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Many systems in life sciences have been modeled by reaction–diffusion equations. However, under some circumstances, these biological systems may experience instantaneous and periodic perturbations (e.g. harvest, birth, release, fire events, etc) such that an appropriate formalism like impulsive reaction–diffusion equations is necessary to analyze them. While several works tackled the issue of traveling waves for monotone reaction–diffusion equations and the computation of spreading speeds, very little has been done in the case of monotone impulsive reaction–diffusion equations. Based on vector-valued recursion equations theory, we aim to present in this paper results that address two main issues of monotone impulsive reaction–diffusion equations. Our first result deals with the existence of traveling waves for monotone systems of impulsive reaction–diffusion equations. Our second result tackles the computation of spreading speeds for monotone systems of impulsive reaction–diffusion equations. We apply our methodology to a planar system of impulsive reaction–diffusion equations that models tree–grass interactions in fire-prone savannas. Numerical simulations, including numerical approximations of spreading speeds, are finally provided in order to illustrate our theoretical results and support the discussion.

International audience; In the first edition of this book, we discussed methods for determining the support of an inhomogeneous media from measured far field data as well as an extensive study of the central role played by the transmission eigenvalue problem in the mathematical development of these methods. In particular, we introduced the generalized linear sampling method (GLSM) and showed that this method provides a mathematical explanation of why it is permissible to use Tikhonov regularization to obtain an approximate solution of the far field equation associated with the linear sampling method.In the five years since the first edition of our book appeared, there has been considerable progress in both the development of GLSM as well as the theory of transmission eigenvalues. In this second edition, in addition to correcting typos in the first edition, we have added several highlights taken from these new developments. In particular, we have included new chapters on 1) the use of modified background media in the nondestructive testing of materials and in particular methods for determining the modified transmission eigenvalues that arise in such applications from measured far field data, 2) a study of a subset of transmission eigenvalues, called non-scattering wave numbers, through the use of techniques taken from the theory of free boundary value problems for elliptic partial differential equations and 3) the duality between scattering poles and transmission eigenvalues which, in addition to their intrinsic mathematical interest, leads to new methods for the numerical computation of scattering poles.

We study the weighted Fock spaces in one and several complex variables. We evaluate the dimension of these spaces in terms of the weight function extending and completing earlier results by Rozenblum–Shirokov and Shigekawa.

Given an arbitrary function in $\boldsymbol{H}({\operatorname{div}})$, we show that the error attained by the global-best approximation by $\boldsymbol{H}({\operatorname{div}})$-conforming piecewise polynomial Raviart–Thomas–Nédélec elements under additional constraints on the divergence and normal flux on the boundary is, up to a generic constant, equivalent to the sum of independent local-best approximation errors over individual mesh elements, without constraints on the divergence or normal fluxes. The generic constant only depends on the shape-regularity of the underlying simplicial mesh, the space dimension, and the polynomial degree of the approximations. The analysis also gives rise to a stable, local, commuting projector in $\boldsymbol{H}({\operatorname{div}})$, delivering an approximation error that is equivalent to the local-best approximation. We next present a variant of the equivalence result, where robustness of the constant with respect to the polynomial degree is attained for unbalanced approximations. These two results together further enable us to derive rates of convergence of global-best approximations that are fully optimal in both the mesh size $h$ and the polynomial degree $p$, for vector fields that only feature elementwise the minimal necessary Sobolev regularity. We finally show how to apply our findings to derive optimal a priori$hp$-error estimates for mixed and least-squares finite element methods applied to a model diffusion problem.

International audience

A new solid-beam approach devoted to thin to very thick beam-like structures is proposed. To meet the beam displacement fields, kinematic relations are directly applied on the solid finite element model which contains several elements through the thickness and width. It is possible to use any beam theory based on kinematic assumptions and three theories are considered in this paper. The classical first-order beam theory, the modified first-order beam theory and a higher-order beam theory lead to the FOSB, the MFOSB and the HOSB models respectively. Linear equations due to kinematic relations are imposed at slave nodes to meet displacement fields throughout the cross-section, resulting in a reduction of the model size. Two examples are presented: a straight beam and a curved beam under distributed loading. The FOSB model gives unsatisfactory results. For a thin beam, the MFOSB model is adequate but leads to moderate errors in the thick case. Compared to the solid model, the HOSB model offers excellent displacements and stresses in thin and thick cases. The final size of these solid-beam models is slightly larger than that of an approach with classical beam elements.

International audience; Two kinds of behaviour are commonly distinguished among porous architectured materials: stretching-dominated and bending-dominated ones. One or the other is favoured depending on the mechanical application. This property has been widely studied for random foams and periodic lattices, but has not yet been considered for quasi-periodic ones. Generally speaking, few studies have investigated their mechanical behaviour, and when this has been done, the impact of the elementary pattern shape has not been considered. In this paper we propose a classification of quasi-periodic structures depending on their dominance type by performing numerical simulations on mesoscopic models, and analysing the proportions of the strain energy stored in stretching and bending. As a result, three different dominance behaviours can be distinguished: completely stretching or bending dominated and varying dominance. Belonging to one or the other category depends mainly on the elementary pattern shape, and thus so does the overall mechanical behaviour.

International audience; The equivalent fluid model (EFM) describes the acoustic properties of rigid porous media by defining the intra-pore fluid phase as a fluid with an effective density and an effective compressibility. Their definitions are based on the dynamic tortuosity α and the normalized dynamic compressibility β. These physical quantities are complex-valued functions depending on the frequency, and can be irrational as in the Johnson-Champoux-Allard-Pride-Lafarge (JCAPL) model. Hence, the system of equations derived from the EFM can involve fractional derivatives in the time domain. This paper presents an approach to formulate the EFM equations described by the JCAPL model in the time domain, leading to an augmented system for which a proof of stability is given. From the EFM, a model for numerical simulation is built with α and β approximated using a multipole model. Sufficient stability conditions are then provided for the multipole-based EFM. Lastly, a numerical analysis is carried out in order to illustrate the theoretical results and a simulation of the impedance tube experiment is presented.

