Implicit mesh discontinuous Galerkin methods and interfacial gauge methods for high-order accurate interface dynamics, with applications to surface tension dynamics, rigid body fluid–structure interaction, and free surface flow: Part I

Saye, Robert I.

doi:10.1016/j.jcp.2017.04.076

Cited by 69 publications

(99 citation statements)

References 58 publications

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“…Numerous numerical approaches also exists, for example using Lagrange multipliers [12], finite cell methods [13,14,15], immersed interface methods [16], the s-version of the finite element method [17,18] and XFEM [19,20,21] to name a few. Discontinuous Galerkin methods have also successfully been used [22,23,24,25], as well as variants using local enrichments [26,27,28]. There are also substantial and important works in domain decomposition methods see for example [29,30] or [31,32] and the references therein.…”

Section: Related Workmentioning

confidence: 99%

Multimesh finite element methods: Solving PDEs on multiple intersecting meshes

Johansson

Kehlet

Larson

et al. 2019

Computer Methods in Applied Mechanics and Engineering

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We present a new framework for expressing finite element methods on multiple intersecting meshes: multimesh finite element methods. The framework enables the use of separate meshes to discretize parts of a computational domain that are naturally separate; such as the components of an engine, the domains of a multiphysics problem, or solid bodies interacting under the influence of forces from surrounding fluids or other physical fields. Such multimesh finite element methods are particularly well suited to problems in which the computational domain undergoes large deformations as a result of the relative motion of the separate components of a multi-body system. In the present paper, we formulate the multimesh finite element method for the Poisson equation. Numerical examples demonstrate the optimal order convergence, the numerical robustness of the formulation and implementation in the face of thin intersections and rounding errors, as well as the applicability of the methodology.In the accompanying paper [1], we analyze the proposed method and prove optimal order convergence and stability.

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Section: Related Workmentioning

confidence: 99%

Multimesh finite element methods: Solving PDEs on multiple intersecting meshes

Johansson

Kehlet

Larson

et al. 2019

Computer Methods in Applied Mechanics and Engineering

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“…For the numerical tests in this work we have used a high-order accurate discontinuous Galerkin (DG) spatial discretization. The methodology and algorithms employed are essentially identical to those detailed in [60,61], and as such, only a brief description is provided here.…”

Section: Spatial Discretizationmentioning

confidence: 99%

“…Briefly, the scheme can be stated as formulating a discrete gradient operator G (using a numerical flux which upwinds from the "left" on internal mesh faces) such that the accompanying discrete divergence operator (using numerical fluxes which upwind from the "right") is the adjoint of G. The primary component of the resulting discrete Laplacian operator then has the form of G G where G is the adjoint of G; boundary data enters in the form of source terms, derived from modifications to the numerical fluxes on boundary mesh faces taking into account Dirichlet and Neumann boundary conditions. In addition, following the discussion in [60], penalty stabilization parameters are included to ensure well-posedness of the discrete system. Ultimately, one obtains a symmetric positive definite linear system for the viscous solve, and a symmetric positive semidefinite linear system for the projection operator.…”

Section: Spatial Discretizationmentioning

confidence: 99%

“…The motivation of this study is to develop higher-order accurate temporal integration methods for the incompressible Navier-Stokes equations in bounded domains and with possibly time-dependent boundary conditions. In [60,61], the second author presents a framework for applying discontinuous Galerkin (DG) methods to problems of interfacial dynamics in incompressible flows. Although the spatial discretization developed in these papers is high-order accurate, only second-order predictor corrector time stepping schemes are considered therein.…”

Section: Introductionmentioning

confidence: 99%

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Higher-order temporal integration for the incompressible Navier–Stokes equations in bounded domains

Minion

Saye

2018

Journal of Computational Physics

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This paper compares and contrasts higher-order, semi-implicit temporal integration strategies for the incompressible Navier-Stokes methods based on spectral deferred corrections applied to certain gauge or auxiliary variable formulations of the equations. Particular focus is placed on the imposition of boundary conditions in the semi-implicit formulation, the accurate treatment of the pressure term, and the smoothness of the numerical solution for different formulations. The main result presented here is the formulation and numerical validation of a single-step, semi-implicit method called spectral deferred pressure corrections, which can in theory obtain arbitrary formal order of accuracy in time. Numerical results demonstrate up to eighth-order accuracy, scenarios where optimal temporal accuracy is attained, and scenarios where order reduction is observed due to time-dependent boundary conditions.

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“…These could include free-volume based models of BMGs [46], plasticity models based on the random first-order transition theory of the glass transition [47], hypo-elastic materials [48,49,50], geophysical models [51,52], and rate-independent plasticity models [53,54,55,56]. We also emphasize that Chorin's projection method represents a first step towards more complex projection-based algorithms such as gauge methods [57,58,59] and pressure-Poisson methods [60,61], and that we have laid the groundwork here to generalize these algorithms to the case of hypo-elastoplasticity.…”

Section: Introductionmentioning

confidence: 99%

Parallel three-dimensional simulations of quasi-static elastoplastic solids

Boffi

Rycroft

2020

Computer Physics Communications

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We extend to three dimensions a new projection method for simulating hypo-elastoplastic solids in the quasi-static limit. The method is based on a surprising mathematical correspondence to the incompressible Navier-Stokes equations, where the projection method of Chorin (1968) is an established numerical technique. We first review the method and present its extension to three dimensions. We discuss the development of a three-dimensional parallel geometric multigrid solver employed to solve a linear system for the quasi-static projection. We test the method by simulating three-dimensional shear band nucleation and growth, a precursor to failure in many materials. Although the method can be applied to any elastoplasticity model, we employ a physical model of a bulk metallic glass based on the shear transformation zone theory. We consider several examples of three-dimensional shear banding, and examine shear band formation in physically realistic materials with heterogeneous initial conditions.

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Implicit mesh discontinuous Galerkin methods and interfacial gauge methods for high-order accurate interface dynamics, with applications to surface tension dynamics, rigid body fluid–structure interaction, and free surface flow: Part I

Cited by 69 publications

References 58 publications

Multimesh finite element methods: Solving PDEs on multiple intersecting meshes

Multimesh finite element methods: Solving PDEs on multiple intersecting meshes

Higher-order temporal integration for the incompressible Navier–Stokes equations in bounded domains

Parallel three-dimensional simulations of quasi-static elastoplastic solids

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