Abstract
Geometric uncertainty refers to the deviation of the geometric boundary from its ideal position, which may have a non-trivial impact on design performance. Since geometric uncertainty is embedded in the boundary which is dynamic and changes continuously in the optimization process, topology optimization under geometric uncertainty (TOGU) poses extreme difficulty to the already challenging topology optimization problems. This paper aims to solve this cutting-edge problem by integrating the latest developments in level set methods, design under uncertainty, and a newly developed mathematical framework for solving variational problems and partial differential equations that define mappings between different manifolds. There are several contributions of this work. First, geometric uncertainty is quantitatively modeled by combing level set equation with a random normal boundary velocity field characterized with a reduced set of random variables using the Karhunen–Loeve expansion. Multivariate Gauss quadrature is employed to propagate the geometric uncertainty, which also facilitates shape sensitivity analysis by transforming a TOGU problem into a weighted summation of deterministic topology optimization problems. Second, a PDE-based approach is employed to overcome the deficiency of conventional level set model which cannot explicitly maintain the point correspondences between the current and the perturbed boundaries. With the explicit point correspondences, shape sensitivity defined on different perturbed designs can be mapped back to the current design. The proposed method is demonstrated with a bench mark structural design. Robust designs achieved with the proposed TOGU method are compared with their deterministic counterparts.
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Abbreviations
- a (x, ω):
-
A continuous random field
- \(\overline{a} (x)\) :
-
The mean function of a (x, ω)
- a i (x):
-
The i-th eigenfunction
- DΩ( ∙ ):
-
The operator of shape derivative
- J(Ω(z), u):
-
Performance functional
- l i·j :
-
The j-th node of the i-th variable
- \(\emph{w}_{i\cdot j}\) :
-
The j-th weight of the i-th variable
- M :
-
The number of random variables
- n :
-
Normal vector
- p(z):
-
Joint probability density function of the random variables z
- u :
-
State/displacement vector
- u(X, t) or U(x, t):
-
Displacement vector connecting the positions of a particle in the undeformed configuration to its counter point in the deformed configuration to its counter point in the deformed configuration
- \(\nabla_{\rm X}\) u :
-
The material displacement gradient tensor
- \(\nabla_{\rm x}\) U :
-
The spatial displacement gradient tensor
- R :
-
Rotation tensor
- t :
-
Time
- U :
-
The right stretch tensor
- V :
-
The left stretch tensor
- V(X):
-
Velocity at a point of the boundary
- V n (X):
-
Normal velocity
- V τ (X):
-
Tangential velocity
- X :
-
Material coordinate
- z :
-
Random variables in the system
- μ :
-
The mean of a response/random variable
- λ i :
-
The ith eigenvalue
- ξ i (ω):
-
Independent random variables with zero mean and unit variance
- ϕ(x):
-
Level set function
- \(\boldsymbol{\Psi} \)(x,t):
-
A mapping between the initial and deformed domains
- τ :
-
The pseudo time
- σ :
-
Standard deviation of a response/random variable
- Ω:
-
Geometry
- Ω(z) or Ωz :
-
Random geometry
References
Abramowitz M, Stegun IA (1965) Handbook of mathematical functions, with formulas, graphs, and mathematical tables. Dover, Mineola
Adalsteinsson D, Sethian J (2003) Transport and diffusion of material quantities on propagating interfaces via level set methods. J Comput Phys 185(1):271–288
Allaire G (2002) Shape optimization by the homogenization method. Springer, New York
Allaire G (2007) Conception optimale de structures. Springer, New York
Allaire G, Jouve F et al (2004) Structural optimization using sensitivity analysis and a level-set method. J Comput Phys 194(1):363–393
Belytschko T, Liu WK et al (2000) Nonlinear finite elements for continua and structures. Wiley, New York
Bertalmío M, Sapiro G et al (1999) Region tracking on level-sets methods. IEEE Trans Med Imag 18(5):448–451
Bertalmío M, Cheng L-T et al (2001) Variational problems and partial differential equations on implicit surfaces. J Comput Phys 174:759–780
Bonet J, Wood RD (2008) Nonlinear continuum mechanics for finite element analysis. Cambridge University Press, Cambridge
Bucher C (2009) Computational analysis of ramdomness in structural mechanics. CRC, London
Burger M (2003) A framework for the construction of level set methods for shape optimization and reconstruction. Interfaces Free Bound 5:301–329
Canuto C, Kozubek T (2007) A fictitious domain approach to the numerical solution of PDEs in stochastic domains. Numer Math 107(2):257–293
Chen S, Merriman B et al (1995) A simple level set method for solving stefan problems. J Comput Phys 135:8–29
Chen S, Wang MY et al (2008) Shape feature control in structural topology optimization. Comput Aided Des 40(9):951–962
Chopp DL (2001) Some improvements of the fast marching method. SIAM J Sci Comput 23(1):230–244
de Gournay F (2006) Velocity extension for the level-set method and multiple eigenvalues in shape optimization. SIAM J Control Optim 45(1):343–367
