Research Papers

Active Contours With Stochastic Fronts and Mechanical Topology Optimization

[+] Author and Article Information
Alireza Kasaiezadeh

Research Associate
e-mail: sakasaie@uwaterloo.ca

Amir Khajepour

Canada Research Chair
e-mail: a.khajepour@uwaterloo.ca
Mechanical and Mechatronic Department,
University of Waterloo,
Ontario, N2L3G1Canada

1Corresponding author.

Contributed by the Design Automation Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received January 18, 2012; final manuscript received February 19, 2013; published online March 28, 2013. Assoc. Editor: Shinji Nishiwaki.

J. Mech. Des 135(4), 041008 (Mar 28, 2013) (12 pages) Paper No: MD-12-1065; doi: 10.1115/1.4023869 History: Received January 18, 2012; Revised February 19, 2013

Active contours with stochastic fronts (ACSF) is developed and investigated in this article to address the dependency of the existing algorithms of topology optimization on the initial guesses. The promising results of ACSF confirms that the use of this approach leads to higher chance of escaping from local solutions compared to the classic level set method. ACSF as a special case of the stochastic active contours (SAC), has a simplified structure that makes its implementation easier, and at the same time it has a rigorous mathematical proof of convergence. Although propitious, there is still a slight chance of trapping scenarios for ACSF that is observed in the presented results.

Copyright © 2013 by ASME
Your Session has timed out. Please sign back in to continue.


