Conservative Remapping of Vectors for Staggered ALE

Milan Kucharik, Czech Technical University in Prague [email protected] Mikhail Shashkov, XCP-4, LANL [email protected] Acknowledgments: This work was carried out under the auspices of the National Nuclear Security Administration of the U.S. Department of Energy at Los Alamos National Laboratory under Contract No. DE-AC52-06NA25396. M. Kucharik have been supported in part by the Czech Ministry of Education grants P201/10/P086 and RVO: 68407700. The authors thank H. Ahn, D. Bailey, A. Barlow, R. Liska, K. Lipnikov, ` ´ R. Loubere, P.-H. Maire, M. Owen, P. Vachal, and J. Velechovsk´y for fruitful discussions and constructive comments.

1

Overview • Introduction. • Remapping algorithm. – Construction of exchange integrals and mass fluxes. – Remap of mass. – Remap of remaining cell-centered quantities. – Construction of nodal mass fluxes. – Remap of nodal quantities. • Approaches for velocity limiting: – Limiting in x,y. – Principal flow direction. – VIP set. – Constrained limiting. • Numerical examples. • Conclusion. 2

Introduction • Eulerian methods – static mesh, mass flux through mesh edges. • Lagrangian methods – mesh moving with fluid, no mass flux. Mesh can tangle. • Combination – ALE – computational mesh moves with the fluid due to Lagrangian solver, Eulerian part (smoothing+remapping) keeps the mesh smooth. • Multimaterial ALE – more than 1 materials allowed in each cell, amount defined by volume (and mass) fractions, and location by material centroid → several complications, as simple as possible. • Often applications in laser plasma – symmetry issues important. • Symmetry violation: limiting of velocity vector.

3

Multimaterial Remapping • Only interested in quadrilateral cells and staggered discretization. • Given: Lagrangian {c} and rezoned {˜ c} meshes, volume fractions αc,k and material centroids ~xc,k in Lagrangian cells. G

• Given: Cell centered mean values gc,k = V c,k of unknown c,k underlying function g = ρ, g = ρ ε, and nodal velocities w~n. R • Goal: Find masses Gc˜,k = g(~x) dV and mean values gc˜,k = c˜,k

Gc˜,k Vc˜,k

in

rezoned cells such that the algorithm is accurate, conservative, and linearity-preserving (∀k). • Goal: Find new volume fractions αc˜,k and centroids ~xc˜,k for each material in new cells. • Goal: Find new nodal velocities w ~ n˜ – at least deBar condition, vector symmetry issues.

4

Flux form of intersections-based remap for meshes with the same connectivity

c0 ∈C 0 (c)

c0 ∈C 0 (c)

c

C 0(c) = C(c) \ c. • Mass update in the flux form Z X m Fc,c m e c˜ = ρ(~x) dV = mc + 0, c˜

c0 ∈C 0 (c)

m m where the fluxes are Fc,c 0 = −Fc0 ,c =

~c’

~c c c

U U U

• Integration by intersections can be written in a flux form [Margolin, Shashkov (JCP,2002)]. ! ! S S S 0 c˜ ∩ c \ • c˜ = c c ∩ c˜0 ,

c’ ~ c’ ~ c"

~ c"

Z

c˜∩c0

Z ρc0 (~x) dV −

ρc(~x) dV

c∩˜ c0

• Flux form −→ conservation guaranteed =⇒ more freedom in flux construction. • Intersection based =⇒ simple generalization for multi-materials. 5

Computation of Exchange Integrals • Computation of integrals up to second-orderR over intersections IPαk = α dV , Pk

for P k =ck ∩ c˜0, c0k ∩ c˜, α = 1, x, y, x2, x y, y 2.

and

• ck denotes polygon of pure material k in c. • Required robust routine for convex polygon intersection. • Integral of polynomial over a polygon – evaluated analytically.

