Introduction to cell-centered Lagrangian schemes Franc¸ois Vilar ´ Institut Montpellierain Alexander Grothendieck Universite´ de Montpellier
15 Mai 2017
Franc¸ois Vilar (IMAG)
Cell-centered Lagrangian schemes
15 Mai 2017
1
Introduction
2
1D gas dynamics system of equations
3
First-order numerical scheme for the 1D gas dynamics
4
High-order extension in the 1D case
5
Numerical results in 1D
6
2D gas dynamics system of equations
7
First-order numerical scheme for the 2D gas dynamics
8
High-order extension in the 2D case
9
Numerical results in 2D Franc¸ois Vilar (IMAG)
Cell-centered Lagrangian schemes
15 Mai 2017
0 / 57
Introduction
Eulerian and Lagrangian formalisms
Eulerian formalism (spatial description) fixed referential attached to the observer fixed observation zone through the fluid flows
Lagrangian formalisme (material description) moving referential attached to the material observation zone moved and deformed as the fluid flows
Lagrangian formalism advantages adapted to problems undergoing large deformations naturally tracks interfaces in multi-material flows avoids the numerical diffusion of the convection terms
Lagrangian formalism drawbacks Robustness issue in the case of strong vorticity or shear flows =⇒ Franc¸ois Vilar (IMAG)
ALE method (Arbitrary Lagrangian-Eulerian) Cell-centered Lagrangian schemes
15 Mai 2017
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Introduction
Cell-centered formulation 1 0 0 1
Staggered formulation 1 0 0 1
1 0 0 1
1 0 0 1 0 1
1 0 0 1
0 1 0 1 0 1 1111111111 0000000000 0 1 0000000000 1111111111 00 11 0000000000 1111111111 00 11 00 11 00 11 0000000000 1111111111 00 11 00 11 11 0000000000 1111111111 00 00 11 00 11 0000000000 1111111111 00 11 0000000000 1111111111 00 11 0000000000 1111111111 00 11 00 11 0000000000 1111111111 00 11 00 11 0000000000 1111111111 00 11 0011 11 00 11 0000000000 1111111111 00 0000000000 1111111111 00 11 0000000000 1111111111 0 1 0000000000 1111111111 0 1 c11 0000000000 1111111111 00 11 0000000000 1111111111 00 00 11 0000000000 1111111111 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11
ρ ε u Ω
11 00 00 11
Eulerian and Lagrangian formalisms
0 1 111111111 000000000 0 1 0 1 000000000 111111111 0 1 000000000 111111111 0 1 000000000 111111111 0000000000 1111111111 0 1 000000000 111111111 0000000000 1111111111 00 11 000000000 111111111 0000000000 1111111111 00 11 000000000 111111111 0000000000 1111111111 000000000 111111111 0000000000 1111111111 00 11 000000000 111111111 0000000000 1111111111 00 11 11 00 000000000 111111111 0000000000 1111111111 11 00 000000000 111111111 0000000000 1111111111 000000000 111111111 00 11 00 11 0000000000 1111111111 p 000000000 111111111 00 11 00 11 0000000000 1111111111 000000000 00 11 00111111111 11 0000000000 1111111111 000000000 111111111 0000000000 1111111111 000000000 111111111 0000000000 1111111111 000000000 111111111 0000000000 1111111111 c 0000000000 1111111111 00 11 0000000000 1111111111 0 1 00 11 00 11 0000000000 1111111111 0 1 00 11 00 11 00 11 00 11 0 1 0 1 1 0 0 1 0 1
1 0 0 1
Franc¸ois Vilar (IMAG)
1 0 0 1
u
Ω
Ω
11 00 00 11
Cell-centered Lagrangian schemes
ρ ε
1 0 0 1
15 Mai 2017
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1D gas dynamics system of equations
1
Introduction
2
1D gas dynamics system of equations
3
First-order numerical scheme for the 1D gas dynamics
4
High-order extension in the 1D case
5
Numerical results in 1D
6
2D gas dynamics system of equations
7
First-order numerical scheme for the 2D gas dynamics
8
High-order extension in the 2D case
9
Numerical results in 2D Franc¸ois Vilar (IMAG)
Cell-centered Lagrangian schemes
15 Mai 2017
2 / 57
1D gas dynamics system of equations
Eulerian description
Definitions ρ the fluid density u the fluid velocity e the fluid specific total energy p the fluid pressure ε = e − 12 u 2 the fluid specific internal energy
Euler system ∂ ρ ∂ ρu + =0 ∂t ∂x ∂ ρu ∂ (ρ u 2 + p) + =0 ∂t ∂x ∂ ρ e ∂ (ρ u e + p u) + =0 ∂t ∂x
Continuity equation Momentum conservation Total energy conservation
Thermodynamical closure p = p(ρ, ε) Franc¸ois Vilar (IMAG)
Equation of state Cell-centered Lagrangian schemes
15 Mai 2017
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1D gas dynamics system of equations
Lagrangian descriptions
Momentum conservation ∂ ρu ∂ (ρ u 2 + p) + =0 ∂t ∂x ∂u ∂u ∂ ρ ∂ ρu ∂p ρ( +u )+u( + )+ =0 ∂t ∂x ∂t ∂x ∂x | {z } =0
∂u ∂u ∂p ρ( +u )+ =0 ∂t ∂x ∂x
Total energy conservation ∂ ρ e ∂ (ρ u e + p u) + =0 ∂t ∂x ∂e ∂e ∂ ρ ∂ ρu ∂pu ρ( +u )+e( + )+ =0 ∂t ∂x ∂x |∂t {z ∂x } =0
∂e ∂e ∂pu ρ( +u )+ =0 ∂t ∂x ∂x Franc¸ois Vilar (IMAG)
Cell-centered Lagrangian schemes
15 Mai 2017
4 / 57
1D gas dynamics system of equations
Lagrangian descriptions
Definitions τ=
1 ρ
the specific volume
U = (τ, u, e)t
the solution vector
F(U) = (−u, p, p u)t
the flux vector
Continuity equation ∂ ρ ∂ ρu + =0 ∂t ∂x ∂ρ ∂ρ ∂u +u +ρ =0 ∂t ∂x ∂x ∂τ ∂τ ∂u ρ( +u )− =0 ∂t ∂x ∂x
Non-conservative form of the gas dynamics system ρ(
∂U ∂U ∂ F(U) +u )+ =0 ∂t ∂x ∂x
Franc¸ois Vilar (IMAG)
Cell-centered Lagrangian schemes
15 Mai 2017
5 / 57
1D gas dynamics system of equations
Lagrangian descriptions
Moving referential X
is the position of a point of the fluid in its initial configuration
x(X , t) is the actual position of this point, moved by the fluid flow
Trajectory equation ∂ x(X , t) = u(x(X , t), t) ∂t x(X , 0) = X
