Article
Mathematics, Applied
Khemraj Shukla, Jesse Chan, Maarten de Hoop
Summary: A new symmetric treatment of anisotropic viscous terms in the viscoelastic wave equation is introduced, resulting in a symmetric system of first-order linear hyperbolic partial differential equations. The discretization is done using a high-order discontinuous Galerkin finite element method, and the accuracy of the numerical scheme is verified through convergence studies and computational experiments in two and three dimensions with various combinations of homogeneous and heterogeneous viscoelastic media.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2021)
Article
Mathematics, Applied
Nicolas Chauchat, Roland Becker, Eric Schall
Summary: The paper discusses the advantages of using entropy variables in computing compressible flows through the discontinuous Galerkin method and generalizing the use of numerical flux. It compares the performance of DG0 discretization based on entropy variables with other methods, and explores the performance of DG1 discretization with different sets of variables.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2021)
Article
Computer Science, Interdisciplinary Applications
Axel Modave, Theophile Chaumont-Frelet
Summary: In this paper, a new hybridizable discontinuous Galerkin (HDG) method called the CHDG method is proposed for solving time-harmonic scalar wave propagation problems. The method utilizes a standard discontinuous Galerkin scheme with upwind numerical fluxes and high-order polynomial bases. Auxiliary unknowns are introduced at the element interfaces to simplify the system, which can be solved using stationary iterative schemes. Numerical results show that the CHDG method improves the properties of the reduced system compared to the standard HDG method and is more suitable for iterative solution procedures.
JOURNAL OF COMPUTATIONAL PHYSICS
(2023)
Article
Mathematics, Applied
Andreas Rupp, Moritz Hauck, Vadym Aizinger
Summary: The method introduced in this work generalizes the enriched Galerkin method with an adaptive two-mesh approach, proving stability and error estimates for a linear advection equation. The analysis technique allows for arbitrary degrees of enrichment on both coarse and fine meshes, covering a wide range of methods from continuous finite element to discontinuous Galerkin with local subcell enrichment. Numerical experiments confirm the analytical results and show good robustness of the proposed method.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2021)
Article
Computer Science, Interdisciplinary Applications
William Doherty, Timothy N. Phillips, Zhihua Xie
Summary: The implementation of a conservative level-set method in the mathematical framework of viscoelastic flow is the main original contribution of this paper. The finite element method is used to discretize the governing equations, and stabilisation of the constitutive equation is achieved using either the discontinuous Galerkin or streamline upwinding method. The discrete elastic viscous stress splitting gradient formulation is also employed in the Navier-Stokes equations. The numerical scheme is validated and shows excellent agreement with published data for both Newtonian and viscoelastic fluids in single and multiphase flows. The behavior of a gas bubble rising in a viscoelastic fluid is studied, considering the influence of polymer concentration, surface tension, fluid elasticity, and shear-thinning behavior on flow features.
JOURNAL OF COMPUTATIONAL PHYSICS
(2023)
Article
Computer Science, Interdisciplinary Applications
Andres M. Rueda-Ramirez, Sebastian Hennemann, Florian J. Hindenlang, Andrew R. Winters, Gregor J. Gassner
Summary: This paper presents two robust entropy stable shock-capturing methods for discontinuous Galerkin spectral element discretizations of compressible magneto-hydrodynamics (MHD) equations. The methods are extended to systems with non-conservative terms, using a hybrid FV/DGSEM scheme for discretization and proving semi-discrete entropy stability on three-dimensional unstructured curvilinear meshes. The second method involves a subcell reconstruction procedure that enhances resolution and ensures entropy stability, with numerical verification on curvilinear meshes and benchmark cases showing robustness and accuracy.
JOURNAL OF COMPUTATIONAL PHYSICS
(2021)
Article
Engineering, Multidisciplinary
Slimane Adjerid, Ivo Babuska, Ruchi Guo, Tao Lin
Summary: This article introduces the first higher degree immersed finite element method for elliptic interface problems with nonhomogeneous jump conditions, demonstrating optimal convergence. It also provides an analysis of the condition numbers of the resulting systems, including optimal upper bounds with respect to mesh size and robustness with respect to small-cut interface elements. The method involves the approximation of jump conditions using basic and enrichment immersed finite elements, which are constructed by solving local Cauchy problems on interface elements.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2023)
Article
Mathematics, Applied
Xinhui Wu, Jesse Chan
Summary: A high-order entropy stable discontinuous Galerkin method has been proposed for addressing nonlinear conservation laws on multi-dimensional domains and networks, using treatments of multi-dimensional interfaces and network junctions to maintain entropy stability when coupling entropy stable discretizations. Numerical experiments confirm the stability of the schemes and show the accuracy of junction treatments in comparisons with fully 2D implementations.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Engineering, Multidisciplinary
Kerstin Weinberg, Christian Wieners
Summary: We propose a new numerical approach for wave induced dynamic fracture. The method combines a discontinuous Galerkin approximation of elastic waves and a phase-field approximation of brittle fracture. The algorithm is staggered in time and uses an implicit midpoint rule for wave propagation and an implicit Euler step for phase-field evolution. Examples in two and three dimensions demonstrate the advantages of this approach in computing crack growth and spalling initiated by reflected and superposed waves.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2022)
Review
Mechanics
M. A. Alves, P. J. Oliveira, F. T. Pinho
Summary: Complex fluids are engineered for specific applications by adding macromolecules to a solvent, imparting viscoelasticity which affects flow instabilities and fluid dynamics. Recent research focuses on numerical methods for simulating viscoelastic fluid flows, particularly the finite-volume method used to assess performance with benchmark flows. Issues in numerical methods and novel applications of viscoelastic fluids requiring further development are also discussed.
