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Numeric Solutions of Thermal Problems Governed by Fractional Diffusion

Numeric Solutions of Thermal Problems Governed by Fractional Diffusion V.R. Voller , D.P Zielinski Department of Civil Engineering, University of Minnesota, Minneapolis, MN 55455 volle001@umn.edu , ziel0064@umn.edu. Objective: Develop approximate solutions for the problem.

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Numeric Solutions of Thermal Problems Governed by Fractional Diffusion

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  1. Numeric Solutions of Thermal Problems Governed by Fractional Diffusion V.R. Voller, D.P Zielinski Department of Civil Engineering, University of Minnesota, Minneapolis, MN 55455 volle001@umn.edu, ziel0064@umn.edu Objective: Develop approximate solutions for the problem Where the flux is modeled as a fractional derivative e.g., Fraction –locality Skew An appropriate model when length-scales of heterogeneities are power-law distributed – e.g., fractal distribution of conductivity

  2. First start by defining the basic LOCAL FLUX via Finite Differrences 0 x n n-1 --- 3 2 w 1 e 0 1st up-stream face gradient local flux Create Finite Difference Scheme from flux balance 0 0

  3. Now define a NON-LOCAL FLUX n n-1 --- 3 2 w 1 e 0 Weighted average of all up-stream face gradients non-local flux Create Finite Difference Scheme from flux balance The Control Volume Weighted Flux Scheme CVWFS 0

  4. What's the Big Deal !! n n-1 --- 3 2 w 1 e 0 Weighted average of all up-stream face gradients non-local flux locality If we chose the power-law weights The left-hand Caputo fractional derivative In limit can be shown that where

  5. So with appropriate choice of weights W We have a scheme for fractional derivative n n-1 --- 3 2 w 1 e 0 Can generalize for right-derivative And Multi-Dimensions

  6. Alternative Monte-Carlo—domain shifting random walk Consider-domain with Dirichlet conditions (T_red and T_blue)—objective find value T_P Note this is the right-hand Levy distribution—fat tail on right associated with left hand Caputo fractional derivative Approach move (shift) centroid of domain by using steps picked from a suitable pdf Until domain crosses point P Then increment boundary counter (blue in case shown) And start over P P After n>>1 realizations—Value at point P can be approximated as

  7. Results: First a simple 1-D problem x = 0 1 integer sol. CVWFS domain shift

  8. Testing of Alternative weighting schemes x = 0 1 CVWFS—Voller, Paola, Zielinski, 2011 Relative Error 0 -0.03 CVWFS G.W Classic Grünwald Weights (GW) L1/L2 L1/L2 Weights: e.g., Yang and Turner, 2011

  9. And a 2-D problem 1 0,0 T=1 T=0 domain shift CVWFS

  10. SO: 1. Fractional Diffusion -a non-local model appropriate in some heterogeneous media 2. Can be numerically modeled using a weighted non-local flux 3. Or with a domain shifting Random walk P 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

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