a mesh-less method for solutions of the stefan problem

A Mesh-less method for solutions of the Stefan Problem

Preliminary Results

Vaughan Voller

If A man knows he is going to be hanged tomorrow it concentrates the mind wonderfully—

Dr Samuel Johnson 1709-1784

0T2

Essence of a Numerical Solution

1. Cover Domain with Field of NODES with locations xi

2. Unknowns are nodal values of dependent variable

iT)(T ix

3. By appropriate APPROXIMATIONSof Governing Equation Obtain a SETOf Discrete Algebraic equationsThat relate the nodal value at P to Values at the neighbors

P

nb

nbnbPP TaTa

Process is facilitated by

DATA Structure of node points

Approximation Process is facilitated by-- DATA Structure of node points

P

The most Straight Forward Is a STRUCTURED GRID OF NODES

Location of anynode point given aRow, Column index

If Grid is “square” immediate application of Taylor Series

0T4TTTTT ijj1ij1i1ij1ij2 Row i

Col. j


A more flexible approach is use Nodes to DefineAn UNSTRUCTURED MESH OF ELEMENTS

In Each Element obtain a Continuous Approximation

1

2

3

332211e T)y,x(NT)y,x(NT)y,x(N)y,x(T

Use this approximation with governing equatione.g. in CVFEM--use

0dAT n

n


The most flexible approach is to have no mesh

Very limited restriction on placement of nodes

But may have to be inventive

In arriving at sound discrete equations

nb

nbnbPP TaTa

---CLOUDS of Near Neighbor Nodes

Increasing Flexibility

GRID---Structure

Easy ApproximationEfficient solution Ax=b —

Difficult to Fit Geometry

MESH---FEM

More Complex approximationLess Efficient solution Ax=b —

Easy to Fit GeometryCould be Difficult to adapt

“CLOUD”---SPH

“Poor” approximationEven Less Efficient solution Ax=b —

Very Easy to Fit GeometryEasy to adapt

Simplistic Summary

Corrective Smooth Particle Method CSPM -related to SPHChen et al IJNME 1999

P

..)yy(yT)xx(

xTT)(T PPP

x

Taylor Series about node P

Multiply by Weighting Factor associated with node P

..)yy(yT)(W)xx(

xT)(WT)(W)(T)(W PPPPPPP

xxxxx

..)yy(yT)(W)xx(


xxxxx

Properties of W

P2hp

nbP Rminh 0.5

Symmetric about point P

Finite region of support

Differentiable

2r,0

2r1,r2h41

1r,rr1h1

)r(W 32

3432

23

2

r=R/h

Char. Length multiple of nearest neig. distance

Corrective Smooth Particle Method CSPM -related to SPHChen et al IJNME 1999

P

..)yy(yT)xx(

xTT)(T PPP

x

Taylor Series about node P

Multiply by Weighting Factor associated with node P

..)yy(yT)(W)xx(


xxxxx

Integrate over support ..)yy(

yT)(W)xx(

xT)(WT)(WdV)(T)(W PPPPPPP

xxxxx

dVW

dV)(T)(WT

P

P

P

xxsimilar manipulations for first derivatives

dVW

dV)(T)(WT

P

P

P

xx

nbP Rminh

P

If Rnb is the radial distance to the neighbors of P

0.5

Corrective Smooth Particle Method CSPM -related to SPHChen et al IJNME 1999 Integrate numerically

Using particles as integration points

nodeskkP

nodeskkkP

P Vol)(W

VolT)(WT

k

k

x

x

2hp

2r,0

2r1,r2h41

1r,rr1h1

)r(W 32

3432

23

2

r=R/h

nbnb

nbnbnbP

1w

,TwT

Critical Feature

Weak point --- 2kk hVol

SolidTi < Tm

a.) time t = 0

Application to The Stefan Problem--- General problem of Interest

SolidT < Tm

b.) time t > 0

LiquidT > Tm

liquid-solid interface T = Tm

T=Ta >Tm

n

At time t>=0 apply a fixed temperatureT=Ta >Tm to a patch of the boundary so as to cause a melt region that grows with time

Objective track the movement of the melt front melt

Initial state Insulated region containing solid atTemperature T; < Tm (melt temp)

l

2

,

Assume constant density and specific heat c — but jump in conductivity K.

