Publications

2026

• Path-following methods for quasi-static crack propagation: Application to phase-field fracture
  
  • Loiseau Flavien
  • Lazarus Véronique
  , 2026. Numerical simulations of quasi-static crack propagation in brittle materials often suffer from numerical instabilities, such as snapback events, due to structural softening. In variational phase-field fracture models, these instabilities manifest as abrupt crack jumps, thereby impeding physical validity, as the energy minimization is performed over a significant crack increment. Moreover, they often prevent incremental force boundary conditions, as they may lead to a loss of force balance as soon as the crack starts propagating. This study introduces path-following methods to mitigate such instabilities in quasi-static phase-field fracture simulations. We evaluate existing methods alongside a novel approach, Control by Maximum Strain Increment Outside the Crack (CMSIOC), which enforces stable crack growth by constraining strain increments in the uncracked region. All studied methods rely on a single scalar control equation that depends solely on displacement fields, enabling integration into classical staggered solvers without major changes. Performance is assessed via three benchmark problems of increasing complexity, with results compared against Linear Elastic Fracture Mechanics (LEFM) references based on Griffith's criterion and the G-max criterion. Our findings show that CMSIOC: 1. Accurately follows the equilibrium path, avoiding abrupt crack jumps while preserving physical validity; 2. Supports force-controlled boundary conditions without loss of force balance during propagation; 3. Ensures uniform incremental crack growth, properly distributing the computational efforts across load steps. These results highlight CMSIOC's robustness as a model- and problem-independent solution for stable phase-field fracture simulations.
• Path-following methods for phase-field simulation of quasi-static crack propagation
  
  • Loiseau Flavien
  • Lazarus Véronique
  International Journal of Solids and Structures, Elsevier, 2026, 334, pp.113974. The variational approach to fracture, particularly through its regularization as a phase-field model, has become a widely used tool for simulating the quasi-static propagation of cracks in structures. However, classic incremental loading can induce unstable crack growth, violating the quasi-static assumption, and in some cases, leads to a loss of force balance, preventing self-consistency and the estimation of dissipated energy during snapback instabilities. To address this challenge, path-following methods are investigated. Their aim is to adjust the applied load so that it stays at the propagation threshold, thereby preserving the quasi-static assumption and ensuring equilibrium solutions. In this work, we apply and evaluate multiple path-following methods within the framework of variational phase-field fracture models, which are developed to regularize linear elastic variational sharp crack evolution problems. Our study pursues two objectives. First, we review several existing path-following methods, with a focus on partitioned strategies based on the displacement field, which decouple the path-following control equation from the rest of the system, facilitating easier integration with staggered solvers. In addition, we introduce a new path-following method whose particularity is to limit the maximum strain increment outside the cracked regions. Second, we use the Γ-convergence to the sharp crack model to evaluate these methods across three crack propagation problems of increasing complexity. The comparison demonstrates that the proposed path-following method offers a simple yet highly effective approach to capture the equilibrium path in phase-field fracture simulations. This method robustly maintains the quasi-static assumption, ensuring physically meaningful results. By enabling accurate estimation of the energy dissipated during snapback instabilities, it paves the way for the rational design of more resistant heterogeneous materials. (10.1016/j.ijsolstr.2026.113974)
  DOI : 10.1016/j.ijsolstr.2026.113974
• Path-Following Methods for Quasi-Static Crack Propagation in Phase-Field Fracture Models
  
  • Loiseau Flavien
  • Lazarus Véronique
  , 2026. Quasi-static crack propagation simulations in brittle materials often suffer from numerical instabilities, such as snapback events, due to structural softening. In phase-field fracture models, these instabilities manifest as abrupt crack jumps, impeding the physical validity as the energy minimization is performed over a significant crack increment. Moreover, they often prevent the imposition of incremental force boundary conditions, as they may lead to a loss of force balance as soon as the crack starts to propagate. This study explores the application of path-following methods to variational phase-field fracture simulation. It aims to identify a method that is robust, model- and problem-independent, and easy to implement in a classic staggered solver. To this aim, we evaluate existing path-following techniques alongside a novel proposition: Control by Maximum Strain Increment Outside the Crack (CMSIOC), which ensures stable crack propagation by limiting strain increments in the uncracked region. All studied methods introduce a single scalar control equation dependent solely on displacement fields, enabling seamless integration into classic staggered solvers without requiring significant modifications. To assess the performance of these methods, we compare them across three numerical benchmark problems of increasing complexity. For reference, we employ a sharp crack model from Linear Elastic Fracture Mechanics (LEFM) based on Griffith's theory and the G-max criterion. Our findings demonstrate that CMSIOC reliably follows the equilibrium path and closely replicates LEFM benchmark solutions. Hence, it enables the capture of snapback instabilities without abrupt crack jumps, also allowing for force-controlled boundary conditions without loss of force balance. Another advantage is that it provides nearly uniform crack growth per increment, thereby balancing the computational cost across load steps. This presentation will: (1) recall the concept of equilibrium paths in fracture mechanics, (2) introduce the CMSIOC, detailing its theoretical foundation and implementation, (3) present the results of the benchmark problems, and (4) conclude on this work with practical recommendations for integrating path-following methods into phase-field fracture simulations.
• Accounting for grain crushing and pore collapse for strain localization in rocks
  