Material derivative f (x, t) is a smooth fluid variable df ∂ f (x(X , t), t) ∂f ∂f ≡ = +u dt ∂t ∂t ∂x
Franc¸ois Vilar (IMAG)
Cell-centered Lagrangian schemes
15 Mai 2017
6 / 57
1D gas dynamics system of equations
Lagrangian descriptions
Updated Lagrangian formulation ρ
d U ∂ F(U) + =0 dt ∂x
Moving configuration
Definitions ∂x ∂X
J= ρ
0
is the Jacobian associated the fluid flow
is the intial fluid density
Mass conservation
R ρ0 dX = ω(t) ρ dx R R ρ dx = ω(0) ρ J dX ω(t) R
ω(0)
ρ J = ρ0
Total Lagrangian formulation ρ0
d U ∂ F(U) + =0 dt ∂X
Franc¸ois Vilar (IMAG)
Fixed configuration Cell-centered Lagrangian schemes
15 Mai 2017
7 / 57
1D gas dynamics system of equations
Mass Lagrangian formulation
Definitions dm = ρ dx = ρ0 dX the mass varaiable ∂ F(U) A(U) = the Jacobian matrix of the system ∂U a = a(ρ, ε) the sound speed
Conservative formulation d U ∂ F(U) + =0 dt ∂m
Non-conservative formulation dU ∂U + A(U) =0 dt ∂m λ(U) = {−ρ a, 0, ρ a}
Franc¸ois Vilar (IMAG)
the eigenvalues of A(U)
Cell-centered Lagrangian schemes
15 Mai 2017
8 / 57
First-order numerical scheme for the 1D gas dynamics
1
Introduction
2
1D gas dynamics system of equations
3
First-order numerical scheme for the 1D gas dynamics
4
High-order extension in the 1D case
5
Numerical results in 1D
6
2D gas dynamics system of equations
7
First-order numerical scheme for the 2D gas dynamics
8
High-order extension in the 2D case
9
Numerical results in 2D Franc¸ois Vilar (IMAG)
Cell-centered Lagrangian schemes
15 Mai 2017
8 / 57
First-order numerical scheme for the 1D gas dynamics
Finite volumes scheme
´ Definitions 0 = t0 < t1 < · · · < tN = T
a partition of the temporal domain [0, T ]
∆t n = t n+1 − t n the n th time step S ω 0 = i=1,l ωi0 the partition of the initial domain ω 0 ωi0 = [Xi− 1 , Xi+ 1 ] a generic cell of size ∆Xi 2
2
n n ωin = [xi− ] 1,x i+ 1 2
the image of ωi0 at time t n through the fluid flow
2
mi = ρ0i ∆Xi = ρni ∆xin Uni
=
(τin ,
uin ,
ein )t
the constant mass of cell ωi
the discrete solution
First-order finite volumes scheme Uin+1 = Uni −
∆t n mi
n
n
Fi+ 21 − Fi− 12
n+1 n xi+ + ∆t n u ni+ 1 1 = x i+ 1 2
2
2
Numerical flux n
Fi+ 12 = (−u ni+ 1 , pni+ 1 , pni+ 1 u ni+ 1 )t 2
Franc¸ois Vilar (IMAG)
2
2
2
Cell-centered Lagrangian schemes
15 Mai 2017
9 / 57
First-order numerical scheme for the 1D gas dynamics
Two-states linearization ∂U dU + A(U) =0 dt ∂m
t
−e zL
dU f ∂U dt + A(UL ) ∂m = 0
fR ) ∂ U = 0 d U + A(U dt ∂m
si m-mi < 0 si m-mi > 0
Simple Riemann problem 0
U
=⇒
Approximate Riemann solver - two-states solver
−
U
UL
+
zeR
UR
U(m, 0) =
UL UR
UL − U U(m, 0) = U+ UR
Relations mi
Riemann problem Franc¸ois Vilar (IMAG)
m
zeL = ρfaL > 0, −
+
u = u = u, Cell-centered Lagrangian schemes
if m-mi < 0 if m-mi > 0 if m-mi < -zeL t if -zeL t < m-mi < 0 if zeR t > m-mi > 0 if m-mi > zeR t
zeR = ρfaR > 0
p− = p+ = p
15 Mai 2017
10 / 57
First-order numerical scheme for the 1D gas dynamics
Approximate Riemann solver - two-states solver
Numerical fluxes u= p=
1 zeL uL + zeR uR − (pR − pL ) zeL + zeR zeL + zeR
zeL zeR zeR pL + zeL pR − (uR − uL ) e e e zL + zR zL + zeR
Intermediate states
u − uL et zeL p u − pL uL e− = eL − zeL τ − = τL +
Acoustic solver
zeL ≡ zL = ρL aL zeR ≡ zR = ρR aR Franc¸ois Vilar (IMAG)
u − uR zeR p u − pR uR e+ = eR + zeR
τ + = τR − et
Left acoustic impedance Right acoustic impedance Cell-centered Lagrangian schemes
15 Mai 2017
11 / 57
First-order numerical scheme for the 1D gas dynamics
Godunov-type scheme
Convex combination Uin+1 = Uni − Uin+1
= (1 −
n ∆t n 1 mi (Fi+ 2
λi )Uni
n
λ− i+ 12
− Ui+ 21
+
λ+ i− 12
+ Ui− 12
2
2
∆t n e− mi (zi+ 21
λ∓ = i± 1 2
2
∆t n e∓ mi zi± 12
λi = λ− + λ+ i+ 1 i− 1
− U i+ 21
+ U i− 12
+ n + zei− 1 )Ui
´ Definitions
− −e zi+ 1
+ zei− 1
t
+
n
n − Fi− 12 ) ± ∆t mi F(Ui ) ±
2
2
n
∓
Ui± 12 = Uni ∓
∆tn
Fi± 12 − F(Uni ) ∓ zei± 1
2
U ni m
CFL condition: λi ≤ 1
mi
mi − e+ 1 zei+ 1 + z i− 2
Scheme illustration Franc¸ois Vilar (IMAG)
∆t n ≤
Cell-centered Lagrangian schemes
∆t n ≤
1 2
∆xin ain
2
∓ n if zei± 1 ≡ zi 2
15 Mai 2017
12 / 57
First-order numerical scheme for the 1D gas dynamics
Godunov-type scheme
Semi-discret first-order scheme mi
d Ui = − F(Ui , Ui+1 ) − F(Ui−1 , Ui ) dt
Gibbs identity T dS = dε + p dτ = de − u du + p dτ
Semi-discret production of entropy d Si d ei d ui d τi = mi + ui mi + pi mi dt dt dt dt d Si − 2 e+ 1 (u i− 1 − ui )2 ≥ 0 mi Ti = zei+ 1 (u i+ 1 − ui ) + z i− 2 2 2 2 dt mi Ti
Positivity of the discrete scheme
F. V ILAR , C.-W. S HU AND P.-H. M AIRE, Positivity-preserving cell-centered Lagrangian schemes for multi-material compressible flows: Form first-order to high-orders. Part I: The 1D case. JCP, 2016. Franc¸ois Vilar (IMAG)
Cell-centered Lagrangian schemes
15 Mai 2017
13 / 57
High-order extension in the 1D case
1
Introduction
2
1D gas dynamics system of equations
3
First-order numerical scheme for the 1D gas dynamics
4
High-order extension in the 1D case
5
Numerical results in 1D
6
2D gas dynamics system of equations
7
First-order numerical scheme for the 2D gas dynamics
8
High-order extension in the 2D case
9
Numerical results in 2D Franc¸ois Vilar (IMAG)
Cell-centered Lagrangian schemes
15 Mai 2017
13 / 57
High-order extension in the 1D case
High-order finite-volumes-type schemes
High-order extension of the finite-volume scheme MUSCL, (W)ENO, DG, . . .
