On the implementation of the eXtended Finite Element Method (XFEM) for interface problems

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1. Apparenté Pas pertinent

   Thirty years of the finite volume method for solid mechanics

   Philip Cardiff, Ismet Demirdžić · 2018 · Analyse numérique · Résumé IA disponible

   Résumé scientifique

   Since early publications in the late 1980s and early 1990s, the finite volume method has been shown suitable for solid mechanics analyses. At present, there are several flavours of the method, which can be classified in a variety of ways, such as grid arrangement (cell-centred vs staggered vs vertex-centred), solution algorithm (implicit vs explicit), and stabilisation strategy (Rhie-Chow vs Jameson-Schmidt-Turkel vs Godunov upwinding). This article gives an overview, historical perspective, comparison and critical analysis of the different approaches where a close comparison with the de facto standard for computational solid mechanics, the finite element method, is given. The article finish…

2. Apparenté Pas pertinent

   Compatible finite element methods for geophysical fluid dynamics

   Colin J. Cotter · 2023 · Analyse numérique · Résumé IA disponible

   Résumé scientifique

   This article surveys research on the application of compatible finite element methods to large scale atmosphere and ocean simulation. Compatible finite element methods extend Arakawa's C-grid finite difference scheme to the finite element world. They are constructed from a discrete de Rham complex, which is a sequence of finite element spaces which are linked by the operators of differential calculus. The use of discrete de Rham complexes to solve partial differential equations is well established, but in this article we focus on the specifics of dynamical cores for simulating weather, oceans and climate. The most important consequence of the discrete de Rham complex is the Hodge-Helmholtz d…

3. Apparenté Pas pertinent

   A robust monolithic solver for phase-field fracture integrated with fracture energy based arc-length method and under-relaxation

   Ritukesh Bharali, Somdatta Goswami, Cosmin Anitescu et al. · 2021 · Analyse numérique · Résumé IA disponible

   Résumé scientifique

   The phase-field fracture free-energy functional is non-convex with respect to the displacement and the phase field. This results in a poor performance of the conventional monolithic solvers like the Newton-Raphson method. In order to circumvent this issue, researchers opt for the alternate minimization (staggered) solvers. Staggered solvers are robust for the phase-field based fracture simulations as the displacement and the phase-field sub-problems are convex in nature. Nevertheless, the staggered solver requires very large number of iterations (of the order of thousands) to converge. In this work, a robust monolithic solver is presented for the phase-field fracture problem. The solver adop…

4. Apparenté Pas pertinent

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

   Eric Cancès, Virginie Ehrlacher, Frederic Legoll et al. · 2018 · Analyse numérique · Pas encore résumé

   Résumé scientifique

   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 p…

5. Apparenté Pas pertinent

   A robust solution strategy for the Cahn-Larché equations

   Erlend Storvik, Jakub Wiktor Both, Jan Martin Nordbotten et al. · 2022 · Analyse numérique · Résumé IA disponible

   Résumé scientifique

   In this paper we propose a solution strategy for the Cahn-Larch\'e equations, which is a model for linearized elasticity in a medium with two elastic phases that evolve subject to a Ginzburg-Landau type energy functional. The system can be seen as a combination of the Cahn-Hilliard regularized interface equation and linearized elasticity, and is non-linearly coupled, has a fourth order term that comes from the Cahn-Hilliard subsystem, and is non-convex and nonlinear in both the phase-field and displacement variables. We propose a novel semi-implicit discretization in time that uses a standard convex-concave splitting method of the nonlinear double-well potential, as well as special treatment…

6. Apparenté Pas pertinent

   Modeling martensitic transformation in shape memory alloys using multi-phase-field elasticity models based on partial rank-one energy relaxation on pairwise interfaces

   Mohammad Sarhil, Oleg Shchyglo, Dominik Brands et al. · 2023 · Analyse numérique · Pas encore résumé

   Résumé scientifique

   To model the mechanically-driven phase transformations, e.g. martensitic transformation, using the phase-field theory, suitable models are needed for describing the mechanical fields of the individual non-vanishing phase-fields in the interface regions in order to obtain the mechanical driving forces of phase-field motion. Quantitative modeling requires satisfying the interfacial static equilibrium and kinematic compatibility conditions which have already been achieved in the literature for dual-phase-field materials by using the rank-one relaxation (or convexification) of the energy density. A direct generalization to the multi-phase-field case is not applicable without breaking these condi…

7. Apparenté Pas pertinent

   A Computationally Efficient Vectorized Implementation of Localizing Gradient Damage Method in MATLAB

   Subrato Sarkar · 2023 · Analyse numérique · Résumé IA disponible

   Résumé scientifique

   In this work, a recently developed fracture modeling method called localizing gradient damage method (LGDM) is implemented in MATLAB. MATLAB is well-known in the computational research community for its simple and easy-to-learn coding interface. As a result, MATLAB is generally preferred for the initial development (prototyping) of computational models by researchers. However, MATLAB-developed codes are seldom used for large-scale simulations (after initial development is complete) due to their computational inefficiency. Hence, a computationally efficient implementation of LGDM using MATLAB vectorization is presented in this work. The choice of LGDM (as the fracture modeling method) is base…

8. Apparenté Pas pertinent

   DelayDiffEq: Generating Delay Differential Equation Solvers via Recursive Embedding of Ordinary Differential Equation Solvers

   David Widmann, Chris Rackauckas · 2022 · Analyse numérique · Résumé IA disponible

   Résumé scientifique

   Traditional solvers for delay differential equations (DDEs) are designed around only a single method and do not effectively use the infrastructure of their more-developed ordinary differential equation (ODE) counterparts. In this work we present DelayDiffEq, a Julia package for numerically solving delay differential equations (DDEs) which leverages the multitude of numerical algorithms in OrdinaryDiffEq for solving both stiff and non-stiff ODEs, and manages to solve challenging stiff DDEs. We describe how compiling the ODE integrator within itself, and accounting for discontinuity propagation, leads to a design that is effective for DDEs while using all of the ODE internals. We highlight som…