Delfour MC, Zolésio J-P (2002) Shapes and geometries: metrics, analysis, differential calculus, and optimization. SIAM, Philadelphia
Engels H (1980) Numerical quadrature and cubature. Academic, London
Ghanem RG, Doostan A (2006) On the construction and analysis of stochastic models: characterization and propagation of the errors associated with limited data. J Comput Phys 217:63–81
Ghanem RG, Spanos PD (1991) Stochastic finite elements: a spectral approach. Springer, New York
Guest JK (2009) Imposing maximum length scale in topology optimization. Struct Multidisc Optim 37(5):463–473
Guest J, Prévost J et al (2004) Achieving minimum length scale in topology optimization using nodal design variables and projection functions. Int J Numer Methods Eng 61:238–254
Gumbert CR, Newman PA et al (2002) Effect of random geometric uncertainty on the computational design of A 3-D flexible wing. In: 20th AIAA applied aerodynamics conference
Haldar A, Mahadevan S (2000) Reliability assessment using stochastic finite element analysis. Wiley, New York
Jin R, Du X et al (2003) The use of metamodeling techniques for optimization under uncertainty. J Struct Multidisc Optim 25(2):99–116
Kalsi M, Hacker K et al (2001) A comprehensive robust design approach for decision trade-offs in complex systems design. J Mech Des 123(1):1–10
Kim NH, Wang H et al (2006) Efficient shape optimization under uncertainty using polynomial chaos expansions and local sensitivities. AIAA J 44(5):1112–1115
Lee S, Chen W (2008) A comparative study of uncertainty propagation methods for black-box type functions. Struct Multidisc Optim 37(3):239–253
Lee SH, Chen W et al (2009) Robust design with arbitrary distributions using Gauss-type quadrature formula. Struct Multidisc Optim 39(3):227–243
Luo J, Luo Z et al (2008) A new level set method for systematic design of hinge-free compliant mechanisms. Comput Methods Appl Mech Eng 198(2):318–331
Memoli F, Sapiro G et al (2003) Solving variational problems and partial differential equations mapping into general target manifolds. J Comput Phys 19:263–292
Murat F, Simon S (1976) Etudes de problemes d’optimal design. Lect Notes Comput Sci 41:54–62 (Berlin, Springer Verlag)
Nouy A, Schoefs F et al (2007) X-SFEM, a computational technique based on X-FEM to deal with random shapes. Eur J Comput Mech 16(2):277–293
Nouy A, Clement A et al (2008) An extended stochastic finite element method for solving stochastic partial differential equations on random domains. Comput Methods Appl Mech Eng 197(51–52):4663–4682
Osher S, Fedkiw R (2003) Level sets methods and dynamic implicit surfaces. Springer, New York
Osher S, Sethian J (1988) Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations. J Comput Phys 79:12–49
Padula SL, Gumbert CR et al (2006) Aerospace applications of optimization under uncertainty. Optim Eng 7(3):317–328
Peng D, Merriman B et al (1999) A pde-based local level set method. J Comput Phys 155:410–438
Pironneau O (1984) Optimal shape design for elliptic systems. Springer, New York
Pons J-P, Hermosillo G et al (2006) Maintaining the point correspondence in the level set framework. J Comput Phys 220(1):339–354
Poulsen T (2003) A new scheme for imposing minimum length scale in topology optimization. Int J Numer Methods Eng 57:741–760
Rahman S, Xu H (2004) A univariate dimension-reduction method for multi-dimensional integration in stochastic mechanics. Probab Eng Mech 19(4):393–408
Sethian JA (1999) Level set methods and fast marching methods. Cambridge University Press, Cambridge
Sigmund O (2007) Morphology-based black and white filters for topology optimization. Struct Multidisc Optim 33:401–424
Sigmund O (2009) Manufacturing tolerant topology optimization. Acta Mech Sin 25:227–239
Sokolowski J, Zolesio JP (1992) Introduction to shape optimization: shape sensitivity analysis. Springer, New York
Stefano G (2009) The stochastic finite element method: past, present and future. Comput Methods Appl Mech Eng 198:1031–1051
Wang M, Wang XM et al (2003) A level set method for structural topology optimization. Comput Methods Appl Mech Eng 192:227–246
Xiu D, Tartakovsky DM (2007) Numerical methods for differential equations in random domains. SIAM J Sci Comput 28(3):1167–1185
Xu H, Rahman S (2004) A generalized dimension-reduction method for multi-dimensional integration in stochastic mechanics. Int J Numer Methods Eng 61:1992–2019
Xu J-J, Zhao H-K (2003) An Eulerian formulation for solving partial differential equations along a moving interface. J Sci Comput 19(1–3):573–594
Zabaras N (2007) Spectral methods for uncertainty quantification. Available online at http://mpdc.mae.cornell.edu/
Zhao Y-G, Ono T (2001) Moment methods for structural reliability. J Struct Saf 23:47–75
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We would like to thank the anonymous reviewers for their helps in greatly improving the quality of this manuscript.
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Chen, S., Chen, W. A new level-set based approach to shape and topology optimization under geometric uncertainty. Struct Multidisc Optim 44, 1–18 (2011). https://doi.org/10.1007/s00158-011-0660-9
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DOI: https://doi.org/10.1007/s00158-011-0660-9