Introduction
Cell-Centered Lagrangian schemes
Lagrangian and Eulerian descriptions
Discretization
Numerical results
Conclusion
High-order cell-centered DG scheme for Lagrangian hydrodynamics F. Vilar1 , P.- H. Maire2 , R. Abgrall3 1
Brown University, Division of Applied Mathematics 182 George Street, Providence, RI 02912 2
CEA CESTA, BP 2, 33 114 Le Barp, France
3
INRIA and University of Bordeaux, Team Bacchus, ´ 351 Cours de la Liberation, 33 405 Talence Cedex, France
September 3rd, 2013
September 3rd, 2013
Franc¸ois Vilar
High-order Cell-Centered DG scheme
1 / 46
Introduction
Cell-Centered Lagrangian schemes
Lagrangian and Eulerian descriptions
1
Introduction
2
Cell-Centered Lagrangian schemes
3
Lagrangian and Eulerian descriptions
4
Discretization
5
Numerical results
6
Conclusion
September 3rd, 2013
Franc¸ois Vilar
Discretization
Numerical results
High-order Cell-Centered DG scheme
Conclusion
1 / 46
Introduction
Cell-Centered Lagrangian schemes
Lagrangian and Eulerian descriptions
Discretization
Numerical results
Discontinuous Galerkin schemes
1
Introduction
2
Cell-Centered Lagrangian schemes
3
Lagrangian and Eulerian descriptions
4
Discretization
5
Numerical results
6
Conclusion
September 3rd, 2013
Franc¸ois Vilar
Conclusion
High-order geometries
High-order Cell-Centered DG scheme
1 / 46
Introduction
Cell-Centered Lagrangian schemes
Lagrangian and Eulerian descriptions
Discretization
Numerical results
Discontinuous Galerkin schemes
Conclusion
High-order geometries
DG schemes Natural extension of Finite Volume method Piecewise polynomial approximation of the solution in the cells High-order scheme to achieve high accuracy
Procedure Local variational formulation Choice of the numerical fluxes (global L2 stability, entropy inequality) Time discretization - TVD multistep Runge-Kutta C.-W. S HU, Discontinuous Galerkin methods: General approach and stability. 2008. Limitation - vertex-based hierarchical slope limiters D. K UZMIN, A vertex-based hierarchical slope limiter for p-adaptive discontinuous Galerkin methods. J. Comp. Appl. Math., 2009.
September 3rd, 2013
Franc¸ois Vilar
High-order Cell-Centered DG scheme
2 / 46
Introduction
Cell-Centered Lagrangian schemes
Lagrangian and Eulerian descriptions
Discretization
Numerical results
Discontinuous Galerkin schemes
Circular polar grid: 10 × 1 cells
Taylor-Green exact motion
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
Conclusion
High-order geometries
0.7
0.8
0.9
1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
V. D OBREV, T. E LLIS , T. KOLEV AND R. R IEBEN, High Order Curvilinear Finite Elements for Lagrangian Hydrodynamics. Part I: General Framework, 2010. Presentation available at https://computation.llnl.gov/casc/blast/blast.html September 3rd, 2013
Franc¸ois Vilar
High-order Cell-Centered DG scheme
3 / 46
Introduction
Cell-Centered Lagrangian schemes
Lagrangian and Eulerian descriptions
1
Introduction
2
Cell-Centered Lagrangian schemes
3
Lagrangian and Eulerian descriptions
4
Discretization
5
Numerical results
6
Conclusion
September 3rd, 2013
Franc¸ois Vilar
Discretization
Numerical results
High-order Cell-Centered DG scheme
Conclusion
3 / 46
Introduction
Cell-Centered Lagrangian schemes
Lagrangian and Eulerian descriptions
Discretization
Numerical results
Conclusion
Finite volume schemes on moving mesh J. K. Dukowicz: CAVEAT scheme A computer code for fluid dynamics problems with large distorsion and internal slip, 1986
´ GLACE scheme B. Despres: Lagrangian Gas Dynamics in Two Dimensions and Lagrangian systems, 2005
P.-H. Maire: EUCCLHYD scheme A cell-centered Lagrangian scheme for two-dimensional compressible flow problems, 2007
G. Kluth: Hyperelasticity Discretization of hyperelasticity with a cell-centered Lagrangian scheme, 2010
S. Del Pino: Curvilinear Finite Volume method A curvilinear finite-volume method to solve compressible gas dynamics in semi-Lagrangian coordinates, 2010
P. Hoch: Finite Volume method on unstructured conical meshes Extension of ALE methodology to unstructured conical meshes, 2011
DG scheme on initial mesh ` R. Loubere: DG scheme for Lagrangian hydrodynamics A Lagrangian Discontinuous Galerkin-type method on unstructured meshes to solve hydrodynamics problems, 2004 September 3rd, 2013
Franc¸ois Vilar
High-order Cell-Centered DG scheme
4 / 46
Introduction
Cell-Centered Lagrangian schemes
Lagrangian and Eulerian descriptions
1
Introduction
2
Cell-Centered Lagrangian schemes
3
Lagrangian and Eulerian descriptions
4
Discretization
5
Numerical results
6
Conclusion
September 3rd, 2013
Franc¸ois Vilar
Discretization
Numerical results
High-order Cell-Centered DG scheme
Conclusion
4 / 46
Introduction
Cell-Centered Lagrangian schemes
Lagrangian and Eulerian descriptions
Discretization
Numerical results
Conclusion
Flow transformation of the fluid The fluid flow is described mathematically by the continuous transformation, Φ, so-called mapping such as Φ : X −→ x = Φ(X , t) n
N Φ x = Φ(X, t)
01
X
0011
Ω
∂ω
∂Ω
ω
Figure: Notation for the flow map.
