arXiv subjects

Frederic Legoll

Publications and source records attributed to Frederic Legoll.

At least 19 recordsLinked to original sources

arXiv2024

Survival probability of structures under fatigue: a data-based approach

This article addresses the probabilistic nature of fatigue life in structures subjected to cyclic loading with variable amplitude. Drawing on the formalisation of Miner's cumulative damage rule that we introduced in the recent article [Cartiaux, Ehrlacher, Legoll, Libal and Reygner, Prob. Eng. Mech. 2023], we apply our methodology to estimate the survival probability of an industrial structure using experimental data. The study considers both the randomness in the initial state of the structure and in the amplitude of loading cycles. The results indicate that the variability of loading cycles can be captured through the concept of deterministic equivalent damage, providing a computationally efficient method for assessing the fatigue life of the structure. Furthermore, the article highlights that the usual combination of Miner's rule and of the weakest link principle systematically overestimates the structure's fatigue life. On the case study that we consider, this overestimation reaches a multiplicative factor of more than two. We then describe how the probabilistic framework that we have introduced offers a remedy to this overestimation.

cs.CE↗

arXiv2023

Embedded corrector problems for homogenization in linear elasticity

In this article, we extend the study of embedded corrector problems, that we have previously introduced in the context of the homogenization of scalar diffusive equations, to the context of homogenized elastic properties of materials. This extension is not trivial and requires mathematical arguments specific to the elasticity case. Starting from a linear elasticity model with highly-oscillatory coefficients, we introduce several effective approximations of the homogenized tensor. These approximations are based on the solution to an embedded corrector problem, where a finite-size domain made of the linear elastic heterogeneous material is embedded in a linear elastic homogeneous infinite medium, the constant elasticity tensor of which has to be appropriately determined. The approximations we provide are proven to converge to the homogenized elasticity tensor when the size of the embedded domain tends to infinity. Some particular attention is devoted to the case of isotropic materials.

math.AP↗

arXiv2023

A variance reduction strategy for numerical random homogenization based on the equivalent inclusion method

Using the equivalent inclusion method (a method strongly related to the Hashin-Shtrikman variational principle) as a surrogate model, we propose a variance reduction strategy for the numerical homogenization of random composites made of inclusions (or rather inhomogeneities) embedded in a homogeneous matrix. The efficiency of this strategy is demonstrated within the framework of two-dimensional, linear conductivity. Significant computational gains vs full-field simulations are obtained even for high contrast values. We also show that our strategy allows to investigate the influence of parameters of the microstructure on the macroscopic response. Our strategy readily extends to three-dimensional problems and to linear elasticity. Attention is paid to the computational cost of the surrogate model. In particular, an inexpensive approximation of the so-called influence tensors (that are used to compute the surrogate model) is proposed.

cs.CE↗

arXiv2023

Probabilistic formulation of Miner's rule and application to structural fatigue

The standard stress-based approach to fatigue is based on the use of S-N curves. They are obtained by applying cyclic loading of constant amplitude $S$ to identical and standardised specimens until they fail. The S-N curves actually depend on a reference probability $p$: for a given cycle amplitude $S$, they provide the number of cycles at which a proportion $p$ of specimens have failed. Based on the S-N curves, Miner's rule is next used to predict the number of cycles to failure of a specimen subjected to cyclic loading with variable amplitude. In this article, we present a probabilistic formulation of Miner's rule, which is based on the introduction of the notion of health of a specimen. We show the consistency of that new formulation with the standard approaches, thereby providing a precise probabilistic interpretation of these. Explicit formulas are derived in the case of the Weibull--Basquin model. We next turn to the case of a complete mechanical structure: taking into account size effects, and using the weakest link principle, we establish formulas for the survival probability of the structure. We illustrate our results by numerical simulations on a I-steel beam, for which we compute survival probabilities and density of failure point. We also show how to efficiently approximate these quantities using the Laplace method.

cs.CE↗

arXiv2022

Combining machine-learned and empirical force fields with the parareal algorithm: application to the diffusion of atomistic defects

