A modified gradient projection method for static and dynamic topology optimization

References

• Allaire, G., C. Dapogny, and P. Frey. 2011. “Topology and Geometry Optimization of Elastic Structures by Exact Deformation of Simplicial Mesh.” Computes Rendus Mathematique 349: 999–1003. doi: 10.1016/j.crma.2011.08.012
   Web of Science ®Google Scholar
• Allaire, G., F. Jouve, and A.-M. Toader. 2004. “Structural Optimization Using Sensitivity Analysis and a Level-Set Method.” Journal of Computational Physics 194: 363–393. doi: 10.1016/j.jcp.2003.09.032
   Web of Science ®Google Scholar
• Amstutz, S., and H. Andrä. 2006. “A New Algorithm for Topology Optimization Using a Level-Set Method.” Journal of Computational Physics 216: 573–588. doi: 10.1016/j.jcp.2005.12.015
   Web of Science ®Google Scholar
• Andreassen, E., A. Clausen, M. Schevenels, B. S. Lazarov, and O. Sigmund. 2011. “Efficient Topology Optimization in MATLAB Using 88 Lines of Code.” Structural & Multidisciplinary Optimization 43: 1–16. doi: 10.1007/s00158-010-0594-7
   Web of Science ®Google Scholar
• Aremu, A., I. Ashcroft, R. Hague, R. Wildman, and C. Tuck. 2010. “Suitability of SIMP and BESO Topology Optimization Algorithms for Additive Manufacture.” Paper presented at the Twenty First International Solid Freeform Fabrication Symposium, Austin, August 9–11.
   Google Scholar
• Bendsøe, M. P. 1989. “Optimal Shape Design as a Material Distribution Problem.” Structural Optimization 1: 193–202. doi: 10.1007/BF01650949
   Google Scholar
• 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: 197–224. doi: 10.1016/0045-7825(88)90086-2
   Web of Science ®Google Scholar
• Blasques, J. P. 2014. “Multi-material Topology Optimization of Laminated Composite Beams with Eigenfrequency Constraints.” Composite Structures 111: 45–55. doi: 10.1016/j.compstruct.2013.12.021
   Web of Science ®Google Scholar
• Buhl, T., C. B. W. Pedersen, and O. Sigmund. 2000. “Stiffness Design of Geometrically Nonlinear Structures Using Topology Optimization.” Structural and Multidisciplinary Optimization 19: 93–104. doi: 10.1007/s001580050089
   Web of Science ®Google Scholar
• Burger, M. 2003. “A Framework for the Construction of Level Set Methods for Shape Optimization and Reconstruction.” Interfaces and Free Boundaries 5: 301–329. doi: 10.4171/IFB/81
   Web of Science ®Google Scholar
• Challis, V. J. 2010. “A Discrete Level-Set Topology Optimization Code Written in Matlab.” Structural & Multidisciplinary Optimization 41 (3): 453–464. doi: 10.1007/s00158-009-0430-0
   Web of Science ®Google Scholar
• Challis, V. J., and J. K. Guest. 2009. “Level Set Topology Optimization of Fluids in Stokes Flow.” International Journal for Numerical Methods in Engineering 79: 1284–1308. doi: 10.1002/nme.2616
   Web of Science ®Google Scholar
• Coffin, P., and K. Maute. 2016. “Level Set Topology Optimization of Cooling and Heating Devices Using a Simplified Convection Model.” Structural and Multidisciplinary Optimization 53: 985–1003. doi: 10.1007/s00158-015-1343-8
   Web of Science ®Google Scholar
• Collet, M., M. Bruggi, and P. Duysinx. 2017. “Topology Optimization for Minimum Weight with Compliance and Simplified Nominal Stress Constraints for Fatigue Resistance.” Structural and Multidisciplinary Optimization 55 (3): 839–855. doi: 10.1007/s00158-016-1510-6
