Numerical methods for FCI B. Despr´ es LJLL-Paris Part II: Hydrodynamics VI+CEA Thanks to PhD: Mazeran, B. Despr´ es Kluth, Franck, LJLL-Paris VI+CEA CEA: Delpino, Labourasse, Carre, Buet, Thanks to . . ., and all PhD: Mazeran, Kluth, Franck, colleagues along these CEA: Delpino, Labourasse, Carre, Buet, . . ., years. and all colleagues along these years. Numerical methods for FCI Part II: Hydrodynamics p. 1 / 41
FCI scenario Introduction PDEs Schemes- construction Numerical results Numerical analysis Hydrodynamic is dominant in the implosion stage Numerical methods for FCI Part II: Hydrodynamics p. 2 / 41
DT capsule in a gold cylinder, heated by laser beams Credit CEA/DAM/DIF Introduction PDEs Schemes- construction Numerical results Numerical analysis The radiation push is not perfectly symetric Numerical methods for FCI Part II: Hydrodynamics p. 3 / 41
Why Lagrangian scheme for compressible flows ? Lagrangian methods are written in the fluid frame Introduction easy discretization of free boundaries (external and internal), naturally adapted for multimaterial flows, PDEs very good accuracy for transport dominated flows Schemes- construction 1 1 "mesh0.Sod" "mesh12.Sod" 0.8 0.8 Numerical results 0.6 0.6 Numerical 0.4 0.4 analysis 0.2 0.2 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 "rho0.Sod" "rho12.Sod" 1 1 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 1 0.9 1 0.8 0.8 0.7 0.6 0.6 0 0 0.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.2 0.4 0.4 0.4 0.3 0.6 0.2 0.2 1 0 0.1 0.8 1 0 Numerical methods for FCI Part II: Hydrodynamics p. 4 / 41
3D examples Introduction PDEs Schemes- construction Numerical results Numerical analysis Numerical methods for FCI Part II: Hydrodynamics p. 5 / 41
Perturbation of a basic flow ICF flows are very sensitive to hydrodynamic perturbation. That is a even very small initial pertubation may have a dramatic influence on the solution. The reason is that ICF flows are quite close to instability, so that the growth rate of the perturbations may be large. Numerical methods are very useful to quantify the Introduction influence these perturbations. Here we illustrate with the result of a very simple numerical simulation done by E. Franck during his Master 2. PDEs Schemes- construction Numerical results Numerical analysis The mesh at t = 0 is displayed on the left. The perturbation is a n = 16 mode. On the right the result at time t f = 10 − 9 s . On this simulation the growth rate of the perturbation is reasonnable. Lagrangian methods are essential for such simulations. Eulerian methods on Cartesian meshes are less efficient. Numerical methods for FCI Part II: Hydrodynamics p. 6 / 41
Plan Introduction PDEs Schemes- construction Partial Differential Equations Numerical Meshes and schemes results Numerical analysis : stability, CFL, convergence (new) Numerical T i − T e discretization analysis Extension to radiation. Development of Hele-Shaw models (for stability of ICF flows, with H. Egly and R. Sentis). Numerical methods for FCI Part II: Hydrodynamics p. 7 / 41
Euler equations The Euler equation for compressible gas dynamics are Introduction 8 > ∂ t ρ + ∇ · ( ρ u ) = 0 , > < ∂ t ( ρ u ) + ∇ · ( ρ u ⊗ u ) + ∇ p = 0 , PDEs > ∂ t ( ρ e ) + ∇ · ( ρ u e + p u ) = 0 , > : Schemes- ∂ t ( ρ S ) + ∇ · ( ρ u S ) ≥ 0 . construction The density is ρ > 0. The velocity is u ∈ R d . The density of total energy is e = ε + 1 Numerical 2 | u | 2 . results Possible EOS Numerical analysis p = ( γ − 1) ρε, p = ( γ − 1) ρε − γ p 0 , p = tabulated , p = analytic . Assumption : there exists a temperature T > 0 such that 1 TdS = d ε + pd τ, τ = . ρ The pressure p = p ( τ, ε ) is provided by the equation of state. The entropy inequality selects the entropy solutions of the Euler system. For a perfect gas EOS, ε = C v T and S = log( ετ γ − 1 ). Numerical methods for FCI Part II: Hydrodynamics p. 8 / 41