Understanding the dynamics underlying fluid transport in tumour tissues is of fundamental importance to assess processes of drug delivery. Here, we analyse the impact of the tumour microscopic properties on the macroscopic dynamics of vascular and interstitial fluid flow by using formal asymptotic techniques.Here, we obtained different macroscopic continuum models that couple vascular and interstitial flows. The homogenization technique allows us to derive two macroscale tissue models of fluid flow that take into account the microscopic structure of the vessels and the interstitial tissue. Different regimes were derived according to the magnitude of the vessel wall permeability and the interstitial hydraulic conductivity. Importantly, we provide an analysis of the properties of the models and show the link between them. Numerical simulations were eventually performed to test the models and to investigate the impact of the microstructure on the fluid transport.Future applications of our models include their calibration with real imaging data to investigate the impact of the tumour microenvironment on drug delivery.

The main objective of this paper is to provide a numerical approach to study thin sheet metals involved in multiphysics manufacturing processes. The proposed approach also accounts for anisotropic plastic deformations in the context of modeling of deep drawing process. In this work, different constitutive models are implemented with a prismatic solid shell element to simulate these applications. Moreover, a new element assembly technique has been developed to permit the assembly of prismatic elements in a tetrahedral element-based finite element code. This technique splits the prism into multiple tetrahedral elements in such a way that all the cross terms are accounted for. The numerical approach has been validated using multiple tests that involve different non-linearities: geometric, material and contact. Then, more complex sheet metal processes have been simulated. The performance of the solid shell element with one element through the thickness is very comparable to the results obtained with more refined mesh of a tetrahedral element. This reduction in the number of elements accelerates the simulation, especially in the deep drawing problem and the coupled magnetic-mechanical simulation used for the magnetic pulse forming process.

International audience; We establish the equivalence between the Multiscale Hybrid-Mixed (MHM) and the Multiscale Hybrid High-Order (MsHHO) methods for a variable diffusion problem with piecewise polynomial source term. Under the idealized assumption that the local problems defining the multiscale basis functions are exactly solved, we prove that the equivalence holds for general polytopal (coarse) meshes and arbitrary approximation orders. We also leverage the interchange of properties to perform a unified convergence analysis, as well as to improve on both methods.

International audience; This paper investigates the primary atomization of airblasted liquid sheet using detailed numerical simulations. The atomization of liquid sheet under airblasting conditions involve complex mechanisms and a thorough understanding is necessary. A planar pre-filming airblast atomization configuration have been chosen to study the breakup of liquid sheet/film injected on a solid flat plate. We have investigated an operating point directly relevant for high altitude relight condition of the aircraft. This configuration has been chosen based on the experimental investigation of Gepperth et al [S. Gepperth, A. Müller, R. Koch, H.-J. Bauer, Ligament and droplet characteristics in pre-filming airblast atomization, Proceedings of the ICLASS, 12th Triennial International Conference on Liquid Atomization and Spray Systems, September 2-6, Heidelberg, Germany, 2012] for the airblast atomization. The numerical simulations have been performed using in-house Navier-Stokes solver that uses consistent mass and momentum flux computation technique. The purpose of this work is to provide a comprehensive database and analyses of the airblast atomization of liquid sheet. This include studies on the effect of velocity profile on the atomization characteristics, occurrence of secondary atomization and drop coalescence, and extraction of near-field atomization characteristics. The qualitative analyses of the results from the simulations showed that there are two major atomization mechanisms of liquid film breakup -- sheet/bag breakup and ligament breakup. The drop diameter and velocity distributions computed from the simulations was found to be of the same order of magnitude although under-predicting the experimental data. Based on the atomized drop data, both the secondary atomization and drop coalescence have been observed to occur in the simulations. The quantitative analyses of the near-field liquid ligaments results revealed the lengths of these ligaments are of the same order of magnitude as the experimental data while an under-prediction in the ligament velocity has been observed. Finally, an excellent agreement between simulations and experimental data has been found for the Sauter Mean Diameter (SMD) of the atomized droplets.

The Cauchy problem for the Dullin-Gottwald-Holm (DGH) equation \[u_t-\alpha^2 u_{xxt}+2\omega u_x +3uu_x+\gamma u_{xxx}=\alpha^2 (2u_x u_{xx} + uu_{xxx})\] with zero boundary conditions (as \(|x|\to\infty\)) is treated by the Riemann-Hilbert approach to the inverse scattering transform method. The approach allows us to give a representation of the solution to the Cauchy problem, which can be efficiently used for further studying the properties of the solution, particularly, in studying its long-time behavior. Using the proposed formalism, smooth solitons as well as non-smooth cuspon solutions are presented.

The logarithmic nonlinearity has been used in many partial differential equations (PDEs) for modeling problems in various applications.Due to the singularity of the logarithmic function, it introducestremendous difficulties in establishing mathematical theories, as well asin designing and analyzing numerical methods for PDEs with such nonlinearity. Here we take the logarithmic Schr\"odinger equation (LogSE)as a prototype model. Instead of regularizing $f(\rho)=\ln \rho$ in theLogSE directly and globally as being done in the literature, we propose a local energy regularization (LER) for the LogSE byfirst regularizing $F(\rho)=\rho\ln \rho -\rho$ locally near $\rho=0^+$ with a polynomial approximation in the energy functional of the LogSE and then obtaining an energy regularized logarithmic Schr\"odinger equation (ERLogSE) via energy variation. Linear convergence is established between the solutions of ERLogSE and LogSE in terms of a small regularization parameter $0, Comment: 31 pages, 10 figures, final version. More explanations and some proofs are more detailed

International audience; This work is devoted to the vibroacoustics of complex systems over a broad-frequency band of analysis. The considered system is composed of a complex structure coupled with an internal acoustic cavity. On one hand, the global displacements are associated with the main stiff part and on the other hand, the local displacements are associated with the preponderant vibrations of the flexible subparts. Such complex structures induce interweaving of these two types of displacements, which introduce an overlap of the usual three frequency bands (low-, medium- and high-frequency bands (LF, MF, and HF). A reduced-order computational vibroacoustic model is constructed by using a classical modal analysis with the elastic and acoustic modes. Nevertheless, the dimension of such reduced-order model (ROM) is still important when there is an overlap for each one of the three frequency bands. A multi-frequency reduced-order model is then constructed for the structure over the LF, MF, and HF bands. The strategy is based on a multilevel projection consisting in introducing three reduced-order bases that are obtained by using a spatial filtering methodology. To filter out the local displacements in the structure, a set of global shape functions is introduced. In addition, a classical ROM using acoustic modes is carried out for the acoustic cavity. Then, the coupling between the multilevel ROM and the acoustic ROM is presented. A nonparametric probabilistic modeling is then proposed to take into account the model uncertainties induced by modeling errors that increase with the frequency. The proposed approach is applied to a large-scale computational vibroacoustic model of a car.