Equation on the mean values i ∆t n h − + + F(U− 1,U 1,U 1 ) − F(U 1) i+ 2 i− 2 i+ 2 i− 2 mi − and Ui+ 1 are the high-order values in ωi at points xi− 1 and xi+ 1
Uin+1 = Uni − U+ i− 1 2
2
2
2
Moving or total formulation ρ
d U ∂ F(U) + =0 dt ∂x
ρ0
ou
∂ U ∂ F(U) + =0 ∂t ∂X
Piecewise polynomial approximation Unh,i (x) the polynomial approximation of the solution on ωin Unh,i (X ) the polynomial approximation of the solution on ωi0 U∓ = Unh,i (xi± 1 ) (moving config.) i± 1 2
Franc¸ois Vilar (IMAG)
2
or
U∓ = Unh,i (Xi± 1 ) (fixed config.) i± 1
Cell-centered Lagrangian schemes
2
2
15 Mai 2017
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Numerical results in 1D
1
Introduction
2
1D gas dynamics system of equations
3
First-order numerical scheme for the 1D gas dynamics
4
High-order extension in the 1D case
5
Numerical results in 1D
6
2D gas dynamics system of equations
7
First-order numerical scheme for the 2D gas dynamics
8
High-order extension in the 2D case
9
Numerical results in 2D Franc¸ois Vilar (IMAG)
Cell-centered Lagrangian schemes
15 Mai 2017
14 / 57
Numerical results in 1D
Smooth solution problem
Initial solution on X ∈ [0, 1]
ρ0 (X ) = 1 + 0.9999995 sin(2πX ),
u 0 (X ) = 0,
p0 (X ) = ρ0 (X )γ
Periodic boundary conditions 1.8
1.5 solution 1st order 3rd order
1.6
solution 1st order 3rd order 1
1.4
1.2
0.5
1 0 0.8
0.6
-0.5
0.4 -1 0.2
0
-1.5 -1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
-1
-0.8
-0.6
(a) Density profiles
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
(b) Velocity profiles
Figure: Solutions at time t = 0.1 on 50 cells for a smooth problem
Franc¸ois Vilar (IMAG)
Cell-centered Lagrangian schemes
15 Mai 2017
15 / 57
Numerical results in 1D
Smooth solution problem
Convergence rates h 1 50 1 100 1 200 1 400 1 800
L1 ELh1 9.69E-5 1.19E-5 1.48E-6 1.85E-7 2.30E-8
qLh1 3.02 3.01 3.00 3.00 -
L2 ELh2 9.31E-5 1.16E-5 1.44E-6 1.80E-7 2.25E-8
qLh2 3.01 3.00 3.00 3.00 -
L∞ ELh∞ 2.75E-4 3.40E-5 4.923E-6 5.26E-7 6.56E-8
qLh∞ 3.01 3.01 3.00 3.00 -
Table: Convergence rates on the pressure for a 3rd order DG scheme
Franc¸ois Vilar (IMAG)
Cell-centered Lagrangian schemes
15 Mai 2017
16 / 57
Numerical results in 1D
Sod shock tube problem
Initial solution on X ∈ [0, 1] (ρ0 , u 0 , p0 ) =
(1, 0, 1), 0 < X < 0.5, (0.125, 0, 0.1), 0.5 < X < 1. 3.2
solution 1st order 3rd order
1
solution 1st order 3rd order
3 0.9
0.8
2.8
0.7
2.6
0.6
2.4
0.5 2.2 0.4 2 0.3 1.8
0.2
0.1
1.6 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
(a) Density profiles
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
(b) Internal energy profiles
Figure: Solutions at time t = 0.2 on 100 cells for a Sod shock tube problem Franc¸ois Vilar (IMAG)
Cell-centered Lagrangian schemes
15 Mai 2017
17 / 57
Numerical results in 1D
Leblanc shock tube problem
Initial solution on X ∈ [0, 9] 0
0
0
(ρ , u , e ) =
(1, 0, 0.1), 0 < X < 3, (0.001, 0, 10−7 ), 3 < X < 9.
1
0.3
solution 1st order 3rd order
solution 1st order 3rd order 0.25
0.1
0.2
0.15
0.01
0.1
0.05
0.001
0
0
1
2
3
4
5
6
(a) Density profiles
7
8
9
0
1
2
3
4
5
6
7
8
9
(b) Internal energy profiles
Figure: Solutions at time t = 6 on 400 cells for a Leblanc shock tube problem
Franc¸ois Vilar (IMAG)
Cell-centered Lagrangian schemes
15 Mai 2017
18 / 57
Numerical results in 1D
Leblanc shock tube problem
Convergence 1
1 solution 200 cells 200 cells 400 cells 800 cells 1600 cells 3200 cells
solution 200 cells 200 cells 400 cells 800 cells 1600 cells 3200 cells
0.1
0.1
0.01
0.01
0.001
0.001 0
1
2
3
4
5
6
7
8
9
0
1
(a) 1st order
2
3
4
5
6
7
8
9
(b) 3rd order
Figure: Convergence at time t = 6 for a Leblanc shock tube problem
Franc¸ois Vilar (IMAG)
Cell-centered Lagrangian schemes
15 Mai 2017
19 / 57
Numerical results in 1D
123 problem - double rarefaction
Initial solution on X ∈ [−4, 4] 0
0
0
(ρ , u , p ) =
(1, −2, 0.4), (1, 2, 0.4),
−4 < X < 0, 0 < X < 4.
