Extensions of a twodimensional Discontinuous ALE (DISCALE) finite volume framework on arbitrary unstructured conical meshes
ω-Lagrange-ALE (Discontinuous) on conical meshes
September 10 2012,
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OUTLINE
1
ω-Lagrangian-ALE (continuous) on unstructured conical meshes Why conics and description Equations in ALE form Continuous ALE: Rezoning over conical cells Continuous ALE: Remapping conical cells
2
(h, ω)-Discontinuous ALE on conical cells Discontinous trajectories equation Main properties
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Curved Edges and geometrical error definition 1
Domain with curved boundaries.
2
Circular Cauchy data on cartesian domain.
3
Instabilites study with curvilinear perturbation of a multi-material line.
4
Mixed cell and sub cell (VOF, MOF ...).
5
Problems with radial symmetries.
6
Third Order (Lagrangian and Lagrangian+ALE).
Definition Geometrical high order based on implicit curve: f (x, y ) = 0. 1 2
order 1: f (x, y) = ax + by + c. Straight 2 2 order 2: f (x, y) = ax + by + cxy + dx + ey + f .
Conics
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Existing curvilinear method
1 V. Dobrev, T. Ellis, T. Kolev, R. Rieben, Energy conserving finite element discretizations of Lagrangian hydrodynamics. Part 1: Theoretical framework, Downloadable presentation of Multimat’09 conference. 2 Cheng J., Shu C.W., A third order Conservative Lagrangian type scheme on curvilinear meshes for the compressible Euler equations, comm. on comput. phys., vol 4, no5, pp 1008-1024, 2008. 3 S. Del Pino, A curvilinear finite-volume method to solve compressible gas dynamics in semi-Lagrangian coordinates, CRAS, Vol 348, num 17-18, pp. 1027-1032, 2010. 4 D. Benson,Y. Bazilevs, G. Scovazzi, M. Shashkov, Isogeometric Analysis of Lagrangian Shock Hydrodynamics, conference US National Cong. Comp. Mech 2011. 5 A. Claisse, B. Despres, E. Labourasse, F. Ledoux, A new exceptional points method with application to cell-centered Lagrangian schemes and curved meshes, soumis JCP.
But what about ALE (standard) and ALE (discontinuous !) (mesh refinement/coarsening, swapping, edge sliding) ? Our work deal with (local) finite volume framework.
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Parameterization: Rational quadratic Bezier (M1 , ω)
(M1 , ω) ω = +∞ ω=3
M ω (q)
ω=1
M0
ω = 0.5
M0
ω=0
M2 ω
M (q) =
M0 (1 − q)2 + M1 2ωq(1 − q) + M2 q 2 , (1 − q)2 + 2ωq(1 − q) + q 2
M2 q ∈ [0, 1],
(M1 , ω) Control Point and associated weight on logical edge (M0 , M2 ), (kind of NURBS). (a) Link with conical arc: 1
ω > 1 : hyperbola
2
ω = 1 : parabola
3
0 < ω < 1 : ellipse, ( then circle..)
4
ω = 0 : straight line
(b) M ω (q) inside simplex (M0 , M1 , M2 ). Exact continuous representation of 2D conical shape, very well suited for finite volumes method September 10 2012,
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Exact formula for area and centroid Two first moment: Z 0 mj = |Cj | = 1dA,
m1j
Cj
Z x dA
=
xj =
Cj
m1j mj0
computed analytically
Green involves: R 1Formula d ω ω (q). M M ω (q) = (x1ω , x2ω )(q). dq M (q)dq. 0 R1 ω d ω xl (q)dq for k, l = 1, 2, l 6= k . (xk (q))2 dq 0 (M1 , ω)
M ω (q)
M0
M
2 For Area: A(M ω (q), M0 , M1 , M2 ) = f (ω) AT (M0 , M1 , M2 ),
f is at least C 1 (IR+ ). September 10 2012,
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Euler equations: ALE
�r+
Mr
�r,r+1 1/ 2
,r +
1
� r,r+1/2
(Mr+1/2 , ωr+1/2 )
� r−
1/
2,
�r
−
(Mr−1/2 , ωr−1/2 )
r
1,
r
�r−1,r−1/2
Mr+1
Mr−1
Cj
exact area and centroid
poly(r,r+1)
Cj
C
poly(r,r+1/2,r+1)
j ALE Formalism, conservation laws integrated on control volume Cj (t), and V arbitrary speed of ∂Cj (t):
R R ˆ Dt C(t) 1 dx − ∂C(t) V .Nds = 0, Geometical Conservation Laws, R R Dt ˆ ρ(U − V ).Nds = 0, Mass, ρ dx + C(t) ∂C(t) R R ˆ + P Nds ˆ ρU dx + ρU(U − V ). N = 0, Momentum, D t C(t) R ∂C(t) D R ˆ ˆ t C(t) ρE dx + ∂C(t) ρE(U − V ).N + PU.Nds = 0, Total Energy. internal energy, P(ρ,) + EOS. Reference: Extension of ALE methodology to unstructured conical meshes, avec B. Boutin, E. Deriaz, P. Navarro, ESAIM proceedings 2011. September 10 2012,
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Splitting V := U − U grid Pure Lagrangian Phase (V = U), R R ˆ Dt C(t) 1 dx − ∂C(t) U.Nds = 0, GCL R Dt ρ dx = 0, mass RC(t) R ˆ D ρU dx + P Nds = 0, momentum t C(t) R R ∂C(t) ˆ Dt C(t) ρE dx + ∂C(t) PU.Nds = 0.