We numerically investigate an adaptive version of the parareal algorithm in the context of molecular dynamics. This adaptive variant has been originally introduced in [F. Legoll, T. Lelievre and U. Sharma, SISC 2022]. We focus here on test cases of physical interest where the dynamics of the system is modelled by the Langevin equation and is simulated using the molecular dynamics software LAMMPS. In this work, the parareal algorithm uses a family of machine-learning spectral neighbor analysis potentials (SNAP) as fine, reference, potentials and embedded-atom method potentials (EAM) as coarse potentials. We consider a self-interstitial atom in a tungsten lattice and compute the average residence time of the system in metastable states. Our numerical results demonstrate significant computational gains using the adaptive parareal algorithm in comparison to a sequential integration of the Langevin dynamics. We also identify a large regime of numerical parameters for which statistical accuracy is reached without being a consequence of trajectorial accuracy.

math.NA↗

arXiv2022

Non-intrusive implementation of Multiscale Finite Element Methods: an illustrative example

Multiscale Finite Element Methods (MsFEM) are finite element type approaches dedicated to multiscale problems. They first compute local, oscillatory, problem-dependent basis functions which generate a specific discretization space, and next perform a Galerkin approximation of the problem on that space. We investigate here how these approaches can be implemented in a non-intrusive way, in order to facilitate their dissemination within industrial codes or non academic environments.

math.NA↗

arXiv2021

A pedagogical review on a posteriori error estimation in Finite Element computations

This article is a review on basic concepts and tools devoted to a posteriori error estimation for problems solved with the Finite Element Method. For the sake of simplicity and clarity, we mostly focus on linear elliptic diffusion problems, approximated by a conforming numerical discretization. The review mainly aims at presenting in a unified manner a large set of powerful verification methods, around the concept of equilibrium. Methods based on that concept provide error bounds that are fully computable and mathematically certified. We discuss recovery methods, residual methods, and duality-based methods for the estimation of the whole solution error (i.e. the error in energy norm), as well as goal-oriented error estimation (to assess the error on specific quantities of interest). We briefly survey the possible extensions to non-conforming numerical methods, as well as more complex (e.g. nonlinear or time-dependent) problems. We also provide some illustrating numerical examples on a linear elasticity problem in 3D.

math.NA↗

arXiv2021

An MsFEM approach enriched using Legendre polynomials

We consider a variant of the conventional MsFEM approach with enrichments based on Legendre polynomials, both in the bulk of mesh elements and on their interfaces. A convergence analysis of the approach is presented. Residue-type a posteriori error estimates are also established. Numerical experiments show a significant reduction in the error at a limited additional off-line cost. In particular, the approach developed here is less prone to resonance errors in the regime where the coarse mesh size $H$ is of the order of the small scale $\varepsilon$ of the oscillations.

math.NA↗

arXiv2021

Mathematical analysis of a coupling method for the practical computation of homogenized coefficients

We present the mathematical study of a computational approach originally introduced by R. Cottereau in [R. Cottereau, IJNME 2013]. The approach aims at evaluating the effective (a.k.a. homogenized) coefficient of a medium with some fine-scale structure. It combines, using the Arlequin coupling method, the original fine-scale description of the medium with an effective description and optimizes upon the coefficient of the effective medium to best fit the response of an equivalent purely homogeneous medium. We prove here that the approach is mathematically well-posed and that it provides, under suitable assumptions, the actual value of the homogenized coefficient of the original medium in the limit of asymptotically infinitely fine structures. The theory presented here therefore usefully complements our numerical developments of [O. Gorynina, C. Le Bris and F. Legoll, SIAM J. Sci. Computing 2021].