with time scale and space scales

Governing Equations

ma

mD

TTTTT

L1

H)TT(cS ma

t

Dimensionless temperature

And dimensionless grouping , H-latent heat, St-Stefan number, L dim. Lat. heat

T=0g=0

g=1

TtT 2

TKtT 2

nvn tS

1TKT

liquid-solid interface

T = 0

n

Two-Domain Stefan Model Diffusive Interface—Single Domain

)TK̂(tH

Phase change occurs smoothly acrossA “narrow” temperature range

gLTH

K)g1(gK̂

Particular Versions

s(t)

T=1 T= 0.5

K=0.25

L=1

one-dimension

Has analytical solution

T=1

Ti = 0

Melting of unit cylinder, initially at phaseChange temperature.

)T(tH

Solve in Cartesian

Check with fine grid FD solutionUsing radial symmetry

)xTK(

xtH

Two Dimension

CSPM Solution

SCPMPTq

SCPMPP

newP qtHH

q)T(tH

Use CSPM approximation of derivative twiceBackward Euler (explicit) in time

PP

nbnbnbP

newP TaTatHH

Can be manipulated into general form

Data Structure

Global Number nodes 1-----n

Identify “cloud” of neighborsassociated with each node----

make a list of nodes (global numbers)that fall within a radial distance 2hP

2r,0

2r1,r2h41

1r,rr1h1

)r(W 32

3432

23

2

P2hp

LHLH

00HH

TnewP

newP

newP

newP

newP

T=1 T=-0.5

K = 0.25

t = 0.002h = 0.033-0.05

One-D solution

Front Movement

o o o CSPM

Analytical

Temperature HistoriesAt ref points

Plateau at phase change temp.A feature in all fixed grid solutions

Temperature Profileat time t =2.8

T=1

Ti = 0

)T(tH

For the 2-D problem Need to consider a means of placing out points

Two Methods

Discretization: Two steps

Put points along boundary of domain—with equal arc spacing r Make a structured mesh with spacing r

Then Lay boundary Mesh Over Structure MeshAdd structured points to SPH node List if they are a distance INSIDE boundary

SPH Nodes are a List i=1 to nbound

4/r

Black Dot:structured mesh point excluded from SPH list

Blue Circle:Boundary Polygon Red Circle:

structured mesh point Included in SPH list

Gives a reasonablyWell spaced grid

Easy to identify “node Clouds”

IDEA stolen from Immersed BoundaryMethods of Fotis

Sotiropoulos

h = r

ResultsRadial Movement of Front with Time

Fine grid radial symmetry solution

Melt pattern atAn intermediate times

Also works when points are“Jostled”

Unstructured Mesh

Delaunay

“Patchy”

Good Results

But sensitiveto choice of h

P

So far very basic calculations

But they show promise—

Need to look at

Adaptivity

Lagrangian

Fans Toes Shoreline

Two Sedimentary Moving Boundary Problems of Interest

Moving Boundaries in Sediment Transport

1km

Examples of Sediment FansMoving Boundary

How does sediment-basement interfaceevolve

Badwater Deathvalley

An Ocean Basin

Melting vs. Shoreline movement

h(x,y,t)

q

bed-rock

ocean

y

shoreline

x = s(t)

land surface

(x,y,t)

A 2-D Front -Limit of Cliff face Shorefront But Account of Subsidence and relative ocean level

0hif),t,y,x(LhH

)h(tH

Enthalpy Sol.

xy

]L/H,[MINfrac 1

Solve on fixed gridin plan view

Track Boundary by calculating in each cell

A 2-D problem Sediment input into an oceanwith an evolving trench driven By hinged subsidence

With Trench

Need to account for Interaction with channels whichcan avulse

Increasing Flexibility

GRID---Structure

Easy ApproximationEfficient solution Ax=b —

Difficult to Fit Geometry

MESH---FEM

More Complex approximationLess Efficient solution Ax=b —

Easy to Fit GeometryCould be Difficult to adapt

“CLOUD”---SPH

“Poor” approximationEven Less Efficient solution Ax=b —

Very Easy to Fit GeometryEasy to adapt

a mesh-less method for solutions of the stefan problem

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