Nonlinear Quantum Graphs are metric graphs, i.e. graphs for which edges are considered with a metric structure, equipped with a nonlinear Schrödinger equation. They are of interest both from the physical point of view, as they are simplified model of the dynamics in elongated structures, and the mathematical point of view, as their analysis poses new and interesting challenges. If nonlinear quantum graphs now attract much attention abroad it is not yet the case in France. The goal of this project is to gather French researchers sharing an interest in this topic and possessing complementary skills. We aim at establishing ourselves as a leading team in the analysis of nonlinear quantum graphs, both from the numerical and the theoretical point of view. An important part of the project will be the development of the topic via the training of young researchers. As when the non-linear Schrödinger equation is placed on the entire space the study of stationary solutions will be the subject of a special attention. The questions that are asked regarding the existence and behavior of these solutions (as their orbital or asymptotic stability), depend on the type of graph considered (compact, non-compact, periodic, of tree type, ...), the choice of vertex conditions (Kirchhoff, Dirac, ...), the type of nonlinearity (mass subcritical, mass supercritical, local or non-local, focusing or defocusing), etc. We intend to attack the analysis of nonlinear quantum graphs with a cross-fertilization of the numerical and analytical approach. Among the directions we are interested in, let us mention : - The relations between various variational characterizations of solutions. - The existence of prescribed mass solutions for metric graphs in the mass supercritical case ; - The existence of sign-changing solutions on metric graphs ; - The dynamics around standing waves ; - The development of novel and efficient numerical methods for the problems listed above, their analysis and their implementation as an open source and easy to use library.

HIPOTHEC is a joint research project between EDF R&D and 4 academic partners, grouped into 2 geographical poles: Inria Lille and the UPHF (Valenciennes) in the North, and the Université de Montpellier and Université Côte d'Azur (Nice) in the South. The aim of the project is to devise, mathematically analyze, and practically implement high-order polyhedral (i.e. applicable on general meshes) methods for the simulation of non-destructive eddy current testing (ECT). ECT is used by EDF to assess the mechanical integrity of steam generator tubes in nuclear plants. In practice, in order to qualify the ECT devices that are employed, engineers rely on modeling and numerical simulation. HIPOTHEC aims at addressing at once 2 important issues faced by EDF in ECT simulations when using lowest-order standard finite element methods: (i) the significant noise on the control signal based on which the presence of a defect is inferred, and (ii) the great complexity of the 3D meshing process, especially when defects have to be accounted for. The project is structured in 3 work-packages (WP). The first WP is concerned with the devising and a priori analysis of (possibly enriched) Hybrid High-Order (HHO) methods for realistic ECT configurations. The second WP revolves around the development of reliable and efficient a posteriori error estimators as well as of efficient and robust multigrid linear solvers for the HHO methods developed in WP1. The third WP is concerned with the proof of concept of our methodology on a target industrial application. The expected outcomes for EDF are substantial: a noticeable gain in accuracy on the calculation of the quantities of interest, and an increased geometrical flexiblity in the 3D meshing process allowing, in particular, for the modeling of complex flaws and for the simplification of the remeshing stage inherent to the displacement of the ECT sensor. From an academic point of view, the project also embeds an important number of mathematical challenges, which justify the need for seasoned researchers expert in numerical analysis of PDEs and scientific computing, as well as for additional non-permanent human resources.