International audience; This study presents a new methodology for estimating the effective properties of solids containing cracks along the inter-granular boundaries, using analytical developments and numerical simulations. The latter are based on the generation of virtual microstructures of such type obtained by superimposing a Voronoï tessellation modeling the granular network with a random dispersion of overlapping spheres in 3-D, or disks in 2-D, which serve to locate the cracks at the inter-granular boundaries. The different features of this microstructure model are studied herein, especially the morphological effects induced by varying the size ratio between grains and spheres/disks. By means of full-field simulations, the effective thermal conductivities of the generated microstructures are estimated and compared with those of uniformly weakened solids (presenting uniform crack dispersion). For the latter microstructures, the Ponte-Castañeda & Willis (1995) upper bound turns out to be close to the full-field results. In addition, the full-field computations show that the spatial distribution of inter-granular cracks induces a dramatic degradation of the effective thermal conductivity. Modifying only the cutoff crack density in the mathematical expression of the Ponte Castañeda and Willis bound provides a relevant analytical estimate of the effective conductivity of solids weakened by inter-granular cracks. This cutoff crack density only depends on the microstructural parameters. This new estimate is shown to improve the one derived by Sevostianov & Kachanov (2019) and based on the differential scheme at least for the microstructures considered herein. Finally, new estimates of the moduli of elasticity for isotropic cracked solids weakened at inter-granular boundaries are also provided. The effective bulk modulus thus estimated for 3-D solids is shown to remain below the upper bound which can also be generated by injecting the effective conductivity predicted by full-field computations into the classical cross-property relations.

International audience; We construct global dissipative solutions on the torus of dimension at most three of the defocusing isothermal Euler-Langevin-Korteweg system, which corresponds to the Euler-Korteweg system of compressible quantum fluids with an isothermal pressure law and a linear drag term with respect to the velocity. In particular, the isothermal feature prevents the energy and the BD-entropy from being positive. Adapting standard approximation arguments we first show the existence of global weak solutions to the defocusing isothermal Navier-Stokes-Langevin-Korteweg system. Introducing a relative entropy function satisfying a Gronwall-type inequality we then perform the inviscid limit to obtain the existence of dissipative solutions of the Euler-Langevin-Korteweg system.

PurposeThis work contributes to the general problem of justifying the validity of the heuristic that underpins medium imaging using topological derivatives (TDs), which involves the sign and the spatial decay away from the true anomaly of the TD functional. The author considers here the identification of finite-sized (i.e. not necessarily small) anomalies embedded in bounded media and affecting the leading-order term of the acoustic field equation.Design/methodology/approachTD-based imaging functionals are reformulated for analysis using a suitable factorization of the acoustic fields, which is facilitated by a volume integral formulation. The three kinds of TDs (single-measurement, full-measurement and eigenfunction-based) studied in this work are given expressions whose structure allows to establish results on their sign and decay properties. The latter are obtained using analytical methods involving classical identities on Bessel functions and Legendre polynomials, as well as asymptotic approximations predicated on spatial scaling assumptions.FindingsThe sign component of the TD imaging heuristic is found to be valid for multistatic experiments and if the sought anomaly satisfies a bound (on a certain operator norm) involving its geometry, its contrast and the operating frequency. Moreover, upon processing the excitation and data by applying suitably-defined bounded linear operatirs to them, the magnitude component of the TD imaging heuristic is proved under scaling assumptions where the anomaly is small relative to the probing region, the latter being itself small relative to the propagation domain. The author additionally validates both components of the TD imaging heuristic when the probing excitation is taken as an eigenfunction of the source-to-measurement operator, with a focusing effect analogous to that achieved in time-reversal based methods taking place. These findings extend those of earlier studies to the case of finite-sized anomalies embedded in bounded media.Originality/valueThe originality of the paper lies in the theoretical justifications of the TD-based imaging heuristic for finite-sized anomalies embedded in bounded media.

International audience; In this work, the influence of aging on the mechanical properties at break of rubber materials are examined. When subjected to physical-chemical aging, the structure of the rubber network is deeply modified through two main mechanisms: crosslinking and chain scission. These aging mechanisms act both on the mechanical behaviour and on the fracture properties of the rubber materials. The goal of this work is to propose a predictive mechanical tool able to give estimates of these mechanical properties. To address this issue, an hyperelastic constitutive model based upon the energy limiter approach, was coupled with physical-chemical parameters in order to capture the whole mechanical behaviour of the damaged materials beyond the fracture. That allows fracture stress and strain estimates. A wide set of materials and experimental data extracted from the literature or obtained in our team were selected based upon the aging mechanism they can exhibit. We found that the elastically active chains concentrations and the swelling rate can be used as relevant indicators of damage either for crosslinking or chain scission process. These parameters are then introduced as damage parameters in the mechanical modelling. The proposed approach leads to very satisfactory predictions, either to capture the whole mechanical behaviour in uniaxial tension or in terms of stress and strain at break estimates.