  • Collins-Craft Nicholas Anton
  • Sulem Jean
  • Stefanou Ioannis
  • Einav Itai
  , 2026. We present a model for crushable granular rocks by embedding breakage mechanics in the Cosserat continuum. This model features a dependence on an enriched set of state variables (the elastic strains and curvatures, the density, the solid fraction, and the breakage state variable), demonstrates a dependence of the yield surface on the Lode angle, breakage and solid fraction, and evolution laws that tightly couple the competing processes. We then outline the notion of linear stability analysis and how we use this technique to obtain both the thickness and orientation of any shear bands that may form in the system. The model is then calibrated from data available in the literature on Fontainebleau sandstone and other similar granular rocks. We compare the model predictions with experimental measures of both the stress-strain response, and the width and angle of shear bands, and find good agreement with the results that have been previously reported.
• Bubble rise dynamics in a quiescent liquid and impact on a cylinder
  
  • Beltran F.
  • Benguigui W.
  • Merigoux N.
  • Bonometti T.
  • Colin C.
  International Journal of Multiphase Flow, Elsevier, 2026, 199 (May), pp.105672. This study investigates the dynamics of a single bubble rising in a quiescent liquid and impacting a fixed cylinder using a resolved two-fluid approach. The resolved two-fluid approach is validated against experimental data and compared with a one-fluid approach through 2D axisymmetric and 3D simulations across a wide range of Reynolds and E & ouml;tvos numbers, and density ratios. The two-fluid model accurately reproduces the bubble shape, terminal velocity, and impact dynamics, showing agreement with both experimental observations and the one-fluid approach. A detailed analysis on the impact force coefficient exerted by the bubble on the cylinder is conducted with the two-fluid approach by varying the bubble's Reynolds and E & ouml;tvos numbers and the density, viscosity, and bubble-to-cylinder diameter ratios (Reb is an element of [1, 80],Eo is an element of [10, 116], rho l/rho g is an element of [25, 1000], mu l/mu g is an element of [10, 100], and db/Dc is an element of [0.5, 1.0]). This study reveals that, when varying one dimensionless number at a time, the bubble Reynolds number has the most significant influence on the impact force coefficient, followed by the bubble-to-cylinder diameter ratio and the E & ouml;tv & ouml;s number, while the effects of viscosity and density ratios are weaker. A correlation on the impact force coefficient (associated to the force applied by the bubble on the cylinder at impact) is proposed and may be useful for Euler-Lagrange point-particle methods. (10.1016/j.ijmultiphaseflow.2026.105672)
  DOI : 10.1016/j.ijmultiphaseflow.2026.105672
• An augmented lagrangian XFEM formulation for stabilizing near-tip singularities in fluid-driven fractures modeling
  
  • Faivre Maxime
  • Martin Alexandre
  • Massin P.
  • Giot Richard
  • Golfier Fabrice
  Computer Methods in Applied Mechanics and Engineering, Elsevier, 2026, 451, pp.118649. This paper enhances a fully-coupled hydromechanical model developed in the XFEM by introducing a new augmented Lagrangian formulation to stabilize the hydrodynamical behavior at interfaces. While the original model stabilized the mechanical part, it inadequately represented pressure fields near high gradients, particularly at the tip. The new formulation projects fields into an appropriate function space, enabling the use of a standard lumping technique on the mass matrix from the discretized time derivative in Reynolds’ lubrication equation. It also incorporates a lumping approach tailored to parabolic PDEs to stabilize the bulk behavior. Since Reynolds’ equation becomes singular near the tip in impermeable porous media, a virtual opening is introduced to regularize the fluid flow. The improved model is validated with analytical solutions, demonstrating its effectiveness in eliminating spurious pressure oscillations near discontinuities. (10.1016/j.cma.2025.118649)
  DOI : 10.1016/j.cma.2025.118649
• Steady-State Algorithm with Structural Periodicity: Application to Computation of Railways’ Ballast Plastic Strains
  