   Web of Science ®Google Scholar
• Díaz, A., and O. Sigmund. 1995. “Checkerboard Patterns in Layout Optimization.” Structural & Multidisciplinary Optimization 10: 40–45. doi: 10.1007/BF01743693
   Web of Science ®Google Scholar
• Du, J., and N. Olhoff. 2007. “Minimization of Sound Radiation from Vibrating Bi-material Structures Using Topology Optimization.” Structural and Multidisciplinary Optimization 33: 305–321. doi: 10.1007/s00158-006-0088-9
   Web of Science ®Google Scholar
• Fleury, C. 1989. “CONLIN: An Efficient Dual Optimizer Based on Convex Approximation Concepts.” Structural Optimization 1: 81–89. doi: 10.1007/BF01637664
   Google Scholar
• Fuchs, M. B., S. Jiny, and N. Peleg. 2005. “The SRV Constraint for 0/1 Topological Design.” Structural and Multidisciplinary Optimization 30: 320–326. doi: 10.1007/s00158-005-0526-0
   Web of Science ®Google Scholar
• Guo, X., W. Zhang, and W. Zhong. 2014. “Doing Topology Optimization Explicitly and Geometrically—A New Moving Morphable Components Based Framework.” Journal of Applied Mechanics 81: 081009-1–081009-12. doi: 10.1115/1.4027609
   Web of Science ®Google Scholar
• Huang, X., and Y. M. Xie. 2007. “A New Look at ESO and BESO Optimization Methods.” Structural and Multidisciplinary Optimization 35: 89–92. doi: 10.1007/s00158-007-0140-4
   Web of Science ®Google Scholar
• Jenkins, N., and K. Maute. 2015. “Level Set Topology Optimization of Stationary Fluid-Structure Interaction Problems.” Structural and Multidisciplinary Optimization 52: 179–195. doi: 10.1007/s00158-015-1229-9
   Web of Science ®Google Scholar
• van Keulen, F., R. T. Haftka, and N. H. Kim. 2005. “Review of Options for Structural Design Sensitivity Analysis. Part 1: Linear Systems.” Computer Methods in Applied Mechanics and Engineering 194: 3213–3243. doi: 10.1016/j.cma.2005.02.002
   Web of Science ®Google Scholar
• Khan, M. R., W. A. Thornton, and K. D. Willmert. 1978. “Optimality Criterion Techniques Applied to Mechanical Design.” Journal of Mechanical Design 100: 319–327. doi: 10.1115/1.3453918
   Web of Science ®Google Scholar
• Kharmanda, G., N. Olhoff, A. Mohamed, and M. Lemaire. 2004. “Reliability-Based Topology Optimization.” Structural & Multidisciplinary Optimization 26: 295–307. doi: 10.1007/s00158-003-0322-7
   Web of Science ®Google Scholar
• Lee, H.-A., and G.-J. Park. 2015. “Nonlinear Dynamic Response Topology Optimization Using the Equivalent Static Loads Method.” Computer Methods in Applied Mechanics and Engineering 283: 956–970. doi: 10.1016/j.cma.2014.10.015
   Web of Science ®Google Scholar
• Osher, S. J., and F. Santosa. 2001. “Level Set Methods for Optimization Problems Involving Geometry and Constraints: I. Frequencies of a Two-Density Inhomogeneous Drum.” Journal of Computational Physics 171: 272–288. doi: 10.1006/jcph.2001.6789
   Web of Science ®Google Scholar
• Parıs, J., I. Muınos, F. Navarrina, I. Colominas, and M. Casteleiro. 2005. “A Minimum Weight FEM Formulation for Structural Topological Optimization with Local Stress Constraints.” Paper presented at the 6th World Congresses of Structural and Multidisciplinary Optimization, Rio de Janeiro, May 30–June 03.