1
1.6 solution 1st order 3rd order
0.9
solution 1st order 3rd order 1.4
0.8 1.2
0.7
0.6 1 0.5 0.8 0.4
0.3
0.6
0.2 0.4 0.1
0
0.2 -6
-4
-2
0
2
(a) Density profiles
4
6
-6
-4
-2
0
2
4
6
(b) Internal energy profiles
Figure: Solutions at time t = 1 on 400 cells for a 123 problem
Franc¸ois Vilar (IMAG)
Cell-centered Lagrangian schemes
15 Mai 2017
20 / 57
Numerical results in 1D
Underwater TNT explosion
Initial solution on X ∈ [0, 1.4] 0
0
0
(ρ , u , p ) =
(1.63 × 10−3 , 0, 8.381 × 103 ), (1.025 × 10−3 , 0, 1),
0 < X < 0.16, 0.16 < X < 3.0.
On [0, 0.3], gaseous product of the explosion (JWL EOS) On [0.3, 1.4], water (stiffened gas EOS) 0.0014
4000
solution 1st order 3rd order
solution 1st order 3rd order 3500
0.0013
3000
0.0012 2500
0.0011 2000
0.001 1500
0.0009 1000
0.0008
500
0.0007
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0
0.2
(a) Density profiles
0.4
0.6
0.8
1
1.2
1.4
(b) Pressure profiles
Figure: Solutions at time t = 0.00025 on 400 cells for a underwater TNT explosion Franc¸ois Vilar (IMAG)
Cell-centered Lagrangian schemes
15 Mai 2017
21 / 57
Numerical results in 1D
Wilkins problem
Initial solution on X ∈ [0, 0.05] 0
ρ (X ) = 2785,
−6
0
p (X ) = 10
0
u (X ) =
,
Aluminium (Mie-Gruneisen EOS) ¨ 3000
800, 0,
0 < X < 0.005, 0.005 < X < 0.05.
7e+09 solution 1st order 3rd order
solution 1st order 3rd order 6e+09
2950 5e+09
4e+09 2900
3e+09
2850 2e+09
1e+09 2800 0
2750
-1e+09 0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
0
0.005
(a) Density profiles
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
(b) Pressure profiles
Figure: Solutions at time t = 5 × 10−6 on 100 cells for a flying plate impact Franc¸ois Vilar (IMAG)
Cell-centered Lagrangian schemes
15 Mai 2017
22 / 57
2D gas dynamics system of equations
1
Introduction
2
1D gas dynamics system of equations
3
First-order numerical scheme for the 1D gas dynamics
4
High-order extension in the 1D case
5
Numerical results in 1D
6
2D gas dynamics system of equations
7
First-order numerical scheme for the 2D gas dynamics
8
High-order extension in the 2D case
9
Numerical results in 2D Franc¸ois Vilar (IMAG)
Cell-centered Lagrangian schemes
15 Mai 2017
22 / 57
2D gas dynamics system of equations
Eulerian description
Euler equations ∂ρ + ∇x ρ u = 0 ∂t ∂ ρu + ∇x (ρ u ⊗ u + p Id ) = 0 ∂t ∂ ρe + ∇x (ρ u e + p u) = 0 ∂t
Trajectory equation d x(X , t) = u(x(X , t), t), dt
x(X , 0) = X
Material derivative d f (x, t) ∂ f (x, t) = + u ∇x f (x, t) dt ∂t
Franc¸ois Vilar (IMAG)
Cell-centered Lagrangian schemes
15 Mai 2017
23 / 57
2D gas dynamics system of equations
Lagrangian descriptions
Definitions U = (τ, u, e)t F(U) = (−u, 1(1) p, 1(2) p, p u)t
where
1(i) = (δi1 , δi2 )t
Updated Lagrangian formulation ρ
dU + ∇x F(U) = 0 dt
Moving configuration
Deformation gradient tensor J = ∇X x
−t
∇X |J|J
with =0
|J| = det J > 0
Piola compatibility condition
Mass conservation ρ |J| = ρ0
Total Lagrangian formulation ρ0
dU + ∇X |J|J−1 F(U) = 0 dt
Franc¸ois Vilar (IMAG)
Cell-centered Lagrangian schemes
Fixed configuration 15 Mai 2017
24 / 57
First-order numerical scheme for the 2D gas dynamics
1
Introduction
2
1D gas dynamics system of equations
3
First-order numerical scheme for the 1D gas dynamics
4
High-order extension in the 1D case
5
Numerical results in 1D
6
2D gas dynamics system of equations
7
First-order numerical scheme for the 2D gas dynamics
8
High-order extension in the 2D case
9
Numerical results in 2D Franc¸ois Vilar (IMAG)
Cell-centered Lagrangian schemes
15 Mai 2017
24 / 57
First-order numerical scheme for the 2D gas dynamics
Finite volumes scheme
´ Definitions 0 = t 0 < t 1 < · · · < t N = T a partition of the time domain [0, T ] S ω 0 = c=1,l ωc0 a partition of the initial domain ω 0 ωcn
the image of ωc0 at time t n through the fluid flow
mc
the constant mass of cell ωc
Unc
= (τcn , ucn , ecn )t
the discrete solution m
p+
ωc
p p−
(a) Straight line edges
p+ ωc
p p−
(b) Conical edges
1100 00 11 1 11 0 00 11 00 00 11 1 0 11 00 + 11 00 0 1 0 0 1 0 1 −1 11 00 11 1 00 0 0 1 00 11 00 11 00 11 00 11 1 0 1 0 0 1 0 1 11 00 11 00
p
p
ωc
p
(c) Polynomial edges
Figure: Generic polygonal cell Franc¸ois Vilar (IMAG)
Cell-centered Lagrangian schemes
15 Mai 2017
25 / 57
First-order numerical scheme for the 2D gas dynamics
Finite volumes scheme
Integration
Z ∆t n F n ds mc ∂ωc Integration of the cell boundary term (analytically, quadrature, . . . ) Ucn+1 = Unc −
General first-order finite volumes scheme Ucn+1 = Unc −
∆t n X Fqc lqc nqc mc q∈Qc
Fqc = (−u q , 1(1) pqc , 1(2) pqc , pqc u q )t xqn+1
numarical flux at point q
= xqn + ∆t n u q
Definitions Qc
the chosen control point set of cell ωc
lqc nqc
some normals to be defined
Franc¸ois Vilar (IMAG)
Cell-centered Lagrangian schemes
15 Mai 2017
26 / 57
First-order numerical scheme for the 2D gas dynamics
Nodal solver
Remark Fqc
is local to the cell ωc
Only
u qc = u q
needs to be continuous, to advect the mesh
Loss of the scheme conservation?