  • Badinier Thibault
  • Maiolino Siegfried
  • Maitournam Habibou
  Geotechnics, MDPI, 2026, 6 (1), pp.29. The geometry of ballasted railway tracks is crucial for ensuring railway safety and efficiency. This paper introduces the use of innovative steady-state algorithms designed to compute plastic strains in linear geotechnical structures like railway ballast layers, within Finite Element Methods (FEMs). Facing the specificities of moving loads, traditional step-by-step algorithms, while simple and adaptable, are computationally expensive and time-consuming. In contrast, the proposed steady-state algorithms leverage an Eulerian approach to describe the movement of loads significantly reducing computational time while maintaining accuracy. This paper proposes these algorithms as a methodological improvement and demonstrates the applicability and efficiency of the method for non-periodic structures, as well as for periodic structures, such as railway tracks with evenly spaced sleepers. This paper demonstrates the applicability and efficiency of theses algorithms through comparative studies with traditional methods on typical railway structures. The results show that the presented algorithm not only matches the accuracy of step-by-step methods but also drastically reduces computation time and data storage requirements. This advancement has practical applications for railway infrastructure managers, enabling more efficient and accurate predictions of track geometry evolution and preventing incidents through improved maintenance strategies. (10.3390/geotechnics6010029)
  DOI : 10.3390/geotechnics6010029
• Determination of the Effective Permeabilities in Partially Saturated Porous Media, Using the Periodic Homogenization Technique
  
  • Bouchard Raphaël
  • Bourbatache Mohamed-Khaled
  • Le Tien Dung
  • Millet Olivier
  • Stefanou Ioannis
  Transport in Porous Media, Springer Verlag, 2026, 153 (4), pp.47. Abstract In this study, we address determination of the effective permeabilities in rigid, partially saturated porous media for an immiscible two-phase Newtonian fluid flow. The periodic homogenization technique is applied to derive macroscopic flow laws for two-phase systems from the pore-scale Navier–Stokes equations governing immiscible fluids. Two distinguished cases are considered: two incompressible fluids (case 1) and an incompressible fluid with a compressible one (case 2). In both cases, the homogenized result shows the independence of the macroscopic laws and the closure problems on the fluid compressibility, except for the macroscopic mass conservation equation. Finally, numerical simulations are performed by solving the closure problems for a given interface position determined from the phase-field simulations, in order to analyze the role of each effective permeability in the generalized Darcy’s law for several fluid mixtures and different porosity. The numerical results offer insights into the influence of microstructure, fluid properties, and capillary bridge distribution on the effective permeabilities. (10.1007/s11242-025-02265-2)
  DOI : 10.1007/s11242-025-02265-2
• Reduced order modelling for shell finite element structures using the direct parametrisation of invariant manifolds: Hardening/softening transition, resonant dynamics and mode selection
  
  • Xia Zixu
  • Touzé Cyril
  • Cong Yu
  • Gu Shuitao
  • Feng Zhi-Qiang
  Thin-Walled Structures, Elsevier, 2026, 220, pp.114329. The accurate simulation of the nonlinear dynamics of thin-walled structures is a critical but computationally demanding task. In this contribution, a 7-parameter solid-shell finite element formulation is combined with the direct parametrisation method for invariant manifolds (DPIM), in order to derive accurate and efficient reduced-order models (ROM) accounting for geometric nonlinearity. The method is illustrated in its ability to be used with different yet complementary purposes. On the one hand, low-order tractable models can be obtained, providing simple ROMs that are amenable to giving physical insights and understanding. On the other hand, higher-order solutions are available within the same framework, hence providing accurate and converged solutions. This scheme is carried out on examples with increasing complexity. First, the transition from hardening to softening behaviour for thin shells with shape imperfections is investigated. The 1:2 resonance as a driver of the change of type of nonlinearity is analysed, and a full understanding of the smooth transition is illustrated. Then, shells with varying thicknesses are investigated, and the case of 1:2 internal resonance is further investigated, showing the emergence of isolated solution branches (isola). In the course of the numerical simulations, it is shown how the reduced basis needs to be enlarged to take into account more and more complex resonance scenarios, and some guidelines are provided in order to help the analyst in selecting the master modes. The numerical results highlight the ability of the reduced-order models to provide a fully comprehensive and integrated framework for the understanding and accurate prediction of thin shells' nonlinear dynamics. (10.1016/j.tws.2025.114329)
  DOI : 10.1016/j.tws.2025.114329
• A harmonic balance normal form parametrisation for single mode reduction of nonlinear vibrating systems
  