where X is the Lagrangian (initial) coordinate, x the Eulerian (actual) coordinate, N the Lagrangian normal and n the Eulerian normal
Deformation Jacobian matrix: deformation gradient tensor F = ∇X Φ = September 3rd, 2013
∂x ∂X
and
J = det F > 0
Franc¸ois Vilar
High-order Cell-Centered DG scheme
5 / 46
Introduction
Cell-Centered Lagrangian schemes
Lagrangian and Eulerian descriptions
Discretization
Numerical results
Conclusion
Trajectory equation dx = U(x, t), dt
x(X , 0) = X
Material time derivative d ∂ f (x, t) = f (x, t) + U ∇x f (x, t) dt ∂t
Transformation formulas FdX = dx
Change of shape of infinitesimal vectors
0
ρ = ρJ JdV = dv
Mass conservation Measure of the volume change
JF−t NdS = nds
Nanson formula
Differential operators transformations ∇x P = J1 ∇X (P JF−t ) ∇x U = September 3rd, 2013
1 J ∇X
(JF
−1
U)
Franc¸ois Vilar
Gradient operator Divergence operator High-order Cell-Centered DG scheme
6 / 46
Introduction
Cell-Centered Lagrangian schemes
Lagrangian and Eulerian descriptions
Discretization
Numerical results
Conclusion
Piola compatibility condition ∇x G = 0, where G = JF−t is the cofactor matrix of F Z Z Z ∇x G dV = G N dS = n ds = 0 Ω
∂Ω
∂ω
Gas dynamics system written in its total Lagrangian form dF − ∇X U = 0 dt d 1 ρ0 ( ) − ∇X (Gt U) = 0 dt ρ dU ρ0 + ∇X (P G) = 0 dt dE + ∇X (Gt PU) = 0 ρ0 dt
Deformation gradient tensor equation Specific volume equation Momentum equation Total energy equation
Thermodynamical closure EOS: September 3rd, 2013
P = P(ρ, ε) where Franc¸ois Vilar
ε = E − 12 U 2 High-order Cell-Centered DG scheme
7 / 46
Introduction
Cell-Centered Lagrangian schemes
DG general framework
Lagrangian and Eulerian descriptions
Deformation gradient tensor
Discretization
1
Introduction
2
Cell-Centered Lagrangian schemes
3
Lagrangian and Eulerian descriptions
4
Discretization
5
Numerical results
6
Conclusion
September 3rd, 2013
Franc¸ois Vilar
Discretization
Control point solvers
Numerical results Limitation
High-order Cell-Centered DG scheme
Conclusion
Initial deformation
7 / 46
Introduction
Cell-Centered Lagrangian schemes
DG general framework
Lagrangian and Eulerian descriptions
Deformation gradient tensor
Discretization
Discretization
Control point solvers
Numerical results Limitation
Conclusion
Initial deformation
(s + 1) th order DG discretization Let {Ωc }c be a partition of the domain Ω into polygonal cells {σkc }k=0...K basis of P s (Ωc ), where K + 1 = φch (X , t) =
K X
(s+1)(s+2) 2
φck (t)σkc (X ) approximate function of φ(X , t) on Ωc
k=0
Definitions Z 1 ρ0 (X ) X dV , mc Ωc where mc is the constant mass of the cell Ωc Z 1 The mean value hφic = ρ0 (X ) φ(X ) dV mc Ω c of the function φ over the cell Ωc Z The associated scalar product (φ ψ)c = ρ0 (X ) φ(X ) ψ(X ) dV Center of mass X c = (Xc , Yc )t =
Ωc
September 3rd, 2013
Franc¸ois Vilar
High-order Cell-Centered DG scheme
8 / 46
Introduction
Cell-Centered Lagrangian schemes
DG general framework
Lagrangian and Eulerian descriptions
Deformation gradient tensor
Discretization
Discretization
Control point solvers
Numerical results Limitation
Conclusion
Initial deformation
Taylor expansion on the cell, located at the center of mass φ(X ) = φ(X c ) +
s X k k−j j X (X − Xc ) (Y − Yc ) ∂k φ (X c ) + o(kX − X c ks ) j!(k − j)! ∂X k−j ∂Y j k=1 j=0
(s + 1) th order scheme polynomial Taylor basis The first-order polynomial component and the associated basis function φc0 = hφic
and
σ0c = 1
The k th -order polynomial components and the associated basis functions
σ ck (k +1) 2
+j
∂k φ (X c ), φck (k +1) +j = (∆Xc )k −j (∆Yc )j ∂X k −j ∂Y j 2 k−j j k−j j X −Xc Y −Yc X −Xc Y −Yc 1 , = j!(k−j)! ∆Xc ∆Yc − ∆Xc ∆Yc c
where 0 < k ≤ s, j = 0 . . . k , ∆Xc =
Xmax −Xmin 2
and ∆Yc =
Ymax −Ymin 2
¨ H. L UO, J. D. B AUM AND R. L OHNER , A DG method based on a Taylor basis for the compressible flows on arbitrary grids. J. Comp. Phys., 2008. September 3rd, 2013
Franc¸ois Vilar
High-order Cell-Centered DG scheme
9 / 46
Introduction
Cell-Centered Lagrangian schemes
DG general framework
Lagrangian and Eulerian descriptions
Deformation gradient tensor
Discretization
Discretization
Control point solvers
Numerical results Limitation
Conclusion
Initial deformation