ωR
ωc3 ωc4
p+
p
q
ωL
ωc2
ωc5
p
ωc1
(a) Face control point
(b) Grid node
Figure: Points neighboring cell sets
1D numerical fluxes pqc = pcn − zeqc (u q − ucn ) nqc
zeqc > 0
local approximation of the acoustic impedance
Franc¸ois Vilar (IMAG)
Cell-centered Lagrangian schemes
15 Mai 2017
27 / 57
First-order numerical scheme for the 2D gas dynamics
Conservation X
mc Un+1 = c
c
X
mc Unc + BC
Nodal solver
?
c
For sake of simplicity, we consider BC = 0 X X Necessary condition: pqc lqc nqc = 0 c
q∈Qc
Example of a solver: LCCDG schemes Conditions suffisantes X − − − + + ∀p ∈ P(ω), ppc lpc npc + p+ pc lpc npc = 0 c∈Cp
=⇒
up =
X
c∈Cp
∀q ∈ Q(ω) \ P(ω), =⇒ Franc¸ois Vilar (IMAG)
Mpc
−1 X Mpc ucn + pcn lpc npc c∈Cp
(pqL − pqR ) lqL nqL = 0 ⇐⇒ pqL = pqR zeqL uLn + zeqR uRn pn − pLn uq = − R nqfpp+ zeqL + zeqR zeqL + zeqR Cell-centered Lagrangian schemes
15 Mai 2017
28 / 57
First-order numerical scheme for the 2D gas dynamics
Godunov-type scheme
Convex combinaison Ucn+1 = Unc −
Ucn+1
= (1 −
X ∆t n ∆t n X Fqc lqc nqc + F(Unc ) lqc nqc mc mc q∈Qc q∈Qc | {z }
λc ) Unc
+
=0
X
λqc Uqc
q∈Qc
Definitions λqc =
∆t n e mc zqc lqc
Uqc = Unc −
CFL condition
Franc¸ois Vilar (IMAG)
F(Unc )
Fqc − zeqc
mc ∆t n ≤ X zeqc lqc q∈Qc
and
=
λc = nqc
acn
|ωcn | X
P
q∈Qc
λqc
if lqc
q∈Qc Cell-centered Lagrangian schemes
zeqc ≡ zcn = ρnc acn 15 Mai 2017
29 / 57
First-order numerical scheme for the 2D gas dynamics
Godunov-type scheme
Semi-discret first-order scheme mc
X d Uc =− Fqc lqc nqc dt q∈Qc
Gibbs identity T dS = dε + p dτ = de − u du + p dτ
Semi-discret production of entropy d Sc d ec d uc d τc = mc + u c mc + pc mc dt dt dt dt X 2 d Sc mc T c = zeqc lqc (u q − uc ) nqc ≥ 0 dt mc T c
q∈Qc
Positivity of the discrete scheme
F. V ILAR , C.-W. S HU AND P.-H. M AIRE, Positivity-preserving cell-centered Lagrangian schemes for multi-material compressible flows: Form first-order to high-orders. Part II: The 2D case. JCP, 2016. Franc¸ois Vilar (IMAG)
Cell-centered Lagrangian schemes
15 Mai 2017
30 / 57
High-order extension in the 2D case
1
Introduction
2
1D gas dynamics system of equations
3
First-order numerical scheme for the 1D gas dynamics
4
High-order extension in the 1D case
5
Numerical results in 1D
6
2D gas dynamics system of equations
7
First-order numerical scheme for the 2D gas dynamics
8
High-order extension in the 2D case
9
Numerical results in 2D Franc¸ois Vilar (IMAG)
Cell-centered Lagrangian schemes
15 Mai 2017
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High-order extension in the 2D case
High-order finite-volumes-type schemes
Mean values equation Ucn+1 = Unc −
∆t n X Fqc lqc nqc mc q∈Qc
In Fqc , the mean values are substituted by the high-order values Uqc in ωc at points q
Updated or total Lagrangian formulation ρ
dU + ∇x F(U) = 0 dt
ρ0
ou
dU + ∇X |J|J−1 F(U) = 0 dt
Piecewise polynomial approximation Unh,c (x) the polynomial approximation of the solution on ωcn Unh,c (X ) the polynomial approximation of the solution on ωc0 Uqc = Unh,c (xq ) (moving config.)
Franc¸ois Vilar (IMAG)
or
Uqc = Unh,c (Xq ) (fixed config.)
