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Design-dependent loads in topology optimization

Published online by Cambridge University Press:  15 September 2003

Blaise Bourdin  and
Antonin Chambolle
Blaise Bourdin
Affiliation:
Courant Institute of Mathematical Science, New York University. : Department of Mathematics, Louisiana State University, Baton Rouge LA 70803-4918, USA; [email protected].
Antonin Chambolle
Affiliation:
CEREMADE, UMR 7534 du CNRS, Université de Paris-Dauphine, 75775 Paris Cedex 16, France; [email protected].
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Abstract

We present, analyze, and implement a new method for the design of the stiffest structure subject to a pressure load or a given field of internal forces. Our structure is represented as a subset S of a reference domain, and the complement of S is made of two other “phases”, the “void” and a fictitious “liquid” that exerts a pressure force on its interface with the solid structure. The problem we consider is to minimize the compliance of the structure S, which is the total work of the pressure and internal forces at the equilibrium displacement. In order to prevent from homogenization we add a penalization on the perimeter of S. We propose an approximation of our problem in the framework of Γ-convergence, based on an approximation of our three phases by a smooth phase-field. We detail the numerical implementation of the approximate energies and show a few experiments.

Keywords

Topology optimizationoptimal designdesign-dependent loadsΓ-convergencediffuse interface method.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 9 , January 2003 , pp. 19 - 48
Copyright
© EDP Sciences, SMAI, 2003