Outcome First moment associated to the basis function σ0c = 1 is the mass averaged value φc0 = hφic The successive moments can be identified as the successive derivatives of the function expressed at the center of mass of the cell φck (k+1) +j = (∆Xc )k−j (∆Yc )j 2
∂k φ (X c ) ∂X k−j ∂Y j
The first basis function is orthogonal to the other ones (σ0c σkc )c = mc δ0k Same basis functions regardless the shape of the cells (squares, triangles, generic polygonal cells)
September 3rd, 2013
Franc¸ois Vilar
High-order Cell-Centered DG scheme
10 / 46
Introduction
Cell-Centered Lagrangian schemes
DG general framework
Lagrangian and Eulerian descriptions
Deformation gradient tensor
Discretization
Discretization
Control point solvers
Numerical results Limitation
Conclusion
Initial deformation
Lagrangian gas dynamics equation type ρ0
dφ + ∇X (Gt f ) = 0, where f is the flux function dt G = JF−t is the cofactor matrix of F
Local variational formulations Z Ωc
ρ0
dφ c σ dV dt j
Z K X d φc
ρ0 σjc σkc dV Ωc k=0 Z Z = f G ∇X σjc dV − =
k
dt
Ωc
f σjc GNdS
∂Ωc
Geometric Conservation Law (GCL) Equation on the first moment of the specific volume Z Z Z dJ d |ωc | dV = = ∇X (Gt U) dV = U GNdS dt Ωc dt Ωc ∂Ωc September 3rd, 2013
Franc¸ois Vilar
High-order Cell-Centered DG scheme
11 / 46
Introduction
Cell-Centered Lagrangian schemes
DG general framework
Lagrangian and Eulerian descriptions
Deformation gradient tensor
Discretization
Discretization
Control point solvers
Numerical results Limitation
Conclusion
Initial deformation
Mass matrix properties Z
ρ0 σjc σkc dV = σjc σkc
Ωc
c
generic coefficient of the symmetric positive
definite mass matrix σ0c σkc c = mc δ0k mass averaged equation is independent of the other polynomial basis components equations
Interior terms Z
f G ∇X σjc dV
is evaluated through the use of a two-dimensional
Ωc
high-order quadrature rule
Boundary terms Z
f σjc GNdS
required a specific treatment to ensure the GCL
∂Ωc
It remains to determine the numerical fluxes September 3rd, 2013
Franc¸ois Vilar
High-order Cell-Centered DG scheme
12 / 46
Introduction
Cell-Centered Lagrangian schemes
DG general framework
Lagrangian and Eulerian descriptions
Deformation gradient tensor
Discretization
Discretization
Control point solvers
Numerical results Limitation
Conclusion
Initial deformation
Requirements Consistency of vector GNdS = nds at the interfaces of the cells Continuity of vector GN at cell interfaces on both sides of the interface Preservation of uniform flows, G = JF−t the cofactor matrix Z Z Z G∇X σjc dV = σjc GNdS ⇐⇒ σjc (∇X G) dV = 0 Ωc
∂Ωc
Ωc
Generalization of the weak form of the Piola compatibility condition
Tensor F discretization Discretization of tensor F by means of a mapping defined on triangular cells Partition of the polygonal cells in the initial configuration into non-overlapping triangles Ωc =
ntri [ i=1
September 3rd, 2013
Ωc Tic
Ti c
Franc¸ois Vilar
High-order Cell-Centered DG scheme
13 / 46
Introduction
Cell-Centered Lagrangian schemes
DG general framework
Lagrangian and Eulerian descriptions
Deformation gradient tensor
Discretization
Discretization
Control point solvers
Numerical results Limitation
Conclusion
Initial deformation
(s + 1) th order continuous mapping function We develop Φ on the Finite Elements basis functions Λiq in Ti of degree s Φih (X , t) =
X
Λiq (X ) Φq (t),
q∈Q(i)
where Q(i) is the Ti control points set, including the vertices {p− , p, p+ } Φq (t) = Φ(X q , t) = x q
X d Φq d = U q =⇒ Fi (X , t) = U q (t) ⊗ ∇X Λiq (X ) dt dt q∈Q(i)
G. K LUTH AND B. D ESPR E´ S, Discretization of hyperelasticity on unstructured mesh with a cell-centered Lagrangian scheme. J. Comp. Phys., 2010.
Outcome Satisfaction of the Piola compatibility condition everywhere Consistency and continuity of the Eulerian normal GN September 3rd, 2013
Franc¸ois Vilar
High-order Cell-Centered DG scheme
14 / 46
Introduction
Cell-Centered Lagrangian schemes
DG general framework
Lagrangian and Eulerian descriptions
Deformation gradient tensor
Discretization
Discretization
Control point solvers
Numerical results Limitation
Conclusion
Initial deformation
Example of the fluid flow mapping in the fourth order case p+
p+ Φ
τi p
−
Ti p−
p
p
Figure: Nodes arrangement for a cubic Lagrange Finite Element mapping.
