Hexahedral meshes based on medial axis

Supervision

Dominique Bechmann (ICube, Strasbourg)

 

Laboratory and team

ICube, Strasbourg - Team IGG

 

Subject description

The construction of a volumetric mesh for a given geometric domain is a complex problem that has been addressed for many years. The generation of purely hexahedral meshes for domains of any shape is still an open problem. Such meshes would be very useful in numerical simulations such as fluid dynamics. As part of the work proposed in this thesis, we aim to develop an efficient and automatic algorithm that, starting from a domain defined by a surface mesh or a point cloud, uses the variational approach [4-HKTB24] to obtain a skeleton, which is then used as a scaffold [2-VKB23] to construct a hexahe-dral volume mesh.

Numerous problems must be solved in order to obtain a complete and integrated solution.

I. A rigorous mathematical demonstration of the robustness of the algorithm could prove useful in ensuring the long-term viability of our method.

II. The remeshing of the internal topology of the skeleton composed of segments (1D) and triangles (2D), obtained by the variational method, will need to be implemented for its coupling with mesh gen-eration. In addition, the management of special cases that we have identified in order to maintain com-patibility with our scaffolding structure needs to be studied rigorously.

III. Particular attention must be paid to preserving the topological properties of meshes, which is necessary if we wish to retain specialised optimisations for simulation. In this context, methods for subdividing and adapting mesh sampling will need to be explored.

IV. Characterisation of the geometric domains that can be represented by skeletons (1D-2D) and then meshed by our algorithm is also required in order to control the domain of validity of the methodology.

V. Finally, validating the results by applying simulation codes to the meshes produced by experts would enable practical validation of the work and might lead to the discovery of new problems to be solved. 

[4-HKTB24] Q. Huang, P. Kraemer, S. Thery, D. Bechmann, Dynamic Skeletonization via Variational Medial Axis Sampling, Full paper at ACM SIGGRAPH ASIA 2024, Tokyo, Japan, décem-bre 2024. 
[2-VKB23] P. Viville, P. Kraemer, D. Bechmann, Meso-Skeleton Guided Hexahedral Mesh Design, Full paper at Pacific Graphics 2023, Computer Graphics Forum, Volume 42, Number 7.

 

Related mathematical skills

The candidate holds a master’s degree in computer science with expertise in computer graphics, specifically geometric modeling. 

He or she possesses the skills necessary to address scientific problems and develop 3D applications (C++ programming and graphics). 

Mathematical skills in geometry would also be a major asset for this position.

Other public funding

Candidates recruited as PhDs will benefit from IRMIA++ funding and will have to follow the Graduate Program "Mathematics and Applications: Research and Interactions" (https://irmiapp.unistra.fr/training/presentation).

IRMIA++ is one of the 15 Interdisciplinary Thematic Institute (ITI) of the University of Strasbourg. It brings together a research cluster and a master-doctorate training program, relying on 12 research teams and 9 master tracks.

It encompasses all mathematicians at Université de Strasbourg, with partners in computer science and physics. ITI IRMIA++ builds on the internationally renowned research in mathematics in Strasbourg, and its well-established links with the socio-economic environment. It promotes interdisciplinary academic collaborations and industrial partnerships.

A core part of the IRMIA++ mission is to realize high-level training through integrated master-PhD tracks over 5 years, with common actions fostering an interdisciplinary culture, such as joint projects, new courses and workshops around mathematics and its interactions.

Selection will rely on the professional project of the candidate, his/her interest for interdisciplinarity and academic results.