ANNUAL REVIEW OF FLUID MECHANICS, VOL 53
(2021)
Article
Computer Science, Interdisciplinary Applications
Mengxia Ma, Jie Ouyang, Xiaodong Wang
Summary: This paper presents a new stabilized high-order discontinuous Galerkin method for simulating highly elastic fluid flows at high Weissenberg numbers. The method is able to accurately and stably simulate different viscoelastic flow problems, and it has high-order accuracy and robustness. Compared with existing schemes, this method is more flexible and easier to implement.
ENGINEERING WITH COMPUTERS
(2023)
Article
Computer Science, Interdisciplinary Applications
Maciej Waruszewski, Jeremy E. Kozdon, Lucas C. Wilcox, Thomas H. Gibson, Francis X. Giraldo
Summary: This work examines a non-conservative balance law formulation that incorporates the rotating, compressible Euler equations for dry atmospheric flows. A semi-discretely entropy stable discontinuous Galerkin method is developed on curvilinear meshes using a generalization of flux differencing for numerical fluxes in fluctuation form. The method utilizes the skew-hybridized formulation of the element operators to ensure entropy stability, even on curvilinear meshes with under-integration. Various atmospheric flow test cases in different dimensions confirm the theoretical entropy stability results and demonstrate the high-order accuracy and robustness of the method.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Physics, Multidisciplinary
Atul Kumar Shukla, Mukesh Kumar Awasthi, Shivam Agarwal
Summary: The present study investigated the stability of a spherical interface formed by a combination of a viscous fluid and an Oldroyd B viscoelastic fluid using linear stability analysis. The study found that the stability of the interface increases with an increase in the viscoelasticity of the fluid.
CHINESE JOURNAL OF PHYSICS
(2023)
Article
Mathematics, Applied
Ramon Codina, Laura Moreno
Summary: This paper presents a numerical analysis of a finite element method for a linearized viscoelastic flow problem, utilizing a linearization of the logarithmic reformulation and a stabilized finite element formulation based on the Variational Multi-Scale concept. The study demonstrates why the logarithmic reformulation outperforms the standard method for high Weissenberg numbers, as reflected in the stability and error estimates provided in the paper.
ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS-MODELISATION MATHEMATIQUE ET ANALYSE NUMERIQUE
(2021)
Article
Computer Science, Interdisciplinary Applications
Anne Kikker, Florian Kummer, Martin Oberlack
Summary: A fully coupled high-order discontinuous Galerkin solver is presented for viscoelastic Oldroyd B fluid flow problems, using LDG formulation and incremental increase of the Weissenberg number. The solver's suitability is demonstrated for a two-dimensional confined cylinder benchmark problem.
INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS
(2021)
Article
Mathematics, Applied
Thomas Hudson, Frederic Legoll, Tony Lelievre
ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS-MODELISATION MATHEMATIQUE ET ANALYSE NUMERIQUE
(2020)
Correction
Mathematics, Applied
Tony Lelievre, Mathias Rousset, Gabriel Stoltz
NUMERISCHE MATHEMATIK
(2020)
Article
Statistics & Probability
Tony Lelievre, Giovanni Samaey, Przemyslaw Zielinski
STOCHASTIC PROCESSES AND THEIR APPLICATIONS
(2020)
Article
Multidisciplinary Sciences
Z. Trstanova, B. Leimkuhler, T. Lelievre
PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES
(2020)
Article
Mathematics, Applied
Sebastien Boyaval
Summary: In this study, multi-dimensional extensions of Maxwell's one-dimensional viscoelastic flow rheological equation were considered. A symmetric hyperbolic system of conservation laws was proposed for compressible flows, with the Upper-Convected Maxwell equation as the causal model. This system includes an additional material metric variable to model viscous effects and can cover various rheological equations depending on the relaxation limit chosen for the material metric variable.
ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS-MODELISATION MATHEMATIQUE ET ANALYSE NUMERIQUE
(2021)
Article
Mathematics, Applied
Tony Lelievre, Dorian Le Peutrec, Boris Nectoux
Summary: This work examines the exit point distribution from a bounded domain of a stochastic process, taking into account the influence of initial conditions on the distribution. The proofs rely on analytical results on the dependency of the exit point distribution on the initial condition, as well as large deviation techniques and results on the genericity of Morse functions.