Curved edges definition using s + 1 control points Projection of the continuous mapping function Φ on the face fpp+ x |pp+ (ζ) = x p λp (ζ) +
X
x q λq (ζ) + x p+ λp+ (ζ),
q∈Q(pp+ )\{p,p+ }
where Q(pp+ ) is the face control points set, ζ ∈ [0, 1] the curvilinear abscissa and λq the 1D Finite Element basis functions of degree s September 3rd, 2013
Franc¸ois Vilar
High-order Cell-Centered DG scheme
15 / 46
Introduction
Cell-Centered Lagrangian schemes
DG general framework
Lagrangian and Eulerian descriptions
Deformation gradient tensor
Discretization
Discretization
Control point solvers
Local variational formulations ntri
Z
ρ0
Ωc
X dφ c σj dV = − dt +
Ti c
G ∇X σjc f dV
p
p∈P(c)
f qc
p
f+ pc Ωc
p+
X Z
Conclusion
Initial deformation
N pp+ f− p+ c
Z
i=1
p+
Numerical results Limitation
f− pc N p− p
f σjc GNdL p−
Polynomial assumptions on face fpp+ f |pp+ (ζ) = f + pc λp (ζ) +
X
f qc λq (ζ) + f − p+ c λp+ (ζ)
q\{p,p+ }
Polynomial properties on face fpp+ X ∂λq ∂x dζ|pp+ × e z = (ζ) (x q × e z ) ∂ζ ∂ζ q X (ζ) = σjc (X p )λp (ζ) + σjc (X q )λq (ζ) + σjc (X p+ )λp+ (ζ) +
G N dL|pp+ (ζ) = n dl|pp+ = σj|c
pp
q\{p,p+ } September 3rd, 2013
Franc¸ois Vilar
High-order Cell-Centered DG scheme
16 / 46
Introduction
Cell-Centered Lagrangian schemes
DG general framework
Lagrangian and Eulerian descriptions
Deformation gradient tensor
Discretization
Discretization
Control point solvers
Numerical results Limitation
Conclusion
Initial deformation
Fundamental assumptions U± pc = U p , PU = P U
∀c ∈ C(p) =⇒
and U qL = U qR = U q
± (PU)± pc = Ppc U p
and (PU)qc = Pqc U q
Procedure Analytical integration + index permutation
Weighted control point normals λp|pp+ (ζ)σj|pp+ (ζ) ∂x dζ × ez | + ∂ζ pp R 1 = 0 λp|p− p (ζ)σj|p− p (ζ) ∂x ∂ζ dζ|p− p × e z
+,j +,j lpc npc = −,j −,j lpc npc
R
1 0
j −,j −,j +,j +,j lpc njpc = lpc npc + lpc npc R 1 j lqc njqc = 0 λq|pp+ (ζ)σj|pp+ (ζ) ∂x dζ |pp+ × e z ∂ζ
j th moment of the subcell forces − −,j −,j + +,j +,j F jpc = Ppc lpc npc + Ppc lpc npc September 3rd, 2013
Franc¸ois Vilar
j and F jqc = Pqc lqc njqc High-order Cell-Centered DG scheme
17 / 46
Introduction
Cell-Centered Lagrangian schemes
DG general framework
Lagrangian and Eulerian descriptions
Deformation gradient tensor
Discretization
Discretization
Control point solvers
Numerical results Limitation
Conclusion
Initial deformation
Semi-discrete equations GCL compatible ntri Z X X X d 1 j j U p lpc ρ0 ( )σjc dV = − U G∇X σjc dV + njpc + U q lqc njqc c dt ρ Ωc i=1 Ti p∈P(c) q\{p,p+ } Z ntri Z X X X dU c F jpc + ρ0 σj dV = PG∇X σjc dV − F jqc c dt Ωc i=1 Ti p∈P(c) q\{p,p+ } Z ntri Z X X X dE c U p F jpc + σj dV = PU G∇X σjc dV − U q F jqc ρ0 c dt Ti Ωc + Z
i=1
q\{p,p }
p∈P(c)
Equation on the first moment of the specific volume d |ωc | = dt
September 3rd, 2013
Z U GNdL = ∂Ωc
X p∈P(c)
Franc¸ois Vilar
0 0 U p lpc npc +
X
0 0 U q lqc nqc
q∈Q(pp+ )\{p,p+ }
High-order Cell-Centered DG scheme
18 / 46
Introduction
Cell-Centered Lagrangian schemes
DG general framework
Lagrangian and Eulerian descriptions
Deformation gradient tensor
Discretization
Discretization
Control point solvers
Numerical results Limitation
Conclusion
Initial deformation
Entropic semi-discrete equation Fundamental assumption P U = P U The use of variational formulations and Piola condition leads to Z Z dη ρ0 θ dV = (P − Ph )(U h − U) GNdS, dt Ωc ∂Ωc where η is the specific entropy and θ the absolute temperature defined by means of the Gibbs identity
Entropic semi-discrete equation Z A sufficient condition to satisfy
ρ0 θ
Ωc
P − Ph = −Z (U − U h )
dη dV ≥ 0 is dt
GN = −Z (U − U h ) n, kGNk
where Z ≥ 0 has the physical dimension of a density times a velocity September 3rd, 2013
Franc¸ois Vilar
High-order Cell-Centered DG scheme
19 / 46
Introduction
Cell-Centered Lagrangian schemes
DG general framework
Lagrangian and Eulerian descriptions
Deformation gradient tensor
Discretization
Discretization
Control point solvers
Numerical results Limitation
Conclusion
Initial deformation
Subcell forces definitions − −,j −,j + +,j +,j F jpc = Ppc lpc npc + Ppc lpc npc
j and F jqc = Pqc lqc njqc
j th moment of the control point subcell forces The use of P = Phc − Zc (U − U ch ) n
to calculate F jpc and F jqc leads to
j F jpc = Phc (X p , t) lpc njpc − Mjpc (U p − U ch (X p , t)), j F jqc = Phc (X q , t) lqc njqc − Mjqc (U q − U ch (X q , t)),
Mjpc = Zc
−,j −,j +,j +,j +,0 lpc npc ⊗ n−,0 and pc + lpc npc ⊗ npc
j Mjqc = Zc lqc njqc ⊗ n0qc
Momentum and total energy conservation X
F 0pc = 0
and
F 0qL + F 0qR = 0
c∈C(p)
September 3rd, 2013
