Design-dependent loads in topology optimization

Bourdin, Blaise; Chambolle, Antonin

ESAIM: Control, Optimisation and Calculus of Variations, Tome 9 (2003), pp. 19-48.

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 $\Gamma$-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.

[1] 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. | MR | Zbl

[2] G. Allaire, É. Bonnetier, G.A. Francfort and F. Jouve, Shape optimization by the homogenization method. Numer. Math. 76 (1997) 27-68. | MR | Zbl

[3] G. Allaire, Shape optimization by the homogenization method. Springer-Verlag, New York (2002). | MR | Zbl

[4] L. Ambrosio and G. Buttazzo, An optimal design problem with perimeter penalization. Calc. Var. Partial Differential Equations 1 (1993) 55-69. | MR | Zbl

[5] H. Attouch, Variational convergence for functions and operators. Applicable Mathematics Series. Pitman (Advanced Publishing Program), Boston, Mass.-London (1984). | MR | Zbl

[6] S. Baldo, Minimal interface criterion for phase transitions in mixtures of Cahn-Hilliard fluids. Ann. Inst. H. Poincaré Anal. Non Linéaire 7 (1990) 67-90. | Numdam | MR | Zbl

[7] A.C. Barroso and I. Fonseca, Anisotropic singular perturbations - the vectorial case. Proc. Roy. Soc. Edinburgh Sect. A 124 (1994) 527-571. | MR | Zbl

[8] M.P. Bendsøe, Optimization of Structural Topology, Shape and Material. Springer Verlag, Berlin Heidelberg (1995). | MR | Zbl

[9] M.P. Bendsøe and O. Sigmund, Material interpolation schemes in topology optimization. Arch. Appl. Mech. 69 (1999) 635-654. | Zbl

[10] É. Bonnetier and A. Chambolle, Computing the equilibrium configuration of epitaxially strained crystalline films. SIAM J. Appl. Math. 62 (2002) 1093-1121. | MR | Zbl

[11] B. Bourdin and A. Chambolle, Implementation of an adaptive finite-element approximation of the Mumford-Shah functional. Numer. Math. 85 (2000) 609-646. | MR | Zbl

[12] B. Bourdin, G.A. Francfort and J.-J. Marigo, Numerical experiments in revisited brittle fracture. J. Mech. Phys. Solids 48 (2000) 797-826. | MR | Zbl

[13] J.W. Cahn and J.E. Hilliard, Free energy of a nonuniform system I - interfacial free energy. J. Chem. Phys. 28 (1958) 258-267.

[14] A. Chambolle, Finite-differences discretizations of the Mumford-Shah functional. ESAIM: M2AN 33 (1999) 261-288. | Numdam | MR | Zbl

[15] B.-C. Chen and N. Kikuchi, Topology optimization with design-dependent loads. Finite Elem. Anal. Des. 37 (2001) 57-70. | Zbl

[16] L.Q. Chen and J. Shen, Application of semi implicit Fourier-spectral method to phase field equations. Comput. Phys. Comm. 108 (1998) 147-158. | Zbl

[17] A. Cherkaev, Variational methods for structural optimization. Springer-Verlag, New York (2000). | MR | Zbl

[18] A. Cherkaev and R.V. Kohn, Topics in the mathematical modelling of composite materials. Birkhäuser Boston Inc., Boston, MA (1997). | MR | Zbl

[19] P.G. Ciarlet, Mathematical elasticity. Vol. I. North-Holland Publishing Co., Amsterdam (1988). Three-dimensional elasticity. | MR | Zbl

[20] G. Dal Maso, An introduction to $\Gamma$-convergence. Birkhäuser, Boston (1993). | MR | Zbl

[21] 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. | MR | Zbl

[22] L.C. Evans and R.F. Gariepy, Measure theory and fine properties of functions. CRC Press, Boca Raton, FL (1992). | MR | Zbl

[23] D. Eyre, Systems of Cahn-Hilliard equations. SIAM J. Appl. Math. 53 (1993) 1686-1712. | MR | Zbl

[24] K.J. Falconer, The geometry of fractal sets. Cambridge University Press, Cambridge (1986). | MR | Zbl

[25] H. Federer, Geometric measure theory. Springer-Verlag, New York (1969). | MR | Zbl

[26] E. Giusti, Minimal surfaces and functions of bounded variation. Birkhäuser, Boston (1984). | MR | Zbl

[27] R.B. Haber, C.S. Jog and M.P. Bendsøe, A new approach to variable-topology shape design using a constraint on the perimeter. Struct. Optim. 11 (1996) 1-12.

[28] V.B. Hammer and N. Olhoff, Topology optimization of continuum structures subjected to pressure loading. Struct. Multidisc. Optim. 19 (2000) 85-92.

[29] R.V. Kohn and G. Strang, Optimal design and relaxation of variational problems I-III. Comm. Pure Appl. Math. 39 (1986) 113-137, 139-182, 353-377. | MR | Zbl

[30] R.V. Kohn and G. Strang, Optimal design in elasticity and plasticity. Internat. J. Numer. Methods Engrg. 22 (1986) 183-188. | MR | Zbl

[31] P.H. Leo, J.S Lowengrub and H.J. Jou, A diffuse interface model for microstructural evolution in elastically stressed solids. Acta Mater. 46 (1998) 2113-2130.

[32] 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. | MR | Zbl

[33] L. Modica and S. Mortola, Un esempio di ${\Gamma }^{-}$-convergenza. Boll. Un. Mat. Ital. B (5) 14 (1977) 285-299. | MR | Zbl

[34] M. Negri, The anisotropy introduced by the mesh in the finite element approximation of the Mumford-Shah functional. Numer. Funct. Anal. Optim. 20 (1999) 957-982. | MR | Zbl

[35] 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. | MR | Zbl

[36] S.J. Osher and F. Santosa, 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. | MR | Zbl

[37] M. Paolini and C. Verdi, Asympto. and numerical analyses of the mean curvature flow with a space-dependent relaxation parameter. Asymptot. Anal. 5 (1992) 553-574. | MR | Zbl

[38] J.A. Sethian and A. Wiegmann, Structural boundary design via level set and immersed interface methods. J. Comput. Phys. 163 (2000) 489-528. | MR | Zbl

[39] R. Temam, Problèmes mathématiques en plasticité. Gauthier-Villars, Paris (1983). | MR | Zbl

[40] W.P. Ziemer, Weakly Differentiable Functions. Springer-Verlag, Berlin (1989). | MR | Zbl