g

f

d

e h

c ~ c

~’ c’ c c b

a

Lagrangian and rezoned meshes

6

Mass Remap • Piecewise linear material density: ρc∗,k (x, y) = ρc∗,k + Scx∗,k (x − xc∗,k ) + Scy∗,k (y − yc∗,k ). R m • Mass flux: FP k = P k ρc∗,k (x, y) =   
± ρc∗,k IP1 k + Scx∗,k IPx k − xc∗,k IP1 k + Scy∗,k IPy k − yc∗,k IP1 k .   P • Mass remap: m e c,k = mc,k + ± Fcm0 ∩˜c − Fcmk ∩˜c0 k

∀c0 ∈C(c)

• In quadrilateral mesh: m e i+ 1 ,j+ 1 ,k = mi+ 1 ,j+ 1 ,k 2 2 2 2  → → m + Fm i+1,j+ 12 ,k − F i,j+ 12 ,k  ↑  ↑ m + Fm i+ 12 ,j+1,k − F i+ 12 ,j,k %  % m + Fm i+1,j+1,k − F i,j,k  m + Fm i,j+1,k − F i+1,j,k

i,j+2

i−1,j+2

i−1/2,j+3/2

i+1,j+2

i+1/2,j+3/2

i+3/2,j+3/2

i,j+1

i−1,j+1

i+2,j+2

i+2,j+1 i+1,j+1

i−1/2,j+1/2

i+1/2,j+1/2

i+3/2,j+1/2 i+2,j

i+1,j

i−1,j i,j i−1/2,j−1/2 i−1,j−1

i+1/2,j−1/2 i,j−1

i+3/2,j−1/2 i+1,j−1

i+2,j−1

7

Volume Fractions, Centroids • Volumes similar to masses   P ± Ic10 ∩˜c − Ic1k ∩˜c0 , α ec,k = Vec,k = Vc,k + k

∀c0 ∈C(c)

ec,k V ec . V

• From α ec,k , status (mixed/pure cell) and materials in new cells can be identified. • Integrals of x and y over materials  in new cell  P x Iec,k ± Icx0 ∩˜c − Icxk ∩˜c0 , = Vc,k xc,k + k ∀c0 ∈C(c)   P y y y Iec,k = Vc,k yc,k + ± Ic0 ∩˜c − Ic ∩˜c0 . ∀c0 ∈C(c)

• New centroids: xc,k =

x Iec,k ec,k , V

yc,k =

k

k

y Iec,k ec,k . V

• Similar flux form for εc,k , pc. • pc,k from εc,k using EOS. 8

Construction of Nodal Mass Fluxes • Initially, nodal mass defined P mc,n. as mn = c∈C(n)

• In order to remap nodal quantities, nodal mass fluxes must be constructed. • 3 fluxes per adjacent cell. • Intersect pure materials in nodal regions – expensive. • Cheap solution: interpolation of inter-nodal fluxes from inter-cell ones.

i,j+1

i−1,j+1 i−1/2,j+1/2

i+1,j+1

i+1/2,j+1/2

i+1,j

i−1,j i,j

i−1/2,j−1/2 i−1,j−1

i+1/2,j−1/2 i,j−1

i+1,j−1

• Using approach of [Pember, Anderson (UCRL-JC-139820, 2000)], extended  by corner fluxesand half-side fluxes for adjacent cells → → → m m m F i+ 1 ,j+ 1 = F i,j+ 1 + F i+1,j+ /4, or optimization-based flux 1 2

4

2

2

construction [Owen, Shashkov (LLNL-PRES-413307, 2011)].

9

Remap of Nodal Quantities – Mass • Total nodal mass    remap in a flux form: m e i,j = mi,j +

→ Fm i+ 21 ,j+ 14

 +  +

↑ Fm i+ 14 ,j+ 12 % Fm i+ 21 ,j+ 12

−

→ Fm i− 21 ,j+ 14



−

↑ Fm i+ 14 ,j− 12



−

% Fm i− 12 ,j− 12

+

→ Fm i+ 12 ,j− 41

 +

↑ Fm i− 14 ,j+ 12

 +

Fm i− 21 ,j+ 21

• Only 3 fluxes remaining.

per



−

↑ Fm i− 41 ,j− 12



−

Fm i+ 21 ,j− 12

i,j+1

i−1,j+1

• Due to nodal mass definition, inter-cell fluxes cancel.