Franc¸ois Vilar
High-order Cell-Centered DG scheme
20 / 46
Introduction
Cell-Centered Lagrangian schemes
DG general framework
Lagrangian and Eulerian descriptions
Deformation gradient tensor
Discretization
Discretization
Control point solvers
Numerical results Limitation
Conclusion
Initial deformation
Nodal velocity X 0 0 Phc (X p , t) lpc npc + M0pc U ch (X p , t) ,
Mp U p =
c∈C(p)
where
Mp =
X
M0pc
is a positive definite matrix
c∈C(p)
Face control point velocity Mq U q = Mq where Mq =
ZL U Lh (X q ) + ZR U R h (X q ) ZL + ZR 1 0 ZR MqR
=
1 0 ZL MqL
! −
PhR (X q ) − PhL (X q ) 0 0 lqL nqL , ZL + ZR
0 0 = lqL nqL ⊗ n0qL is positive semi-definite
1D approximate Riemann problem solution (U q
n0qL )
September 3rd, 2013
=
ZL U Lh (X q ) + ZR U R h (X q ) ZL + ZR Franc¸ois Vilar
! n0qL −
PhR (X q ) − PhL (X q ) ZL + ZR
High-order Cell-Centered DG scheme
21 / 46
Introduction
Cell-Centered Lagrangian schemes
DG general framework
Lagrangian and Eulerian descriptions
Deformation gradient tensor
Discretization
Discretization
Control point solvers
Numerical results Limitation
Conclusion
Initial deformation
Tangential component of the face control point velocity (U q
t 0qL )
=
ZL U Lh (X q ) + ZR U R h (X q ) ZL + ZR
! t 0qL
Face control point velocity Uq =
P R (X q ) − PhL (X q ) 0 ZL U Lh (X q ) + ZR U R h (X q ) − h nqL ZL + ZR ZL + ZR
Deformation tensor X d U Q ⊗ ∇X ΛiQ Fi = dt Q∈Q(i)
Interior points velocity U Q = U ch (X Q , t) September 3rd, 2013
1 0 0 1 0 1
11 00
0 1 1 0 0 1
Φ
1 0 0 1 11 00 00 11
1 0 11 00 00 11
11 00 00 11 00 11
0 1 1 0 0 1
11 00
11 00 00 11
11 00 11 00
1 0 11 00 00 11 1 0 1 0
Franc¸ois Vilar
11 00
1 0 1 0
1 0 0 1
Ωc
11 00 11 00
1 0 1 0
11 00 00 11
1 0 0 1
1 0 1 0 1 0
11 00 1 0 00 11 11 00 00 11
11 00 00 11 00 11
0 1 1 0 0 1
1 0
11 00
ωc
1 0 1 0 1 0
High-order Cell-Centered DG scheme
1 0
11 00 00 11 11 00 11 00
1 0 0 1 0 1 11 00
1 0
22 / 46
Introduction
Cell-Centered Lagrangian schemes
DG general framework
Lagrangian and Eulerian descriptions
Deformation gradient tensor
Riemann invariants differentials
Discretization
Discretization
Control point solvers
Numerical results Limitation
Conclusion
Initial deformation
Mean value linearization c = U ch t αt,h
dαt = dU t dα− = d( ρ1 ) − dα+ = d( ρ1 ) +
1 ρa 1 ρa
dU n
c 1 Zc U h n ( ρ1 )ch + Z1c U ch n Ehc − U c0 U ch + P0c
c = ( ρ1 )ch − α−,h
dU n
c = α+,h
dαE = dE − U dU + P d( ρ1 )
c αE,h =
( ρ1 )ch
where Zc = a0c ρc0
a denotes the sound speed
System variables polynomial approximation components c c ( ρ1 )ck = 21 (α+,k + α−,k ) c c c U ck = 12 Zc (α+,k − α−,k )n + αt,k t c c c c c c Ekc = αE,k + 21 Zc (α+,k − α−,k )U c0 n + αt,k U c0 t − 12 P0c (α+,k + α−,k )
Unit direction ensuring symmetry preservation n=
U c0 kU c0 k
September 3rd, 2013
and
t = ez × Franc¸ois Vilar
U c0 kU c0 k High-order Cell-Centered DG scheme
23 / 46
Introduction
Cell-Centered Lagrangian schemes
DG general framework
Lagrangian and Eulerian descriptions
Deformation gradient tensor
Discretization
Discretization
Control point solvers
Numerical results Limitation
Conclusion
Initial deformation
Composed derivatives Ωc
FT = ∇Xr ΦT (X r , t)
X
= ∇X ΦH (X , t) ◦ ∇Xr Φ0 (X r )
= FH F0
ΦH (X, t)
Ωrc
JT (X r , t) = JH (X , t) J0 (X r )
Φ0(X r )
Xr x
Mass conservation
ωc
ρ0 J0 = ρ JT
ΦT (X r , t)
Modification of the mass matrix Z K X d ψhc d ψk σj dω = ρ0 J0 σj σk dΩr time rate of change of r dt dt ωc Ω c k=0 successive moments of function ψ New definitions of mass matrix, of mass averaged value and of the associated scalar product Z
ρ
September 3rd, 2013
Franc¸ois Vilar
High-order Cell-Centered DG scheme
24 / 46
Introduction
Cell-Centered Lagrangian schemes
Lagrangian and Eulerian descriptions
Discretization
Numerical results
Second-order scheme
1
Introduction Discontinuous Galerkin schemes High-order geometries
2
Cell-Centered Lagrangian schemes
3
Lagrangian and Eulerian descriptions
4
Discretization DG general framework Deformation gradient tensor Discretization Control point solvers Limitation Initial deformation
5
Numerical results Second-order scheme Third-order scheme
6
Conclusion
September 3rd, 2013
Conclusion
Third-order scheme
Franc¸ois Vilar
High-order Cell-Centered DG scheme
24 / 46
Introduction
Cell-Centered Lagrangian schemes
Lagrangian and Eulerian descriptions
Discretization
Numerical results
Second-order scheme
Conclusion
Third-order scheme
Sedov point blast problem on a Cartesian grid 5.5
solution 2nd order
6
5
1
4.5
5
4
0.8
4 3.5 3
0.6
3 2.5 2
0.4
2
1.5 1
0.2
1
0.5 0
0 0
0.2
0.4
0.6
0.8
1
(a) Second-order scheme.