Cell-centered Lagrangian schemes
15 Mai 2017
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Numerical results in 2D
2nd order scheme
1
Introduction
2
1D gas dynamics system of equations
3
First-order numerical scheme for the 1D gas dynamics
4
High-order extension in the 1D case
5
Numerical results in 1D
6
2D gas dynamics system of equations
7
First-order numerical scheme for the 2D gas dynamics
8
High-order extension in the 2D case
9
Numerical results in 2D Franc¸ois Vilar (IMAG)
Cell-centered Lagrangian schemes
15 Mai 2017
31 / 57
Numerical results in 2D
2nd order scheme
Sedov point blast problem 60
solution 2nd order
6 1
50
5
0.8
40
4
0.6
30
3
0.4
20
2
0.2
10
1
0 0 0
0.2
0.4
0.6
0.8
1
1.2
0
0.2
(a) Pressure field
0.4
0.6
0.8
1
1.2
1.4
(b) Density profiles
Figure : Solution at time t = 1 for a Sedov problem on a 30 × 30 Cartesian mesh
Franc¸ois Vilar (IMAG)
Cell-centered Lagrangian schemes
15 Mai 2017
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Numerical results in 2D
2nd order scheme
Sedov point blast problem 60
solution 2nd order
6 1
50
5
0.8
40
4
0.6
30
3
0.4
20
2
0.2
10
1
0 0 0
0.2
0.4
0.6
0.8
1
1.2
0
0.2
(a) Pressure field
0.4
0.6
0.8
1
1.2
1.4
(b) Density profiles
Figure : Solution at time t = 1 for a Sedov problem on a 30 × 30 Cartesian mesh
Franc¸ois Vilar (IMAG)
Cell-centered Lagrangian schemes
15 Mai 2017
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Numerical results in 2D
2nd order scheme
Sedov point blast problem
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0 0
0.2
0.4
0.6
0.8
1
(c) Triangular grid - 1110 cells
1.2
0
0.2
0.4
0.6
0.8
1
1.2
(d) Polygonal grid - 775 cells
Figure : Initial unstructured grids for Sedov point blast problem Franc¸ois Vilar (IMAG)
Cell-centered Lagrangian schemes
15 Mai 2017
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Numerical results in 2D
2nd order scheme
Sedov point blast problem 5.5
solution 2nd order
6
5
1
4.5
5
4
0.8
4 3.5 3
0.6
3 2.5 0.4
2
2
1.5 0.2
1
1
0.5 0
0 0
0.2
0.4
0.6
0.8
1
1.2
0
0.2
(e) Density field
0.4
0.6
0.8
1
1.2
1.4
(f) Density profiles
Figure : Solution at time t = 1 for a Sedov problem on a grid made of 1110 triangular cells
Franc¸ois Vilar (IMAG)
Cell-centered Lagrangian schemes
15 Mai 2017
34 / 57
Numerical results in 2D
2nd order scheme
Sedov point blast problem solution 2nd order
5
6 4.5 1 4
5
3.5
0.8
4 3 0.6
2.5
3
2 0.4
2 1.5 1
0.2
1
0.5 0
0 0
0.2
0.4
0.6
0.8
1
1.2
0
0.2
(g) Density field
0.4
0.6
0.8
1
1.2
1.4
(h) Density profiles
Figure : Solution at time t = 1 for a Sedov problem on a grid made of 775 polygonal cells
Franc¸ois Vilar (IMAG)
Cell-centered Lagrangian schemes
15 Mai 2017
35 / 57
Numerical results in 2D
2nd order scheme
Underater TNT explosion 0.0013 solution 1st order 2nd order
−3
x 10
0.0012
1.6
0.8
0.0011
0.7 1.5
0.001
0.6 0.0009
1.4 0.5 0.0008
0.4
1.3
0.3
0.0007
0.0006
1.2 0.2
0.0005
1.1
0.1
0.0004
0.0003
0 0
0.2
0.4
0.6
0.8
1
1.2
0
(i) Density field - 2nd order
0.2
0.4
0.6
0.8
1
1.2
(j) Density profiles
Figure : Solution at time t = 2.5 × 10−4 for a underwater TNT explosion on a 120 × 9 polar mesh
Franc¸ois Vilar (IMAG)
Cell-centered Lagrangian schemes
15 Mai 2017
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Numerical results in 2D
2nd order scheme
Underater TNT explosion 0.0013 solution 1st order 2nd order
−3
x 10
0.0012
1.6
0.8
0.0011
0.7 1.5
0.001
0.6 0.0009
1.4 0.5 0.0008
0.4
1.3
0.3
0.0007
0.0006
1.2 0.2
0.0005
1.1
0.1
0.0004
0.0003
0 0
0.2
0.4
0.6
0.8
1
1.2
0
(i) Density field - 2nd order
0.2
0.4
0.6
0.8
1
1.2
(j) Density profiles
Figure : Solution at time t = 2.5 × 10−4 for a underwater TNT explosion on a 120 × 9 polar mesh
Franc¸ois Vilar (IMAG)
Cell-centered Lagrangian schemes
15 Mai 2017
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Numerical results in 2D
2nd order scheme
Aluminium projectile impact problem 1.5
2820
1
2800 2780
0.5 2760 0 0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
(k) Density field
Figure : Solution at time t = 0.05 for a projectile impact problem on a 100 × 10 Cartesian mesh
Franc¸ois Vilar (IMAG)
Cell-centered Lagrangian schemes
15 Mai 2017
37 / 57
Numerical results in 2D
2nd order scheme
Aluminium projectile impact problem 1.5
2820
1
2800 2780
0.5 2760 0 0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
(k) Density field
Figure : Solution at time t = 0.05 for a projectile impact problem on a 100 × 10 Cartesian mesh
Franc¸ois Vilar (IMAG)
Cell-centered Lagrangian schemes
15 Mai 2017
37 / 57
Numerical results in 2D
2nd order scheme
Taylor-Green vortex 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
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0
0.1
(l) 2nd order
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
(m) Exact solution
Figure : Final deformed grids at time t = 0.75, on a 10 × 10 Cartesian mesh Franc¸ois Vilar (IMAG)
Cell-centered Lagrangian schemes
15 Mai 2017
38 / 57
Numerical results in 2D
2nd order scheme
Taylor-Green vortex 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
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0
0.1
(l) 2nd order
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
(m) Exact solution
Figure : Final deformed grids at time t = 0.75, on a 10 × 10 Cartesian mesh Franc¸ois Vilar (IMAG)
Cell-centered Lagrangian schemes
15 Mai 2017
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Numerical results in 2D
2nd order scheme
Convergence rates h 1 10 1 20 1 40 1 80 1 160
L1 ELh1 5.06E-3 1.32E-3 3.33E-4 8.35E-5 2.09E-5
qLh1 1.94 1.98 1.99 2.00 -
L2 ELh2 6.16E-3 1.62E-3 4.12E-4 1.04E-4 2.60E-5
qLh2 1.93 1.97 1.99 2.00 -
L∞ ELh∞ 2.20E-2 5.91E-3 1.53E-3 3.86E-4 9.69E-5
qLh∞ 1.84 1.95 1.98 1.99 -
Table: Convergence rates on the pressure for a 2nd order DG scheme
Franc¸ois Vilar (IMAG)
Cell-centered Lagrangian schemes
15 Mai 2017
39 / 57
Numerical results in 2D
3rd order scheme
Taylor-Green vortex 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
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0
0.1
(n) 3rd order
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
(o) Exact solution
Figure : Final deformed grids at time t = 0.75, on a 10 × 10 Cartesian mesh Franc¸ois Vilar (IMAG)
Cell-centered Lagrangian schemes
