The discipline of shape and topology optimization has aroused a growing enthusiasm among mathematicians, physicists and engineers since the seventies,
fostered by its impressive technological and industrial achievements.
Nowadays, problems pertaining to fields so diverse as mechanical engineering, fluid mechanics or quantum chemistry
are currently tackled with such techniques, and raise new, challenging issues.
Schedule:
The course is divided into 7 lectures, with unequal lengths.
In each case below, you will find the slides of the lectures, as well as sample codes written in FreeFem++.
These personal implementations are designed from the sole knowledge of the authors, and for pedagogical purposes mainly, rather than computational efficiency.
All conclusions drawn from their use should be mitigated with these points in mind!
Here is also a link to a series of videos (in French) corresponding to the lectures (course taught for the University of Tripoli, Lebanon).
Part I: Introduction
1 lecture presenting the physical motivations, and the main framework of the course.
Each attendant is expected to select one of the following themes, in close connection with the lectures, and to contact both instructors by email to give a list of three themes (sorted by rank of preference). Themes will then be attributed on a first come first served basis, and the corresponding articles will be sent to you.
Additive manufacturing and shape and topology optimization
G. Allaire and L. Jakabcin, Taking into account thermal residual stresses in topology optimization of structures built by additive manufacturing
M. Langelaar, Topology optimization of 3D self-supporting structures for additive manufacturing
J. Liu et al, Current and future trends in topology optimization for additive manufacturing
The fixed point method based on topological derivatives for shape and topology optimization
S. Amstutz, Analysis of a level set method for topology optimization
S. Amstutz and H. Andra, A new algorithm for topology optimization using a level-set method
S. Amstutz, Sensitivity analysis with respect to a local perturbation of the material property
Existence and non existence of optimal designs
F. Murat, Contre-exemples pour divers problèmes où le contrôle intervient dans les coefficients
A. Henrot and M. Pierre, Variation et optimisation de formes (Chap. 4)
Shape optimization via the level set method
G. Allaire, F. Jouve and A.-M. Toader, Structural optimization using sensitivity analysis and a level-set method
G. Allaire and F. Jouve, A level-set method for vibration and multiple loads structural optimization
Variants of the level set method for shape optimization
G. Pingen, M. Waidmann, A. Evgrafov and K. Maute, A parametric level-set approach for topology optimization of flow domains
T. Yamada, K. Izui, S. Nishiwaki and A. Takezawa, A topology optimization method based on the level set method incorporating a fictitious interface energy
The Moving Morphable Components (MMC) method for shape optimization
X. Guo, W. Zhang and W. Zhong, Doing Topology Optimization Explicitly and Geometrically – A New Moving Morphable Components Based Framework
W. Zhang, J. Yuan, J. Zhang and X. Guo A new topology optimization approach based on Moving Morphable Components (MMC) and the ersatz material model
About the homogenization theory
A. Cherkaev and R.V. Kohn (Eds), Topics in the Mathematical Modelling of Composite Materials (Chaps. 1 and 3)
Bloch-wave homogenization
C. Conca and M. Vanninathan, Homogenization of periodic structures via Bloch decomposition
C. Conca and M. Vanninathan, Fourier approach to homogenization problems
Mesh update strategies for shape optimization
G. Allaire and O. Pantz, Structural optimization with FreeFem++
A. N. Christiansen, M. Nobel-Jorgensen, N. Aage, O. Sigmund and J. A. Baerentzen Topology optimization using an explicit interface representation
G. Allaire, C. Dapogny and P. Frey, Shape optimization with a level set based mesh evolution method
Applications of homogenization to numerical shape and topology optimization
M. P. Bendsoe and N. Kikuchi, Generating optimal topologies in structural design using a homogenization method
G. Allaire, E. Bonnetier, G. Francfort and F. Jouve, Shape optimization by the homogenization method
Multi-phase shape and topology optimization
G. Allaire, C. Dapogny, G. Delgado and G. Michailidis, Multi-phase structural optimization using a level set method
O. Pantz, Sensibilité de l’équation de la chaleur aux sauts de conductivité
M. Wang and X. Wang, ‘‘Color’’ level sets: a multi-phase method for structural topology optimization with multiple materials
The Solid Isotropic Material with Penalization (SIMP) method
M. P. Bendsoe and O. Sigmund, Material interpolation schemes in topology optimization
O. Sigmund and J. Petersson, Numerical instabilities in topology optimization:
A survey on procedures dealing with checkerboards, mesh-dependencies and local minima
O. Sigmund, A 99 line topology optimization code written in Matlab
Applications of shape optimization in computer graphics
H. Zhao, S. Osher and R. Fedkiw, Fast Surface Reconstruction Using the Level Set Method
G. Peyre, J. Fadili and J. Rabin, Wasserstein Active Contours
Each theme contains two or three research articles (depending on the length and difficulty), and the attendants are required to make a synthesis out of them, resulting in
A two-page report, to be sent to both instructors before Sunday, January 28th, 11.59pm;
An oral presentation of 12 minutes (followed by 3 minutes of questions), taking place live on January, 29th, 2024, 2 pm in Room H103 (Ensimag).
Here is the schedule for the presentations:
2.00 pm. M. Benedetti: Mesh update strategies for shape optimization
2.15 pm. K. Killias: Applications of shape optimization in computer graphics
2.30 pm. Y. Liu: About the homogenization theory
2 45 pm. H. Mazloum: The fixed point method based on topological derivatives for shape and topology optimization
3.00 pm. M. Renard: Shape optimisation via the level set method
3.15 pm. L. Vandame: Variants of the level set method for shape optimization
G. Allaire, C. Dapogny and F. Jouve, Shape and topology optimization:
a comprehensive book chapter, accounting for most of the material in the course.
G. Allaire and O. Pantz, Structural Optimization with FreeFem++: an educational article describing a user-friendly implementation of geometric shape optimization.