International audience; We present some sharper finite extinction time results for solutions of a class of damped nonlinear Schrödinger equations when the nonlinear damping term corresponds to the limit cases of some ``saturating non-Kerr law'' $F(|u|^2)u=\frac{a}{\varepsilon+(|u|^2)^\alpha}u,$ with $a\in\mathbb{C},$ $\varepsilon\geqslant0,$ $2\alpha=(1-m)$ and $m\in[0,1).$ To carry out the improvement of previous results in the literature we present in this paper a careful revision of the existence and regularity of weak solutions under very general assumptions on the data. We prove that the problem can be solved in the very general framework of the maximal monotone operators theory, even under a lack of regularity of the damping term. This allows us to consider, among other things, the singular case $m=0.$ We replace the above approximation of the damping term by a different one which keeps the monotonicity for any $\varepsilon\geqslant0$. We prove that, when $m=0,$ the finite extinction time of the solution arises for merely bounded right hand side data $f(t,x).$ This is specially useful in the applications in which the Schrödinger equation is coupled with some other functions satisfying some additional equations.

We present a sharp interface approach for compressible two-phase flows that preserves its accuracy in the low Mach regime. The interface between both fluids is captured by a Level Set function via the Ghost Fluid method. In the low Mach regime, classic Finite Volume compressible solvers lose accuracy on quadrangle or hexahedral meshes and a low Mach correction is then necessary to reduce the numerical dissipation. The sharp interface that separates both fluids, may induce an important jump in the medium properties that induces additional challenges for the design of a numerical scheme. Indeed, the well-known low Mach fixes in the literature could lead to significant truncation errors when the flow involves large density and sound speed ratios between both phases. To preserve the accuracy of the numerical method, we propose a specific low Mach fix for this interface problem that yields a uniform truncation error with respect to the Mach number. Numerical results show significant evidances of the efficiency and accuracy of the proposed low Mach correction.

International audience; We consider a general class of diffusive Kermack-McKendrick SIR epidemic models with an age-structured protection phase with limited duration, for example due to vaccination or drugs with temporary immunity. A saturated incidence rate is also considered which is more realistic than the bilinear rate. The characteristics method reduces the model to a coupled system of a reaction-diffusion equation and a continuous difference equation with a time-delay and a nonlocal spatial term caused by individuals moving during their protection phase. We study the existence and non-existence of non-trivial traveling wave solutions. We get almost complete information on the threshold and the minimal wave speed that describes the transition between the existence and non-existence of non-trivial traveling waves that indicate whether the epidemic can spread or not. We discuss how model parameters, such as protection rates, affect the minimal wave speed. The difficulty of our model is to combine a reaction-diffusion system with a continuous difference equation. We deal with our problem mainly by using Schauder's fixed point theorem. More precisely, we reduce the problem of the existence of non-trivial traveling wave solutions to the existence of an admissible pair of upper and lower solutions.

International audience; The mechanical behavior of fibrous metamaterials is mainly determined by the interactions between the fibers composing the architecture. These interactions are usually of two different kinds: the ones directly depending on the positions of the fibers and those that need mediators, usually consisting in hinges, either inelastic or perfect, inducing restrictions on the kinematics of the fiber joints. In cases of interest, it has been observed that hinges can either have a certain torsional stiffness or behave as perfect joints, simply ensuring that the fibers remain interconnected, but not applying any constraint on the relative rotations between them. Here the effect of torsional stiffness of inelastic hinges is studied in two shear tests for a selected fibrous metamaterial. It is shown that the stiffness of hinges can be tailored to avoid, or at least reduce, out-of-plane deformations. Moreover, it is shown that, after reaching a threshold, permanent deformations are observed. This phenomenon is treated in a simplified way, by introducing damage in the continuum model.

International audience; The search space of full-waveform inversion (FWI) can be extended via a relaxation of the wave equation to increase the linear regime of the inversion. This wave-equation relaxation is implemented by solving jointly (in a least-squares sense) the wave equation weighted by a penalty parameter and the observation equation such that the reconstructed wavefields closely match the data, hence preventing cycle skipping at receivers. Then, the subsurface parameters are updated by minimizing the temporal and spatial source extension generated by the wave-equation relaxation to push back the data-assimilated wavefields toward the physics. This extended formulation of FWI has been efficiently implemented in the frequency domain with the augmented Lagrangian method in which the overdetermined systems of the data-assimilated wavefields can be solved separately for each frequency with linear-algebra methods and the sensitivity of the optimization to the penalty parameter is mitigated through the action of the Lagrange multipliers. Applying this method in the time domain is, however, hampered by two main issues: the computation of data-assimilated wavefields with explicit time-stepping schemes and the storage of the Lagrange multipliers capturing the history of the source residuals in the state space. These two issues are solved by recognizing that the source residuals on the right side of the extended wave equation, when formulated in a form suitable for explicit time stepping, are related to the extended data residuals through an adjoint equation. This relationship first allows us to relate the extended data residuals to the reduced data residuals through a normal equation in the data space. Once the extended data residuals have been estimated by solving (exactly or approximately) this normal equation, the data-assimilated wavefields are computed with explicit time-stepping schemes by cascading an adjoint and a forward simulation. Second, the relationship between the source and the data residuals allows us to project the Lagrange multipliers tied to the wave-equation constraint in the data space at the expense of additional adjoint simulations. This leads to a formulation of the extended FWI that is very similar to that of the classic reduced-space FWI. The computational cost of the extended FWI is two times that of the classic reduced-space FWI, and the storage requirement is the same. Applications to different synthetic benchmarks involving large contrasts confirm the resilience of the method to cycle skipping even when the quadratic problem for the extended data residuals is solved approximately with one inner iteration of a steepest-descent method.

In this paper, we present a new and efficient strategy to perform 3D computational fracture modeling from computed tomography (CT) images. The image-based fracture modeling creates a new way to more accurately predict fracture in realistic structures. However, it is currently complex and expensive to perform such simulations. In this work, we propose to use a matrix-free type preconditioned conjugate gradient solver based multigrid method to achieve the fastest numerical performance for phase field modeling of fracture. Several specific methods are investigated to enhance the robustness and to improve the efficiency of the proposed strategy. An automatic load control strategy to control crack propagation is investigated. High performance computing is applied using a hybrid MPI/OpenMP strategy. With all of the proposed procedures, the phase field modeling of fracture can be performed in the real 3D microstructure of heterogeneous materials using tomographic images.