−

→ Fm i− 21 ,j− 14

i−1/2,j+1/2

. i+1,j+1

i+1/2,j+1/2

cell i+1,j

i−1,j i,j

i−1/2,j−1/2 i−1,j−1

i+1/2,j−1/2 i,j−1

i+1,j−1

10

Remap of Nodal Quantities – Momenta • Nodal momentum  remap in a flux form: µ ei,j =



→ ∗ mi,j ui,j + ui+ 1 ,j+ 1 F m i+ 12 ,j+ 14 2 4  → ∗ + ui+ 1 ,j− 1 F m i+ 12 ,j− 41 2 4

→ ∗ − ui− 1 ,j+ 1 F m i− 21 ,j+ 14 2 4  → − u∗i− 1 ,j− 1 F m i− 21 ,j− 41 2 4

↑ ∗ + ui+ 1 ,j+ 1 F m i+ 14 ,j+ 12 4 2  ↑ ∗ + ui− 1 ,j+ 1 F m i− 14 ,j+ 12 4 2

↑ ∗ − ui+ 1 ,j− 1 F m i+ 41 ,j− 12 4 2  ↑ − u∗i− 1 ,j− 1 F m i− 41 ,j− 21 4 2

% ∗ + ui+ 1 ,j+ 1 F m i+ 12 ,j+ 12 2 2  ∗ + ui− 1 ,j+ 1 F m i− 12 ,j+ 12 2 2

% ∗ − ui− 1 ,j− 1 F m i− 12 ,j− 21 2 2  − u∗i+ 1 ,j− 1 F m i+ 21 ,j− 21 . 2 2









• The same for νe in y direction. • New velocity: u ei,j = µ ei,j /m e i,j , vei,j = νei,j /m e i,j . • Obviously satisfies deBar condition. • Need to define all u∗ and v ∗. 11

Approaches for Flux Velocity ∗ • Various approaches for u∗n,n0 and vn,n 0:

– – – –

Low order donor (left or right, depending on sign(F~n,n0 )). Bilinear reconstruction in cell between n and n0. Linear reconstruction in c or dual cell corresponding to n. Optimization-based kinetic energy preserving approach [Bailey et al. (JCAM, 2010)].

• All these approaches keep symmetry, but high order methods can violate bounds – by components and in magnitude. • Limiting by components helps, but violates symmetry. • Two approaches recently trying to limit and keep symmetry: – VIP Set [Luttwak, Falcovitz (IJNMF, 2011)]. – Limiting in principal flow direction [Maire (habilitation, 2011)]. • Ideas from both approaches ⇒ third alternative: constrained limiting. 12

Limiting in x, y

• Looking only at situation in a single cell c.

∂u ∂u • uc(x, y) = uc + ∂x c(x − xc) + ∂y c(y − yc),

∂v ∂v vc(x, y) = vc + ∂x c(x − xc) + ∂y c(y − yc). • uc, vc are “cell mean values” – in practice average.   • Slopes ∂u,v computed in least-squares sense. ∂x,y c

• Limiting: uc(x, y) = uc + Φxc vc(x, y) = vc + Φyc

∂u ∂x
c (x − xc ) ∂v ∂x c (x − xc )




+ Φxc ∂u ∂y
c (y − yc ), ∂v + Φyc ∂y (y − yc). c


• Φxc, Φyc – standard BJ limiter. • Violates symmetry, even for common limiter min(Φxc, Φyc ). 13

Limiting in Principal Flow Directions • Approach of [Maire (habilitation, 2011)]. • Local deformation matrix Fc, either from velocity or mesh motion. • Local (right) Cauchy-Green tensor   xx 2 yx 2 xx xy yx yy (f ) + (f ) f f +f f Gc = FcT Fc = . f xx f xy + f yx f yy (f xy )2 + (f yy )2 • Normalized eigenvectors of Gc → principal flow directions ξ~c, ~ηc.   ξc,x ξc,y • Orthonormal projection matrix Ac = , projects ηc,x ηc,y arbitrary vector to ξ~c, ~ηc. • Applied to reconstruction: Ac w ~ c(x, y) = Ac w ~ c + Ac (∇w) ~ c (~x − ~xc) or w ~ cξ,η (x, y) = w ~ cξ,η + Ac (∇w) ~ c (~x − ~xc).

