Shape Optimization via The Level Set Method
Collaborators: Youngjean Jung, Salvatore Torquato
Schoen G triply periodic, minimal surface generated by minimizing the total surface area under a constraint in the voulme fractions of the two regions separated by the surface. The optimization procedure used to generate this surface is based on a variational level set method approach. The simulation that generated this figure was implemented using LSMLIB. (Image generated by Y. Jung)
The level set method is naturally suited to shape optimization problems where the objective function and constraints are representable (possibly implicitly) as a functionals of the level set function. In these situations, it is possible to use variational calculus to devise a constrained, steepest descent optimization procedure. In addition to yielding a computational procedure, the variational level set method approach also provides us with a means to theoretically characterize optimal surfaces.
Using a variational, level set-based shape optimization approach, we investigated the properties of triply periodic microstructures. Specifically, we explored the structure of triply periodic surfaces that minimize the total surface area while satisfying a constraint on the volume fractions of the two phases it separates. The main outcomes of this work were
- a proof that surfaces that optimize the total surface area under a volume fraction constraint are precisely those possessing a constant mean curvature;
- a computational exploration of the properties of optimal structures as a function of the the volume fractions of the two phases separated by surface.
References
- Jung, Y., Chu, K. T., & Salvatore, T. (2007). A Variational Level Set Approach for Surface Area Minimization of Triply Periodic Surfaces. Journal of Computational Physics, 223(2), 711–730. https://doi.org/10.1016/j.jcp.2006.10.007 [pdf]