Nodal Discontinuous Galerkin Methods

Hesthaven, Jan S.; Warburton, Tim

doi:10.1007/978-0-387-72067-8

Cited by 1,229 publications

(1,049 citation statements)

References 0 publications

Supporting

Mentioning

1,039

Contrasting

Unclassified

Order By: Relevance

“…The discontinuous galerkin approach relies on approximation functions that are discontinuous at cells interfaces and hence, the weak form of equation (5) is written on each element of the grid [3]:…”

Section: Weak Form Of Equation Of Motionmentioning

confidence: 99%

“…Furthermore, the MPM space discretization leads to a decrease of the Courant number that prevents the capture of discontinuities. The discontinuous galerkin finite element method (DGFEM) has been developed for the neutron transport equation [2] and is now used in fluid and solid mechanics [3]. This extension of classical finite element method is based on the discontinuous Galerkin (DG) approximation in which shape functions are discontinuous across elements boundaries.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A Discontinuous Galerkin Material Point Method (Dgmpm) for the Simulation of Impact Problems in Solid Mechanics

Renaud¹,

Heuzé²

2017

Proceedings of the 6th International Conference on Computational Methods in Structural Dynamics and Earthquake Engineering (COM

View full text Add to dashboard Cite

Abstract. The material point method is extended in this work to the Discontinuous Galerkin approximation framework for the simulation of impacts on elastic and hyperelastic solids. The formulation is based on the weak form of conservation laws on each cell of an eulerian grid in which volume integrals are discretized on a set of material points lying in that cell, and on the computation of Godunov fluxes at cells faces. The resulting method is first derived within the small strains framework and illustrated on a one-dimensional and a two-dimensional problem of impact on an elastic media. Then a one-dimensional hyperelastic problem of a solid undergoing large strains is presented, and a comparison is performed with an analytical solution.

show abstract

Section: Weak Form Of Equation Of Motionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A Discontinuous Galerkin Material Point Method (Dgmpm) for the Simulation of Impact Problems in Solid Mechanics

Renaud¹,

Heuzé²

2017

Proceedings of the 6th International Conference on Computational Methods in Structural Dynamics and Earthquake Engineering (COM

View full text Add to dashboard Cite

show abstract

“…For the Euler equations, a system of three equations, U i (x, t) is a vector of three polynomials U i,j (x, t), representing the conserved variables ρ, ρu and E. At each cell interface the solution is not unique but defined twice for each variable, because the cells are closed intervals. A detailed derivation and analysis of the Discontinuous Galerkin method can be found in [8].…”

Section: Discontinuous Galerkin Methodsmentioning

confidence: 99%

“…If M is chosen too small, the local dissipation and order reduction are increased. If, on the other hand, M is chosen too large, oscillations occur [8]. The first step in the limiting procedure is to look for cells in which oscillations might occur and mark them as so-called troubled cells.…”

Section: Limitermentioning

confidence: 99%

See 1 more Smart Citation

A Monolithic Fluid-Structure Interaction Method, Application to a Piston Problem

Ischinger¹,

Anthonissen²,

Koren³

2016

Proceedings of the VII European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS Congress 2016)

View full text Add to dashboard Cite

Abstract. A monolithic FSI method is presented. A standard piston problem is considered as test case. The piston problem's fluid domain is represented by a closed tube filled with air. One end of the fluid tube is formed by a piston connected to a spring. We use the Euler equations of gas dynamics as well as a linear simplification of these, the acoustic equations, to model the gas dynamics in the tube. A Discontinuous Galerkin method is applied to discretize the fluidflow equations, together with an immersed-boundary method to account for the moving piston. A monolithic formulation of the coupled system is derived and analyzed. It is proven that the semidiscrete formulation is stable, if two correction terms are used at the coupling interface. We use Lyapunov functions to prove stability of the semi-discrete monolithic formulation. Further, different time-integration methods are considered, analyzed and tested. The numerical results are very accurate; they correspond very well to analytical approximations. The theoretical prediction of the eigenfrequency can be reproduced very accurately. Moreover, the amplitude of the spring oscillation is conserved very well.

show abstract

Stereographic projection for three‐dimensional global discontinuous Galerkin atmospheric modeling

Blaise

Lambrechts

Deleersnijder

2015

J Adv Model Earth Syst

View full text Add to dashboard Cite

A method to solve the three-dimensional compressible Navier-Stokes equations on the sphere is suggested, based on a stereographic projection with a high-order mapping of the elements from the stereographic space to the sphere. The projection is slightly modified, in order to take into account the domain thickness without introducing any approximation about the aspect ratio (deep-atmosphere). In a discontinuous Galerkin framework, the elements alongside the equator are exactly represented using a nonpolynomial geometry, in order to avoid the numerical issues associated with the seam connecting the two hemispheres. This is an crucial point, necessary to avoid mass loss and spurious deviations of the velocity. The resulting model is validated on idealized three-dimensional atmospheric test cases on the sphere, demonstrating the good convergence properties of the scheme, its mass conservation, and its satisfactory behavior in terms of accuracy and low numerical dissipation. A simulation is performed on a variable resolution unstructured grid, producing accurate results despite a substantial reduction of the number of elements.

show abstract

Nodal Discontinuous Galerkin Methods

Cited by 1,229 publications

References 0 publications

A Discontinuous Galerkin Material Point Method (Dgmpm) for the Simulation of Impact Problems in Solid Mechanics

A Discontinuous Galerkin Material Point Method (Dgmpm) for the Simulation of Impact Problems in Solid Mechanics

A Monolithic Fluid-Structure Interaction Method, Application to a Piston Problem

Stereographic projection for three‐dimensional global discontinuous Galerkin atmospheric modeling

Contact Info

Product

Resources

About