NUMERIC SOLUTIONS OF THERMAL PROBLEMS GOVERNED BY FRACTIONAL DIFFUSIONV.R. Voller, D.P Zielinski
Department of Civil Engineering, University of Minnesota, Minneapolis, MN [email protected], [email protected]
Objective: Develop approximate solutions for the problem
0 qWhere the flux is modeled as a fractional derivative e.g.,
)(2
1
2
1
x
T
x
Tqx
11
10
Fraction –locality
Skew An appropriate model when length-scales of heterogeneities are power-law distributed –e.g., fractal distribution of conductivity
n n-1 --- 3 2 w 1 e 0
x
TTqw
12
local flux 1st up-stream face gradient
First start by defining the basic LOCAL FLUX via Finite Differrences
Create Finite Difference Scheme from flux balance
0x
qq ew
0
"
"
"
"
"
"
"
'
**
***
***
***
***
***
***
**
sT
0
0
x
0 x
n n-1 --- 3 2 w 1 e 0
1
1
1n
j
jjjw x
TTWqnon-local flux
Weighted average of all up-stream face gradients
Now define a NON-LOCAL FLUX
Create Finite Difference Scheme from flux balance
0x
qq ew
0
"
"
"
"
"
"
"
'
********
********
*******
******
*****
****
***
**
sT
0
x
The
Control Volume Weighted Flux Scheme
CVWFS
n n-1 --- 3 2 w 1 e 0
1
1
1n
j
jjjw x
TTWqnon-local flux
Weighted average of all up-stream face gradients
What's the Big Deal !!
If we chose the power-law weights
xxjW j ]))[(1(
1)1(10,101
where 0
0.1
0.2
0.3
-10 -8 -6 -4 -2 0locality
x
w
x
w
w
xn x
Td
Txq
)2()1(lim00
In limit can be shown that The left-hand Caputo fractional derivative
n n-1 --- 3 2 w 1 e 0
w
n
j
jjjw x
T
x
TTWq
)2(1
1
1
So with appropriate choice of weights W We have a scheme for fractional derivative
x
0
"
"
"
"
"
"
"
'
********
********
*******
******
*****
****
***
**
sT
0x
qq ew
Can generalize for right-derivative
w
e x
Tq
)(
)2(
And Multi-Dimensions
Alternative Monte-Carlo—domain shifting random walk
Consider-domain with Dirichlet conditions (T_red and T_blue)—objective find value T_P
Approach move (shift) centroid of domain by using steps picked from a suitable pdf
P
P
Until domain crosses point P
Then increment boundary counter(blue in case shown)
And start over
After n>>1 realizations—Value at point P can be approximated as redred
blueblue
P Tn
nT
n
nT
Note this is the right-hand Levydistribution—fat tail on rightassociated with left hand Caputofractional derivative
Results: First a simple 1-D problem
0
x
T
x1T
0T
1,5. CVWFS
domain shift
integer sol.
x = 0 1
1
1
1n
j
jjjw x
TTWq
xxjW j ]))[(1(
1)1(101
0
x
T
x1T
0T
Testing of Alternative weighting schemes
CVWFS—Voller, Paola, Zielinski, 2011
k
iik
Gk xigW
1
1)2(
kk
gk
k
21
2)1(
Classic Grünwald Weights (GW)
1112/1 )1()2( xkkW LLk
L1/L2 Weights: e.g., Yang and Turner, 2011
Relative Error
L1/L2
G.W
CVWFS0
-0.03
x = 0 1
And a 2-D problem1
0,0 T=0T=1
5.,0
y
T
yx
T
x
0
0.2
0.4
0.6
0.8
1
1.2
-0.5 -0.45 -0.4 -0.35 -0.3 -0.25Location
f
CVWFS
Domain Shift
domain shift
CVWFS
0
0.2
0.4
0.6
0.8
1
1.2
0.25 0.3 0.35 0.4 0.45 0.5Location
f
CVWFS
Domain Shift
5.,0
y
T
yx
T
x
SO:1. Fractional Diffusion -a non-local model appropriate in some heterogeneous media
2. Can be numerically modeled using a weighted non-local flux
1
1
1n
j
jjjw x
TTWq
3. Or with a domain shifting Random walkP
4. Gives accurate and consistent solutions
5. Approach Can and Has been extended to transient case
6. Work is on-going for a FEM implementations of the CVWFS