We show that the Kantorovich-Rubinstein quasi-metrics d KR and d KR a of Part I extend naturally to various spaces of previsions, and in particular not just the linear previsions (roughly, measures) of Part I. There are natural isomorphisms between the Hoare and Smyth powerdomains, as used in denotational semantics, and spaces of discrete sublinear previsions, and of discrete normalized superlinear previsions, respectively. Turning to the corresponding hyperspaces, namely the same powerdomains, but equipped with the lower Vietoris and upper Vietoris topologies instead, this turns into homeomorphisms with the corresponding space of previsions, equipped with the so-called weak topology. Through these isomorphisms again, the two powerdomains inherit quasi-metrics d H and d Q , respectively, that are reminiscent of the well-known Hausdorff metric. Then we show that the Hoare and Smyth powerdomains of an algebraic complete quasi-metric space are again algebraic complete, with those quasi-metrics, and similarly that the corresponding powerdomains of continuous complete quasi-metric spaces are continuous complete. Furthermore, in the continuous complete case, the d H -Scott topology coincides with the lower Vietoris topology, and the d Q -Scott topology coincides with the upper Vietoris topology.

We consider the first exit point distribution from a bounded domain $$\Omega $$ of the stochastic process $$(X_t)_{t\ge 0}$$ solution to the overdamped Langevin dynamics $$\begin{aligned} d X_t = -\nabla f(X_t) d t + \sqrt{h} \ d B_t \end{aligned}$$ starting from deterministic initial conditions in $$\Omega $$ , under rather general assumptions on f (for instance, f may have several critical points in $$\Omega $$ ). This work is a continuation of the previous paper [14] where the exit point distribution from $$\Omega $$ is studied when $$X_0$$ is initially distributed according to the quasi-stationary distribution of $$(X_t)_{t\ge 0}$$ in $$\Omega $$ . The proofs are based on analytical results on the dependency of the exit point distribution on the initial condition, large deviation techniques and results on the genericity of Morse functions.

We look for solutions to the Schr\"{o}dinger-Poisson-Slater equation $$- \Delta u + \lambda u - \gamma (|x|^{-1} * |u|^2) u - a |u|^{p-2}u = 0 \quad \text{in} \quad \mathbb{R}^3, $$ which satisfy \begin{equation*} \int_{\mathbb{R}^3}|u|^2 \, dx = c \end{equation*} for some prescribed $c>0$. Here $ u \in H^1(\mathbb{R}^3)$, $\gamma \in \mathbb{R},$ $ a \in \mathbb{R}$ and $p \in (\frac{10}{3}, 6]$. When $\gamma >0$ and $a > 0$, both in the Sobolev subcritical case $p \in (\frac{10}{3}, 6)$ and in the Sobolev critical case $p=6$, we show that there exists a $c_1>0$ such that, for any $c \in (0,c_1)$, the equation admits two solutions $u_c^+$ and $u_c^-$ which can be characterized respectively as a local minima and as a mountain pass critical point of the associated {\it Energy} functional restricted to the norm constraint. In the case $\gamma >0$ and $a < 0$, we show that, for any $p \in (\frac{10}{3},6]$ and any $c>0$, the equation admits a solution which is a global minimizer. Finally, in the case $\gamma 0$ and $p=6$ we show that it does not admit positive solutions., Comment: This version is the final one, corresponding to the paper now published in Journal of Differential Equations

International audience; We design an unfitted hybrid high-order (HHO) method for the wave equation. The wave propagates in a domain where a curved interface separates subdomains with different material properties. The key feature of the discretization method is that the interface can cut more or less arbitrarily through the mesh cells. We address both the second-order formulation in time of the wave equation and its reformulation as a first-order system. We prove H 1-error estimates for a space semi-discretization in space of the second-order formulation, leading to optimal convergence rates for smooth solutions. Numerical experiments illustrate the theoretical findings and show that the proposed numerical schemes can be used to simulate accurately the propagation of acoustic waves in heterogeneous media, with meshes that are not fitted to the geometry. For the second-order formulation, the implicit, second-order accurate Newmark scheme is used for the time discretization, whereas (diagonally-implicit or explicit) Runge-Kutta schemes up to fourth-order accuracy are used for the first-order formulation. In the explicit case, we study the CFL condition on the time step and observe that the unfitted approach combined with local cell agglomeration leads to a comparable condition as when using fitted meshes.

In this paper, a general numerical framework for the modeling of ductile fracture in 3D meshes is introduced. The strategy is inspired from the phase field model and adaptive remeshing tools. The phase field model was introduced as a continuous model for predicting the initiation and propagation of cracks in materials. However, the model has a limitation on the choice of the characteristic length scale that controls the width of the cracked region. This work contributes to the full modeling of transition between the continuous damage using the phase field model to the discontinuous crack initiation and propagation within a unified numerical framework called CIPFAR. The contributions of the work include: (i). identification of the crack surface on arbitrary mesh topologies ; (ii). intersection by a Sequence Agnostic Partitioning strategy which is introduced to adapt the mesh to the computed crack surface; (iii). a nodal duplication by virtual non-manifold patch repair to open the mesh. Combining all the mentioned algorithms with adaptive remeshing allows modeling the initiation and propagation of cracks in materials efficiently. Different numerical examples are presented to prove the ability of the developed algorithm to model ductile fracture cases without the need to predefine the crack initiation region.

The intrinsic performance of epoxy adhesives seems to designate them as the ideal solution for assembling certain materials. Pore can have an influence on the load at which the first crack is initiated and on the final strength. In the context of damage tolerance, this work aims to explore the effect of small voids that cannot be detected by conventional non-destructive testing (NDT). To attain this goal, a method based on a criterion coupling strength and toughness is used. 2D finite-element calculations are performed on simple configurations to understand the steps leading to failure when pores are present. When the defects are far from the edges, the loss in load at the initiation of the first crack is a function of the minimum distance between two pores. When the location of the defect interacts with the stress concentration due to a free edge, this loss is a function of the minimum spacing between pores and the free edge. The knock-down factor on the load at failure for a bonded joint with epoxy adhesives remains small (close to the defect area divided by the total area of the bondline). The presence of pores may change the fracture surface (from an adhesive one without pore to a cohesive one with pores).

