A Level Set Method for Structural Topology Optimization and Its Applications”. Advances in Engineering Software 35 (7): 415–441. doi:10.1016/j.advengsoft.2004.06.004.
  • Milton, G. W. 1992. “Composite Materials with Poisson’s Ratios Close to –1”. Journal of the Mechanics and Physics of Solids 40 (5): 1105–1137. doi:10.1016/0022-5096(92)90063-8.
  • Nicolaou, Z. G., and A. E. Motter. 2012. “Mechanical Metamaterials with Negative Compressibility Transitions”. Nature Materials 11: 608–613. doi:10.1038/nmat3331.
  • Osher, S., and R. Fedkiw. 2003. Level Set Methods and Dynamic Implicit Surfaces. New York: Springer.
  • Osher, S., and J. A. Sethian. 1988. “Fronts Propagating with Curvature-Dependent Speed: Algorithms Based on Hamilton-Jacobi Formulations”. Journal of Computational Physics 79 (1): 12–49. doi:10.1016/0021-9991(88)90002-2.
  • Otomori, M., T. Yamada, K. Izui, S. Nishiwaki, and J. Andkjær. 2012. “A Topology Optimization Method Based on the Level Set Method for the Design of Negative Permeability Dielectric Metamaterials”. Computer Methods in Applied Mechanics and Engineering 237–240: 192–211. doi:10.1016/j.cma.2012.04.022.
  • Scarpa, F., and G. Tomlinson. 2000. “Sandwich Structures with Negative Poisson’s Ratio for Deployable Structures”. IUTAM-IASS Symposium on Deployable Structures: Theory and Applications 80: 335–343, Dordrecht: Springer.
  • Sethian, J. A. 1999. Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science. Cambridge: Cambridge University Press.
  • Sethian, J. A., and A. Wiegmann. 2000. “Structural Boundary Design via Level Set and Immersed Interface Methods”. Journal of Computational Physics 163 (2): 489–528. doi:10.1006/jcph.2000.6581.
  • Sigmund, O. 2001. “Design of Multiphysics Actuators Using Topology Optimization—Part I: One-Material Structures”. Computer Methods in Applied Mechanics and Engineering 190 (49–50): 6577–6604. doi: 10.1016/S0045-7825(01)00251-1.
  • Sigmund, O., and S. Torquato. 1996. “Composites with Extremal Thermal Expansion Coefficients”. Applied Physics Letters 69 (21): 3203–3205. doi:10.1063/1.117961.
  • Sigmund, O., and S. Torquato. 1997. “Design of Materials with Extreme Thermal Expansion Using a Three-Phase Topology Optimization Method”. Journal of the Mechanics and Physics of Solids 45 (6): 1037–1067. doi:10.1016/S0022-5096(96)00114-7.
  • Smith, D., J. Pendry, and M. Wiltshire. 2004. “Metamaterials and Negative Refractive Index”. Science 305 (5685): 788–792. doi:10.1126/science.1096796.
  • Svanberg, K. 1987. “The Method of Moving Asymptotes—A New Method for Structural Optimization”. International Journal for Numerical Methods in Engineering 24: 359–373. doi:10.1002/nme.1620240207.
  • Tavakoli, R. 2014. “Multimaterial Topology Optimization by Volume Constrained Allen-Cahn System and Regularized Projected Steepest Descent Method”. Computer Methods in Applied Mechanics and Engineering 276: 534–565. doi:10.1016/j.cma.2014.04.005.
  • Tavakoli, R., and S. M. Mohseni. 2014. “Alternating Active-Phase Algorithm for Multimaterial Topology Optimization Problems: A 115-Line MATLAB Implementation”. Structural and Multidisciplinary Optimization 49 (4): 621–642. doi:10.1007/s00158-013-0999-1.
  • Veselago, V. G. 1968. “The Electrodynamics of Substances with Simultaneously Negative Values of Ε and μ”. Physics-Uspekhi 10 (4): 509–514. doi:10.1070/PU1968v010n04ABEH003699.