math.AP↗

arXiv2019

Goal-oriented error estimation and adaptivity in MsFEM computations

We introduce a goal-oriented strategy for multiscale computations performed using the Multiscale Finite Element Method (MsFEM). In a previous work, we have shown how to use, in the MsFEM framework, the concept of Constitutive Relation Error (CRE) to obtain a guaranteed and fully computable a posteriori error estimate in the energy norm (as well as error indicators on various error sources). Here, the CRE concept is coupled with the solution of an adjoint problem to control the error and drive an adaptive procedure with respect to a given output of interest. Furthermore, a local and non-intrusive enrichment technique is proposed to enhance the accuracy of error bounds. The overall strategy, which is fully automatic and robust, enables to reach an appropriate trade-off between certified reliability and computational cost in the MsFEM context. The performances of the proposed method are investigated on several illustrative numerical test cases. In particular, the error estimation is observed to be very accurate, yielding a very efficient adaptive procedure.

math.NA↗

arXiv2018

An embedded corrector problem for homogenization. Part II: Algorithms and discretization

This contribution is the numerically oriented companion article of the work [E. Canc\`es, V. Ehrlacher, F. Legoll, B. Stamm and S. Xiang, arxiv preprint 1807.05131]. We focus here on the numerical resolution of the embedded corrector problem introduced in [E. Canc\`es, V. Ehrlacher, F. Legoll and B. Stamm, CRAS 2015; E. Canc\`es, V. Ehrlacher, F. Legoll, B. Stamm and S. Xiang, arxiv preprint 1807.05131] in the context of homogenization of diffusion equations. Our approach consists in considering a corrector-type problem, posed on the whole space, but with a diffusion matrix which is constant outside some bounded domain. In [E. Canc\`es, V. Ehrlacher, F. Legoll, B. Stamm and S. Xiang, arxiv preprint 1807.05131], we have shown how to define three approximate homogenized diffusion coefficients on the basis of the embedded corrector problems. We have also proved that these approximations all converge to the exact homogenized coefficients when the size of the bounded domain increases. We show here that, under the assumption that the diffusion matrix is piecewise constant, the corrector problem to solve can be recast as an integral equation. In case of spherical inclusions with isotropic materials, we explain how to efficiently discretize this integral equation using spherical harmonics, and how to use the fast multipole method (FMM) to compute the resulting matrix-vector products at a cost which scales only linearly with respect to the number of inclusions. Numerical tests illustrate the performance of our approach in various settings.

math.NA↗

arXiv2018

An embedded corrector problem for homogenization. Part I: Theory

This article is the first part of a two-fold study, the objective of which is the theoretical analysis and numerical investigation of new approximate corrector problems in the context of stochastic homogenization. We present here three new alternatives for the approximation of the homogenized matrix for diffusion problems with highly-oscillatory coefficients. These different approximations all rely on the use of an embedded corrector problem (that we previously introduced in [Canc\`es, Ehrlacher, Legoll and Stamm, C. R. Acad. Sci. Paris, 2015]), where a finite-size domain made of the highly oscillatory material is embedded in a homogeneous infinite medium whose diffusion coefficients have to be appropriately determined. The motivation for considering such embedded corrector problems is made clear in the companion article [Canc\`es, Ehrlacher, Legoll, Stamm and Xiang, in preparation], where a very efficient algorithm is presented for the resolution of such problems for particular heterogeneous materials. In the present article, we prove that the three different approximations we introduce converge to the homogenized matrix of the medium when the size of the embedded domain goes to infinity.

math.NA↗

arXiv2017

Multiscale Finite Element methods for advection-dominated problems in perforated domains

We consider an advection-diffusion equation that is advection-dominated and posed on a perforated domain. On the boundary of the perforations, we set either homogeneous Dirichlet or homogeneous Neumann conditions. The purpose of this work is to investigate the behavior of several variants of Multiscale Finite Element type methods, all of them based upon local functions satisfying weak continuity conditions in the Crouzeix-Raviart sense on the boundary of mesh elements. In the spirit of our previous works [Le Bris, Legoll and Lozinski, CAM 2013 and MMS 2014] introducing such multiscale basis functions, and of [Le Bris, Legoll and Madiot, M2AN 2017] assessing their interest for advection-diffusion problems, we present, study and compare various options in terms of choice of basis elements, adjunction of bubble functions and stabilized formulations.

