The phase-field method in Optimal Design
In collaboration with Antonin Chambolle (École Polytechnique, France)The phase-field method was originally introduced in [BC03]. It is a versatile, robust, and rigorous framework for topology optimization problems. It is based on the penalization of the variation of the properties of designs (i.e. perimeter penalization), and its variational approximation. It uses a smooth function, the phase-field, to represent the materials involved in the device or the system. Consider the following optimal design problem of finding p materials occupying p disjoint regions D1 , . . . , Dp of a ground domain Ω, and minimizing an objective function F , under a perimeter constraint:
In the phase field framework,
one uses a single differentiable function ρ =
(ρ1 , . . . ,
ρp)
to represent all the materials. For any ε >
0, one defines the following problem:
where W
is a p–wells
function, such that W (ρ) =
0 if one and only one of the components of
ρ
is equal to 1, and strictly positive otherwise. Then, under
some technical conditions on F
, one can prove that when ε
→
0, the solutions of problem (Pε)
converge in some sense to that of (P)
. Moreover, since (Pε)
is a well-posed problem, whose arguments are classical
differential functions, the convergence result suggests a
numerical algorithm, that is to solve (Pε)
for a “small enough” ε.
This framework has been applied to multiple problem in
optimal design. The following figures are example taken
from the literature
Optimal design of a pressurized structure. From left to
right: schematic of the problem, phase field ρ for
the initial design on a half domain
(blue corresponds to a liquid under pressure, magenta to
some elastic material and yellow to the void), and the
final design (From [BC03]).
Optimal design for maximum stiffness: the wheel problem. A
vertical load is applied in the center of the lower edge of
the domain while both ends of the lower edge are clamped.
The yellow color corresponds to the void and the magenta to
the structure. Credit: A. Anantharaman and A. Griveau,
under A. Chambolle's supervision.
Optimal design for maximum stiffness: the 3 materials MBB
Beam. A load is applied at the center of the upper edge of
the domain while the vertical motion of the lower edges of
the domain is proscribed. The softest material is
represented on green, the soft one in blue and the stiff
one in green. The void is in black. (From [ZW07]).
Optimal design for maximum stiffness with a maximum stress
constraint: another beam. The leftmost figure represents
the distribution of material (in red) and void (in blue)
while the rightmost one show the Von-Misses stress in the
structure. (From [BS06])
References
[BC03] B. Bourdin and A. Chambolle, Design-dependent loads in topology optimization, ESAIM Control, Optimization and Calculus of Variations, 9 (2003) pp. 19-48. [Preprint] [DOI: 10.1051/cocv:2002070][BC05] The Phase field method in optimal design, IUTAM Symposium on Topological Design Optimization of Structures, Machines and Material (M.P. Bendsøe, N. Olhoff, and O. Sigmund eds.), Solid Mechanics and its Applications, Springer, June 2006, pp. 207–216. [Preprint]
[BS06] M. Burger and R. Stainko, Phase-field relaxation of topology optimization with local stress constraints, SIAM J. Control Optim. 45 (2006), no. 4, 1447–1466. [DOI: 10.1137/05062723X]
[ZW07] S. W. Zhou and M. Y. Wang, Multimaterial structural topology optimization with a generalized cahn-hilliard model of multiphase transition, Struct. Multidisc. Optim. 33 (2007), no. 2, 89–111. [DOI: 10.1007/s00158-006-0035-9]
Links
Martin Burger, University of Linz (Austria)Antonin Chambolle, École Polytechnique (France)
Michael Wang, City University of Hong Kong