Abstract
An optimization approach for black-and-white and hinge-removal topology designs is studied. To achieve this motive, an optimal topology allowing grey boundaries is found firstly. When a suitable design has been obtained, this solution is then used as a starting point for the follow-up optimization with the goal to free unfavorable intermediate elements. For this purpose, an updated optimality criterion in which a threshold factor is introduced to gradually suppress elements with low density is proposed. The typical optimality method and new technique proposed are applied to the design procedure sequentially. Besides, to circumvent the one-point hinge connection problem producing in the process of freeing intermediate elements, a hinge-removal strategy is also proposed. During the optimization, the binary constraints on design variables are relaxed based on the scheme of solid isotropic material with penalization. Meanwhile, the mesh-independency filter is employed to ensure the existence of a solution and remove well-known checkerboards. In this way, a solution that has few intermediate elements and is free of one-point hinge connections is obtained. Finally, different numerical examples including the compliance minimization, compliant mechanisms and vibration problems demonstrate the validity of the proposed approach.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
S. Bureerat and J. Limtragool, Performance enhancement of evolutionary search for structural topology optimisation, Finite Elements in Analysis and Design, 42 (2006) 547–66.
B. Baumann and B Kost, Structure assembling by stochastic topology optimization, Computers & Structures, 83 (2005) 2175–84.
G. Anagnostou and E. M. RHnquist and A. T. Patera, A computational procedure for part design, Computer Methods in Applied Mechanics and Engineering, 97 (1992) 33–48.
C. Ghaddar, Y. Maday and A. T. Patera,, Analysis of a part design procedure, Numerische Mathematik, 71 (1995) 465–510.
M. Werme, Using the sequential linear integer programming method as a post-processor for stress-constrained topology optimization problems, International Journal for Numerical Methods in Engineering, 76 (2008) 1544–1567.
K. Svanberg and M. Werme, Sequential integer programming methods for stress constrained topology optimization, Structural and Multidisciplinary Optimization, 34 (2007) 277–299.
M. Beckers, Topology optimization using a dual method with discrete variables, Structural Optimization, 17 (1999) 14–24.
C. S. Jog, A dual algorithm for the topology optimization of non-linear elastic structures, International Journal for Numerical Methods in Engineering, 77 (2009) 502–517.
M. Stolpe and K. Svanberg, Modelling topology optimization problems as linear mixed black-and-white programs, International Journal for Numerical Methods in Engineering, 57 (2003) 723–739.
M. Stolpe and T. Stidsen, A hierarchical method for discrete structural topology design problems with local stress and displacement constraints, International Journal for Numerical Methods in Engineering, 69 (2007) 1060–1084.
M. P. Bendsøe and N. Kikuchi, Generating optimal topologies in structural design using a homogenization method, Computer Methods in Applied Mechanics and Engineering, 71 (1988) 197–224.
S. Nishiwaki, M. I. Frecker, S. Min and N. Kikuchi, Topology optimization of compliant mechanisms using the homogenization method, International Journal for Numerical Methods in Engineering, 42 (1998) 535–559.
M. P. Bendsøe and O. Sigmund, Material interpolation schemes in topology optimization, Archive of Applied Mechanics, 69 (1999) 635–654.
C. B. W. Petersen, T. Buhl and O. Sigmund, Topology synthesis of large-displacement compliant mechanisms, International Journal for Numerical Methods in Engineering, 50 (2001) 2683–2705.
G. I. N. Rozvany, Aims, scope, methods, history and unified terminology of computer-aided topology optimization in structural mechanics, Structural and Multidisciplinary Optimization, 21 (2001) 90–108.
O. Sigmund, A 99 line topology optimization code written in Matlab. Structural and Multidisciplinary Optimization, 21 (2001) 120–127.
M. P. Bendsøe, Optimal shape design as a material distribution problem, Structural Optimization, 1 (1989) 193–202.
H. P. Mlejnik and R. Schirrmacher, An engineering approach to optimal material distribution and shape finding, Computer Methods in Applied Mechanics and Engineering, 106 (1993) 1–26.
A. A. Groenwold and L. F. P. Etman, A simple heuristic for gray-scale suppression in optimality criterion-based topology optimization, Structural and Multidisciplinary Optimization (2009) DOI 10.1007/s00158-008-0337-1.
M. P. Bendsøe and O. Sigmund, Topology optimization: theory, methods, and applications, Springer-Verlag, Berlin (2003).
H. A. Eschenauer and N. Olhoff, Topology optimization of continuum structures: A review, Applied Mechanics Reviews, 54(4) (2001) 331–390.
R. B. Haber, A new approach to variable-topology shape design using a constraint on perimeter, Structural Optimization, 11 (1996) 1–12.
J. Petersson and O. Sigmund, Slope constrained topology optimization, International Journal for Numerical Methods in Engineering, 41 (1998) 1417–1434.
O. Sigmund, Design of material structures using topology optimization, Ph.D. Thesis, Department of Solid Mechanics, Technical University of Denmark, Denmark (2006).
O. Sigmund, On the design of compliant mechanisms using topology optimization, Mechanics of Structures and Machines, 25(4) (1997) 493–524.
Y. Q. Fu and X. M. Zhang, Topology extraction in the topology optimization design for structures and compliant mechanisms, Advances in Mechanics, 36(1) (2006) 75–84.
Y. K. Sui, D. Q. Yang and B. Wang, Topological optimization of continuum structure with stress and displacement constraints under multipul loading cases, Acta Mechanica Sinica, 32(2) (2000) 171–179.
N. Kikuchi, S. Nishiwaki, J. S. O. Fonseca and E. C. N. Silva, Design optimization method for compliant mechanisms and material microstructure, Computer Methods in Applied Mechanics and Engineering, 151 (1998) 401–417.
