巴黎萨克雷大学

Lp spaces of mappings taking values in arbitrary metric spaces, which we call nonlinear Lebesgue spaces, play an important role in several fields of mathematics. For instance, membership in these spaces is typically required for transport maps in optimal transport theory and for stochastic processes in probability theory. Nonlinear Lebesgue spaces also arise naturally in applications such as medical imaging, where the physical signals at play often exhibit little regularity and take their values in nonlinear spaces. Yet, these spaces remain little studied in the literature, likely due to their lack of differential structure outside the case where mappings are valued in a linear space. This paper is the first in a series devoted to the study of geometric and analytic properties of nonlinear Lebesgue spaces. The present article exposes a systematic treatment of their measure-theoretic properties, unifying and refining scattered results from the literature while also extending classical results from the linear setting to this broader nonlinear framework—including the characterizations of their completeness and their separability as well as the density of some of their subspaces: the spaces of simple, continuous and smooth mappings.

A Physics-Augmented Neural Network is introduced to learn a shared isotropic hyperelastic constitutive representation from full-field experiments on several material families. Each experimental instance is assigned a spatially homogeneous constitutive law at the modeling scale, while deformation may be heterogeneous. Both nearly incompressible and compressible responses are considered, namely, an isochoric kinematic completion is used for two-dimensional displacement data, whereas three-dimensional bulk data retain measured volume changes. The complete database comprises 30 experiments quantified by image correlation on various materials such as elastomers, cork, foam, and architected metamaterials. The database includes several specimen geometries, loading paths, mesh resolutions, and measurement modalities.The model consists of a shared input-convex backbone that maps strain invariants to convex energy modes. Material dependence is introduced only through non-negative modal amplitudes, yielding a compact constitutive embedding while preserving a common admissible backbone. Among the investigated embedding dimensions, the four-mode representation provides the best post-calibration performance, which corresponds to a reduction of 66% relative to independently calibrated Neo-Hookean models, while remaining less accurate than independently trained single-dataset PANNs.The learned amplitudes exhibit family-level organization in exploratory low-dimensional projections. The frozen backbone further enables reduced calibration of experimental instances withheld from backbone training, using only four non-negative coefficients. These results show that heterogeneous full-field experiments can be used collectively to learn a reusable, physically constrained constitutive basis, rather than being only treated as independent identification problems.

The geometry and ventilation of upper deep-water masses in the Pacific sector of the Southern Ocean are critical to glacial carbon storage, yet their evolution since the Last Glacial Maximum remains poorly constrained. Here, we present the foraminiferal neodymium isotope composition (εNd) from core MD97-2115 (eastern Chatham Rise, 2160 m water depth), which is located at the present-day boundary between Upper Circumpolar Deep Water (UCDW) and modified North Pacific Deep Water (mNPDW) in the Southwest Pacific, to reconstruct the upper deep-water mass changes since 30 ka. Combining quantitative binary mixing calculations with vertical εNd gradients relative to Lower Circumpolar Deep Water, we suggest substantial shifts in the geometry and mixing of UCDW and mNPDW since 30 ka. The variation phases include reduced UCDW influence during 30–20 ka, enhanced UCDW contribution between 20 and 14.3 ka, slightly decreased UCDW during 14.3–11.4 ka, and stable Holocene conditions. The glacial dominance of mNPDW is consistent with basin-wide evidence for enhanced storage of respired carbon in Pacific-sourced deep waters, likely contributing to the maintenance of low atmospheric CO2 during the glacial period. Crucially, the early rapid increase in UCDW at ∼20 ka preceded the penetration of North Atlantic-sourced water into Southern Ocean and the deglacial CO2 increase by several millennia, implying a critical role for Antarctic and Southern Ocean processes in establishing the oceanic precondition for glacial termination. During the last deglacial period, the decoupling between our reconstruction and atmospheric CO2 evolution suggests that water-mass geometry alone is insufficient to drive the full deglacial CO2 increase, pointing to additional roles for processes such as regional outgassing and variability in air–sea gas exchange efficiency.

General boundary conditions are implemented within a fast Fourier transform framework for linear and non-linear mechanical problems using small or finite transformation formulations. In the context of parallel computing (distributed memory), we present a framework that enables the combination of periodic and non-periodic (Dirichlet or Neumann) boundary conditions. Taking advantage of the link between non-periodic boundary conditions and the symmetries of the relevant components of the fluctuation displacement and stress fields, discrete trigonometric transforms are employed to adapt the classical Moulinec–Suquet fast Fourier transform approach. The present study employs an original displacement-based fixed-point algorithm in combination with a convergence acceleration method in order to solve boundary value problems. Finite difference approaches are used to build the discrete Green operators associated with a pre-conditioner (reference material), whose choice depends on the loading type and the small or finite transformation frameworks. The newly developed double tetrahedron scheme is employed to investigate non-periodic problems. Outcomes are compared to those of the classical hexahedral scheme. The robustness and computational efficiency of the presented parallel solver is demonstrated through numerical experiments of non-trivial loading scenarios (tension, bending, normal-mixed loading, torsion-bending), complex and densely discretized microstructures and diverse behavior laws (elasticity, isotropic plasticity, crystal plasticity), within small and finite transformation frameworks.

The geological disposal of radioactive waste is under investigation in several countries, especially in Europe. In this field, clay-rich sedimentary rocks are promising natural barriers to limit and delay the release of radionuclides toward the biosphere, due to their diffusion-driven transport and retention properties. In addition to laboratory studies, field studies in Underground Research Laboratories (URL) represent a significant value to strengthen safety demonstrations. In this context, the DIR experiment (Diffusion of Inert and Reactive tracers) was developed and implanted in the Meuse/Haute-Marne URL located in the Callovo-Oxfordian clay-rich formation. Over 6 years, several diffusion experiments were performed with radioactive tracers: HTO (tritiated water), 36Cl−, 125I−, 22Na+, 134Cs+, assessing the behaviour of neutral, anionic and cationic species. The evolution of each radiotracer activity was monitored in the injection borehole, from at least 100 days up to 3 years. Overcoring of the rock at the end of the experiments allowed the accurate characterization of radiotracer profiles. An overview of the DIR experiment is provided here, illustrating the large amount of data gathered in 4 specific boreholes at two different depths: −445 and − 495 m, within the Callovo-Oxfordian geological layer. The modelling of experimental results allowed the quantification of diffusion properties parallel to the bedding of the rock for the studied radionuclides. Diffusion parameters for HTO are in agreement with those determined on samples using the conventional through diffusion method. Anions are partially excluded from porosity with quantified values 0.3 < εa(anion)/ε(H2O) < 0.6. On contrary, the diffusion of cations is limited by their adsorption on clay surfaces with the evidence of surface diffusion processes. The sensitivity analysis on adjusted parameters provided additional insights on the local properties, such as homogeneity of the diffusion plume and the limited extension of the damaged zone surrounding boreholes: LEDZ ~ 1–3 cm. Still, further field experiments are necessary to measure diffusion in such EDZ and to improve accuracy on the diffusion anisotropy ratio.