Although the diverse uncertainty quantification tools they provide, there are still settings where Gaussian processes would be relevant but are not necessarily explored. This project focuses on simulation and identification problems where random fields are inputs in mechanical systems. Our contributions are mainly fivefold. First, we seek to encode expert knowledge into kernels modeling mechanical fields. For instance, we consider applications describing spatial flows in porous media and wear topographies in automotive brake pads, both applications referring to major current issues in engineering related to the reduction of costs and nuisances. There, since datasets are often not rich enough to allow capturing the dynamics of simulations, the choice of the kernel is key to avoid model misspecification and to reduce predictions errors. We propose to construct new kernels relying on dynamical principles (e.g. by Darcy's law) and/or properties from the kernel design theory (e.g. composite kernels). The resulting models can then be used to generate a large number of virtual experiments to be processed through machine learning techniques aiming to characterize other random events (e.g. the injection accidents in component manufacturing). Second, we focus on mechanical systems where inputs are fields, and outputs are functions, more precisely, time-series. For instance, we consider outputs describing acoustic pressures when modeling friction-induced vibrations. To the best of our knowledge, there are no previous works on this topic. We also address the case where inputs are probability distribution functions. This latter case is inspired by a mechanical application where different experimental tests with the same input parametrization can lead to different output profiles. There, we propose to associate an empirical probability distribution to the set of experimental realizations with the same input parametrization. The resulting probability distribution is then passed as the input of the mechanical system. This implies further investigating proper distances over probability distributions and Gaussian process models with distribution inputs and functional outputs. Third, we provide theoretical guarantees of estimators for Gaussian processes with (multivariate or distribution) functional inputs, guarantees that can be used for the certification of the proposed Gaussian process frameworks. Fourth, we tackle the inverse problem of identifying parameters associated to mechanical fields from measurable quantities. This identification task allows characterizing pessimistic scenarios of mechanical problems in manufacturing and comfort. In this step, inverse models are fed by virtual simulations generated in the previous steps. Finally, we provide R/Python codes which will be disseminated to the scientific community as part of an open-source toolbox. The successful achievement of our contributions allows modeling random fields in our two mechanical applications within a unified framework where both simulation and identification problems are addressed. The ANR JCJC GAME project proposes both innovative theoretical and numerical contributions that are of interest in research fields such as applied mathematics, machine learning and mechanics (but not limited to them). Our theoretical contributions are based on Gaussian process models with (multivariate or distribution) functional inputs and functional outputs. Since they may be independent of the mechanical applications, the resulting models can then be used in other research fields (e.g. coastal engineering). The extensive numerical tests proposed throughout this project will lead to further recommendations that can be useful in mechanics and other engineering domains.

Amorce-MultiMaterial-Ceramic-3D is a seed project to develop and prepare the MultiMaterial-Ceramic-3D project (ceramics multimaterial 3D printing and additive manufacturing development for aeronautics, medical and energy). The main objective of Amorce-MultiMaterial-Ceramic-3D is to validate the main beneficiaries and to extend the partnership to industrial and academic partners with the aim of parity. MultiMaterial-Ceramic-3D will be a proposal in response to the call: Horizon europe, Action Marie Skłodowska-Curie: Doctoral networks (DN): HORIZON-MSCA-2023-DN-01-01 MultiMaterial-Ceramic-3D will be an innovative doctoral network (combining training and research) composed of approximately (50%) academic participants, non-profit associations and approximately (50%) industrialists (large companies and SMEs) focused on CERAMIC-METAL and CERAMIC-CERAMIC MULTIMATERIAL 3D PRINTING applied to the medical, aeronautical and energy fields. The penetration of 3D printing processes in the ceramic multi-material industry remains low. Technological advances remain to be made, but research activities are limited. Indeed, the additive manufacturing community is mainly aimed at metallic or polymeric materials and their composites, and the ceramic multi-material industry lacks qualified researchers capable of exploiting academic proofs of concept to innovations. Training is therefore a key issue for supporting innovation in the ceramic multi-materials industry and for European competitiveness. This is why a European doctoral network combining training and research is the most relevant to take up the challenge. The training program created for this project will be sustained after the project through associative networking. The certification and assessment of the qualifications of PhD students and early career researchers on this subject will be created by the consortium with the methodologies developed by the partners. The research methodology will be based on multidisciplinary and cross-sectoral collaborations between project participants. The work packages will meet the needs of each VALUE CHAIN: (i) Raw material, (ii) Printer, (iii) Product, (iv) Standardization. The expected innovations are: (i) multi-structured and adapted raw materials (ii) a printer specially designed for multi-materials (iii) ceramic-metal and/or ceramic-ceramic products of innovative design (iv) actions standardization. From a Technology Readiness Level (TRL) perspective, Multi-Ceramic-3D will bridge the “valley of death” (TRL 3 to 7) from scientific TRL to commercial TRL. The expected positive results require a joint effort of universities and industry facilitated by the proposed synergistic work plan.