total energy
Extension (Despres-Mazeran)[GLACE] or (Maire et al)[EUCCLHYD] : ω−GLACE and ω−EUCCLHYD. Pure advection Phase (V = −U grid ), R R ˆ Dt C(t) 1 dx + ∂C(t) 1 U grid .Nds = 0, GCL. R R D grid ˆ ρ dx + ρ U . Nds = 0, t C(t) R R ∂C(t) grid ˆ Dt RC(t) ρU dx + R ∂C(t) ρU U .Nds = 0, D ˆ ρE dx + ρE U grid .Nds = 0. t C(t)
∂C(t)
System of advection equations, ω-ALE. September 10 2012,
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Cjr,ω
j
C r−
,r+1 /2
Nr
/2,r
2,ω
N r −1
r+1/
Cj (Mr+1/2 , ωr+1/2 )
1/ 2, ω
ω-GLACE, ω-EUCCLHYD
Geometrical Normals:
(Mr−1/2 , ωr−1/2 )
N
Mr
r+
1
−
1
Nr ,r +1
Mr+1
Nr−1,r−1/2
Cj
r,r
1/2 ,
N
r+
Mr−1
Proposition 1
Cjr Formalism for areas: |Cj | =
2
Cj
r +1/2,ω
r ,ω Cj
1 2
P
r
Crj ,ω .OMr +
P
r +1/2
r +1/2,ω
Cj
.OMr +1/2
f (ωr +1/2 )
= (Nr ,r +1/2 + Nr +1/2,r +1 ) 2 1 = (1 − f (ωr −1/2 ))Nr −1,r + (1 − f (ωr +1/2 ))Nr ,r +1 2 +f (ωr −1/2 )Nr −1/2,r + f (ωr +1/2 )Nr ,r +1/2 September 10 2012,
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From GLACE .... to ... EUCCLHYD ... r+1/2,ω+
Nr
,r+ /2 +1
Nj (Mr+1/2 , ωr+1/2 ) Nr+1/2,ω− j 2 1/
1
r+ N r,
Nr+1,ω− j
Nr,ω+ j Mr Nr+1,r
Mr+1
Cj
Proposition By setting
r ,ω− := (1 − f (ωr −1/2 ))Nr −1,r + f (ωr −1/2 )Nr −1/2,r , Nj Nr ,ω+ := (1 − f (ωr +1/2 ))Nr ,r +1 + f (ωr +1/2 )Nr ,r +1/2 , j r +1/2,ω− := f (ωr +1/2 )Nr ,r +1/2 , Nj Nr +1/2,ω+ := f (ω r +1/2 )Nr +1/2,r +1 . j
Identity for (“GLACE vectors”) and vector”): ( (“EUCCLHYD Crj ,ω = 21 (Nrj ,ω− + Nrj ,ω+ ), r +1/2,ω
Cj
=
r +1/2,ω− 1 2 (Nj
r +1/2,ω+ ). September 10 2012,
+ Nj
| PAGE 10/29
Conical Hydrodynamic schemes Ck
Ck Ur+1/2 (Mr+1/2 , ωr+1/2 )
(Mr+1/2 , ωr+1/2 ) Ur+1/2 pr+1/2 j
Mr+1 Ur+1 pr+1 j
prj
Cj
r+1/2,+
pj
Mr Ur
pr+1,− j Mr+1 Ur+1
r+1/2,−
pj
pr,+ j Mr
Cj
Ur pr,− j
P P r +1/2,ω r ,ω , Ur +1/2 ), τj (t) = , Ur ) + r +1/2 (Cj Mj τj0 (t) = r (Cj P P r +1/2,ω r +1/2 r ,ω r 0 ω − GLACE pj , Mj Uj (t) = − r Cj pj − r +1/2 Cj r +1/2 r +1/2,ω M E 0 (t) = − P (Cr ,ω , U )pr − P , Ur +1/2 )pj . r j j j r +1/2 (Cj r j
1 , ρj (t)
ω-EUCCLHYD: P P r +1/2,ω+ r +1/2,ω− r ,ω− Mj τj0 (t) = 12 + Nrj ,ω+ ).Ur + 12 + Nj ).Ur +1/2 , τj (t) = ρ 1(t) , r (Nj r +1/2 (Nj j P P r +1/2,ω− r +1/2,− r +1/2,ω+ r +1/2,+ r ,ω− r ,− r ,ω+ r ,+ 1 Mj U0 (t) = − 1 pj + Nj pj ) − 2 (Nj pj + Nj pj ), j r (N r +1/2 2 j P r ,ω− .Ur )pjr ,− + (Nrj ,ω+ .Ur )pjr ,+ Mj Ej0 (t) = − 12 r (Nj r +1/2,ω− r +1/2,− r +1/2,ω+ r +1/2,+ −1 P .Ur +1/2 )pj + (Nj .Ur +1/2 )pj . r +1/2 (Nj 2 September 10 2012,
