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We study several structure aspects of functor categories from a small additive category to a module category, in particular the category F(A,K) of functors from finitely generated free modules over a commutative ring A to vector spaces over a field K -- such functors are sometimes called \textit{generic representations} of linear groups over A with coefficients in K. We are especially interested with finitely generated functors of F(A,K) taking finite dimensional values. We prove that they can, under a mild extra assumption (always satisfied if the ring A is noetherian), be built from much better understood functors, namely polynomial functors (in the sense of Eilenberg-MacLane), or factorising at the source through reduction modulo a cofinite ideal of A. We deduce that such functors are always noetherian et that, if the ring A is finitely generated, they have finitely generated projective resolutions.Our methods rely mainly on the study of weight decompositions of functors and their cross-effects, our recent previous work with Vespa (Ann. ENS 2023) and elementary commutative algebra., Comment: in French language

One of the challenge that face the effectiveness of Polyvinyl Chloride geomembranes (PVC GMs) as a hydraulic barrier is the capacity to withstand unexpected mechanical actions, particularly tensile forces, during installation and throughout their lifespan. These forces pose risks of premature failure and impermeability degradation. In this study, the characterization of the short and long-term mechanical response of PVC GMs to uniaxial tensile forces has been investigated. Uniaxial tensile test have been performed for tensile rates spanning several orders of magnitude. Analysis of the true stress-strain curves reveals a significant decrease in tensile modulus, strength, and strain at failure at low strain rates, which are relatively close to those applied in situ. Long-term investigations have been conducted as well, through relaxation tests. Our key results unveil two distinct characteristic times in stress relaxation, with the fast relaxation occurring over the first four hours. During this phase, the pre-relaxation loading rate affects the relaxation behavior. Beyond this phase, the relaxation behavior becomes independent from the pre-relaxation loading rate. Burger's rheological model is proposed to measure the stress relaxation at different rates. The model's results validate the existence of two characteristic times.

The direct parametrisation method for invariant manifolds is adjusted to consider a varying parameter. More specifically, the case of systems experiencing a Hopf bifurcation in the parameter range of interest are investigated, and the ability to predict the amplitudes of the limit cycle oscillations after the bifurcation is demonstrated. The cases of the Ziegler pendulum and Beck's column, both of which have a follower force, are considered for applications. By comparison with the eigenvalue trajectories in the conservative case, it is advocated that using two master modes to derive the ROM, instead of only considering the unstable one, should give more accurate results. Also, in the specific case where an exceptional bifurcation point is met, a numerical strategy enforcing the presence of Jordan blocks in the Jacobian matrix during the procedure, is devised. The ROMs are constructed for the Ziegler pendulum having two and three degrees of freedom, and then Beck's column is investigated, where a finite element procedure is used to space discretize the problem. The numerical results show the ability of the ROMs to correctly predict the amplitude of the limit cycles up to a certain range, and it is shown that computing the ROM after the Hopf bifurcation gives the most satisfactory results. This feature is analyzed in terms of phase space representations, and the two proposed adjustments are shown to improve the validity range of the ROMs.

We compute certain Ext and Tor groups in the category of all functors from an Z/p-linear additive category A to vector spaces in terms of Ext and Tor computed in the full subcategory of additive functors from A to vector spaces. We thus obtain group homology computations for general lineargroups., Comment: This preprint reworks a part of the material of our previous preprint "Functor homology over an additive category" (hal-03432824 or arXiv:2111.09719), which is being completely reorganized

We generalize the strong comparison theorem of Franjou, Friedlander, Scorichenko and Suslin to the setting of Fp-linear additive categories. Our results have a strong impact in terms of explicit computations of functor homology, and they open the way to new applications to stable homology of groups or to K-theory. As an illustration, we prove comparison theorems between cohomologies of classical algebraic groups over infinite perfect fields, in the spirit of a celebrated result of Cline, Parshall, Scott et van der Kallen for finite fields., Comment: This preprint reworks a part of the material of our previous preprint ''Functor homology over an additive category'' (hal-03432824 or arXiv:2111.09719), which is being completely reorganized. 48 pages

This paper presents the first application of the direct parametrisation method for invariant manifolds to a fully coupled multiphysics problem involving the nonlinear vibrations of deformable structures subjected to an electrostatic field. The formulation proposed is intended for model order reduction of electrostatically actuated resonating Micro-Electro-Mechanical Systems (MEMS). The continuous problem is first rewritten in a manner that can be directly handled by the parametrisation method, which relies upon automated asymptotic expansions. A new mixed fully Lagrangian formulation is thus proposed which contains only explicit polynomial nonlinearities, which is then discretised in the framework of finite element procedures. Validation is performed on the classical parallel plate configuration, where different formulations using either the general framework, or an approximation of the electrostatic field due to the geometric configuration selected, are compared. Reduced-order models along these formulations are also compared to full-order simulations operated with a time integration approach. Numerical results show a remarkable performance both in terms of accuracy and wealth of nonlinear effects that can be accounted for. In particular, the transition from hardening to softening behaviour of the primary resonance while increasing the constant voltage component of the electric actuation, is recovered. Secondary resonances leading to superharmonic and parametric resonances are also investigated with the reduced-order model.