STOCHASTICS AND PARTIAL DIFFERENTIAL EQUATIONS-ANALYSIS AND COMPUTATIONS
(2022)
Article
Statistics & Probability
Benjamin Jourdain, Tony Lelievre, Pierre-Andre Zitt
Summary: By drawing a parallel between metadynamics and self interacting models for polymers, the study focuses on the longtime convergence of the original metadynamics algorithm in the adiabatic setting, and discusses the bias introduced when the adiabatic assumption does not hold.
ANNALS OF APPLIED PROBABILITY
(2021)
Article
Mathematics, Applied
Tony Lelievre, Lise Maurin, Pierre Monmarche
Summary: The study proposes an investigation into the robustness of the Adaptive Biasing Force method under generic (possibly non-conservative) forces. The researchers ensure the satisfaction of the flat histogram property and establish the existence of a stationary state for both the Adaptive Biasing Force and Projected Adapted Biasing Force algorithms. Using classical entropy techniques, the study proves the exponential convergence of the biasing force and law over time for both methods.
ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS
(2022)
Article
Chemistry, Physical
Zineb Belkacemi, Paraskevi Gkeka, Tony Lelievre, Gabriel Stoltz
Summary: Free energy biasing methods are powerful in accelerating molecular conformational changes simulation, but they usually require prior knowledge of collective variables. Machine learning and dimensionality reduction algorithms can be used to identify these collective variables. A new iterative method involving autoencoders, FEBILAE, is introduced in this paper to ensure optimization of the same loss at each iteration and achieve collective variable convergence.
JOURNAL OF CHEMICAL THEORY AND COMPUTATION
(2022)
Article
Mathematics, Applied
Frederic Legoll, Tony Lelievre, Upanshu Sharma
Summary: The aim of this article is to design parareal algorithms for thermostated molecular dynamics simulations. The traditional parareal algorithm is not suitable for molecular dynamics due to its limitations. This article proposes a modified version of the parareal algorithm that is better suited for molecular dynamics simulations. However, the modified algorithm still has some limitations, including intermediate trajectory blow-up, encounters with undefined values, and no computational advantage in long time horizons. Through numerical experiments, this article demonstrates that the adaptive algorithm overcomes the limitations of the standard algorithm and achieves significant improvements.
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2022)
Article
Statistics & Probability
Tony Lelievre, Mouad Ramil, Julien Reygner
Summary: This study investigates the properties of the Langevin process on a bounded-in-position domain, proving compactness of its semigroup and the existence of a unique quasi-stationary distribution. A spectral interpretation of the QSD is provided, along with exponential convergence of the process towards the QSD under non-absorption conditions. An explicit formula for the first exit point distribution from the domain, starting from the QSD, is given.
STOCHASTIC PROCESSES AND THEIR APPLICATIONS
(2022)
Article
Mathematics, Applied
Tony Lelievre, Mouad Ramil, Julien Reygner
Summary: This article focuses on classical solutions to the kinetic Fokker-Planck equation within a bounded domain O in position, utilizing the Langevin diffusion process with absorbing boundary conditions to obtain probabilistic representations of the solutions. Important results such as the Harnack inequality, maximum principle, and the smooth transition density for the absorbed Langevin process are provided on the phase-space cylindrical domain D = O x R-d. The study also examines the continuity and positivity of the transition density at the boundary of D.
JOURNAL OF EVOLUTION EQUATIONS
(2022)
Article
Statistics & Probability
Manon Baudel, Arnaud Guyader, Tony Lelievre
Summary: We show how the Hill relation and the concept of quasi-stationary distribution can be applied to analyze biasing errors in various numerical procedures used in molecular dynamics to compute mean reaction times between metastable states for Markov processes. Theoretical findings are demonstrated on different examples, highlighting the precision of biasing error analysis and the applicability of our study to elliptic diffusions.
STOCHASTIC PROCESSES AND THEIR APPLICATIONS
(2023)
Article
Materials Science, Multidisciplinary
Mouad Ramil, Tony Lelievre, Julien Reygner
Summary: Molecular dynamics methods are used to study the time evolution of complex molecular systems and their transitions between stable states. The Parallel Replica algorithm is a powerful tool to efficiently sample rare events in these systems. This research letter establishes the existence of a quasi-stationary distribution for the Langevin dynamics involved in the Parallel Replica algorithm and provides insight into the overdamped limit behavior of the dynamics.
MRS COMMUNICATIONS
(2022)
Article
Mathematics, Interdisciplinary Applications
Frederic Legoll, Tony Lelievre, Keith Myerscough, Giovanni Samaey
COMPUTING AND VISUALIZATION IN SCIENCE
(2020)
