In the field of structural mechanics, simulated systems often involve deformable structures that may come into contact. In numerical models, this generally translates into kinematic constraints on the unknown of the problem (i.e. the displacement field), which are dealt with by the introduction of so-called dual unknowns that ensure the non-interpenetration of contacting structures. This leads to the resolution of so-called 'saddle-point' linear systems, for which the matrix is 'indefinite' (it has positive and negative eigenvalues) and 'sparse' (the vast majority of terms in this matrix are zero).

In the context of high-performance parallel computing, we're turning to 'iterative' methods for solving linear systems, which, unlike 'direct' methods, can perform well for highly refined numerical models when using a very large number of parallel computing processors. But for this to happen, they need to be carefully designed and/or adapted to the problem at hand.

While iterative methods for solving 'positive definite' linear systems (which are obtained in the absence of kinematic constraints) are relatively well mastered, solving linear point-saddle systems remains a major difficulty [1]. A relatively abundant literature proposes iterative methods adapted to the treatment of the 'Stokes problem', emblematic of incompressible fluid mechanics. But the case of point-saddle problems arising from contact constraints between deformable structures is still a relatively open problem.

The proposed thesis consists in proposing iterative methods adapted to the resolution of linear 'saddle-point' systems arising from contact problems between deformable structures, in order to efficiently handle large-scale numerical models. The target linear systems have a size of several hundred million unknowns, distributed over several thousand processes, and cannot currently be solved efficiently, either by direct methods, or by 'basic' preconditioned iterative methods. In particular, we will validate the approach proposed by Nataf and Tournier [2] and adapt it to cases where the constraints do not act on all the primal unknowns.

The work carried out can be applied to numerous industrial problems, particularly in the nuclear industry. One example is the case of fuel pellets, which expand under the effect of temperature and the generation of fission products, and come into contact with the metal cladding of the fuel rod, which can lead to cladding failure [3].

This thesis is in collaboration with the LIP6 laboratory (Sorbonne-université).

An internship can be arranged in preparation for thesis work, depending on the candidate's wishes.

[1] Benzi, M., Golub, G. H., & Liesen, J. (2005). Numerical solution of saddle point problems. Acta numerica, 14, 1-137. (https://page.math.tu-berlin.de/~liesen/Publicat/BenGolLie05.pdf)
[2] Nataf, F., & Tournier, P. H. (2023). A GenEO Domain Decomposition method for Saddle Point problems. Comptes Rendus. Mécanique, 351(S1), 1-18. (https://doi.org/10.5802/crmeca.175)
[3] Michel, B., Nonon, C., Sercombe, J., Michel, F., & Marelle, V. (2013). Simulation of pellet-cladding interaction with the pleiades fuel performance software environment. Nuclear Technology, 182(2), 124-137. (https://hal.science/hal-04060973/document)