C.-Y. Lin and F.-M. Sheu, Adaptive volume constraint algorithm for stress limit-based topology optimization, Computer-Aided Design, 41 (2009) 685–694.
T. Borrvall and J. Pertersson, Topology optimization using regularized intermediate density control, Computer Methods in Applied Mechanics and Engineering, 190 (2001) 4911–4928.
M. Werme, Using the sequential linear integer programming method as a post-processor for stress-constrained topology optimization problems, International Journal for Numerical Methods in Engineering, 76 (2008) 1544–1567.
O. Sigmund, Morphology-based black and white filters for topology optimization, Structural and Multidisciplinary Optimization, 33 (2007) 401–424.
M. Y. Wang and S. Wang, Bilateral filtering for structural topology optimization, International Journal for Numerical Methods in Engineering, 63 (2005) 1911–1938.
T. E. Bruns, A reevaluation of the SIMP method with filtering and an alternative formulation for solid-void topology optimization, Structural and Multidisciplinary Optimization, 30 (2005) 428–436.
D. N. Chu, Y. M. Xie, A. Hira and G. P. Steven, Evolutionary structural optimization for problems with stiffness constraints. Finite Elements in Analysis and Design, 21 (1996) 239–251.
M. Sauter, G. Kress, M. Giger and P. Ermanni, Complexshaped beam element and graph-based optimization of compliant mechanisms, Structural and Multidisciplinary Optimization, 36 (2008) 429–442.
N. P. Garcia-Lopez, M. Sanchez-Silva, A. L. Medaglia and A. Chateauneuf, A hybrid topology optimization methodology combining simulated annealing and SIMP, Computers & Structures, 89 (2011) 1512–1522.
G.-W. Jang, M.-J. Kim and Y. Y. Kim, Design optimization of compliant mechanisms consisting of standardized elements, ASME Journal of Mechanical Design, 131 (2009) 121006-1–121006-8.
Z. Luo, L. Y. Tong, M. Y. Wang and S. Y. Wang, Shape and topology optimization of compliant mechanisms using a parameterization level set method, Journal of Computational Physics, 227 (2007) 680–705.
K. A. James and J. R. R. A. Martins. An isoparametric approach to level set topology optimization using a bodyfitted finite-element mesh, Computers & Structures, 90–91 (2012) 97–106.
N. F. Wang and K. Tai, Design of grip-and-move manipulators using symmetric path generating compliant mechanisms, ASME Journal of Mechanical Design, 130 (2008) 112305-1–112305-9.
H. Zhou and K.-L. Ting, Geometric optimization of spatial compliant mechanisms using three-dimensional wide curves, ASME Journal of Mechanical Design, 131 (2009) 051002-1–051002-7.
J. Du and N. Olhoff, Topological design of freely vibrating continuum structures for maximum values of simple and multiple eigenfrequencies and frequency gaps, Structural and Multidisciplinary Optimization, 34 (2007) 91–110.
N. L. Pedersen, Maximization of eigenvalues using topology optimization, Structural and Multidisciplinary Optimization, 20 (2000) 2–11.
X. M. Zhang, Topology optimization of compliant mechanisms, Chinese Journal of Mechanical Engineering, 39(1) (2003) 47–51.
M. I. Frecker, S. Kota and N. Kikuchi, Use of penalty function in topological synthesis and optimization of strain energy density of compliant mechanisms, Proceedings of the 1997 ASME Design Engineering Technical Conferences, DETC97/DAC-3760 (1997).
M. I. Frecker, N. Kikuchi and S. Kotqa, Topology optimization of compliant mechanisms with multiple outputs, Structural Optimization, 17 (1999) 269–278.
M. I. Frecker, G. K. Ananthasuresh, S. Nishiwaki and S. Kota, Topological synthesis of compliant mechanisms using multi-criteria optimization, ASME Journal of Mechanical Design, 119(2) (1997) 238–245.
M. I. Frecker and S. Canfield, Design of efficient compliant mechanisms from ground structure based optimal topologies, Proceedings of the 2000 ASME Design Engineering Technical Conferences DETC200/MECH-14142 (2000).
G. K. Ananthasuresh, S. Kota and N. Kikuchi, Stratrgies for systematic synthesis of compliant MEMS, Proceedings of the 1994 ASME Winter Annual Meeting on Dynamic Systems and Control, IL, Chicago, DSC 55(2) (1994) 677–686.
J. A. Hetrick and S. Kota, Topological and geometric synthesis of compliant mechanisms, Proceedings of the 2000 ASME Design Engineering Technical Conferences DETC200/MECH-14140 (2000).
M. Bendsøe, Optimization of structural topology, shape and material, Springer-Verlag, Berlin (1995).
X. M. Zhang, Optimal design of flexible mechanisms with frequency constraints, Mechanism and Machine Theory, 30(1) (1995) 131–139.
X. M. Zhang, Integrated optimal design of flexible mechanism and vibration control. International Journal of Mechanical Sciences, 46 (2004) 1607–1620.
K. T. Zuo, Research of theory and application about topology optimization of continuum structure, Ph.D. Thesis, Huazhong University of Science and Technology, Wuhan, China (2004).
Author information
Authors and Affiliations
Corresponding author
Additional information
Recommended by Associate Editor Jeonghoon Yoo
Yongqing Fu received her Ph.D. in Mechanical Engineering from South China University of Technology, China, in 2009. She is an associate professor in the School of Design of South China University of Technology, China. Her research interests include topology optimization and topology extraction.
Rights and permissions
About this article
Cite this article
Fu, Y., Zhang, X. An optimization approach for black-and-white and hinge-removal topology designs. J Mech Sci Technol 28, 581–593 (2014). https://doi.org/10.1007/s12206-013-1191-7
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12206-013-1191-7