Lecturers:

• Grégoire Allaire
• Charles Dapogny

Shape and topology optimization is a timely issue, spurred by the modern progress in scientific computing and by the need to adapt the industrial design process to the resource constraints. In a nutshell, it consists in optimizing the design of a device -- whether a mechanical structure, a fluid device, an electromagnetic component -- so that it best satisfies a set of physical requirements. Over the last decades, multiple strategies have been thought of to address such questions. These lectures, presented at parts of the online Topology Optimization Webinar series, are meant as an introduction to one efficient and versatile framework for modeling and treating such problems: the Level Set Method.

Schedule:

• 05/13/2026 16h00 -- 17h00.
• 05/20/2026 16h00 -- 17h00.
• 05/28/2026 16h00 -- 17h00.

Program:

The course is composed of 3 lectures. You will find below the slides of the lectures, as well as useful references.

• Part I: Introduction to the Level Set Method for shape and topology optimization.

This lecture presents the basic mathematical and numerical ingredients of the Level Set method for shape and topology optimization. Among other topics, it skims through the "Hadamard" notion of shape derivative and the general Level Set algorithm for interface tracking.



The Level Set Method for interface tracking.	Deployment of the Level Set Method for shape and topology optimization.




• Download the slides of the lecture.
• Go to the recording of the lecture.



• Part II: Mathematical and technical details of the Level Set Method

This lecture goes into slightly more details about the Level Set Method for shape and topology optimization. In particular, it details
• The calculation of shape derivatives by Céa's adjoint method,
• The Hilbertian method for extracting descent directions from shape derivatives,
• The Null-Space constrained optimization algorithm,
• The deployment of the Level Set Method on unstructured computational meshes,
• The optional use of topological derivatives.




The Null-Space optimization algorithm.	Topological derivatives.	Use of the Level Set Method on an unstructured mesh.




• Download the slides of the lecture.
• Go to the recording of the lecture.



• Part III: The body-fitted Level Set Method, based on remeshing

This lecture is devoted to a recent variant of the Level Set Method, which additionally features an exact mesh of the optimized shape, thus allowing for accurate and non intrusive mechanical computations, while still benefitting from the robust account of arbitrary motions allowed by the Level Set Method. The lecture more particularly deals with:
• A taste of meshing and remeshing,
• An extensive description of the method,
• Multiple examples in multi-physics applications and beyond.

A taste of remeshing.	Optimized 2d section of a two-fluid heat exchanger.	Optimized boundary conditions of an electro-osmotic mixer.




• Download the slides of the lecture.
• Go to the recording of the lecture.