math.NA↗

arXiv2017

A posteriori error estimation and adaptive strategy for the control of MsFEM computations

We introduce quantitative and robust tools to control the numerical accuracy in simulations performed using the Multiscale Finite Element Method (MsFEM). First, we propose a guaranteed and fully computable a posteriori error estimate for the global error measured in the energy norm. It is based on dual analysis and the Constitutive Relation Error (CRE) concept, with recovery of equilibrated fluxes from the approximate MsFEM solution. Second, the estimate is split into several indicators, associated to the various MsFEM error sources, in order to drive an adaptive procedure. The overall strategy thus enables to automatically identify an appropriate trade-off between accuracy and computational cost in the MsFEM numerical simulations. Furthermore, the strategy is compatible with the offline/online paradigm of MsFEM. The performances of our approach are demonstrated on several numerical experiments.

math.NA↗

arXiv2016

On the best constant matrix approximating an oscillatory matrix-valued coefficient in divergence-form operators

We approximate an elliptic problem with oscillatory coefficients using a problem of the same type, but with constant coefficients. We deliberately take an engineering perspective, where the information on the oscillatory coefficients in the equation can be incomplete. A theoretical foundation of the approach in the limit of infinitely small oscillations of the coefficients is provided, using the classical theory of homogenization. We present a comprehensive study of the implementation aspects of our method, and a set of numerical tests and comparisons that show the potential practical interest of the approach. The approach detailed in this article improves on an earlier version briefly presented in [C. Le Bris, F. Legoll and K. Li, C. R. Acad. Sci. Paris 2013].

math.OC↗

arXiv2016

Stable approximation of the advection-diffusion equation using the invariant measure

We consider an advection-diffusion equation that is both non-coercive and advection-dominated. We present a possible numerical approach, to our best knowledge new, and based on the invariant measure associated to the original equation. The approach has been summarized in [C. Le Bris, F. Legoll and F. Madiot, C. R. Acad. Sci. Paris, Serie I, vol. 354, 799-803 (2016)]. We show that the approach allows for an unconditionally well-posed finite element approximation. We provide a numerical analysis and a set of comprehensive numerical tests showing that the approach can be stable, as accurate as, and more robust than a classical stabilization approach.

math.NA↗

arXiv2016

Pathwise estimates for an effective dynamics

Starting from the overdamped Langevin dynamics in $\mathbb{R}^n$, $$ dX_t = -\nabla V(X_t) dt + \sqrt{2 \beta^{-1}} dW_t, $$ we consider a scalar Markov process $\xi_t$ which approximates the dynamics of the first component $X^1_t$. In the previous work [F. Legoll, T. Lelievre, Nonlinearity 2010], the fact that $(\xi_t)_{t \ge 0}$ is a good approximation of $(X^1_t)_{t \ge 0}$ is proven in terms of time marginals, under assumptions quantifying the timescale separation between the first component and the other components of $X_t$. Here, we prove an upper bound on the trajectorial error $\mathbb{E} \left( \sup_{0 \leq t \leq T} \left| X^1_t - \xi_t \right| \right)$, for any $T > 0$, under a similar set of assumptions. We also show that the technique of proof can be used to obtain quantitative averaging results.

math.PR↗

arXiv2016

Examples of computational approaches to accommodate randomness in elliptic PDEs

We overview a series of recent works addressing numerical simulations of partial differential equations in the presence of some elements of randomness. The specific equations manipulated are linear elliptic, and arise in the context of multiscale problems, but the purpose is more general. On a set of prototypical situations, we investigate two critical issues present in many settings: variance reduction techniques to obtain sufficiently accurate results at a limited computational cost when solving PDEs with random coefficients, and finite element techniques that are sufficiently flexible to carry over to geometries with random fluctuations. Some elements of theoretical analysis and numerical analysis are briefly mentioned. Numerical experiments, although simple, provide convincing evidence of the efficiency of the approaches.

math.NA↗