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ω-GLACE 1 2
r +1/2,ω
Project. on acoust Riemann inv. dir. Crj ,ω et Cj P P r +1/2 r +1/2,ω Conservation j pjr Crj ,ω = 0 et j pj Cj =0 1 2
Mr : Ar Ur = br always invertible, unique velocity, then pressure pjr using acoustic relation. r +1/2 r +1/2 Mr +1/2 ⇒ non invertible BUT continuity of pressure pj = pk : 1D Riemann problem r+1/2,ω
Ur+1/2 .nj
pr +1/2 =
r+1/2,ω
=
ρj cj Uj .nj
ρj cj pj +ρk ck pk ρj cj +ρk ck
+
r+1/2,ω
+ρk ck Uk .nj
ρj cj +ρk ck ρ j cj ρ k ck (Uj ρj cj +ρk ck
−
1 ρj cj +ρk ck r+1/2,ω . Uk ).nj
+
(pj − pk ),
To obtain a 2D velocity (tangential part): Gallilean Invariance r+1/2,ω r+1/2,ω , if Ur+1/2 .nj > 0, Uj .τj r+1/2,ω r+1/2,ω r+1/2,ω Uk .τj , if Ur+1/2 .nj < 0, Ur+1/2 .τj = ρj cj Uj +ρk ck Uk .τ r+1/2,ω , else. ρj cj +ρk ck
j
September 10 2012,
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ω-EUCCLHYD 1 2
r +1/2,ω±
Proj. on acoust. Riemann inv. dir. Nrj ,ω± and Nj P Conservation j pjr + Nrj ,ω+ + pjr − Nrj ,ω− = 0 et P r +1/2+ r +1/2,ω+ r +1/2− r +1/2,ω− Nj + pj Nj =0 j pj
Mr : Ar Ur = br always invertible, unique velocity. Mr +1/2 : Ar +1/2 Ur+1/2 = br+1/2 , 2 Cases :
1 2
1 2 r+1/2,ω
Ur+1/2 .nj
Invertible: velocity Ur+1/2 unique. Non-invertible: r+1/2,ω
=
ρj cj Uj .nj
r+1/2,ω
+ρk ck Uk .nj
ρj cj +ρk ck
+
1 ρj cj +ρk ck
(pj − pk )
To obtain a 2D velocity (tangential part): Gallilean Invariance r+1/2,ω r+1/2,ω U .τ , if Ur+1/2 .nj > 0, j j r+1/2,ω r+1/2,ω r+1/2,ω Uk .τj , if Ur+1/2 .nj < 0, Ur+1/2 .τj = ρj cj Uj +ρk ck Uk .τ r+1/2,ω , else. ρj cj +ρk ck
3
r +1/2±
then pressure pjr ± , pj
j
using acoustic relation September 10 2012,
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Second order extension using centroids
Centroid xc are give (analytically), so that second (or higher) order scheme can be obtain. Q 1 (x) = QC + ∇QC (x − xc ),
(Pressure and Velocity )
(1)
∇Qc computed using least-square method (ONLY centroid of cells are needed, not their topology or geometry). Limitor strategies involve evaluation at vertices AND shoulder point (mid-edge). dP = ρc dU evaluated with reconstruction (1) at DOF. For ALE advection equation same reconstruction, local iterative over-limitor can be used to obtain local min/max principle.
