[PPT] - High-Order Adaptive ALE Calculations of Solidification B. T. PowerPoint Presentation

SLIDE 1

High-Order Adaptive ALE Calculations of Solidification

B. T. Helenbrook
Mech. & Aero. Eng. Dept.

Clarkson University Potsdam, NY USA

SLIDE 2

Motivation

Clarkson

U N I V E R S I T Y

Th Tc u, Th u, dT/dx = 0 Th liquid solid

◮ Materials manufacturing,

◮ Horizontal ribbon growth ◮ Spin-casting ◮ Aluminum smelters

◮ Welding

SLIDE 3

Field Equations

Clarkson

U N I V E R S I T Y

◮ Solid

∂ρscp,sT ∂t + ∂ρsujcp,sT ∂xj + ∂ ∂xj

−ks

∂T ∂xj

= 0

◮ Liquid

∂ρlcp,lT ∂t + ∂ρlujcp,lT ∂xj + ∂ ∂xj

−kl

∂T ∂xj

= 0

∂ρlui ∂t + ∂ρlujui ∂xj = ∂p ∂xi + ∂τi,j ∂xj

SLIDE 4

Discrete Field Equations

Clarkson

U N I V E R S I T Y

x = x(ξ, η, τ) y = y(ξ, η, τ) t = τ

Ω

v

∂JρcpT ∂τ

− ∂v

∂ξ E − ∂v ∂η F

dξdη +
Γ

v (E, F) · ndΓ +

nel

n=1

Ωn

∂v ∂ξ aE + ∂v ∂η aF

¯

τ ∂JρcpT ∂τ + ∂ ∂ξ E + ∂ ∂ηF

dΩ = 0

◮ E, F = contravariant fluxes in ξ η coordinates:

ξ y ,-x

F =

ρcpT(u − xτ) − k ∂T

∂x

yξ−
ρcpT(v − yτ) − k ∂T

∂y

xξ

◮ aE, aF convective velocities

aF = (u − xτ)yξ − (v − yτ)xξ

SLIDE 5

Continued...

Clarkson

U N I V E R S I T Y

◮ Stabilization constant

h = 4 (p + 1)2 A hmax hmax = maximum edge length of element ¯ τ = 1 J h

a2

E + a2 F + ν/h + h/∆t ◮ Time derivatives: 3rd order accurate

A-Stable DIRK(Williams et. al 2002)

◮ Implicit time step ∆t ◮ Implicit equations solved using either geometric multgrid or

SuperLU/Petsc

SLIDE 6

Continued...

Clarkson

U N I V E R S I T Y

◮ Triangular elements ◮ C 0 Continuous FEM ◮ Modified Dubiner basis ◮ Isoparametric mappings

0.83 0.50 0.17

0.17
0.50
0.83

Vertex Modes Side 1 Side 2 Side 3 Interior Modes P 1 2 3 4

SLIDE 7

Interface Equations

Clarkson

U N I V E R S I T Y

◮ Normal Motion

ρs(us − xt, vs − yt) · ns = ρl(u − xt, v − yt) · ns = ˙ ms→l [ [(ρcpTm(u − xt, v − yt) − k∇T) · ns] ] = ˙ ms→lLf

◮ Temperature

T = Tm

◮ Velocity (Let ρs = ρl)

u = us v = vs

SLIDE 8

Discrete Interface Equations (Galerkin)

Clarkson

U N I V E R S I T Y

◮ Let η be constant on interface

ξ y ,-x

n = yξ

i − xξ j

◮ Interface equation is jump in contravariant flux

[ [(ρcpTm(u − xt, v − yt) − k∇T) · ns] ] = [ [F] ] = ˙ ms→lLf

◮ Enforce usual energy equation with additional source term Ω

v

∂JρcpT ∂τ

− ∂v

∂ξ E − ∂v ∂η F

dξdη +
Γ

v (E, F) · ndΓ +

ΓI

v ˙ ms→lLf dΓI = 0

◮ Energy equation corresponding to nodes at interface

determines interface motion, not T.