International audience; The results obtained in traditional topology optimization with a single density design variable assigned to each element, are typically improved through h-refinement where the resolution of the FE mesh is scaled up along with the number of density elements/voxels, without increasing the order of the finite element interpolation. Higher order elements allow a more realistic simulation of the physics without some of the spurious results associated with lower order elements, including improved numerical instability as a tertiary advantage. In this work, we conduct a detailed investigation of p-refinement in the context of SIMP topology optimization including optimized topologies, compliance values and CPU clock time, for various 2D classical benchmark problems. © 2021 Institution of Structural Engineers

A fully nonlinear potential Numerical Wave Tank (NWT) is developed in two dimensions, using a combination of the Harmonic Polynomial Cell (HPC) method for solving the Laplace problem on the wave potential and the Immersed Boundary Method (IBM) for capturing the free surface motion. This NWT can consider fixed, submerged or wall-sided surface piercing, bodies. To compute the flow around the body and associated pressure field, a novel multi overlapping grid method is implemented. Each grid having its own free surface, a two-way communication is ensured between the problem in the body vicinity and the larger scale wave propagation problem. Pressure field and nonlinear loads on the structure are computed by solving a boundary value problem on the time derivative of the potential. The stability and convergence properties of the solver are studied basing on extensive tests with standing waves of large to extreme wave steepness, up to $H/\lambda=0.2$ ($H$ is the crest-to-trough wave height and $\lambda$ the wavelength). Ranges of optimal time and spatial discretizations are determined and high-order convergence properties are verified, first without using any filter. For cases with either high level of nonlinearity or long simulation duration, the use of mild Savitzky-Golay filters is shown to extend the range of applicability of the model. Then, the NWT is tested against two wave flume experiments, analyzing forces on bodies in various wave conditions. First, nonlinear components of the vertical force acting on a small horizontal circular cylinder with low submergence below the mean water level are shown to be accurately simulated up to the third order in wave steepness. The second case is a dedicated experiment with a floating barge of rectangular cross-section. This very challenging case (body with sharp corners in large waves) allows to examine the behavior (...to continue...), Comment: 47 pages, 26 figures

In this note, we consider the derivation of continuum models for the interactions between a cloud of particles and a viscous fluid. We review recent results on the derivation of so-called Stokes–Brinkman models. We give some insights into analytical tools that are required for such results and discuss a possible extension to the Darcy regime.

Digital Twins, which tend to intervene over the entire life cycle of products from early design phase to predictive maintenance through optimization processes, are increasingly emerging as an essential component in the future of industries. To reduce the computational time reduced-order modeling (ROM) methods can be useful. However, the spread of ROM methods at an industrial level is currently hampered by the difficulty of introducing them into commercial finite element software, due to the strong intrusiveness of the associated algorithms, preventing from getting robust and reliable tools all integrated in a certified product. This work tries to circumvent this issue by introducing a weakly-invasive reformulation of the LATIN-PGD method which is intended to be directly embedded into Simcenter Samcef$$^{\hbox {TM}}$$TMfinite element software. The originality of this approach lies in the remarkably general way of doing, allowing PGD method to deal with not only a particular application but with all facilities already included in such softwares—any non-linearities, any element types, any boundary conditions...—and thus providing a new high-performance all-inclusive non-linear solver.

This paper outlines a rigorous variational-based multilevel Global-Local formulation for ductile fracture. Here, a phase-field formulation is used to resolve failure mechanisms by regularizing the sharp crack topology on the local state. The coupling of plasticity to the crack phase-field is realized by a constitutive work density function, which is characterized through a degraded stored elastic energy and the accumulated dissipated energy due to plasticity and damage. Two different Global-Local approaches based on the idea of multiplicative Schwarz' alternating method are proposed: (i) A global constitutive model with an elastic-plastic behavior is first proposed, while it is enhanced with a single local domain, which, in turn, describes an elastic-plastic fracturing response. (ii) The main objective of the second model is to introduce an adoption of the Global-Local approach toward the multilevel local setting. To this end, an elastic-plastic global constitutive model is augmented with two distinct local domains; in which, the first local domain behaves as an elastic-plastic material and the next local domain is modeled due to the fracture state. To further reduce the computational cost, predictor-corrector adaptivity within Global-Local concept is introduced. An adaptive scheme is devised through the evolution of the effective global plastic flow (for only elastic-plastic adaptivity), and through the evolution of the local crack phase-field state (for only fracture adaptivity). Thus, two local domains are dynamically updated during the computation, resulting with two-way adaptivity procedure. The overall response of the Global-Local approach in terms of accuracy/robustness and efficiency is verified using single-scale problems. The resulting framework is algorithmically described in detail and substantiated with numerical examples., Comment: 50 pages, 24 Figures, 4 Tables

We investigate the numerical reconstruction of the missing thermal boundary data on a part of the boundary for the steady-state heat conduction equation in anisotropic solids from the knowledge of exact or noisy Cauchy data on the remaining and accessible boundary. This inverse boundary value problem is tackled by applying and adapting to the anisotropic case the algorithm based on the fading regularization method, originally proposed by Cimetiere, Delvare, and Pons (Comptes Rendus de l’Academie des Sciences - Serie IIb - Mecanique, 328 639–644 2000), and Cimetiere, Delvare, et al. (Inverse Probl., 17 553–570 2001) for the isotropic heat conduction equation. The numerical implementation is realised for 2D homogeneous solids by using the boundary element method, whilst the numerical solution is stabilized/regularized by stopping the iterative process based on an L-curve type criterion (Hansen 1998).