   Google Scholar
• Park, G. J., and B. S. Kang. 2003. “Validation of a Structural Optimization Algorithm Transforming Dynamic Loads into Equivalent Static Loads.” Journal of Optimization Theory and Applications 118: 191–200. doi: 10.1023/A:1024799727258
   Web of Science ®Google Scholar
• Querin, O. M., G. P. Steven, and Y. M. Xie. 1998. “Evolutionary Structural Optimisation (ESO) Using a Bidirectional Algorithm.” Engineering Computations 15: 1031–1048. doi: 10.1108/02644409810244129
   Web of Science ®Google Scholar
• Querin, O. M., V. Young, G. P. Steven, and Y. M. Xie. 2000. “Computational Efficiency and Validation of Bi-directional Evolutionary Structural Optimization.” Computer Methods in Applied Mechanics and Engineering 189: 559–573. doi: 10.1016/S0045-7825(99)00309-6
   Web of Science ®Google Scholar
• Rosen, J. B. 1960. “The Gradient Projection Method for Nonlinear Programming. Part I. Linear Constraints.” Journal of the Society for Industrial and Applied Mathematics 8: 181–217. doi: 10.1137/0108011
   Google Scholar
• Rosen, J. B. 1961. “The Gradient Projection Method for Nonlinear Programming. Part II. Nonlinear Constraints.” Journal of the Society for Industrial and Applied Mathematics 9: 514–532. doi: 10.1137/0109044
   Google Scholar
• Rozvany, G. I. N. 1972. “Grillages of Maximum Strength and Maximum Stiffness.” International Journal of Mechanical Sciences 14: 651–666. doi: 10.1016/0020-7403(72)90023-9
   Web of Science ®Google Scholar
• Rozvany, G. I. N. 1998. “Exact Analytical Solutions for Some Popular Benchmark Problems in Topology Optimization.” Structural & Multidisciplinary Optimization 15: 42–48. doi: 10.1007/BF01197436
   Web of Science ®Google Scholar
• Rozvany, G. I. N., M. P. Bendsøe, and U. Kirsch. 1995. “Layout Optimization of Structures.” Applied Mechanics Reviews 48: 41–119. doi: 10.1115/1.3005097
   Google Scholar
• Rozvany, G. I. N., and O. M. Querin. 2002. “Combining ESO with Rigorous Optimality Criteria.” International Journal of Vehicle Design 28: 294–299. doi: 10.1504/IJVD.2002.001991
   Web of Science ®Google Scholar
• Rozvany, G. I. N., O. M. Querin, Z. Gaspar, and V. Pomezanski. 2003. “Weight-Increasing Effect of Topology Simplification.” Structural & Multidisciplinary Optimization 25: 459–465. doi: 10.1007/s00158-003-0334-3
   Web of Science ®Google Scholar
• Rozvany, G. I. N., M. Zhou, and T. Birker. 1992. “Generalized Shape Optimization Without Homogenization.” Structural Optimization 4: 250–252. doi: 10.1007/BF01742754
   Web of Science ®Google Scholar
• Shu, L., M. Y. Wang, Z. Fang, Zhengdong Ma, and Peng Wei. 2011. “Level Set Based Structural Topology Optimization For Minimizing Frequency Response.” Journal of Sound and Vibration 330: 5820–5834. doi: 10.1016/j.jsv.2011.07.026
   Web of Science ®Google Scholar
• Sigmund, O. 1997. “On the Design of Compliant Mechanisms Using Topology Optimization.” Mechanics of Structures and Machines 25: 493–524. doi: 10.1080/08905459708945415
   Web of Science ®Google Scholar
• Sigmund, O. 2001a. “A 99 Line Topology Optimization Code Written in Matlab.” Structural and Multidisciplinary Optimization 21: 120–127. doi: 10.1007/s001580050176
   Web of Science ®Google Scholar
• Sigmund, O. 2001b. “Design of Multiphysics Actuators Using Topology Optimization – Part I: One-Material Structures.” Computer Methods in Applied Mechanics and Engineering 190: 6577–6604. doi: 10.1016/S0045-7825(01)00251-1