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Numerical results: Sedov GLACE
1.2
1.2×100
1
1.0×100
0 −3.2×10−17
1.0×100
−1.5×10−16 −3.3×10−17
1.0×100
September 10 2012,
polygonal GLACE and ω-GLACE
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Numerical Results: Sedov EUCCLHYD
1.2
1.2×100
1
1.0×100
0 −3.2×10−17
1.0×100
−1.2×10−16 −3.2×10−17
1.0×100
September 10 2012,
polygonal EUCCLHYD and ω-EUCCLHYD
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Numerical Results 6
6
5
5
4
4
3
3
2
2
1
1
0
0 0
1
1.2
0
1
1.2
Density versus rayon: First order ω-GLACE with 30 × 30 (left) and 60 × 60 (right)
Remarks ω-GLACE do not show diagonal instability, (cf. corner matrix). Unlike to polygonal GLACE, ω-GLACE possess an edge pressure with is continuous. Unlike to polygonal EUCCLHYD, ω-EUCCLHYD possess corner pressure. ω-GLACE et ω-EUCCLHYD tends toward their polygonal version ω → 0. Comparison with “all circle” GLACE work [Claisse, Despres, Labourasse, Ledoux]. September 10 2012,
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Sod: (Polar) first and second order (density)
0.999
0.999
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
0.1249
0.1249 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Figure: Left: Glace
0.9
0.1
Right: Eucclhyd
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
(sector=21, layer=51)
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continuous Rezoning over conical cells
1 Moved vertices using logical polygonal connectivity without control point r−1
r−2 r+2
Cj r
r+1
⇒ Able to re-use all polygonal rezoning algo. (RJM, Escobar, Winslow etc) with/or without physical weights and also nodal mesh quality using short/medium range.
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Move control points and associated weight ω 2 Moving control points: Predictor: Predicted displacement is halfsum of extremities vertices displacement (already computed): ∆pred = 12 (∆0 + ∆2 ) 1 Corrector: Predicted control point position is (eventualy) projected on a new position for which no intersection occurs. (M1e,pred , ω) ∆cort 1 (M1e,N ,
ω)
e
g ed al
fin
M0e,N M2e,N
Control Point Correction by diminushing the distance (toward the logical mid edge
e,N 1 2 (M0
+ M2e,N )) in
case of non-valid cell (curvature diminishing): always converge if polygon is valid.
3 Re-computation of “optimal” ω weight to obtain a geometrical continuous re-adaptation of boundary cells. September 10 2012,
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Conservative advection over conical cells Integrating PDE (t n , t n+1 ): |C(t
n+1
¯ n+1 = |C(t n )|Q ¯n − )|Q
Z
t n+1
!
Z QU
tn
grid
ˆ ds .N
dt.
∂C(t)
area swept by conical edges rezoning exact computation
Remark 1
Q ≡ 1 give DGCL, exact (necessary condition for conservative advection).
2
Rezoning is sum of displacement and “deformation”: Ugrid = Udisplacement + Urepresentation . September 10 2012,
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Two remapping schemes Extension based on volume fluxing: (M1 (tn+1 ), ω(tn+1 )) M2 (tn+1 ) area swept region
M2 (tn )
M0 (tn+1 )
N
N
O
O
Qj |Cj | = Qj |Cj | +
X
M0 (tn )
∗
Qe δVe .
e
(M1 (tn ), ω(tn )) Extension: self-intersections (fourth order algebraic equation)
q2,1 q1,1 q2,2 q1,2
N
N
O
O
Qj |Cj | = Qj |Cj | +
X #subareas X e
∗
Qe,k δVe,k .
k
September 10 2012,
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Symmetry preservation for advection Circular indicator function 1r