1.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
(b) Density profile.
Figure: Point blast Sedov problem on a Cartesian grid made of 30 × 30 cells: density.
September 3rd, 2013
Franc¸ois Vilar
High-order Cell-Centered DG scheme
25 / 46
Introduction
Cell-Centered Lagrangian schemes
Lagrangian and Eulerian descriptions
Discretization
Numerical results
Second-order scheme
Conclusion
Third-order scheme
Sedov point blast problem on unstructured grids
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
(a) Polygonal grid.
1
1.2
0
0.2
0.4
0.6
0.8
1
1.2
(b) Triangular grid.
Figure: Unstructured initial grids for the point blast Sedov problem. September 3rd, 2013
Franc¸ois Vilar
High-order Cell-Centered DG scheme
26 / 46
Introduction
Cell-Centered Lagrangian schemes
Lagrangian and Eulerian descriptions
Discretization
Numerical results
Second-order scheme
Conclusion
Third-order scheme
Sedov point blast problem a polygonal grid 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
(a) Second-order scheme.
1.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
(b) Density profile.
Figure: Point blast Sedov problem on an unstructured grid made of 775 polygonal cells: density map.
September 3rd, 2013
Franc¸ois Vilar
High-order Cell-Centered DG scheme
27 / 46
Introduction
Cell-Centered Lagrangian schemes
Lagrangian and Eulerian descriptions
Discretization
Numerical results
Second-order scheme
Conclusion
Third-order scheme
Sedov point blast problem on a triangular grid 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
(a) Second-order scheme.
1.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
(b) Density profile.
Figure: Point blast Sedov problem on an unstructured grid made of 1100 triangular cells: density map.
September 3rd, 2013
Franc¸ois Vilar
High-order Cell-Centered DG scheme
28 / 46
Introduction
Cell-Centered Lagrangian schemes
Lagrangian and Eulerian descriptions
Discretization
Numerical results
Second-order scheme
Conclusion
Third-order scheme
Noh problem 16
exact solution 2nd order
16 0.5
14
14
12
12
10
10
0.4
0.3
8 8
6
0.2 6
4 4
0.1
2 2 0
0 0
0.1
0.2
0.3
0.4
0.5
(a) Second-order scheme.
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
(b) Density profile.
Figure: Noh problem on a Cartesian grid made of 50 × 50 cells: density.
September 3rd, 2013
Franc¸ois Vilar
High-order Cell-Centered DG scheme
29 / 46
Introduction
Cell-Centered Lagrangian schemes
Lagrangian and Eulerian descriptions
Discretization
Numerical results
Second-order scheme
Conclusion
Third-order scheme
Taylor-Green vortex problem, introduced by R. Rieben (LLNL) 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
0.1
0.2
0.3
0.4
0.5
0.6
0.7
(a) Second-order scheme.
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) Exact solution.
Figure: Motion of a 10 × 10 Cartesian mesh through a T.-G. vortex, at t = 0.75. September 3rd, 2013
Franc¸ois Vilar
High-order Cell-Centered DG scheme
30 / 46
Introduction
Cell-Centered Lagrangian schemes
Lagrangian and Eulerian descriptions
Discretization
Numerical results
Second-order scheme
Conclusion
Third-order scheme
Taylor-Green vortex problem L1 h 1 10 1 20 1 40 1 80 1 160
L2
L∞
ELh1
qLh1
ELh2
qLh2
ELh∞
5.06E-3 1.32E-3 3.33E-4 8.35E-5 2.09E-5
1.94 1.98 1.99 2.00 -
6.16E-3 1.62E-3 4.12E-4 1.04E-4 2.60E-5
1.93 1.97 1.99 2.00 -
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: Rate of convergence computed on the pressure at time t = 0.1.
September 3rd, 2013
Franc¸ois Vilar
High-order Cell-Centered DG scheme
31 / 46
Introduction
Cell-Centered Lagrangian schemes
Lagrangian and Eulerian descriptions
Discretization
Numerical results
Second-order scheme
1
Introduction Discontinuous Galerkin schemes High-order geometries
2
Cell-Centered Lagrangian schemes
3
Lagrangian and Eulerian descriptions
4
Discretization DG general framework Deformation gradient tensor Discretization Control point solvers Limitation Initial deformation
5
Numerical results Second-order scheme Third-order scheme
6
Conclusion
September 3rd, 2013
Conclusion
Third-order scheme
Franc¸ois Vilar
High-order Cell-Centered DG scheme
31 / 46
Introduction
Cell-Centered Lagrangian schemes
Lagrangian and Eulerian descriptions
Discretization
Numerical results
Second-order scheme
Conclusion
Third-order scheme
Polar grids 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
(a) Non-uniform grid.