   Web of Science ®Google Scholar
• Sigmund, O. 2007. “Morphology-Based Black and White Filters for Topology Optimization.” Structural and Multidisciplinary Optimization 33: 401–424. doi: 10.1007/s00158-006-0087-x
   Web of Science ®Google Scholar
• Sigmund, O., and J. Petersson. 1998. “Numerical Instabilities in Topology Optimization: A Survey on Procedures Dealing with Checkerboards, Mesh-Dependencies and Local Minima.” Structural Optimization 16: 68–75. doi: 10.1007/BF01214002
   Web of Science ®Google Scholar
• Stolpe, M., and K. Svanberg. 2001. “An Alternative Interpolation Scheme for Minimum Compliance Topology Optimization.” Structural and Multidisciplinary Optimization 22: 116–124. doi: 10.1007/s001580100129
   Web of Science ®Google Scholar
• Suzuki, K., and N. Kikuchi. 1991. “A Homogenization Method for Shape and Topology Optimization.” Computer Methods in Applied Mechanics and Engineering 93: 291–318. doi: 10.1016/0045-7825(91)90245-2
   Web of Science ®Google Scholar
• 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
   Web of Science ®Google Scholar
• Tanskanen, P. 2002. “The Evolutionary Structural Optimization Method: Theoretical Aspects.” Computer Methods in Applied Mechanics and Engineering 191: 5485–5498. doi: 10.1016/S0045-7825(02)00464-4
   Web of Science ®Google Scholar
• 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: 227–246. doi: 10.1016/S0045-7825(02)00559-5
   Web of Science ®Google Scholar
• Xie, Y. M., and G. P. Steven. 1993. “A Simple Evolutionary Procedure for Structural Optimization.” Computers and Structures 49: 885–896. doi: 10.1016/0045-7949(93)90035-C
   Web of Science ®Google Scholar
• Yulin, M., and W. Xiaoming. 2004. “A Level Set Method for Structural Topology Optimization and its Applications.” Advances in Engineering Software 35: 415–441. doi: 10.1016/j.advengsoft.2004.06.004
   Web of Science ®Google Scholar
• Zegard, T., and G. H. Paulino. 2016. “Bridging Topology Optimization and Additive Manufacturing.” Structural and Multidisciplinary Optimization 53: 175–192. doi: 10.1007/s00158-015-1274-4
   Web of Science ®Google Scholar
• Zhang, W., J. Chen, X. Zhu, Jianhua Zhou, Dingchuan Xue, Xin Lei, and Xu Guo. 2017. “Explicit Three Dimensional Topology Optimization via Moving Morphable Void (MMV) Approach.” Computer Methods in Applied Mechanics and Engineering 322: 590–614. doi: 10.1016/j.cma.2017.05.002
   Web of Science ®Google Scholar
• Zhang, W., J. Yuan, J. Zhang, and X. Guo. 2016. “A New Topology Optimization Approach Based on Moving Morphable Components (MMC) and the Ersatz Material Model.” Structural and Multidisciplinary Optimization 53: 1243–1260. doi: 10.1007/s00158-015-1372-3
   Web of Science ®Google Scholar
• Zhao, J., and C. Wang. 2016. “Dynamic Response Topology Optimization in the Time Domain Using Model Reduction Method.” Structural and Multidisciplinary Optimization 53: 101–114. doi: 10.1007/s00158-015-1328-7
   Web of Science ®Google Scholar
• Zhou, M., Y. K. Shyy, and H. L. Thomas. 2001. “Checkerboard and Minimum Member Size Control in Topology Optimization.” Structural and Multidisciplinary Optimization 21: 152–158. doi: 10.1007/s001580050179
   Web of Science ®Google Scholar
• Zhu, J.-H., W.-H. Zhang, and L. Xia. 2016. “Topology Optimization in Aircraft and Aerospace Structures Design.” Archives of Computational Methods in Engineering 23 (4): 595–622. doi: 10.1007/s11831-015-9151-2
   Web of Science ®Google Scholar