15 Mai 2017
40 / 57
Numerical results in 2D
3rd order scheme
Taylor-Green vortex 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
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0
0.1
(n) 3rd order
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
(o) Exact solution
Figure : Final deformed grids at time t = 0.75, on a 10 × 10 Cartesian mesh Franc¸ois Vilar (IMAG)
Cell-centered Lagrangian schemes
15 Mai 2017
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Numerical results in 2D
3rd order scheme
Convergence rates h 1 10 1 20 1 40 1 80 1 160
L1 ELh1 2.67E-4 3.43E-5 4.37E-6 5.50E-7 6.91E-8
qLh1 2.96 2.97 2.99 2.99 -
L2 ELh2 3.36E-7 4.36E-5 5.59E-6 7.06E-7 8.87E-8
qLh2 2.94 2.96 2.98 2.99 -
L∞ ELh∞ 1.21E-3 1.66E-4 2.18E-5 2.80E-6 3.53E-7
qLh∞ 2.86 2.93 2.96 2.99 -
Table: Convergence rates on the pressure for a 3rd order DG scheme
Franc¸ois Vilar (IMAG)
Cell-centered Lagrangian schemes
15 Mai 2017
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Numerical results in 2D
3rd order scheme
Polar meshes - symmetry preservation 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
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0
0.1
(p) 100 × 3
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
(q) 100 × 1
Figure : Curvilinear grids defined in polar coordinates
Franc¸ois Vilar (IMAG)
Cell-centered Lagrangian schemes
15 Mai 2017
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Numerical results in 2D
3rd order scheme
Sod shock tube problem - symmetry preservation 1
1
0.9
0.9
0.8
1
0.9 0.9 0.8
0.8 0.7
0.8 0.7
0.7 0.6
0.7 0.6
0.6 0.5
0.6 0.5
0.5 0.4 0.4
0.3
0.2
0.3
0.1
0
0.5
0.4
0.2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.4
0.3
0.2
0.3
0.1
0.2
0
0
0.1
0.2
(r) 1st order
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
(s) 2nd order
Figure : Density fields with 1st and 2nd order schemes on a 3rd mesh
Franc¸ois Vilar (IMAG)
Cell-centered Lagrangian schemes
15 Mai 2017
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Numerical results in 2D
3rd order scheme
Sod shock tube problem - symmetry preservation 1
1
0.9 0.9 1.1 solution 3rd order
0.8 0.8
1
0.7 0.7 0.6
0.9 0.8
0.6 0.5
0.7
0.5
0.4
0.3
0.4
0.2
0.3
0.1
0.2
0.6 0.5 0.4 0.3
0
0.2 0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
(t) Density field
0.2
0.4
0.6
0.8
1
(u) Density profiles
Figure : 3rd order solution for a Sod shock tube problem on a 100 × 3 polar grid
Franc¸ois Vilar (IMAG)
Cell-centered Lagrangian schemes
15 Mai 2017
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Numerical results in 2D
3rd order scheme
Sod shock tube problem - symmetry preservation 1
1
0.9
0.9 1.1
0.8
solution 3rd order
0.8 1
0.7 0.7
0.9
0.6 0.8
0.6 0.5
0.7
0.5 0.4 0.4
0.3
0.6
0.5
0.4
0.2
0.3 0.3
0.1
0.2
0.2
0.1
0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
(v) Density field
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
(w) Density profiles
Figure : 3rd order solution for a Sod shock tube problem on a 100 × 1 polar grid
Franc¸ois Vilar (IMAG)
Cell-centered Lagrangian schemes
15 Mai 2017
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Numerical results in 2D
3rd order scheme
Sod shock tube problem - symmetry preservation 1
1
0.9
0.9 1.1
0.8
solution 3rd order
0.8 1
0.7 0.7
0.9
0.6 0.8
0.6 0.5
0.7
0.5 0.4
0.3
0.4
0.2
0.3
0.6
0.5
0.4
0.3
0.1
0.2
0.2
0.1
0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
(v) Density field
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
(w) Density profiles
Figure : 3rd order solution for a Sod shock tube problem on a 100 × 1 polar grid
Franc¸ois Vilar (IMAG)
Cell-centered Lagrangian schemes
15 Mai 2017
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Numerical results in 2D
3rd order scheme
Gresho-like vortex problem 0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
−0.1
−0.1
−0.2
−0.2
−0.3
−0.3
−0.4
−0.4
−0.5 −0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
−0.5 −0.5
−0.4
(a) 1st order
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
(b) 2nd order
Figure : Final deformed grids at time t = 1, on a 20 × 18 polar mesh Franc¸ois Vilar (IMAG)
Cell-centered Lagrangian schemes
15 Mai 2017
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Numerical results in 2D
3rd order scheme
Gresho-like vortex problem 0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
−0.1
−0.1
−0.2
−0.2
−0.3
−0.3
−0.4
−0.4
−0.5 −0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
−0.5 −0.5
−0.4
(a) 1st order
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
(b) 2nd order
Figure : Final deformed grids at time t = 1, on a 20 × 18 polar mesh Franc¸ois Vilar (IMAG)
Cell-centered Lagrangian schemes
15 Mai 2017
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Numerical results in 2D
3rd order scheme
Gresho-like vortex problem 0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
−0.1
−0.1
−0.2
−0.2
−0.3
−0.3
−0.4
−0.4
−0.5 −0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
−0.5 −0.5
−0.4
(c) 3rd order
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
(d) Exact solution
Figure : Final deformed grids at time t = 1, on a 20 × 18 polar mesh Franc¸ois Vilar (IMAG)
Cell-centered Lagrangian schemes
15 Mai 2017
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Numerical results in 2D
3rd order scheme
Gresho-like vortex problem 0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
−0.1
−0.1
−0.2
−0.2
−0.3
−0.3
−0.4
−0.4
−0.5 −0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
−0.5 −0.5
−0.4
(c) 3rd order
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
(d) Exact solution
Figure : Final deformed grids at time t = 1, on a 20 × 18 polar mesh Franc¸ois Vilar (IMAG)
Cell-centered Lagrangian schemes
15 Mai 2017
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Numerical results in 2D
3rd order scheme
Gresho-like vortex problem 1
5.6 solution 1st order 2nd order 3rd order
0.9
solution 1st order 2nd order 3rd order 5.5
0.8
0.7 5.4 0.6
0.5
5.3
0.4 5.2 0.3
0.2 5.1 0.1
0
5 0
0.2
0.4
0.6
0.8
1
0
(e) Velocity profiles
0.2
0.4
0.6
0.8
1
(f) Pressure profiles
Figure : Velocity and pressure profiles at time t = 1, on a 20 × 18 polar grid
Franc¸ois Vilar (IMAG)
Cell-centered Lagrangian schemes