0.8
0.9
1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
(b) One angular cell grid.
Figure: Polar initial grids for the Sod shock tube problem.
September 3rd, 2013
Franc¸ois Vilar
High-order Cell-Centered DG scheme
32 / 46
Introduction
Cell-Centered Lagrangian schemes
Lagrangian and Eulerian descriptions
Discretization
Numerical results
Second-order scheme
Conclusion
Third-order scheme
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
(a) First-order scheme.
1
0.4
0.3
0.2
0.3
0.1
0.2
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
(b) Second-order scheme.
Figure: Sod shock tube problem on a polar grid made of 100 × 3 non-uniform cells.
September 3rd, 2013
Franc¸ois Vilar
High-order Cell-Centered DG scheme
33 / 46
Introduction
Cell-Centered Lagrangian schemes
Lagrangian and Eulerian descriptions
Discretization
Numerical results
Second-order scheme
Conclusion
Third-order scheme
Symmetry preservation 1
1
1.1 solution 3rd order
0.9 0.9 0.8
1 0.9
0.8 0.7
0.8 0.7
0.6
0.7 0.6
0.5
0.6 0.5
0.4
0.5
0.3
0.4
0.4
0.2
0.3
0.3
0.1
0.2
0
0.2 0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
(a) Density map.
1
0
0.2
0.4
0.6
0.8
1
(b) Density profile.
Figure: Third-order DG solution for a Sod shock tube problem on a polar grid made of 100 × 3 non-uniform cells. September 3rd, 2013
Franc¸ois Vilar
High-order Cell-Centered DG scheme
34 / 46
Introduction
Cell-Centered Lagrangian schemes
Lagrangian and Eulerian descriptions
Discretization
Numerical results
Second-order scheme
Conclusion
Third-order scheme
One angular cell polar Sod shock tube problem 1
1
1.1 solution 3rd order
0.9
0.9
0.8 0.8 0.7
1 0.9 0.8
0.7 0.6
0.7 0.6
0.5
0.6 0.5
0.4
0.5
0.3
0.4
0.4
0.2
0.3
0.3
0.1
0.2
0
0.2 0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
(a) Density map.
1
0
0.2
0.4
0.6
0.8
1
(b) Density profile.
Figure: Third-order DG solution for a Sod shock tube problem on a polar grid made of 100 × 1 cells. September 3rd, 2013
Franc¸ois Vilar
High-order Cell-Centered DG scheme
35 / 46
Introduction
Cell-Centered Lagrangian schemes
Lagrangian and Eulerian descriptions
Discretization
Numerical results
Second-order scheme
Conclusion
Third-order scheme
Variant of the incompressible Gresho 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
(a) First-order scheme.
0.3
0.4
0.5
−0.5 −0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
(b) Second-order scheme.
Figure: Motion of a polar grid defined in polar coordinates by (r , θ) ∈ [0, 1] × [0, 2π], with 40 × 18 cells at t = 1: zoom on the zone (r , θ) ∈ [0, 0.5] × [0, 2π]. September 3rd, 2013
Franc¸ois Vilar
High-order Cell-Centered DG scheme
36 / 46
Introduction
Cell-Centered Lagrangian schemes
Lagrangian and Eulerian descriptions
Discretization
Numerical results
Second-order scheme
Conclusion
Third-order scheme
Variant of the incompressible Gresho 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
(a) Third-order scheme.
0.3
0.4
0.5
−0.5 −0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
(b) Exact solution.
Figure: Motion of a polar grid defined in polar coordinates by (r , θ) ∈ [0, 1] × [0, 2π], with 40 × 18 cells at t = 1: zoom on the zone (r , θ) ∈ [0, 0.5] × [0, 2π]. September 3rd, 2013
Franc¸ois Vilar
High-order Cell-Centered DG scheme
37 / 46
Introduction
Cell-Centered Lagrangian schemes
Lagrangian and Eulerian descriptions
Discretization
Numerical results
Second-order scheme
Conclusion
Third-order scheme
Variant of the Gresho vortex problem 5.6
1 solution 1st order 2nd order 3rd order
solution 1st order 2nd order 3rd order
0.9
5.5 0.8
0.7 5.4 0.6
5.3
0.5
0.4 5.2 0.3
0.2 5.1 0.1
5
0 0
0.2
0.4
0.6
(a) Pressure profile.
0.8
1
0
0.2
0.4
0.6
0.8
1
(b) Velocity profile.
Figure: Gresho variant problem on a polar grid defined in polar coordinates by (r , θ) ∈ [0, 1] × [0, 2π], with 40 × 18 cells at t = 1.
September 3rd, 2013
Franc¸ois Vilar
High-order Cell-Centered DG scheme
38 / 46
Introduction
Cell-Centered Lagrangian schemes
Lagrangian and Eulerian descriptions
Discretization
Numerical results
Second-order scheme
Conclusion
Third-order scheme
Variant of the Gresho 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
0.2
0.4
0.6
0.8
1
Figure: Gresho variant problem on a polar grid defined in polar coordinates by (r , θ) ∈ [0, 1] × [0, 2π], with 40 × 18 cells at t = 1: density profile.
