Phase-fields
It is well known that optimal design problems can be rendered well-posed by penalizing the perimeter of the designs. This approach was not widely accepted for the lack of an efficient numerical scheme, which is precisely what I devised with A. Chambolle for the more complicated problem of optimal design under pressure forces (Bourdin & Chambolle, 2003) (Bourdin & Chambolle, 2006). As for the variational model of fracture, I devised a “phase–field” model where the unknown structure is represented by a smooth function \(\varphi\) depending on a length parameter \(\ell\), namely
\[\begin{equation} \label{eq:mincompliancePerell} \min_{u,\varphi} \int_\Omega \varphi^2 f \cdot u \, dx +\frac{\lambda}{c_W}\int_\Omega \frac{W(\varphi)}{\ell} + \ell |\nabla \varphi|^2\, dx, \end{equation}\]\(W\) being a positive function vanishing only at \(0\) and \(1\), and \(c_W = \int_0^1\sqrt{W(s)}\, ds\) a normalization constant.
Due to its versatility, this approach has grown in popularity, illustrated by reference (Bourdin & Chambolle, 2003) being cited over 100 times.
The figure below from (Bourdin & Chambolle, 2003) represent the optimal design of a piston, a challenging problem where the loads depend on the unknown geometry of the interfaces between a pressurized liquid (in blue) and a structure (in pink in the simulation and grey in the schematic of the problem).
Optimal design of a pressurized structure (half domain), for different values of \(\lambda\).
References
- Bourdin, B., & Chambolle, A. (2003). Design-dependent loads in topology optimization. ESAIM Contr. Optim. Ca., 9, 19–48. DOI:10.1051/cocv:2002070 Download
- Bourdin, B., & Chambolle, A. (2006). The phase-field method in optimal design. In O. S. M.P. Bendsøe & N. Olhoff (Eds.), IUTAM Symposium on Topological Design Optimization of Structures, Machines and Materials (pp. 207–216). Springer. Download
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