Abstract
In level set based structural optimization, semi-Lagrange method has an advantage to allow for a large time step without the limitation of Courant–Friedrichs–Lewy (CFL) condition for numerical stability. In this paper, a line search algorithm and a sensitivity modulation scheme are introduced for the semi-Lagrange method. The line search attempts to adaptively determine an appropriate time step in each iteration of optimization. With consideration of some practical characteristics of the topology optimization process, incorporating the line search into semi-Lagrange optimization method can yield fewer design iterations and thus improve the overall computational efficiency. The sensitivity modulation is inspired from the conjugate gradient method in finite-dimensions, and provides an alternative to the standard steepest descent search in level set based optimization. Two benchmark examples are presented to compare the sensitivity modulation and the steepest descent techniques with and without the line search respectively.
Similar content being viewed by others
References
Allaire G, Jouve F, Toader A-M (2004) Structural optimization using sensitivity analysis and a level-set method. J Comput Phys 194(1):363–393. doi:10.1016/j.jcp.2003.09.032
Bargteil AW, Goktekin TG, O’Brien JF, Strain JA (2006) A semi-Lagrangian contouring method for fluid simulation. ACM Trans Graph 25(1). doi:10.1145/1187 112.1187281
Bendsoe MP, Kikuchi N (1988) Generating optimal topologies in structural design using a homogenization method. Comput Methods Appl Mech Eng 71:197–224. doi:10.1016/0045-7825(88)90086-2
Bendsøe MP, Sigmund O (1999) Material interpolation schemes in topology optimization. Arch Appl Mech 69:635–654. doi:10.1007/s004190050248
Bendsoe MP, Sigmund O (2003) Topology optimization-theory, methods and applications. Springer, Berlin
Burger M (2003) Infinite-dimensional optimization and optimal design. Lecture Notes, 285J, Department of Mathematics, UCLA. doi:10.1.1.6.881
Courant R, Isaacson E, Rees M (1952) On the solution of nonlinear hyperbolic differential equations by finite differences. Commun Pure Appl Math 5:243–249. doi:10.1002/cpa.3160050303
Courant R, Friedrichs K, Lewy H (1967) On the partial difference equations of mathematical physics. IBM J Res Dev 11(2): 215–234. doi:10.1175/1520-0493(1970)098<0001:OPDEIM>2.3.CO;2
Dai YH, Yuan Y (1999) A nonlinear conjugate gradient method with a strong global convergence property. SIAM J Optim 10(1):177–182. doi:10.1.1.46.3325
Eshenauer HA, Olhoff N (2001) Topology optimization of continuum structures: a review. Appl Mech Rev 54(4):332–390. doi:10.1115/1.1388075
Fletcher R (1987) Practical methods of optimization, unconstrained optimization. Wiley, New York
Fletcher R, Reeves C (1964) Function minimization by conjugate gradients. Comput J 7(2):149–154. doi:10.1093/comjnl/7.2.149
Guo X, Cheng GD (2010) Recent development in structural design and optimization. Acta Mech Sin 26(6):807–823. doi:10.1007/s10409-010-0395-7
Haftkaa RT, Grandhib RV (1986) Structural shape optimization–a survey. Comput Methods Appl Mech Eng 57(1):91–106. doi:10.1016/0045-7825(86)90072-1
Hestenes MR, Stiefel EL (1952) Methods of conjugate gradients for solving linear systems. J Res Natl Bur Stand 49(6):409–432
Liu L, Storey CS (1991) Efficient generalized conjugate gradient algorithms, part 1: theory. J Optim Theory Appl 69(1):129–137. doi:10.1007/BF00940464