Quasi-Lagrange in 2D Introduction It writes 8 PDEs < ρ D t τ − ∇ · u = 0 , ρ D t u + ∇ p = 0 , : Schemes- ρ D t e + ∇ · ( p u ) = 0 , construction where the material derivative is D t = ∂ t + u · ∇ . Numerical Or also with an artificial viscosity q results 8 Numerical ρ D t τ − ∇ · u = 0 , < analysis ρ D t u + ∇ ( p + q ) = 0 , : ρ D t ε + ( p + q ) ∇ · u = 0 . By construction ρ Td t S = ρ D t ε + p ρ D t τ = − q ∇ · u ≥ 0 A possible artificial viscosity is q = C ∆ x max(0 , −∇ · u ) ≥ 0. I will not discuss these methods which are non conservative. The seminal idea’s comes from Von Neumann. Numerical methods for FCI Part II: Hydrodynamics p. 9 / 41
Reminder • A non linear system of conservation laws with an entropy writes ∂ t U + ∂ x f ( U ) + ∂ y g ( U ) = 0 . Introduction PDEs • Assume the system is endowed with a strictly convex entropy b S ( U ) ∈ R : smooth solutions satisfy Schemes- ∂ t b S ( U ) + ∂ x F ( U ) + ∂ y G ( U ) = 0. construction • Then the system is hyperbolic, which implies stability around constant, and well posedness of the Cauchy problem for smooth initial data in finite time 0 < t < T . Numerical • Discontinuous solutions satisfy the Rankine Hugoniot relation results Numerical − σ ( U R − U L ) + n x ( f ( U R ) − f ( U L )) n y ( g ( U R ) − g ( U L )) = 0 analysis and the entropy inequality “ ” b S ( U R ) − b − σ S ( U L ) + n x ( F ( U R ) − F ( U L )) n y ( G ( U R ) − G ( U L )) ≥ 0 . Problem : the Quasi-Lagrange formulations are not conservative. Question : how to write pure conservative Lagrange PDE’s ? cf Godlewski-Raviart, Serre, . . . Question : how about hyperbolicity for Lagrangian systems ? Numerical methods for FCI Part II: Hydrodynamics p. 10 / 41
From Euler to Lagrange To shorten the notations we set Introduction x = ( x , y ) and X = ( X , Y ) . PDEs Let us consider the Euler-to-Lagrange change of frame defined by Schemes- construction ∂ t x ( X , t ) = u ( x ( X , t ) , t ) with initialization x ( X , 0) = X . Numerical results 1 1 "mesh0.Sod" "mesh12.Sod" Numerical 0.8 0.8 analysis 0.6 0.6 0.4 0.4 0.2 0.2 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Principle : A Lagrange formulation is written in the reference frame coordinate X. We need a tool to rewrite the equations in the X frame. Numerical methods for FCI Part II: Hydrodynamics p. 11 / 41
Piola’s transformations Introduction Rule 1 A smooth change of variable x �→ X ( x ) is such that n d σ = cof ( ∇ X x ) n X d σ X . where cof ( M ) ∈ R d × d is the comatrix of M . PDEs Schemes- construction Rule 2 A system of conservation laws ∇ x F ( U ) = 0 written in the eulerian variable is transformed in a lagrangian system written with the X variable Numerical results ∇ X . [ F ( U )cof ( ∇ X x )] = 0 . Numerical analysis This is a consequence of rule 1. Rule 3 One must not forget the Piola’s identity ∇ X . cof ( ∇ X x ) = 0 . A reason is that the system contains a new unknown cof ( ∇ X x ). Therefore we need a new equation to close the system.. Numerical methods for FCI Part II: Hydrodynamics p. 12 / 41
PDEs in 1D We consider the change of coordinates ( x , t ) �→ X , t ) in R 2 . The gradient of the space-time transformation „ « „ « 1 0 J − u with the Jacobian J = ∂ x is ∂ X . The comatrix is cof = . u J 0 1 The Lagrangian system is Introduction 2 0 1 «3 0 1 0 1 „ PDEs ρ ρ u ρ J 0 J − u ρ u 2 + p 4 @ A 5 = ∂ t @ A + ∂ X @ A = 0 . ∇ X , t · ρ Ju p ρ u 0 1 Schemes- ρ Je pu ρ e ρ ue + pu construction Numerical The Piola identity writes results ∂ t J − ∂ X u = 0 . Numerical It is usual to define the mass variable analysis dm = ρ ( x , t ) dx = ρ ( X , 0) dX which is independent of the time, to eliminate the density ρ J = ρ ( X , 0) and to get the system of conservation laws in the mass variable 8 ∂ t τ − ∂ m u = 0 , < ∂ t u + ∂ m p = 0 , : ∂ t e + ∂ m ( pu ) = 0 . Notice that ρ > 0 is necessary for the validity of the transformation. Numerical methods for FCI Part II: Hydrodynamics p. 13 / 41
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