A Discontinuous Galerkin Material Point Method for the solution of impact problems in solid dynamics

An extension of the Material Point Method [1] based on the Discontinuous Galerkin approximation (DG) [2] is presented here. A solid domain is represented by a collection of particles that can move and carry the fields of the problem inside an arbitrary computational grid in order to provide a Lagran...

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Vydané v:	Journal of computational physics Ročník 369; s. 80 - 102
Hlavní autori:	Renaud, Adrien, Heuzé, Thomas, Stainier, Laurent
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	Cambridge Elsevier Inc 15.09.2018 Elsevier Science Ltd Elsevier
Predmet:	Approximation Computational grids Computational physics Computer simulation Conservation laws Deformation Discontinuous Galerkin approximation Engineering Sciences Finite deformation Finite element analysis Finite element method Fluxes Galerkin method Hyperelasticity Impact Inverse problems Material Point Method Mechanics Meshless methods Multidimensional methods Numerical analysis Numerical methods Riemann solver Solid mechanics Structural mechanics Tangling Impact Solid mechanics Material Point Method Finite deformation Discontinuous Galerkin approximation Hyperelasticity finite deformation hyperelasticity solid mechanics impact
ISSN:	0021-9991, 1090-2716
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Abstract	An extension of the Material Point Method [1] based on the Discontinuous Galerkin approximation (DG) [2] is presented here. A solid domain is represented by a collection of particles that can move and carry the fields of the problem inside an arbitrary computational grid in order to provide a Lagrangian description of the deformation without mesh tangling issues. The background mesh is then used as a support for the Discontinuous Galerkin approximation that leads to a weak form of conservation laws involving numerical fluxes defined at element faces. Those terms allow the introduction of the characteristic structure of hyperbolic problems within the numerical method by using an approximate Riemann solver [3]. The Discontinuous Galerkin Material Point Method, which can be viewed as a Discontinuous Galerkin Finite Element Method (DGFEM) with modified quadrature rule, aims at meeting advantages of both mesh-free and DG methods. The method is derived within the finite deformation framework for multidimensional problems by using a total Lagrangian formulation. A particular attention is paid to one specific discretization leading to a stability condition that allows to set the CFL number at one. The approach is illustrated and compared to existing or developed analytical solutions on one-dimensional problems and compared to the finite element method on two-dimensional simulations. •Development of the Discontinuous Galerkin Material Point method.•von Neumann stability analysis of the DGMPM.•Analytical solution of a one-dimensional hyperbolic problem for a hyperelastic Saint–Venant–Kirchhoff material.
AbstractList	An extension of the Material Point Method [1] based on the Discontinuous Galerkin approximation (DG) [2] is presented here. A solid domain is represented by a collection of particles that can move and carry the fields of the problem inside an arbitrary computational grid in order to provide a Lagrangian description of the deformation without mesh tangling issues. The background mesh is then used as a support for the Discontinuous Galerkin approximation that leads to a weak form of conservation laws involving numerical fluxes defined at element faces. Those terms allow the introduction of the characteristic structure of hyperbolic problems within the numerical method by using an approximate Riemann solver [3]. The Discontinuous Galerkin Material Point Method, which can be viewed as a Discontinuous Galerkin Finite Element Method (DGFEM) with modified quadrature rule, aims at meeting advantages of both mesh-free and DG methods. The method is derived within the finite deformation framework for multidimensional problems by using a total Lagrangian formulation. A particular attention is paid to one specific discretization leading to a stability condition that allows to set the CFL number at one. The approach is illustrated and compared to existing or developed analytical solutions on one-dimensional problems and compared to the finite element method on two-dimensional simulations. •Development of the Discontinuous Galerkin Material Point method.•von Neumann stability analysis of the DGMPM.•Analytical solution of a one-dimensional hyperbolic problem for a hyperelastic Saint–Venant–Kirchhoff material. An extension of the Material Point Method based on the Discontinuous Galerkin approximation (DG) is presented here. A solid domain is represented by a collection of particles that can move and carry the fields of the problem inside an arbitrary computational grid in order to provide a Lagrangian description of the deformation without mesh tangling issues. The background mesh is then used as a support for the Discontinuous Galerkin approximation that leads to a weak form of conservation laws involving numerical fluxes defined at element faces. Those terms allow the introduction of the characteristic structure of hyperbolic problems within the numerical method by using an approximate Riemann solver. The Discontinuous Galerkin Material Point Method, which can be viewed as a Discontinuous Galerkin Finite Element Method with modified quadrature rule, aims at meeting advantages of both mesh-free and DG methods. The method is derived within the finite deformation framework for multidimensional problems by using a total Lagrangian formulation. A particular attention is paid to one specific discretization leading to a stability condition that allows to set the CFL number at one. The approach is illustrated and compared to existing or developed analytical solutions on one-dimensional problems and compared to the finite element method on two-dimensional simulations. An extension of the Material Point Method [1] based on the Discontinuous Galerkin approximation (DG) [2] is presented here. A solid domain is represented by a collection of particles that can move and carry the fields of the problem inside an arbitrary computational grid in order to provide a Lagrangian description of the deformation without mesh tangling issues. The background mesh is then used as a support for the Discontinuous Galerkin approximation that leads to a weak form of conservation laws involving numerical fluxes defined at element faces. Those terms allow the introduction of the characteristic structure of hyperbolic problems within the numerical method by using an approximate Riemann solver [3]. The Discontinuous Galerkin Material Point Method, which can be viewed as a Discontinuous Galerkin Finite Element Method (DGFEM) with modified quadrature rule, aims at meeting advantages of both mesh-free and DG methods. The method is derived within the finite deformation framework for multidimensional problems by using a total Lagrangian formulation. A particular attention is paid to one specific discretization leading to a stability condition that allows to set the CFL number at one. The approach is illustrated and compared to existing or developed analytical solutions on one-dimensional problems and compared to the finite element method on two-dimensional simulations.
Author	Heuzé, Thomas Stainier, Laurent Renaud, Adrien
Author_xml	– sequence: 1 givenname: Adrien surname: Renaud fullname: Renaud, Adrien email: adrien.renaud@ec-nantes.fr – sequence: 2 givenname: Thomas surname: Heuzé fullname: Heuzé, Thomas email: thomas.heuze@ec-nantes.fr – sequence: 3 givenname: Laurent surname: Stainier fullname: Stainier, Laurent email: laurent.stainier@ec-nantes.fr
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Keywords	Impact Solid mechanics Material Point Method Finite deformation Discontinuous Galerkin approximation Hyperelasticity finite deformation hyperelasticity solid mechanics impact
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Snippet	An extension of the Material Point Method [1] based on the Discontinuous Galerkin approximation (DG) [2] is presented here. A solid domain is represented by a... An extension of the Material Point Method based on the Discontinuous Galerkin approximation (DG) is presented here. A solid domain is represented by a...
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SubjectTerms	Approximation Computational grids Computational physics Computer simulation Conservation laws Deformation Discontinuous Galerkin approximation Engineering Sciences Finite deformation Finite element analysis Finite element method Fluxes Galerkin method Hyperelasticity Impact Inverse problems Material Point Method Mechanics Meshless methods Multidimensional methods Numerical analysis Numerical methods Riemann solver Solid mechanics Structural mechanics Tangling
Title	A Discontinuous Galerkin Material Point Method for the solution of impact problems in solid dynamics
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