SLIDE 9

Interface Discussion

Clarkson

U N I V E R S I T Y

◮ Advantage: global energy conservation Ω

∂JρcpT ∂τ dξdη +

Γ

(E, F) · ndΓ +

ΓI

˙ ms→lLf dΓI = 0

◮ Disadvantage: needs interface stabilization?

SLIDE 10

Rigid Body Translation of Solid

Clarkson

U N I V E R S I T Y

◮ Let T = Tm everywhere ◮ Horizontal translation of rigid body solid vs = 0 ◮ Only vertical motion of mesh with, xτ = 0 everywhere ◮ Gov. Eq. becomes

ΓI

v ˙ ms→lLf dΓI =

ΓI

ρsLf v(u −xτ, v −yτ)·(yξ, −xξ)dΓI = 0 Simplfies to

ΓI

v (yτxξ + usyξ) dΓI =

ΓI

v (yτ + usyx) dx = 0

◮ Wave equation with Galerkin?

SLIDE 11

Discrete Interface Equations (SUPG)

Clarkson

U N I V E R S I T Y

◮ 1D formulation

ΓI
v + aI

∂v ∂ξ τI

[[

[F] ] − ˙ ms→lLf ] dΓI = 0 aI = (u − xτ)xξ + (v − yτ)yξ ¯ τI = 1 J2 h (|aI| + h/∆t) h = J (p + 1)2

◮ Upwinding, but no global energy conservation ◮ Should thermal conductivity affect stabilization?

SLIDE 12

Third Implementation Possible?

Clarkson

U N I V E R S I T Y

◮ Let FI and be represented by continuous 1D polynomials on

interface (trace of temperature space on interface)

◮ Find FI such that Ω

v

∂JρcpT ∂τ

− ∂v

∂ξ E − ∂v ∂η F

dξdη +
Γ

v (E, F) · ndΓ +

nel

n=1

Ωn

∂v ∂ξ aE + ∂v ∂η aF

¯

τ ∂JρcpT ∂τ + ∂ ∂ξ E + ∂ ∂ηF

dΩ

+

ΓI

vFIdΓ = 0

◮ Determines interface jump in flux FI

SLIDE 13

Third Implementation...

Clarkson

U N I V E R S I T Y

◮ Now use jump in flux in interface equation

ΓI
v + aI

∂v ∂ξ τI

[FI − ˙

ms→lLf ] dΓI = 0

◮ Method is globally energy conservative (set v = 1), Ω

∂JρcpT ∂τ

dξdη +
Γ

(E, F) · ndΓ +

ΓI

FIdΓ = 0 and

ΓI

[FI − ˙ ms→lLf ] dΓI = 0

◮ Conservative and stabilized

SLIDE 14

Mesh Movement

Clarkson

U N I V E R S I T Y

◮ Tangential interface positions

ΓI

∂v ∂ξ ks

x2

ξ + y2 ξ dξ = L(v) ◮ For L(v) = 0 and ks = constant, gives uniform element size

∂

x2

ξ + y2 ξ

∂ξ = 0

◮ Interior vertex movement by spring method nneighbor

j=1

ks,i−j(xi − xj) = Xi

nneighbor

j=1

ks,i−j(yi − yj) = Yi

SLIDE 15

Mesh Deformation & Adaptation

Clarkson

U N I V E R S I T Y

x y

0.5

0.5

x y

0.5

0.5

◮ Deform: Spring method ◮ Swap: maximize minimum angle to create Delaunay mesh ◮ Coarsen: collapse edge to vertex closest to center of area,

then edge swap to maintain Delaunay

◮ Refine: Rebay point placement, with Bowyer-Watson insertion

algorithm

SLIDE 16

Test Problem 1

Clarkson

U N I V E R S I T Y

0.5 1

1
0.8
0.6
0.4
0.2

0.2 0.4 0.6 0.8 1 Solid Liquid

◮ Parameters

◮ ρl, µl, cp,l, kl ◮ ρs, cp,s, ks ◮ Lf

◮ Boundary Conditions

◮ Periodic in x ◮ Bottom:

T = Tm, u, v = (us, 0)

◮ Top: T = Tm ◮ Solid moves at speed us

◮ ρl = ρs: velocity is

continuous at interface (u, v = us, 0)

◮ Translation of liquid and

solid and constant T.