  • Wang, Y., Z. Luo, Z. Kang, and N. Zhang. 2015. “A Multi-material Level Set-Based Topology and Shape Optimization Method”. Computer Methods in Applied Mechanics and Engineering 283: 1570–1586. doi:10.1016/j.cma.2014.11.002.
  • Wang, Y., Z. Luo, X. Zhang, and Z. Kang. 2014a. “Topological Design of Compliant Smart Structures with Embedded Movable Actuators”. Smart Material Structures 23: 984–986. doi:10.1088/0964-1726/23/4/045024.
  • Wang, Y., Z. Luo, N. Zhang, and Z. Kang. 2014b. “Topological Shape Optimization of Microstructural Metamaterials Using a Level Set Method”. Computational Materials Science 87: 178–186. doi:10.1016/j.commatsci.2014.02.006.
  • Wang, M. Y., and X. Wang. 2004. “‘Color’ Level Sets: A Multi-phase Method for Structural Topology Optimization with Multiple Materials”. Computer Methods in Applied Mechanics and Engineering 193 (6–8): 469–496. doi:10.1016/j.cma.2003.10.008.
  • Wang, M. Y., and X. Wang. 2005. “A Level-Set Based Variational Method for Design and Optimization of Heterogeneous Objects”. Computer-Aided Design 37 (3): 321–337. doi:10.1016/j.cad.2004.03.007.
  • Wang, M. Y., X. Wang, and D. Guo. 2003. “A Level Set Method for Structural Topology Optimization”. Computer Methods in Applied Mechanics and Engineering 192 (1–2): 227–246. doi:10.1016/S0045-7825(02)00559-5.
  • Wei, P., and M. Y. Wang. 2009. “Piecewise Constant Level Set Method for Structural Topology Optimization”. International Journal for Numerical Methods in Engineering 78 (4): 379–402. doi:10.1002/nme.2478.
  • Wendland, H. 2006. “Computational Aspects of Radial Basis Function Approximation”. Studies in Computational Mathematics 12: 231–256. doi:10.1016/S1570-579X(06)80010-8.
  • Xie, Y. M., and G. P. Steven. 1993. “A Simple Evolutionary Procedure for Structural Optimization”. Computers & Structures 49 (5): 885–896. doi:10.1016/0045-7949(93)90035-C.
  • Yamada, T., K. Izui, S. Nishiwaki, and A. Takezawa. 2010. “A Topology Optimization Method Based on the Level Set Method Incorporating a Fictitious Interface Energy”. Computer Methods in Applied Mechanics and Engineering 199 (45–48): 2876–2891. doi:10.1016/j.cma.2010.05.013.
  • Yang, W., Z. M. Li, W. Shi, B. H. Xie, and M. B. Yang. 2004. “Review on Auxetic Materials”. Journal of Materials Science 39 (10): 3269–3279. doi: 10.1023/B:JMSC.0000026928.93231.e0
  • Zhou, S., W. Li, Y. Chen, G. Sun, and Q. Li. 2011. “Topology Optimization for Negative Permeability Metamaterials Using Level-Set Algorithm”. Acta Materialia 59 (7): 2624–2636. doi:10.1016/j.actamat.2010.12.049.
  • Zhou, M., and G. I. N. Rozvany. 1991. “The COC Algorithm, Part II: Topological, Geometrical and Generalized Shape Optimization”. Computer Methods in Applied Mechanics and Engineering 89 (1–3): 309–336. doi:10.1016/0045-7825(91)90046-9.
  • Zhou, S., and M. Y. Wang. 2007. “Multimaterial Structural Topology Optimization with a Generalized Cahn–Hilliard Model of Multiphase Transition”. Structural & Multidisciplinary Optimization 33: 89–111. doi: 10.1007/s00158-006-0035-9.
  • Zhu, J., W. Zhang, and P. Beckers. 2009. “Integrated Layout Design of Multi-component System”. International Journal for Numerical Methods in Engineering 78 (6): 631–651. doi:10.1002/nme.2499.

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