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.

References

References
1.
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
.10.1006/jcph.2000.6581
2.
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
.
3.
Osher
,
S.
, and
Fedkiw
,
R. P.
,
2003
,
Level Set Methods And Dynamic Implicit Surfaces
,
Springer-Verlag Inc.
,
New York
.
4.
Sokolowski
,
J.
, and
Zochowski
,
A.
,
2003
, “
Optimality Conditions For Simultaneous Topology and Shape Optimization
,”
SIAM J. Control and Optimization
,
42
(
4
), pp.
1198
1221
.10.1137/S0363012901384430
5.
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
.10.1016/j.jcp.2007.01.003
6.
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
.10.1016/j.jcp.2005.12.015
7.
Rouhi
,
M.
,
Rais-Rohani
,
M.
, and
Williams
,
T.
,
2010
, “
Element Exchange Method For Topology Optimization
,”
Structural and Multidisciplinary Optimization
,
42
(
2
), pp.
215
231
.10.1007/s00158-010-0495-9
8.
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
.10.1016/j.compstruc.2010.11.003
9.
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
.10.1016/j.cma.2007.11.026
10.
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
.
11.
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
.10.1007/s11263-006-6849-5
12.
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
.10.1007/s00158-011-0660-9
13.
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
.10.1007/s00158-009-0449-2
14.
Walsh
,
J. B.
,
1986
, “
An Introduction To Stochastic Partial Differential Equations
,”
Lecture Notes in Math
, Vol.
1180
,
Springer Verlag
,
Berlin
.
15.
Yip
,
N. K.
,
1998
, “
Stochastic Motion by Mean Curvature
,”
Arch. Rat. Mech. Anal.
,
144
(
4
), pp.
331
355
.10.1007/s002050050120
16.
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
.10.1016/S0764-4442(00)00583-8
17.
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
.10.1016/S0764-4442(00)01597-4
18.
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
.
19.
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
.10.1016/j.anihpc.2010.11.002
20.
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
.10.1109/TIP.2008.2005823
21.
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
.
22.
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
.
23.
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
.
24.
Challis
,
V.
,
2010
, “
A Discrete Level-Set Topology Optimization Code Written In Matlab
,”
J. Struct. Multidisciplinary Optimiz.
,
41
(
3
), pp.
453
464
.10.1007/s00158-009-0430-0
25.
Howell
,
L. L.
,
2001
,
Compliant Mechanisms
,
John Wiley & Sons
,
New York
.
26.
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
.
27.
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
.
28.
Kunita
,
H.
,
1990
,
Stochastic Flows and Stochastic Differential Equations, Series: Cambridge Studies in Advanced Mathematics
, Vol.
24
,
Cambridge University Press
,
Cambridge, New York
.
29.
Higham.
,
D. J.
,
2001
, “
An Algorithmic Introduction To Numerical Simulation Of Stochastic Differential Equations
,”
J. SIAM Rev.
,
43
(
2
), pp.
525
546
.10.1137/S0036144500378302
30.
Schaffter
,
T.
,
2010
, “
Numerical Integration of SDEs: A Short Tutorial
,”
technical report, Swiss Federal Institute of Technology in Lausanne (EPFL)
.
31.
Kasaiezadeh
,
A.
,
2012
, “
Developing a Class of Global Optimization Methods for Engineering Applications
,”
Ph.D. thesis
,
University of Waterloo
,
Ontario, Canada
.
You do not currently have access to this content.