Lu T, Neittaanmaki P, Tai XC (1991) A parallel splitting up method and its application to Navier–Stokes equations. Appl Math Lett 4:25–29. doi:10.1016/0893–9659(91)90161-N
Luo Z, Wang MY, Wang S, Wei P (2007) A level set-based parameterization method for structural shape and topology optimization. Int J Numer Methods Eng 76:1–26. doi:10.1002/nme.2092
Luo J, Luo Z, Chen L, Tong L, Wang MY (2008) A semi-implicit level set method for structural shape and topology optimization. J Comput Phys 227(11):5561–5581. doi:10.1016/j.jcp.2008.02.003
Nocedal J, Wright SJ (2006) Numerical optimization. Springer-Verlag, New York
Osher S, Sethian JA (1988) Fronts propagating with curvature dependent speed: algorithms based on Hamilton–Jacobi formulations. J Comput Phys 79(1):12–49. doi:10.1.1.46.1266
Osher SJ, Fedkiw RP (2002) Level set methods and dynamic implicit surfaces. Springer
Polyak BT (1969) The conjugate gradient method in extreme problems. USSR Comput Math Math Phys 9(4):94–112. doi:10.1016/0041-5553(69)90035-4
Ritchie H (1986) Eliminating the interpolation associated with the semi-Lagrangian scheme. Mon Weather Rev 114:135–14. doi:10.1175/1520–0493(1986)114<0135:ETIAWT>2.0.CO;2
Sethian JA (1996) A fast marching level set method for monotonically advancing fronts. Proc Natl Acad Sci 93:1591–1595. doi:10.1073/pnas.93.4.1591
Sethian JA, Wiegmann A (2000) Structural boundary design via level set and immersed interface methods. J Comput Phys 163(2):489–528. doi:10.1006/jcph.2000.6581
Staniforth A, Cote J (1991) Semi-Lagrangian integration schemes for atmospheric models-a review. Mon Weather Rev 119:2206–2223. doi:10.1175/1520–0493(1991)119<2206:SLISFA>2.0.CO;2
Strain JA (1999) Semi-Lagrangian methods for level set equations. J Comput Phys 151:498–533. doi:10.1006/jcph.1999.6194
Strain JA (2000) A fast modular semi-Lagrangian method for moving interfaces. J Comput Phys 161:512–536. doi:10.1006/jcph.2000.6508
Strain JA (2001) A fast semi-Lagrangian contouring method for moving interfaces. J Comput Phys 169:1–22. doi:10.1006/jcph.2001.6740
Wang MY, Wang XM, Guo DM (2003) A level set method for structural topology optimization. Comput Methods Appl Mech Eng 192(1–2):227–246. doi:10.1016/S0045-7825(02)00559-5
Wang MY, Wang XM (2004) PDE-driven level sets, shape sensitivity, and curvature flow for structural topology optimization. Comput Model Eng Sci 6(4):373–395. doi:10.3970/cmes.2004.006.373
Weickert J, Romeny BH, Viergever MA (1998) Efficient and reliable schemes for nonlinear diffusion filtering. IEEE Trans Image Process 7:398–409. doi:10.1109/83.661190
Xia Q, Wang MY, Wang SY, Chen SK (2006) Semi-Lagrange method for level-set based structural topology and shape optimization. Struct Multidisc Optim 31(6):419–429. doi:10.1007/s00158-005-0597-y
Xie YM, Steven GP (1993) A simple evolutionary procedure for structural optimization. Comput Struct 49:885–896. doi:10.1016/0045-7949(93)90035-C
Zhou M, Rozvany GIN (1991) The COC algorithm, part II: topological, geometry and generalized shape optimization. Comput Methods Appl Mech Eng 89:197–224. doi:10.1016/0045-7825(91)90046-9
Zhou A, Zhu Z, Fan H, Qing Q (2011) Three new hybrid conjugate gradient methods for optimization. Appl Math 2(3):303–308. doi:10.4236/am.2011.23035
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zhou, M., Wang, M.Y. A semi-Lagrangian level set method for structural optimization. Struct Multidisc Optim 46, 487–501 (2012). https://doi.org/10.1007/s00158-012-0842-0
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00158-012-0842-0