Traditional turbulence models are derived for single-phase flow. Extension of the family of two-equation turbulence models for two-phase flow is obtained via scaling the transport equations by the density. In the special case of two-phase flow with a sharp interface, jump conditions exist. Two types of jump conditions are found: (1) jump in the partial differential equation (PDE) physical quantities such as density and viscosity and (2) jump in the turbulence frequency. We first derive and clarify the jump in the equations. The jump in the turbulence frequency is proportional to the kinematic viscosity ratio, which is approximately $10$ in the case of air-water. Then a new field, the inverse turbulence area, is considered to model the turbulence effects instead of the turbulence frequency. For the system of air and water, the effect of the jump of the kinematic viscosity is always greater than the effect arising from the jump of velocity gradient. This approximation leads to the assumption of a continuous inverse turbulence area scale. Validation versus experimental measurements from the literature is then presented to demonstrate the improvement of the model. In particular, the wave breaking phenomenon is simulated in two conditions: spilling and plunging wave breakers. The proposed model shows its ability to predict the turbulence in the surf zone accurately. Finally, it explains the low values of the time-averaged turbulent kinetic energy in the surf zone which is caused by the increase of the turbulence frequency in the air.

The direct parametrisation method for invariant manifold is a model-order reduction technique that can be applied to nonlinear systems described by PDEs and discretised e.g. with a finite element procedure in order to derive efficient reduced-order models (ROMs). In nonlinear vibrations, it has already been applied to autonomous and non-autonomous problems to propose ROMs that can compute backbone and frequency-response curves of structures with geometric nonlinearity. While previous developments used a first-order expansion to cope with the non-autonomous term, this assumption is here relaxed by proposing a different treatment. The key idea is to enlarge the dimension of the parametrising coordinates with additional entries related to the forcing. A new algorithm is derived with this starting assumption and, as a key consequence, the resonance relationships appearing through the homological equations involve multiple occurrences of the forcing frequency, showing that with this new development, ROMs for systems exhibiting a superharmonic resonance, can be derived. The method is implemented and validated on academic test cases involving beams and arches. It is numerically demonstrated that the method generates efficient ROMs for problems involving 3:1 and 2:1 superharmonic resonances, as well as converged results for systems where the first-order truncation on the non-autonomous term showed a clear limitation.

This paper presents a novel derivation of the direct parametrisation method for invariant manifolds able to build simulation-free reduced-order models for nonlinear piezoelectric structures, with a particular emphasis on applications to Micro-Electro-Mechanical-Systems. The constitutive model adopted accounts for the hysteretic and electrostrictive response of the piezoelectric material by resorting to the Landau-Devonshire theory of ferroelectrics. Results are validated with full-order simulations operated with a harmonic balance finite element method to highlight the reliability of the proposed reduction procedure. Numerical results show a remarkable gain in terms of computing time as a result of the dimensionality reduction process over low dimensional invariant sets. Results are also compared with experimental data to highlight the remarkable benefits of the proposed model order reduction technique.

Let A be a finitely-generated commutative ring and k a noetherian commutative ring. We show that, in the category of functors from finitely-generated projective A-modules to k-modules, each finitely-generated polynomial functor is noetherian and has a finitely-generated projective resolution., Comment: 15 pages, in French language

We uncover several general phenomenas governing functor homology over additive categories. In particular, we generalize the strong comparison theorem of Franjou Friedlander Scorichenko and Suslin to the setting of Fp-linear additive categories. Our results have a strong impact in terms of explicit computations of functor homology, and they open the way to new applications to stable homology of groups or to K-theory. As an illustration, we prove comparison theorems between cohomologies of classical algebraic groups over infinite perfect fields, in the spirit of a celebrated result of Cline, Parshall, Scott et van der Kallen for finite fields.

