Instructor:

• Charles Dapogny

Optimal design is about finding the best "shape" of a device with respect to a physical criterion such as mechanical robustness, vibration... This historical concern has recently aroused a growing enthusiasm among mathematicians, physicists and engineers, fostered by the urgent need for robust and efficient designs, that operate under minimum material consumption. The present course is an introduction to the basic features of the discipline, in the simplified "parametric" setting, where the shape under optimization is a priori described by a collection of parameters in a fixed vector space; it aims to provide background material, fundamental concepts (the adjoint method, the density-based paradigm) and basic numerical methods (notably, gradient-based parametric optimization). It serves as a foundation for more advanced expositions, focused on e.g. theoretical questions about existence and non existence of an optimal design, and the notions of shape and topological derivatives.

Schedule:

• 13/04/2026 09h30 -- 11h00.
• 14/04/2026 11h30 -- 13h00.
• 15/04/2026 11h30 -- 13h00.
• 15/04/2026 16h30 -- 18h00.

Syllabus:

The course is composed of 3 lectures and a hands-on session. You will find below the slides of the lectures, as well as sample codes written in FreeFem.

• Part I: Some basic material from scientific computing

1 lecture devoted to basic concepts from scientific computing, notably the Finite Element Method for solving partial differential equations, and gradient-based methods for (infinite-dimensional) optimization problems.
• Download the slides of the lecture.
• The FreeFem codes for the treatment of the L^2- and ROF-denoising image models:
  
  

  Noisy input image.	L^2 denoising of the image.	Denoising of the image by the Rudin-Osher-Fatemi model
  	Download the code.	Download the code.



• Part II: Theoretical ingredients for parametric optimization

1 lecture mainly dealing with the adjoint method for calculating the derivative of a PDE-constrained objective function.
• Download the slides of the lecture.


• Part III: Numerical methods for parametric optimization

1 lecture about the practical implementation of parametric optimization problems, and their extension as the density-based topology optimization paradigm.
• Download the slides of the lecture.
• The FreeFem codes associated to the examples of the lectures:
  
  

  Tartar's radiator without Hilbertian regularization.	Tartar's radiator using Hilbertian regularization.	Optimization of the design of a heat lens.
  Download the code.	Download the code.	Download the code.



• Hands-on session: Presentation of FreeFem and implementation of the SIMP method

A primer about the resolution of partial differential equations using FreeFem, and a guide through the implementation of the SIMP topology optimization method.
• Download the subject of the practical session.
• The FreeFem code to generate the mesh of the exercise in Section 1.2.1.
• The FreeFem codes for the resolution of the topology optimization problems involved in there:
  
  

  Optimization of the design of a heat lens.	Optimization of a cantilever beam.	Optimization of the design of a gripping mechanism.
  Download the code.	Download the code.	Download the code.




References and resources:

• G. Allaire, Optimal design of structures: webpage of the course taught by G. Allaire at Ecole Polytechnique.
• 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.
• C. Dapogny, An introduction to shape and topology optimization: webpage of a previous version of the course, containing a set of programming exercises (including a primer about FreeFem).
• C. Dapogny, P. Frey, F. Omnès and Y. Privat, Geometrical shape optimization in Fluid Mechanics using FreeFem++: an educational article describing a similar framework in the context of fluid mechanics.
• E. Andreassen, A. Clausen, M. Schevenels, B. S. Lazarov and O. Sigmund, Efficient topology optimization in MATLAB using 88 lines of code: an educational article associated to a Matlab implementation of the SIMP method.
• P. Frey and Y. Privat, Aspects théoriques et numériques pour les fluides incompressibles: webpage of a similar course, given at Sorbonne Universités.