15 Mai 2017
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Numerical results in 2D
3rd order scheme
Gresho-like vortex problem 1.06 solution 1st order 2nd order 3rd order
1.05
1.04
1.03
1.02
1.01
1
0.99
0.98
0.97 0
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1
Figure : Density profiles at time t = 1, on a 20 × 18 polar grid
Franc¸ois Vilar (IMAG)
Cell-centered Lagrangian schemes
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Numerical results in 2D
3rd order scheme
Kidder isentropic compression 1
1
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0
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(g) 1st order
0.2
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(h) 2nd order
Figure : Intial and final grids for a Kidder problem on a 10 × 5 polar mesh Franc¸ois Vilar (IMAG)
Cell-centered Lagrangian schemes
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Numerical results in 2D
3rd order scheme
Kidder isentropic compression Ri: exact solution Ri: 1st order Ri: 2nd order Re: exact solution Re: 1st order Re: 2nd order
1
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0.001
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Figure : Interior and exterior shell radii evolution for a Kidder problem on a 10 × 5 polar mesh Franc¸ois Vilar (IMAG)
Cell-centered Lagrangian schemes
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Numerical results in 2D
3rd order scheme
Kidder isentropic compression 1
Ri: exact solution Ri: 3rd order Re: exact solution Re: 3rd order
1 0.9
0.8
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0.7
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0.8
(i) Initial and final grids
0.9
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0.003
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0.007
(j) Shell radii evolution
Figure : 3rd order solution for a Kidder compression problem on a 10 × 3 polar grid
Franc¸ois Vilar (IMAG)
Cell-centered Lagrangian schemes
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Numerical results in 2D
3rd order scheme
Accuracy and computational time for a Taylor-Green vortex D.O.F 600 2400
N 24 × 25 48 × 50
ELh1 2.67E-2 1.36E-2
ELh2 3.31E-2 1.69E-2
ELh∞ 8.55E-2 4.37E-2
time (sec) 2.01 11.0
Table: 1st order scheme
D.O.F 630 2436
N 14 × 15 28 × 29
ELh1 2.76E-3 7.52E-4
ELh2 3.33E-3 9.02E-4
ELh∞ 1.07E-2 2.73E-3
time (sec) 2.77 11.3
Table: 2nd order scheme
D.O.F 600 2400
N 10 × 10 20 × 20
ELh1 2.67E-4 3.43E-5
ELh2 3.36E-4 4.36E-5
ELh∞ 1.21E-3 1.66E-4
time (sec) 4.00 30.6
Table: 3rd order scheme
Franc¸ois Vilar (IMAG)
Cell-centered Lagrangian schemes
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Numerical results in 2D
3rd order scheme
Sedov point blast problem - spurious deformations 5.5
solution 3rd order
6 5 1 4.5
5
4 0.8 3.5
4
3 0.6
3 2.5 2 0.4
2 1.5 1
0.2
1
0.5
0
0 0
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0.4
0.6
0.8
1
1.2
0
0.2
(k) Density field
0.4
0.6
0.8
1
1.2
1.4
(l) Density profiles
Figure : Third-order solution at time t = 1 for a Sedov problem on a 30 × 30 Cartesian mesh
Franc¸ois Vilar (IMAG)
Cell-centered Lagrangian schemes
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Numerical results in 2D
Even higher-order scheme
Taylor-Green vortex MESH FOR A TAYLOR-GREEN PROBLEM WITH A 5th ORDER SCHEME
MESH FOR A TAYLOR-GREEN PROBLEM WITH A 3rd ORDER SCHEME 1
1
0.9
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0
0 0
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1
0
(m) 3rd order
0.2
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0.6
0.8
1
(n) 5th order
Figure : Final deformed grids at time t = 0.6, for 16 triangular cells meshes
Franc¸ois Vilar (IMAG)
Cell-centered Lagrangian schemes
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Numerical results in 2D
Even higher-order scheme
Taylor-Green vortex MESH FOR A TAYLOR-GREEN PROBLEM WITH A 3rd ORDER SCHEME
MESH FOR A TAYLOR-GREEN PROBLEM WITH A 5th ORDER SCHEME
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
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0.3
0.2
0.2
0.1
0.1
0
0 0
0.2
0.4
0.6
0.8
1
0
(m) 3rd order
0.2
0.4
0.6
0.8
1
(n) 5th order
Figure : Final deformed grids at time t = 0.6, for 16 triangular cells meshes
Franc¸ois Vilar (IMAG)
Cell-centered Lagrangian schemes
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Numerical results in 2D
Even higher-order scheme
Sod shock tube problem - symmetry preservation DENSITY FOR A POLAR SOD PROBLEM WITH A 4th ORDER SCHEME
1.1
1
solution 4th order
1
1 0.9 0.9
0.9
0.8 0.8
0.8
0.7 0.7
0.7
0.6 0.6
0.5
0.6
0.5
0.5
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0 0
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1
0
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0.2
(o) Density field
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
(p) Density profiles
Figure : 4th order solution for a Sod shock tube problem on a polar grid made of 308 triangular cells
Franc¸ois Vilar (IMAG)
Cell-centered Lagrangian schemes
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Numerical results in 2D
Even higher-order scheme
F. V ILAR , P.-H. M AIRE AND R. A BGRALL, Cell-centered discontinuous Galerkin discretizations for two-dimensional scalar conservation laws on unstructured grids and for one-dimensional Lagrangian hydrodynamics. CAF, 2010. F. V ILAR, Cell-centered discontinuous Galerkin discretization for two-dimensional Lagrangian hydrodynamics. CAF, 2012. F. V ILAR , P.-H. M AIRE AND R. A BGRALL, A discontinuous Galerkin discretization for solving the two-dimensional gas dynamics equations written under total lagrangian formulation on general unstructured grids. JCP, 2014. F. V ILAR , C.-W. S HU AND P.-H. M AIRE, Positivity-preserving cell-centered Lagrangian schemes for multi-material compressible flows: Form first-order to high-orders. Part I: The 1D case. JCP, 2016. F. V ILAR , C.-W. S HU AND P.-H. M AIRE, Positivity-preserving cell-centered Lagrangian schemes for multi-material compressible flows: Form first-order to high-orders. Part II: The 2D case. JCP, 2016. Franc¸ois Vilar (IMAG)
Cell-centered Lagrangian schemes
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