We give sufficient conditions which ensure that a functor of finite length from an additive category to finite-dimensional vector spaces has a projective resolution whose terms are finitely generated. For polynomial functors, we study also a weaker homological finiteness property, which applies to twisted homological stability for matrix monoids. This is inspired by works by Schwartz and Betley-Pirashvili, which are generalised; this also uses decompositions {\`a} la Steinberg over an additive category that we recently got with Vespa. We show also, as an application, a finiteness property for stable homology of linear groups on suitable rings., Comment: in French language

This paper investigates model-order reduction methods for geometrically nonlinear structures. The parametrisation method of invariant manifolds is used and adapted to the case of mechanical systems expressed in the physical basis, so that the technique is directly applicable to problems discretised by the finite element method. Two nonlinear mappings, respectively related to displacement and velocity, are introduced, and the link between the two is made explicit at arbitrary order of expansion. The same development is performed on the reduced-order dynamics which is computed at generic order following the different styles of parametrisation. More specifically, three different styles are introduced and commented: the graph style, the complex normal form style and the real normal form style. These developments allow making better connections with earlier works using these parametrisation methods. The technique is then applied to three different examples. A clamped-clamped arch with increasing curvature is first used to show an example of a system with a softening behaviour turning to hardening at larger amplitudes, which can be replicated with a single mode reduction. Secondly, the case of a cantilever beam is investigated. It is shown that the invariant manifold of the first mode shows a folding point at large amplitudes which is not connected to an internal resonance. This exemplifies the failure of the graph style due to the folding point, whereas the normal form style is able to pass over the folding. Finally, A MEMS micromirror undergoing large rotations is used to show the importance of using high-order expansions on an industrial example., Comment: 43 pages, 11 figures, 3 tables

This paper aims at reviewing nonlinear methods for model order reduction of structures with geometric nonlinearity, with a special emphasis on the techniques based on invariant manifold theory. Nonlinear methods differ from linear based techniques by their use of a nonlinear mapping instead of adding new vectors to enlarge the projection basis. Invariant manifolds have been first introduced in vibration theory within the context of nonlinear normal modes (NNMs) and have been initially computed from the modal basis, using either a graph representation or a normal form approach to compute mappings and reduced dynamics. These developments are first recalled following a historical perspective, where the main applications were first oriented toward structural models that can be expressed thanks to partial differential equations (PDE). They are then replaced in the more general context of the parametrisation of invariant manifold that allows unifying the approaches. Then the specific case of structures discretized with the finite element method is addressed. Implicit condensation, giving rise to a projection onto a stress manifold, and modal derivatives, used in the framework of the quadratic manifold, are first reviewed. Finally, recent developments allowing direct computation of reduced-order models (ROMs) relying on invariant manifolds theory are detailed. Applicative examples are shown and the extension of the methods to deal with further complications are reviewed. Finally, open problems and future directions are highlighted., Comment: 64 pages, 8 figures

This paper develops a novel method to detect cold pools from atmospheric soundings over tropical oceans and applies it to sounding data from the EUREC4A field campaign, which took place south and east of Barbados in January-February 2020. The proposed method exploits the fact that the air in a cold pool is denser than the air above it. It leads us to define cold pool soundings as those for which the mixed-layer height is smaller than 400 m. We first test this criterion by verifying its consistency with surface temperature and precipitation in a realistic high-resolution simulation over the western tropical Atlantic. Applying it to EUREC4A data, we then identify 7% of EUREC4A dropsondes and radiosondes as cold pool soundings. In two selected case studies, we find that cold pools soundings coincide with mesoscale cloud arcs and temperature drops in the surface time series. Statistics for the entire campaign further characterize the signature of cold pools in temperature, humidity and wind profiles. In the presence of wind shear, we show in particular that the spreading of cold pools is favored downshear, suggesting downward momentum transport by unsaturated downdrafts. These results support the robustness of our simple method in different environmental conditions and illustrate the new insights it offers for the characterization of cold pools and their environment.

Dimensionality reduction in mechanical vibratory systems poses challenges for distributed structures including geometric nonlinearities, mainly because of the lack of invariance of the linear subspaces. A reduction method based on direct normal form computation for large finite element (FE) models is here detailed. The main advantage resides in operating directly from the physical space, hence avoiding the computation of the complete eigenfunctions spectrum. Explicit solutions are given, thus enabling a fully non-intrusive version of the reduction method. The reduced dynamics is obtained from the normal form of the geometrically nonlinear mechanical problem, free of non-resonant monomials, and truncated to the selected master coordinates, thus making a direct link with the parametrisation of invariant manifolds. The method is fully expressed with a complex-valued formalism by detailing the homological equations in a systematic manner, and the link with real-valued expressions is established. A special emphasis is put on the treatment of second-order internal resonances and the specific case of a 1:2 resonance is made explicit. Finally, applications to large-scale models of Micro-Electro-Mechanical structures featuring 1:2 and 1:3 resonances are reported, along with considerations on computational efficiency., Comment: 34 pages, 10 figures