Solving a magnetic diffusion equation by the Mixed Hybrid Finite Element method.

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Solving a magnetic diffusion equation by the Mixed Hybrid Finite Element method. Corinne Aymard, Christophe Fochesato CEA, DAM, DIF, F-91297 Arpajon CEA/DAM/DIF 6 juillet 2010 1

Outline Objectives Model Discretization Results Conclusion CEA/DAM/DIF 6 juillet 2010 2

Outline Objectives Model Discretization Results, soon Conclusion CEA/DAM/DIF 6 juillet 2010 3

Objectives to add a MHD model in an existing multimaterial lagrangian hydrodynamical code to study the behaviour of materials under electromagnetic conditions magnetoforming zpinch type applications CEA/DAM/DIF 6 juillet 2010 4

Model CEA/DAM/DIF 6 juillet 2010 5

Model Hydrodynamics Euler s equations Electromagnetics Maxwell s equations CEA/DAM/DIF 6 juillet 2010 6

Model classical hypotheses for MHD => convective/magnetic diffusion equation modelling choice: 2D configurations axisymmetrical flow with azimutal magnetic field B θ transverse plane flow with axial current B z transverse plane flow with axial magnetic field j z splitting in time: hydro / convective evolution of B / diffusive evolution of B CEA/DAM/DIF 6 juillet 2010 7

Model: diffusive part axisymmetrical flow with azimutal magnetic field transverse plane flow with axial current transverse plane flow with axial magnetic field + global constraint by material for fixing temporal function Ec(t) CEA/DAM/DIF 6 juillet 2010 8

Model: generic diffusion equation Diffusive system for the different configurations + initial conditions + Dirichlet (imposed q) and Neumann boundary conditions (imposed X.n) + eventually a global constraint on q properties of the diffusion equation discontinuous coefficients large ratios between different materials coefficients distorted mesh by hydrodynamical flow CEA/DAM/DIF 6 juillet 2010 9

Discretization CEA/DAM/DIF 6 juillet 2010 10

Discretization: Finite Element Method Weak form of the diffusive system: looking for and on with on Approximation by Mixed Hybrid Finite Element in in Ω m Г a CEA/DAM/DIF 6 juillet 2010 11

Discretization: MHFE Discrete weak form of the diffusive system with local boundaries conditions if internal edge if Neumann boundary for any if Dirichlet boundary Degrees of freedom X.n q λ CEA/DAM/DIF 6 juillet 2010 12

Discretization: basis functions for RT0 square reference element 1 η -1 1 ξ basis functions on the reference element -1 Piola transform CEA/DAM/DIF 6 juillet 2010 13

Discretization: equation for the fluxes in each element with mass matrix we get CEA/DAM/DIF 6 juillet 2010 14

Discretization: equation for q discrete divergence obtained from Green s formula in a matrix-like writing CEA/DAM/DIF 6 juillet 2010 15

Discretization: local system local system for each element with inter-element continuity for edges not in CEA/DAM/DIF 6 juillet 2010 16

Discretization: elimination of the fluxes local inversion of the mass matrix eliminating fluxes unknowns X m if if we get the system m edge mk k CEA/DAM/DIF 6 juillet 2010 17

Discretization: time scheme and matrix system with implicit Euler s time scheme Diagonal system for 1st equation: easily inverted => system in Λ to be solved with the matrix is definite positive solved by a preconditioned conjugate gradient method CEA/DAM/DIF 6 juillet 2010 18

Solving: bordering algorithm for the constraint the constraint is discretized use of the bordering algorithm for an arrow matrix with A V = a C and A W = B A (sparse) a C X = B we get : ( a a R V ) x = ( b a R W ) a R a x b X = W V x CEA/DAM/DIF 6 juillet 2010 19

Coupling with hydrodynamics Euler s equations u Q1-Q0 approach + leapfrog in time ρ, p, e CEA/DAM/DIF 6 juillet 2010 20

adding MHD coupling terms Euler s equations in Ω Laplace Laplace force needed at nodes Joule Joule effect needed at center CEA/DAM/DIF 6 juillet 2010 21

Coupling terms using all the information from the MHFE discretization of the diffusion equation Laplace Force deduced from the weak form of the definition of j for example : the axisymmetrical case Joule effect CEA/DAM/DIF 6 juillet 2010 22

Conclusion In the modelling context, need for a generic planar 2D diffusion solver 3 configurations global constraint eventually Discretization of electromagnetic part in a Lagrangian hydrodynamical code Mixed Finite Element Method for the diffusion equation Euler s implicit scheme in time preconditionned conjugate gradient method for the linear system same discretization for coupling terms : Laplace s Force and Joule s effect bordering algorithm for the additional constraint hoping for results soon if problem with distorted quadrilateral meshes use of modified RT element by Boffi & Gastaldi 2009 : adding a quadratic function to the basis functions CEA/DAM/DIF 6 juillet 2010 23

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