Sethian, J. A., and Wiegmann, A., 1999, “Structural Boundary Design Via Level Set And Immersed Interface Methods,” J. Comput. Phys., 163(2), pp. 489–528. [CrossRef]
Allaire, G., and Jouve, F., 2006, “Coupling the Level Set Method and the Topological Gradient in Structural Optimization, Springer, Netherlands,” M. P.Bendsoe, N.Olhoff, O.Sigmund, eds., IUTAM Symposium on Topological Design Optimization of Structures, Machines and Materials, Solid Mechanics and Its Applications, Vol. 137, pp. 3–12.
Osher, S., and Fedkiw, R. P., 2003, Level Set Methods And Dynamic Implicit Surfaces, Springer-Verlag Inc., New York.
Sokolowski, J., and Zochowski, A., 2003, “Optimality Conditions For Simultaneous Topology and Shape Optimization,” SIAM J. Control and Optimization, 42(4), pp. 1198–1221. [CrossRef]
He, L., Kao, C. Y., and Osher, S., 2007, “Incorporating Topological Derivatives Into Shape Derivatives Based Level Set Methods,” J. Comput. Phys., 225(1), pp. 891–909. [CrossRef]
Amstutz, S., and Andra, H., 2006, “A New Algorithm For Topology Optimization Using A Level-Set Method,” J. Comput. Phys., 216(2), pp. 573–588. [CrossRef]
Rouhi, M., Rais-Rohani, M., and Williams, T., 2010, “Element Exchange Method For Topology Optimization,” Structural and Multidisciplinary Optimization, 42(2), pp. 215–231. [CrossRef]
Jia, H., Beom, H. G., Wang, Y., Lin, S., and Liu, B., 2011, “Evolutionary Level Set Method For Structural Topology Optimization,” Comput. Struct., 89(5–6), pp. 445–454. [CrossRef]
Rong, J. H., and Liang, Q. Q., 2008, “A Level Set Method For Topology Optimization Of Continuum Structures With Bounded Design Domains,” Comput. Methods Appl. Mech. Eng., 197(17-18), pp. 1447–1465. [CrossRef]
Kasaiezadeh, A., Khajepour, A., and Waslander, S. L., 2010, “Spiral Bacterial Foraging Optimization Method,” American Control Conference, ACC2010, Baltimore, MD, June 30–July 2, pp. 4845–4850.
Juan, O., Keriven, R., and Postelnicu, G., 2006, “Stochastic Motion And The Level Set Method In Computer Vision: Stochastic Active Contours,” Int. J. Comput. Vision, 69(1), pp. 7–25. [CrossRef]
Chen, S., and Chen, W., 2011, “A New Level-Set Based Approach To Shape And Topology Optimization Under Geometric Uncertainty,” J. Struct. Multidisciplinary Optimiz., 44(1), pp. 1–18. [CrossRef]
Chen, S., Chen, W., and Lee, S., 2010, “Level Set Based Robust Shape And Topology Optimization Under Random Field Uncertainties,” J. Struct. Multidisciplinary Optimiz., 41(4), pp. 507–524. [CrossRef]
Walsh, J. B., 1986, “An Introduction To Stochastic Partial Differential Equations,” Lecture Notes in Math, Vol. 1180, Springer Verlag, Berlin.
Yip, N. K., 1998, “Stochastic Motion by Mean Curvature,” Arch. Rat. Mech. Anal., 144(4), pp. 331–355. [CrossRef]
Lions, P., and Souganidis, P., 2000(a), “Fully Nonlinear Stochastic Partial Differential Equations With Semilinear Stochastic Dependence,” C. R. Acad. Sci. Paris Ser. I Math, 331(8), pp. 617–624. [CrossRef]
Lions, P., and Souganidis, P. E., 2000(b), “Uniqueness Of Weak Solutions Of Fully Nonlinear Stochastic Partial Differential Equations,” C.R. Acad. Sci. Paris Ser. I Math, 331(10), pp. 783–790. [CrossRef]
Lang, A., 2007, “Simulation of Stochastic Partial Differential Equations and Stochastic Active Contours,” Ph.D. dissertation, University at Mannheim, Institut fur Mathematik, Lehrstuhl fur Mathematik, V. Mannheim, Germany.
Caruana, M., Friz, P. K., and Oberhauser, H., 2011, “A (Rough) Pathwise Approach To A Class Of Non-Linear Stochastic Partial Differential Equations,” Annales de l'Institut Henri Poincaré. Analyse Non Linéaire; Rainer Buckdahn (Brest), 28(1), pp. 27–46. [CrossRef]
Law, Y., Lee, H., and, Yip, A., 2008, “A Multi-Resolution Stochastic Level Set Method for Mumford-Shah Image Segmentation,” IEEE Trans. Image Processing, 17(12), pp. 2289–2300. [CrossRef]
Mumford, D., and Shah, J., 1985, “Boundary Detection by Minimizing Functional,” Proceedings of International Conference on Computer Vision and Pattern Recognition, San Francisco, CA, pp. 22–26.
Chen, S., and Radke, R. J., 2009, “Markov Chain Monte Carlo Shape Sampling Using Level Sets,” Second Workshop on Non-Rigid Shape Analysis and Deformable Image Alignment (NORDIA), in conjunction with International Conference on Computer Vision 2009, Sept. 27–Oct. 4, pp. 296–303.
Pan, Y., Birdwell, J. D., and Djouadi, S. M., 2005, “Probabilistic Curve Evolution Using Particle Filters,” Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference 2005, Seville, Spain, December 12–15, pp. 6335–6340.
Challis, V., 2010, “A Discrete Level-Set Topology Optimization Code Written In Matlab,” J. Struct. Multidisciplinary Optimiz., 41(3), pp. 453–464. [CrossRef]
Howell, L. L., 2001, Compliant Mechanisms, John Wiley & Sons, New York.
Chen, S., 2007, “Compliant Mechanisms With Distributed Compliance And Characteristic Stiffness: A Level Set Approach,” Ph.D. thesis, The Chinese University of Hong Kong, Shatin, Hong Kong.
Karatzas, I., and Shreve, S. E., 1991, Brownian Motion and Stochastic Calculus, Series: Graduate Texts in Mathematics, Vol. 113, 2nd ed. Springer-Verlag, New York.
Kunita, H., 1990, Stochastic Flows and Stochastic Differential Equations, Series: Cambridge Studies in Advanced Mathematics, Vol. 24, Cambridge University Press, Cambridge, New York.
Higham., D. J., 2001, “An Algorithmic Introduction To Numerical Simulation Of Stochastic Differential Equations,” J. SIAM Rev., 43(2), pp. 525–546. [CrossRef]
Schaffter, T., 2010, “Numerical Integration of SDEs: A Short Tutorial,” technical report, Swiss Federal Institute of Technology in Lausanne (EPFL).
Kasaiezadeh, A., 2012, “Developing a Class of Global Optimization Methods for Engineering Applications,” Ph.D. thesis, University of Waterloo, Ontario, Canada.


Grahic Jump Location
Fig. 1

A general load case including external and body forces

Grahic Jump Location
Fig. 2

Deflections at input and output ports due to the external force

Grahic Jump Location
Fig. 3

Boundary condition of the Bridge problem

Grahic Jump Location
Fig. 4

Convergence of the ACSF algorithm with annealing scheme for the Bridge problem shown in Table 3

Grahic Jump Location
Fig. 5

Boundary conditions of case study 1

Grahic Jump Location
Fig. 6

Boundary conditions of case study 2

Grahic Jump Location
Fig. 7

Boundary conditions of case study 3

Grahic Jump Location
Fig. 8

Convergence pattern of ACSF for case study 1

Grahic Jump Location
Fig. 9

Convergence pattern of ACSF for case study 2

Grahic Jump Location
Fig. 10

Convergence pattern of ACSF for case study 3



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In