14

Limiting in Principal Flow Directions

• Whole equation in ξ~c, ~η be done in ξ~c, ~ηc in a standard c , limiting can  Φξc 0 ξ,η ξ,η way w ~ c (x, y) = w ~c + Ac (∇w) ~ c (~x − ~xc). 0 Φηc T • Projecting back tox, y – multiplying by A c:  Φξc 0 > w ~ c(x, y) = w ~ c + Ac Ac (∇w) ~ c (~x − ~xc) 0 Φηc =w ~ c + Lc (∇w) ~ c (~x − ~xc) = w ~ c + Hc (~x − ~xc).

• We can understand Lc as generalized limiter and Hc as generalized (limited) velocity gradient.

15

Limiting by VIP • Vector Image Polygon [Luttwak, Falcovitz (IJNMF, 2011)], degenerates to standard limiter in 1D. • VIP set = convex hull of velocities defining local bounds. • Limiting = moving unlimited value anywhere to the VIP set; practically closest point (least dissipative) or point on line connecting with mean value (linear transition from donor, direction of unlimited). • Example for radial cell and polar velocity.

1.6

1.6

1.4

1.4

1.2

1.2

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.8

1

1.2

1.4

1.6

1.8

2

closest

2.2

2.4

2.6

2.8

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

connection with donor 16

2.8

Constrained Limiting • Adopted ideas from both previous approaches.   x Φc 0 (∇w) ~ c (~x − ~xc). • Limiting in x, y: w ~ c(x, y) = w ~c + 0 Φyc • Projection to ξ, η using Ac:   x Φc 0 ξ,η ξ,η w ~ c (x, y) = w ~ c + Ac 0 Φyc

∂u ∂x ∂v ∂x

∂u ∂y ∂v ∂y

! (~x − ~xc). c

• Evaluated in all n ∈ N (c), bound preservation then (wcξ )min ≤ w ~ cξ (xn, yn) ≤ (wcξ )max, (wcη )min ≤ w ~ cη (xn, yn) ≤ (wcη )max. • Four constraints for each n: ξ, η and min, max, so finally 16 (for x y quadrilateral cell) linear inequalities for Φ , Φ c c , for example     0 ≤ axx

∂u (x − x ) + ∂u (y − y ) n c n c ∂x ∂y

Φx c + axy

∂v ∂v ∂x (xn − xc ) + ∂y (yn − yc )

Φyc + w ~ cξ − (wcξ )min .

• Intersection of 16 halfplanes ⇒ feasible limiters (reusing the infrastructure from feasible set rezoning). 17

Constrained Limiting

1

1

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6 Φy

Φy

• Choosing the limiter: closest to [1, 1] (least diffusive) or on diagonal (linear transition from donor).

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 Φx

0.6

0.7

0.8

0.9

1

0

0.1

0.2

0.3

0.4

0.5 Φx

0.6

0.7

0.8

0.9

1

closest connection with donor • Linear transition from donor useful ⇒ Φxc = Φyc ⇒ whole derivation reduces to 1D – simple and fast ! 18

Numerical Example – 1 Cell • Polar cell (r ∈ h1, 2i, ϕ ∈ hπ/10, π/5i). • Radial velocity – points from origin, kwk ~ = 3 at internal edge, kwk ~ = 1 at external edge. • Comparing bilinear, unlimited linear, BJ in x, y, BJ in ξ, η, VIP, and constrained limiting. • kw ~ c(~xn)k – norm of velocity in all points.

13

reconstructed 9

5 8

4

• dϕ – discrepancy between velocity angle and analytic angle given by position.

3 12

10

7

1

6

• For remapper to be symmetric – dϕ must be the same for axisymmetric pairs of points.

2

11

19

Numerical Example – 1 Cell kw ~ c(~xn)k method bilin. unlim. x, y ξ, η VIP, int. VIP, cl. con., cl. con., diag.