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References

G. Alberti, Variational models for phase transitions, an approach via $\Gamma$ -convergence, in Calculus of Variations and Partial Differential Equations, edited by G. Buttazzo et al. Springer-Verlag (2000) 95-114.
Allaire, G., Bonnetier, É., Francfort, G.A. and Jouve, F., Shape optimization by the homogenization method. Numer. Math. 76 (1997) 27-68. CrossRef
G. Allaire, Shape optimization by the homogenization method. Springer-Verlag, New York (2002).
Ambrosio, L. and Buttazzo, G., An optimal design problem with perimeter penalization. Calc. Var. Partial Differential Equations 1 (1993) 55-69. CrossRef
H. Attouch, Variational convergence for functions and operators. Applicable Mathematics Series. Pitman (Advanced Publishing Program), Boston, Mass.-London (1984).
Baldo, S., Minimal interface criterion for phase transitions in mixtures of Cahn-Hilliard fluids. Ann. Inst. H. Poincaré Anal. Non Linéaire 7 (1990) 67-90. CrossRef
Barroso, A.C. and Fonseca, I., Anisotropic singular perturbations - the vectorial case. Proc. Roy. Soc. Edinburgh Sect. A 124 (1994) 527-571. CrossRef
M.P. Bendsøe, Optimization of Structural Topology, Shape and Material. Springer Verlag, Berlin Heidelberg (1995).
Bendsøe, M.P. and Sigmund, O., Material interpolation schemes in topology optimization. Arch. Appl. Mech. 69 (1999) 635-654.
Bonnetier, É. and Chambolle, A., Computing the equilibrium configuration of epitaxially strained crystalline films. SIAM J. Appl. Math. 62 (2002) 1093-1121.
Bourdin, B. and Chambolle, A., Implementation of an adaptive finite-element approximation of the Mumford-Shah functional. Numer. Math. 85 (2000) 609-646. CrossRef
Bourdin, B., Francfort, G.A. and Marigo, J.-J., Numerical experiments in revisited brittle fracture. J. Mech. Phys. Solids 48 (2000) 797-826. CrossRef
Cahn, J.W. and Hilliard, J.E., Free energy of a nonuniform system I - interfacial free energy. J. Chem. Phys. 28 (1958) 258-267. CrossRef
Chambolle, A., Finite-differences discretizations of the Mumford-Shah functional. ESAIM: M2AN 33 (1999) 261-288. CrossRef
Chen, B.-C. and Kikuchi, N., Topology optimization with design-dependent loads. Finite Elem. Anal. Des. 37 (2001) 57-70. CrossRef
Chen, L.Q. and Shen, J., Application of semi implicit Fourier-spectral method to phase field equations. Comput. Phys. Comm. 108 (1998) 147-158. CrossRef
A. Cherkaev, Variational methods for structural optimization. Springer-Verlag, New York (2000).
A. Cherkaev and R.V. Kohn, Topics in the mathematical modelling of composite materials. Birkhäuser Boston Inc., Boston, MA (1997).
P.G. Ciarlet, Mathematical elasticity. Vol. I. North-Holland Publishing Co., Amsterdam (1988). Three-dimensional elasticity.
G. Dal Maso, An introduction to $\Gamma$ -convergence. Birkhäuser, Boston (1993).
I. Ekeland and R. Témam, Convex analysis and variational problems. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, English Edition (1999). Translated from the French.
L.C. Evans and R.F. Gariepy, Measure theory and fine properties of functions. CRC Press, Boca Raton, FL (1992).
Eyre, D., Systems of Cahn-Hilliard equations. SIAM J. Appl. Math. 53 (1993) 1686-1712. CrossRef
K.J. Falconer, The geometry of fractal sets. Cambridge University Press, Cambridge (1986).
H. Federer, Geometric measure theory. Springer-Verlag, New York (1969).
E. Giusti, Minimal surfaces and functions of bounded variation. Birkhäuser, Boston (1984).
Haber, R.B., Jog, C.S. and Bendsøe, M.P., A new approach to variable-topology shape design using a constraint on the perimeter. Struct. Optim. 11 (1996) 1-12. CrossRef
Hammer, V.B. and Olhoff, N., Topology optimization of continuum structures subjected to pressure loading. Struct. Multidisc. Optim. 19 (2000) 85-92. CrossRef
Kohn, R.V. and Strang, G., Optimal design and relaxation of variational problems I-III. Comm. Pure Appl. Math. 39 (1986) 113-137, 139-182, 353-377. CrossRef
Kohn, R.V. and Strang, G., Optimal design in elasticity and plasticity. Internat. J. Numer. Methods Engrg. 22 (1986) 183-188.
Leo, P.H., Lowengrub, J.S and Jou, H.J., A diffuse interface model for microstructural evolution in elastically stressed solids. Acta Mater. 46 (1998) 2113-2130. CrossRef
L. Modica and S. Mortola. Il limite nella $\Gamma$ -convergenza di una famiglia di funzionali ellittici. Boll. Un. Mat. Ital. A (5) 14 (1977) 526-529.
Modica, L. and Mortola, S., Un esempio di $\Gamma\sp{-}$ -convergenza. Boll. Un. Mat. Ital. B (5) 14 (1977) 285-299.
Negri, M., The anisotropy introduced by the mesh in the finite element approximation of the Mumford-Shah functional. Numer. Funct. Anal. Optim. 20 (1999) 957-982. CrossRef
R.H. Nochetto, S. Rovida, M. Paolini and C. Verdi, Variational approximation of the geometric motion of fronts, in Motion by mean curvature and related topics (Trento, 1992) de Gruyter, Berlin (1994) 124-149.
Osher, S.J. and Santosa, F., Level set methods for optimization problems involving geometry and constraints. I. Frequencies of a two-density inhomogeneous drum. J. Comput. Phys. 171 (2001) 272-288. CrossRef
Paolini, M. and Verdi, C., Asympto. and numerical analyses of the mean curvature flow with a space-dependent relaxation parameter. Asymptot. Anal. 5 (1992) 553-574.
Sethian, J.A. and Wiegmann, A., Structural boundary design via level set and immersed interface methods. J. Comput. Phys. 163 (2000) 489-528. CrossRef
R. Temam, Problèmes mathématiques en plasticité. Gauthier-Villars, Paris (1983).
W.P. Ziemer, Weakly Differentiable Functions. Springer-Verlag, Berlin (1989).