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

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

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   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…

2. 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…

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   Provably stable numerical method for the anisotropic diffusion equation in toroidally confined magnetic fields

   Dean Muir, Kenneth Duru, Matthew Hole et al. · 2023 · Analyse numérique · Résumé IA disponible

   Résumé scientifique

   We present a novel numerical method for solving the anisotropic diffusion equation in toroidally confined magnetic fields which is efficient, accurate and provably stable. The continuous problem is written in terms of a derivative operator for the perpendicular transport and a linear operator, obtained through field line tracing, for the parallel transport. We derive energy estimates of the solution of the continuous initial boundary value problem. A discrete formulation is presented using operator splitting in time with the summation by parts finite difference approximation of spatial derivatives for the perpendicular diffusion operator. Weak penalty procedures are derived for implementing …

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   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…

5. Apparenté Pas pertinent

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   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…

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   Convergence Analysis for Nonlinear GMRES

   Yunhui He · 2025 · Analyse numérique · Résumé IA disponible

   Résumé scientifique

   In this work, we revisit nonlinear generalized minimal residual method (NGMRES) applied to nonlinear problems. NGMRES is used to accelerate the convergence of fixed-point iterations, which can substantially improve the performance of the underlying fixed-point iterations. We consider NGMRES with a finite window size $m$, denoted as NGMRES($m$). However, there is no convergence analysis for NGMRES($m$) applied to nonlinear systems. We prove that for general $m>0$, the residuals of NGMRES($m$) converge r-linearly under some conditions. For $m=0$, we prove that the residuals of NGMRES(0) converge q-linearly.

7. Apparenté Pas pertinent

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

   Thomas Carraro, Sven Wetterauer · 2015 · Analyse numérique · Résumé IA disponible

   Résumé scientifique

   The eXtended Finite Element Method (XFEM) is used to solve interface problems with an unfitted mesh. We present an implementation of the XFEM in the FEM-library deal.II. The main parts of the implementation are (i) the appropriate quadrature rule; (ii) the shape functions for the extended part of the finite element formulation; (iii) the boundary and interface conditions. We show how to handle the XFEM formulation providing a code that demonstrates the solution of two exemplary interface problems for a strong and a weak discontinuity respectively. In the weak discontinuity case, the loss of conformity due to the blending effect and its remedy are discussed. Furthermore, the optimal convergen…

8. Apparenté Pas pertinent

   Gradual Variation Analysis for Groundwater Flow

   Li Chen · 2010 · Analyse numérique · Pas encore résumé

   Résumé scientifique

   Groundwater flow in Washington DC greatly influences the surface water quality in urban areas. The current methods of flow estimation, based on Darcy's Law and the groundwater flow equation, can be described by the diffusion equation (the transient flow) and the Laplace equation (the steady-state flow). The Laplace equation is a simplification of the diffusion equation under the condition that the aquifer has a recharging boundary. The practical way of calculation is to use numerical methods to solve these equations. The most popular system is called MODFLOW, which was developed by USGS. MODFLOW is based on the finite-difference method in rectangular Cartesian coordinates. MODFLOW can be vie…