  • Grolet Aurélien
  • Touzé Cyril
  • de Figueiredo Stabile André
  • Thomas Olivier
  Communications in Nonlinear Science and Numerical Simulation, Elsevier, 2026, 157, pp.109708. This paper introduces a model-order reduction technique for lightly damped nonlinear vibrating systems. By combining calculation details that are specific to the harmonic balance method, the asymptotic numerical method, and the normal form style parametrisation for invariant manifolds, a complete procedure that can cope with single-mode reduction is detailed. Introducing harmonic decomposition in the process allows for a different treatment of the temporal information of the solution, which comes with advantages as compared to normal form expansions based on polynomial expansions. The computation proceeds with two nested loops on both the harmonics and the polynomial degree expansion. A decisive advantage of the procedure is its ability to compute a new expansion from a known solution, which allows the derivation of amplitudedependent piecewise reduced order models (ROMs), together with an integrated procedure that can switch from the invariant manifolds computation attached to either fixed points or limit cycles. Once the validity limit of a first expansion is met, the procedure can restart from a point where convergence is reached and produce a new ROM. This feature has the potential to overcome the well-known limitations of asymptotic expansions associated with the parametrisation method for invariant manifolds, and is derived here only for conservative systems. The whole analysis also clearly establishes the links existing between the normal form approach and computations based on the harmonic balance combined with the asymptotic numerical method. Examples of increasing complexity, starting from a Duffing equation, a two-degree-of-freedom system and a finite element beam model, are analysed, and comparisons with existing techniques are provided. (10.1016/j.cnsns.2026.109708)
  DOI : 10.1016/j.cnsns.2026.109708
• Review of "Asymptotic approaches for dealing with distorted crack geometries
  
  • Lazarus Véronique
  , 2026.
• Normal form computation of nonlinear dispersion relationship for locally resonant metamaterial
  
  • Wang Tao
  • Touzé Cyril
  • Li Haiqin
  • Ding Qian
  Physica D: Nonlinear Phenomena, Elsevier, 2026, 488, pp.135115. This article is devoted to the application of the parametrisation method for invariant manifold with a complex normal form style (CNF), for the derivation of higher-order approximations of underdamped nonlinear dispersion relationships for periodic structures, more specifically by considering the case of a locally resonant metamaterial chain incorporating damping and various nonlinear stiffnesses. Two different strategies are proposed to solve the problem. In the first one, Bloch's assumption is first applied to the equations of motion. The nonlinear change of coordinates provided by the complex normal form style in the parametrisation method is applied. This direct procedure, which applies first the wave dependency to the original physical coordinates of the problem, is referred to as CNF-BP (for CNF applied with Bloch's assumption on physical coordinates). In the second strategy, the nonlinear change of coordinates provided by the parametrisation method, which relates the physical coordinates to the so-called normal coordinates, is first applied. Then the periodic assumption is used, thus imposing a Bloch wave ansatz on the normal coordinates. This method will be referred to as CNF-PN (for CNF with a periodic assumption on normal coordinates). In the conservative case, the two CNF calculation strategies are first verified by comparing with the results from existing literature. Subsequently, two carefully selected examples demonstrate that the CNF-PN strategy exhibits superior capability in capturing complex wave propagation phenomena, whereas the CNF-BP strategy encounters limitations in handling non-fundamental harmonics and the nonlinear interactions between host oscillators. The influence of truncation order on the accuracy of CNF-PN is further examined, demonstrating its effectiveness in extending the validity limit. For underdamped systems, the CNF-PN is systematically compared against numerical techniques, a classical analytical perturbation technique (the method of multiple scales), and direct numerical time integration of annular chain structures. The results confirm the exceptional accuracy of the CNF-PN in predicting nonlinear dispersion relationships, damping ratios, invariant manifolds, and wave attenuation characteristics, as long as the validity limit of the asymptotic expansion is not reached. This advancement provides a novel and efficient analytical and numerical tool for studying nonlinear metamaterials. (10.1016/j.physd.2026.135115)
  DOI : 10.1016/j.physd.2026.135115
• Reduced order modelling of Hopf bifurcations for the Navier-Stokes equations through invariant manifolds
  
  • Colombo Alessio
  • Vizzaccaro Alessandra
  • Touzé Cyril
  • de F. Stabile André
  • Pastur Luc
  • Frangi Attilio
  Physical Review E, American Physical Society (APS), 2026, 113 (3), pp.034202. This work introduces a parametric simulation-free reduced order model for incompressible flows undergoing a Hopf bifurcation, leveraging the parametrisation method for invariant manifolds. Unlike data-driven approaches, this method operates directly on the governing equations, eliminating the need for full-order simulations. The proposed model is computed at a single value of the bifurcation parameter, yet remains valid over a range of values. The approach systematically constructs an invariant manifold and embedded dynamics, providing an accurate and efficient reduction of the original system. The ability to capture pre-critical steady states, the bifurcation point, and post-critical limit cycle oscillations is demonstrated by a strong agreement between the reduced-order model and full-order simulations, while achieving significant computational speed-up. (10.1103/w6z2-m5wk)
  DOI : 10.1103/w6z2-m5wk