1. A Mesh-less method for solutions of the Stefan Problem
Vaughan Voller
If A man knows he is going to be hanged tomorrow it concentrates the mind wonderfully—
Dr Samuel Johnson
Preliminary Results

2. Essence of a Numerical Solution
Cover Domain with
Field of NODES with locations xi
2. Unknowns are nodal values
of dependent variable
3. By appropriate APPROXIMATIONS
of Governing Equation Obtain a SET
Of Discrete Algebraic equations
That relate the nodal value at P to
Values at the neighbors
Process is facilitated by
DATA Structure of node points P

3. The most Straight Forward Is a STRUCTURED GRID OF NODES
Approximation Process is facilitated by-- DATA Structure of node points P
Location of any
node point given a
Row, Column index
The most Straight Forward
Is a STRUCTURED GRID OF NODES
If Grid is “square” immediate application of Taylor Series
Row i
Col. j

4. A more flexible approach is use Nodes to Define
Approximation Process is facilitated by-- DATA Structure of node points
A more flexible approach is use Nodes to Define
An UNSTRUCTURED MESH OF ELEMENTS n 3
In Each Element
obtain a
Continuous
Approximation
Use this approximation with governing equation
e.g. in CVFEM--use 2 1

5. Approximation Process is facilitated by-- DATA Structure of node points
The most flexible approach is to have no mesh
---CLOUDS of Near Neighbor Nodes
Very limited restriction on
placement of nodes
But may have to be inventive
In arriving at sound discrete equations

6. Easy to adapt Simplistic Summary Increasing Flexibility
GRID---Structure
MESH---FEM
“CLOUD”---SPH
More Complex approximation
Less Efficient solution Ax=b —
Easy to Fit Geometry
Could be Difficult to adapt
“Poor” approximation
Even Less Efficient solution Ax=b —
Very Easy to Fit Geometry
Easy to adapt
Easy Approximation
Efficient solution Ax=b —
Difficult to Fit Geometry

7. Corrective Smooth Particle Method CSPM -related to SPH
Chen et al IJNME 1999 P
Taylor Series about node P
Multiply by Weighting Factor associated with node P

8. Properties of W Symmetric about point P Finite region of support
Differentiable P
2hp
1.5 > a > 0.5
Char. Length multiple of nearest neig. distance
r=R/h

9. similar manipulations for first derivatives
Corrective Smooth Particle Method CSPM -related to SPH
Chen et al IJNME 1999 P
Taylor Series about node P
Multiply by Weighting Factor associated with node P
Integrate over support
similar manipulations for first derivatives

10. Integrate numerically
Corrective Smooth Particle Method CSPM -related to SPH
Chen et al IJNME 1999
Integrate numerically
Using particles as integration points P
2hp
1.5 > a > 0.5
If Rnb is the radial distance to the neighbors of P
r=R/h
Weak point ---
Critical Feature

11. Application to The Stefan Problem--- General problem of Interest
Solid
T < Tm
b.) time t > 0
Liquid
T > Tm
liquid-solid interface
T = Tm
T=Ta >Tm n
At time t>=0 apply a fixed temperature
T=Ta >Tm to a patch of the boundary so
as to cause a melt region that grows with time
Initial state Insulated region containing solid at
Temperature T; < Tm (melt temp)
a.) time t = 0
Solid
Ti < Tm
Objective track the movement of the melt front Gmelt

12. Governing Equations
Assume constant density and specific heat c — but jump in conductivity K.
with time scale and space scales
Dimensionless temperature
And dimensionless grouping
, DH-latent heat, St-Stefan number, L dim. Lat. heat
liquid-solid interface
T = 0 n
Two-Domain Stefan Model
Diffusive Interface—Single Domain
T=0
g=0
g=1
Phase change occurs smoothly across
A “narrow” temperature range

13. Particular Versions Two Dimension one-dimension s(t) K=0.25 T= 0.5
Melting of unit cylinder, initially at phase
Change temperature.
Has analytical solution
Solve in Cartesian
Check with fine grid FD solution
Using radial symmetry

14. CSPM Solution
Use CSPM approximation of derivative twice
Backward Euler (explicit) in time P
2hp
Data Structure
Global Number nodes n
Identify “cloud” of neighbors
associated with each node----
make a list of nodes (global numbers)
that fall within a radial distance 2hP
Can be manipulated into general form

15. Dt = 0.002 One-D solution Front Movement K = 0.25 T=1 T=-0.5
o o o CSPM
Analytical
h =
Dt = 0.002
Temperature Profile
at time t =2.8
Temperature Histories
At ref points
Plateau at phase change temp.
A feature in all fixed grid solutions

16. Ti = 0
T=1
For the 2-D problem Need to consider a means of placing out points
Two Methods

17. Discretization: Two steps
Put points along boundary of domain—
with equal arc spacing
Make a structured mesh with spacing
SPH Nodes are a List i=1 to nbound
Then Lay boundary Mesh Over Structure Mesh
Add structured points to SPH node List if they are a distance INSIDE boundary

18. Black Dot:
structured mesh point
excluded from SPH list
Blue Circle:
Boundary Polygon
Red Circle:
structured mesh point
Included in SPH list
IDEA stolen from Immersed Boundary
Methods of
Fotis Sotiropoulos
Gives a reasonably
Well spaced grid
Easy to identify “node Clouds”
h = Dr

19. Results Radial Movement of Front with Time
Fine grid radial symmetry solution
Melt pattern at
An intermediate
times

20. Also works when points are
“Jostled”

21. “Patchy”
Unstructured Mesh
Delaunay
Good Results
But sensitive
to choice of h P

22. So far very basic calculations
But they show promise—
Need to look at
Adaptivity
Lagrangian

23. Moving Boundaries in Sediment Transport
Two Sedimentary Moving Boundary Problems of Interest
Shoreline
Fans Toes

24. Badwater Deathvalley
Examples of Sediment Fans
Moving Boundary
1km
How does sediment-
basement interface
evolve

25. Melting vs. Shoreline movement
An Ocean Basin

26. Enthalpy Sol. A 2-D Front -Limit of Cliff face Shorefront But
Account of Subsidence and relative ocean level
Enthalpy Sol.
land surface
shoreline
ocean
h(x,y,t)
x = s(t)
G(x,y,t)
Solve on fixed grid
in plan view
bed-rock y
Track Boundary by calculating in each cell x y

27. A 2-D problem
Sediment input into an ocean
with an evolving trench driven
By hinged subsidence

28. With Trench

33. Need to account for
Interaction with channels which
can avulse

34. Easy to adapt Increasing Flexibility GRID---Structure MESH---FEM
“CLOUD”---SPH
Easy Approximation
Efficient solution Ax=b —
Difficult to Fit Geometry
More Complex approximation
Less Efficient solution Ax=b —
Easy to Fit Geometry
Could be Difficult to adapt
“Poor” approximation
Even Less Efficient solution Ax=b —
Very Easy to Fit Geometry
Easy to adapt