1 2.000 2.097 2.104 2.087 2.097 2.097 2.079 2.085

2 1.982 2.087 2.089 2.077 2.087 2.087 2.073 2.077

3 1.975 2.084 2.081 2.074 2.084 2.084 2.074 2.074

4 1.982 2.087 2.077 2.077 2.087 2.087 2.080 2.077

5 2.000 2.097 2.079 2.087 2.097 2.097 2.092 2.085

6 2.963 3.059 3.032 2.963 2.963 2.963 2.959 2.963

7 2.469 2.572 2.556 2.519 2.572 2.572 2.516 2.519

8 1.482 1.596 1.606 1.630 1.596 1.596 1.631 1.630

9 0.988 1.108 1.132 1.185 1.108 1.108 1.189 1.185

10 3.000 3.063 3.025 2.967 2.966 2.967 2.968 2.966

11 3.000 3.063 3.046 2.967 2.966 2.967 2.957 2.966

12 1.000 1.151 1.171 1.226 1.423 1.143 1.223 1.219

13 1.000 1.151 1.161 1.226 1.423 1.143 1.228 1.219

1 0.000 0.045 0.054 0.044 0.045 0.045 0.051 0.054

2 0.000 0.023 0.026 0.023 0.023 0.023 0.027 0.028

3 0.000 0.000 0.003 0.000 0.000 0.000 0.001 0.000

4 0.000 0.023 0.031 0.023 0.023 0.023 0.024 0.028

5 0.000 0.045 0.058 0.044 0.045 0.045 0.049 0.054

6 0.000 0.000 0.018 0.000 0.000 0.000 0.010 0.000

7 0.000 0.000 0.012 0.000 0.000 0.000 0.007 0.000

8 0.000 0.000 0.012 0.000 0.000 0.000 0.006 0.000

9 0.000 0.000 0.039 0.000 0.000 0.000 0.020 0.000

10 0.000 0.106 0.129 0.104 0.110 0.104 0.097 0.109

11 0.000 0.106 0.093 0.104 0.110 0.104 0.117 0.109

12 0.000 0.118 0.047 0.101 0.000 0.000 0.108 0.079

13 0.000 0.118 0.127 0.101 0.000 0.000 0.067 0.079

dϕ method bilin. unlim. x, y ξ, η VIP, int. VIP, cl. con., cl. con., diag.

• Conclusion: all 3 limiting methods comparable. • Sanity check: rotation by non-trivial angle → OK. 20

Numerical Example – Full Remap • Full remap in 4 × 4 polar (r ∈ h0.75, 1.75i, ϕ ∈ hπ/10, π/5i).

mesh 0.9 0.8 0.7

• 1 rezone step: internal angular edges 0.6 move by half of cell. 0.5 • Jump from kwk ~ = 3 to kwk ~ = 1.

0.4 0.3

|w| ~

• Measuring L1 : discrepancy of velocity magnitude. • Same results, except closest point in VIP set, which violates symmetry. • In general, regarding symmetry, all methods are acceptable and comparable.

0.2 0.6

0.8

method bilin. unlim. x, y ξ, η VIP, int. VIP, cl. con., cl. con., diag.

1

1.2

1.4

1.6

|w| ~

L1 0.658e-13 0.108e-12 0.960e-2 0.137e-12 0.151e-12 0.225e0 0.597e-2 0.148e-12

21

Note: FCT in ξ, η • Cell based reconstruction can not degenerate to donor for 0 limiter, so nodal-based reconstruction more suitable. • Limiting impractical, reconstruction needs explicit construction of (non-convex) dual cells, FCT approach more suitable. • [Velechovsky´ et al. (in preparation, 2012)] – FCT adapted to ξ, η coordinates. • Whole FCT derivation performed in ξ, η (using Ac), constraints also in ξ, η. • Significantly reduces overshoots, keeps symmetry.

22

Conclusion • Work still in progress. • New vector limiting method with emphasis on symmetry. • Significantly reduces local-bound violations. • Applicable in the context of multi-material flux-based remap. • Comparison with VIP and Principal Flow Direction methods for simple 1 cell example and full remap, similar results.

23