September 3rd, 2013
Franc¸ois Vilar
High-order Cell-Centered DG scheme
39 / 46
Introduction
Cell-Centered Lagrangian schemes
Lagrangian and Eulerian descriptions
Discretization
Numerical results
Second-order scheme
Conclusion
Third-order scheme
Kidder isentropic compression 1
250
9
0.4 0.9 8
0.35
0.8
200
7 0.7
0.3 6
0.6
150
0.25 5 0.5
0.2 4 0.4
100 0.15
0.3
3
0.2
2
0.1 50 0.05
0.1
1
0
0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
(a) At time t = 0.
1
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
(b) At time t = 0.9τ .
Figure: Third-order DG solution for a Kidder isentropic compression problem on a polar grid made of 10 × 3 cells: pressure map.
September 3rd, 2013
Franc¸ois Vilar
High-order Cell-Centered DG scheme
40 / 46
Introduction
Cell-Centered Lagrangian schemes
Lagrangian and Eulerian descriptions
Discretization
Numerical results
Second-order scheme
Conclusion
Third-order scheme
Kidder isentropic compression 0.055 solution 3rd order 0.05
0.045
0.04
0.035
0.03
0.025
0.02
0.015
0.01
0.005 0.39
0.395
0.4
0.405
0.41
0.415
0.42
0.425
0.43
0.435
Figure: Third-order DG solution for a Kidder isentropic compression problem on a polar grid made of 10 × 3 cells: density profile.
September 3rd, 2013
Franc¸ois Vilar
High-order Cell-Centered DG scheme
41 / 46
Introduction
Cell-Centered Lagrangian schemes
Lagrangian and Eulerian descriptions
Discretization
Numerical results
Second-order scheme
Conclusion
Third-order scheme
Taylor-Green vortex problem 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
0.1
0.2
0.3
0.4
0.5
0.6
0.7
(a) Third-order scheme.
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) Exact solution.
Figure: Motion of a 10 × 10 Cartesian mesh through a T.-G. vortex, at t = 0.75. September 3rd, 2013
Franc¸ois Vilar
High-order Cell-Centered DG scheme
42 / 46
Introduction
Cell-Centered Lagrangian schemes
Lagrangian and Eulerian descriptions
Discretization
Numerical results
Second-order scheme
Conclusion
Third-order scheme
Taylor-Green vortex problem L1 h 1 10 1 20 1 40 1 80 1 160
L2
L∞
ELh1
qLh1
ELh2
qLh2
ELh∞
2.67E-4 3.43E-5 4.37E-6 5.50E-7 6.91E-8
2.96 2.97 2.99 2.99 -
3.36E-4 4.36E-5 5.59E-6 7.06E-7 8.87E-8
2.94 2.96 2.98 2.99 -
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: Rate of convergence computed on the pressure at time t = 0.1.
September 3rd, 2013
Franc¸ois Vilar
High-order Cell-Centered DG scheme
43 / 46
Introduction
Cell-Centered Lagrangian schemes
Lagrangian and Eulerian descriptions
Discretization
Numerical results
Second-order scheme
Conclusion
Third-order scheme
Taylor-Green vortex problem 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: First-order DG scheme at time t = 0.1.
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: Second-order DG scheme without limitation at time t = 0.1.
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: Third-order DG scheme without limitation at time t = 0.1. September 3rd, 2013
Franc¸ois Vilar
High-order Cell-Centered DG scheme
44 / 46
Introduction
Cell-Centered Lagrangian schemes
Lagrangian and Eulerian descriptions
Discretization
Numerical results
Second-order scheme
Conclusion
Third-order scheme
Sedov point blast problem on a Cartesian grid 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
0.2
0.4
0.6
0.8
1
(a) Third-order scheme.
1.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
(b) Density profile.
Figure: Point blast Sedov problem on a Cartesian grid made of 30 × 30 cells: density.
September 3rd, 2013
Franc¸ois Vilar
High-order Cell-Centered DG scheme
45 / 46
Introduction
Cell-Centered Lagrangian schemes
Lagrangian and Eulerian descriptions
1
Introduction
2
Cell-Centered Lagrangian schemes
3
Lagrangian and Eulerian descriptions
4
Discretization
5
Numerical results
6
Conclusion
September 3rd, 2013
Franc¸ois Vilar
Discretization
Numerical results
High-order Cell-Centered DG scheme
Conclusion
45 / 46
Introduction
Cell-Centered Lagrangian schemes
Lagrangian and Eulerian descriptions
Discretization
Numerical results
Conclusion
Conclusions Development of 2nd and 3rd order DG schemes for the 2D gas dynamics system in a total Lagrangian formalism GCL and Piola compatibility condition ensured by construction Dramatic improvement of symmetry preservation by means of third-order DG scheme Riemann invariants limitation
Perspectives High-order limitation Positivity preserving limitation WENO limiter
Code parallelization Development of a 3rd order DG scheme on moving mesh Extension to 3D Extension to ALE and solid dynamics September 3rd, 2013
Franc¸ois Vilar
High-order Cell-Centered DG scheme
46 / 46
Introduction
Cell-Centered Lagrangian schemes
Lagrangian and Eulerian descriptions
Discretization
Numerical results
Conclusion
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. Computers and Fluids, 2010. F. V ILAR, Cell-centered discontinuous Galerkin discretization for two-dimensional Lagrangian hydrodynamics. Computers and Fluids, 2012. F. V ILAR , P.-H. M AIRE AND R. A BGRALL, Third order Cell-Centered DG scheme for Lagrangian hydrodynamics on general unstructured Bezier grids. Article in preparation.
September 3rd, 2013
Franc¸ois Vilar
High-order Cell-Centered DG scheme
46 / 46