International audience; The Partition of Unity Finite Element Method (PUFEM) shows promise for modeling wave-like problems in the mid-to-high frequency range, allowing to capture several wavelengths in a single element. Despite the increasing attention it received in acoustics and in structural dynamics, its efficacy to deal with coupled problems has not been addressed. The main challenge in this case is to be able to represent different types of physical waves accurately, knowing that the wavelengths can be very different and vary differently, exemplified by the dispersion of flexural waves in a solid. Without a proper handling of the coupling between the coupled media, at best the number of degrees of freedom (DoF) will not be optimal, at worst the coupled model will not converge. Techniques like mesh refinement, wave enrichment and compatible or incompatible meshes might offer a potential solution to the problem, but the model usually needs to be adjusted through a time consuming trial-and-error procedure. To tackle the problem, this paper considers a 2D coupled vibro-acoustic problem, in which the structural and acoustic domains, modeled with PUFEM, are coupled using compatible and incompatible meshes based on different coupling strategies. Numerical analyses show that the proposed method outperforms the classical finite element method by several orders of magnitude in terms of number of DoF. Recommendations are proposed on the technique to choose depending on the frequency range of interest in relation to the critical frequency of the structure to ensure the best convergence rate. Finally, an application example is presented to highlight the performance of the proposed method.

We describe completely 2-solitary waves related to the ground state of the nonlinear damped Klein–Gordon equation $$\begin{aligned} \partial _{tt}u+2\alpha \partial _{t}u-\Delta u+u-|u|^{p-1}u=0 \end{aligned}$$ on $$\mathbb {R}^N$$ , for $$1\leqslant N\leqslant 5$$ and energy subcritical exponents $$p>2$$ . The description is twofold. First, we prove that 2-solitary waves with same sign do not exist. Second, we construct and classify the full family of 2-solitary waves in the case of opposite signs. Close to the sum of two remote solitary waves, it turns out that only the components of the initial data in the unstable direction of each ground state are relevant in the large time asymptotic behavior of the solution. In particular, we show that 2-solitary waves have a universal behavior: the distance between the solitary waves is asymptotic to $$\log t$$ as $$t\rightarrow \infty $$ . This behavior is due to damping of the initial data combined with strong interactions between the solitary waves.

International audience; Abstract Solutions of partial differential equations can exhibit multiple time scales. Standard discretization techniques are constrained to capture the finest scale to accurately predict the response of the system. In this paper, we provide an alternative route to circumvent prohibitive meshes arising from the necessity of capturing fine-scale behaviors. The proposed methodology is based on a time-separated representation within the standard Proper Generalized Decomposition, where the time coordinate is transformed into a multi-dimensional time through new separated coordinates, each representing one scale, while continuity is ensured in the scale coupling. For instance, when considering two different time scales, the governing Partial Differential Equation is commuted into a nonlinear system that iterates between the so-called microtime and macrotime, so that the time coordinate can be viewed as a 2D time. The macroscale effects are taken into account by means of a finite element-based macro-discretization, whereas the microscale effects are handled with unidimensional parent spaces that are replicated throughout the time domain. The resulting separated representation allows us a very fine time discretization without impacting the computational efficiency. The proposed formulation is explored and numerically verified on thermal and elastodynamic problems.; Solutions of partial differential equations can exhibit multiple time scales. Standard discretization techniques are constrained to capture the finest scale to accurately predict the response of the system. In this paper, we provide an alternative route to circumvent prohibitive meshes arising from the necessity of capturing fine-scale behaviors. The proposed methodology is based on a time-separated representation within the standard Proper Generalized Decomposition, where the time coordinate is transformed into a multi-dimensional time through new separated coordinates, each representing one scale, while continuity is ensured in the scale coupling. For instance, when considering two different time scales, the governing Partial Differential Equation is commuted into a nonlinear system that iterates between the so-called microtime and macrotime, so that the time coordinate can be viewed as a 2D time. The macroscale effects are taken into account by means of a finite element-based macro-discretization, whereas the microscale effects are handled with unidimensional parent spaces that are replicated throughout the time domain. The resulting separated representation allows us a very fine time discretization without impacting the computational efficiency. The proposed formulation is explored and numerically verified on thermal and elastodynamic problems.

There have been several existence results for the standing waves of FitzHugh-Nagumo equations. Such waves are the connecting orbits of an autonomous second-order Lagrangian system and the corresponding kinetic energy is an indefinite quadratic form in the velocity terms. When the system has two stable hyperbolic equilibria, there exist two stable standing fronts, which will be used in this paper as building blocks, to construct stable standing waves with multiple fronts in case the equilibria are of saddle-focus type. The idea to prove existence is somewhat close in spirit to [Buffoni-Sere, CPAM 49, 285-305]. However several differences are required in the argument: facing a strongly indefinite functional, we need to perform a nonlo-cal Lyapunov-Schmidt reduction; in order to justify the stability of multiple front standing waves, we rely on a more precise variational characterization of such critical points. Based on this approach, both stable and unstable standing waves are established., Final version to appear in Journal of Differential Equations

International audience We derive results on equivalence of piecewise polynomial approximations of a given function in the Sobolev space H(curl). We namely show that the global-best approximation of a given H(curl) function in a H(curl)-conforming piecewise polynomial space imposing the continuity of the tangential trace can be bounded above and below by the Hilbertian sum of the respective local approximations from the elementwise spaces without any inter-element continuity requirement. In other words, the approximation of a H(curl) function by tangential-trace-continuous and discontinuous piecewise polynomials has comparable precision. We consider approximations of the curl of the target function in the L 2-norm, as well as approximations of the target function in the L 2-norm with a constraint on the curl; in the latter case, the constraint is removed in the local approximations. These best-approximation localizations hold under the minimal H(curl) regularity, on arbitrary shape-regular tetrahedral meshes, and involve imposition of conditions on a part of the boundary. They extend to the H(curl) context some recent results from the H 1 and H(div) spaces and have direct applications to a priori and a posteriori error analysis of numerical discretizations related to the H(curl) space, namely the Maxwell equations.