SLIDE 17

Numerical Results for Model Problem

Clarkson

U N I V E R S I T Y

time step amplitude 5 10 15 20 25 30 35 40 0.0045 0.005 0.0055 0.006

◮ Amplitude versus time for ∆t = 1/320, Pe = 1, 4 points per

wavelength, p = 1.

◮ Galerkin formulation undamped, singular for steady problems

SLIDE 18

Accuracy for Model Problem

Clarkson

U N I V E R S I T Y

10 10

1

10

−8

10

−6

10

−4

10

−2

k ∆ x ||y−yexact||

◮ × = SUPG, line = Galerkin ◮ Accuracy is similar between two methods

SLIDE 19

Solid Sheet Manufacturing

Clarkson

U N I V E R S I T Y

x=0 h Liquid Solid Heat removal profile

◮ Increase pull speed and see effect on grown sheet ◮ Heat equation solved for point at leading edge ◮ Leading edge point constrained to move horizontally

SLIDE 20

Results for Galerkin

Clarkson

U N I V E R S I T Y

◮ Calculations stop because of grid-size kind at leading edge ◮ Final non-dimensional pull-speed: 0.028

SLIDE 21

Results for SUPG

Clarkson

U N I V E R S I T Y

◮ Calculations again stop, but with more resolved cusp ◮ Final non-dimensional pull-speed: 0.047

SLIDE 22

Comparison of Shapes

Clarkson

U N I V E R S I T Y

Galerkin

0.1
0.05

0.05 0.1 0.15 0.2

0.26
0.24
0.22
0.2
0.18
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02

x y

0.1
0.05

0.05 0.1 0.15 0.2

0.26
0.24
0.22
0.2
0.18
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02

x y

pull speed = 0.016 SUPG

0.1
0.05

0.05 0.1 0.15 0.2

0.26
0.24
0.22
0.2
0.18
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02

x y

0.1
0.05

0.05 0.1 0.15 0.2

0.26
0.24
0.22
0.2
0.18
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02

x y

pull speed = 0.016

SLIDE 23

Comparison of Final Shapes

Clarkson

U N I V E R S I T Y

Galerkin

0.1
0.05

0.05 0.1 0.15 0.2

0.26
0.24
0.22
0.2
0.18
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02

x y

667 time steps, pull speed = 0.029

0.1
0.05

0.05 0.1 0.15 0.2

0.26
0.24
0.22
0.2
0.18
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02

x y

603 time steps, pull speed = 0.029 SUPG

0.1
0.05

0.05 0.1 0.15 0.2

0.26
0.24
0.22
0.2
0.18
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02

x y

1702 time steps, pull speed = 0.047

0.1
0.05

0.05 0.1 0.15 0.2

0.26
0.24
0.22
0.2
0.18
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02

x y

475 time steps, pull speed = 0.023

SLIDE 24

Simplified Test Problem

Clarkson

U N I V E R S I T Y

◮ Rotated normal propagation

ˆ y = y cos(θ) + xxin(θ)

◮ v = 0 - normal velocity towards interface and

tangential velocity along interface

◮ Fixed temperatures at ˆ

y = ±1

◮ Non-adiabatic walls, but no convective flux ◮ Perfect solutions

1
0.8
0.6
0.4
0.2

0.2 0.4 0.6 0.8 1 x y

SLIDE 25

Conclusions

Clarkson

U N I V E R S I T Y

◮ Conservative formulation

◮ Energy/mass conservative ◮ Stable for unsteady ◮ Less robust for steady problems (odd/even decoupling)

◮ Non-conservative

◮ SUPG stabilized ◮ Should be more robust for steady and unsteady ◮ non-conservative