In the recent years, the domain of fast flow field prediction has been vastly dominated by pixel-based convolutional neural networks. Yet, the recent advent of graph convolutional neural networks (GCNNs) have attracted a considerable attention in the computational fluid dynamics (CFD) community. In this contribution, we proposed a GCNN structure as a surrogate model for laminar flow prediction around twodimensional (2D) obstacles. Unlike traditional convolution on image pixels, the graph convolution can be directly applied on body-fitted triangular meshes, hence yielding an easy coupling with CFD solvers. The proposed GCNN model is trained over a data set composed of CFD-computed laminar flows around 2,000 random 2D shapes. Accuracy levels are assessed on reconstructed velocity and pressure fields around outof-training obstacles, and are compared with that of standard U-net architectures, especially in the boundary layer area.

International audience; We develop a two-dimensional high-order numerical scheme that exactly preserves and captures the moving steady states of the shallow water equations with topography or Manning friction. The high-order accuracy relies on a suitable polynomial reconstruction, while the well-balancedness property is based on the first-order scheme from [Michel-Dansac et al., 2016 & Michel-Dansac et al., 2017], extended to two space dimensions. To get both properties, we use a convex combination between the high-order scheme and the first-order well-balanced scheme. By adequately choosing the convex combination parameter following a very simple steady state detector, we ensure that the resulting scheme is both high-order accurate and well-balanced. The method is then supplemented with a MOOD procedure to eliminate the spurious oscillations coming from the high-order polynomial reconstruction and to guarantee the physical admissibility of the solution. Numerical experiments show that the scheme indeed possesses the claimed properties. The simulation of the 2011 Japan tsunami, on real data, further confirms the relevance of this technique.

International audience; An orthotropic adhesion model is proposed based on the bi-potential method to solve adhesive contact problems with orthotropic interface properties between hyperelastic bodies. The model proposes a straightforward description of interface adhesion with orthotropic adhesion stiffness, whose components are conveniently expressed according to the local coordinate system. Based on this description, a set of extended unilateral and tangential contact laws has been formulated. Furthermore, we use an elementwise scalar parameter β to characterize the strength of interface adhesive bonds, and the effects of damage. Therefore, complete cycles of bonding and de-bonding of adhesive links with the account for orthotropic interface effects can be modelled. The proposed model has been tested on cases involving both tangential and unilateral contact kinematics. The test cases allowed emergence of orthotropic interface effects between elastomer bodies involving hyperelasticity. Meanwhile, the model can be implemented with minimum effort, and provides inspiration for the modelling of adhesive interface effects in areas of applications such as biomechanics.

International audience; This paper deals with the coupling between one-dimensional heat and wave equations in unbounded subdomains, as a simplified prototype of fluid-structure interaction problems. First we apply appropriate artificial boundary conditions that yield an equivalent problem, but with bounded subdomains, and we carry out the stability analysis for this coupled problem in truncated domains. Then we devise an optimized Schwarz-in-time (or Schwarz Waveform Relaxation) method for the numerical solving of the coupled equations. Particular emphasis is made on the design of optimized transmission conditions. Notably, for this setting, the optimal transmission conditions can be expressed analytically in a very simple manner. This result is illustrated by some numerical experiments.

International audience; Based on the blossoming theory, in this work we develop a new method for deriving Hermite-Padé approximants of certain hypergeometric series. Its general principle consists in building identities generalising the Hermite identity for exponentials, and in then applying their blossomed versions to appropriate tuples to simultaneously produce explicit expressions of the approximants and explicit integral representations of the corresponding remainders. For binomial series we use classical blossoms while for q-hypergeometric series we have to use q-blossoms.

International audience; The present paper is devoted to the convergence analysis of a class of asymptotic preserving particle schemes [Filbet & Rodrigues, SIAM J. Numer. Anal., 54 (2) (2016)] for the Vlasov equation with a strong external magnetic field. In this regime, classical Particle-In-Cell (PIC) methods are subject to quite restrictive stability constraints on the time and space steps, due to the small Larmor radius and plasma frequency. The asymptotic preserving discretization that we are going to study removes such a constraint while capturing the large-scale dynamics, even when the discretization (in time and space) is too coarse to capture fastest scales. Our error bounds are explicit regarding the discretization, stiffness parameter, initial data and time.

The paper is devoted to the mathematical analysis of the Cauchy problem for general cross-diffusion systems without any assumption about its entropic structure. A global existence result of nonnegative solutions is obtained by applying a classical Schauder fixed point theorem. The proof is upgraded for enhancing the regularity of the solution, namely its gradient belongs to the space L r ( ( 0 , T ) × Ω ) for some r > 2 . To this aim, the Schauder's strategy is coupled with an extension of Meyers regularity result for linear parabolic equations. We show how this approach allows to prove the well-posedness of the problem using only assumptions prescribing and admissibility range for the ratios between the diffusion and cross-diffusion coefficients. The results are compared with those that are reachable with an additional regularity assumption on the parabolic operator, namely a small BMO assumption for its coefficients. Finally, the question of the maximal principle is also addressed, especially when source terms are incorporated in the equation in order to ensure the confinement of the solution.

International audience; This paper addresses simulation of heat dominated compressible flows in a closed cavity using a pressure-based lattice Boltzmann (LB) method, in which thermal effects are modeled by applying a pressure-featured zero-order moment of distribution functions. A focus is made on the conservation of mass at boundary nodes, which is a challenging issue that significantly complicated by the density-decoupled zero-order moment here. The mass leakage at boundary nodes is mathematically quantified, which enables an efficient local mass correction scheme. The performance of this solver is assessed by simulating buoyancy-driven flows in a closed deferentially heated cavity with large temperature differences (non-Boussinesq) at Rayleigh numbers ranging from 103 to 107. Simulations show that mass leakage at solid walls in such configurations is a critical issue to obtain reliable solutions, and it eventually leads to simulations overflow when the cavity is inclined. The proposed mass correction scheme is, however, shown to be effective to control the mass leakage and get accurate solutions. Thus, associated with the proposed mass conservation scheme, the pressure-based LB method becomes reliable to study natural convection dominated flows at large temperature differences in closed geometries with mesh aligned boundaries or not