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MESA LAB CTRW MATLAB Toolbox Applied Fractional Calculus Workshop Series Tomas Oppenheim MESA LAB MESA (Mechatronics, Embedded Systems and Automation)

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Presentation on theme: "MESA LAB CTRW MATLAB Toolbox Applied Fractional Calculus Workshop Series Tomas Oppenheim MESA LAB MESA (Mechatronics, Embedded Systems and Automation)"— Presentation transcript:

1 MESA LAB CTRW MATLAB Toolbox Applied Fractional Calculus Workshop Series Tomas Oppenheim MESA LAB MESA (Mechatronics, Embedded Systems and Automation) LAB School of Engineering, University of California, Merced E: toppenheim@ucmerced.edu Phone: 310-853-9234 toppenheim@ucmerced.edu Lab: CAS Eng 820 (T: 228-4398) June 16, 2014. Monday 4:00-6:00 PM Applied Fractional Calculus Workshop Series @ MESA Lab @ UCMerced

2 MESA LAB Outline 1.Continuous Time Random Walk (CTRW) Background 2.CTRW MATLAB Basic Equations 3.1D Forward Modeling: Generating Temporal/Spatial Profiles 4.1D Inverse Problem: Fitting CTRW Solutions to Data Sets 5.Conclusions and Future Work AFC Workshop Series @ MESALAB @ UCMerced 06/16/2014

3 MESA LAB CTRW Background AFC Workshop Series @ MESALAB @ UCMerced 06/16/2014 Based on Book: Klages et.al. Anomalous Transport. Foundations and Applications. Wiley (2008). “CTRW’s are rather good and general phenomenological models of diffusion, including processes of anomalous transport.” PHYSICAL REVIEW E 84, 051139 (2011)

4 MESA LAB CTRW Background AFC Workshop Series @ MESALAB @ UCMerced 06/16/2014 Based on Book: Klages et.al. Anomalous Transport. Foundations and Applications. Wiley (2008). “CTRW’s are rather good and general phenomenological models of diffusion, including processes of anomalous transport.” Probability of the particle being in point x at instant t.

5 MESA LAB CTRW Background AFC Workshop Series @ MESALAB @ UCMerced 06/16/2014 Based on paper: Berkowitz et. al. Modeling Non-Fickian Transport in Geological Formations as a Continuous Time Random Walk. Review of Geophysics, 44 (2006). http://www.newcastle.edu.au/research-and-innovation/centre/cgmm/research/thermal-transport-in-composites-and-porous-media Porous MediaDye Flowing Through Porous Media Use CTRW to model flow of non-Fickian (anomalous) and Fickian (Gaussian) transport Different velocities due to heterogeneities in Porous media

6 MESA LAB CTRW Background AFC Workshop Series @ MESALAB @ UCMerced 06/16/2014 Based on paper: Berkowitz et. al. Modeling Non-Fickoan Transport in Geoligical Formations as a Continuous Time Random Walk. Review of Geophysics, 44 (2006). Hu Sheng, YangQuan Chen, TianShuang Qiu. Fractional Processes and Fractional-Order Signal Processing, Techniques and Applications. Springer (2012) Gaussian or Anomalous Transport (PDF)?

7 MESA LAB CTRW Background AFC Workshop Series @ MESALAB @ UCMerced 06/16/2014 Based on paper: Berkowitz et. al. Modeling Non-Fickian Transport in Geological Formations as a Continuous Time Random Walk. Review of Geophysics, 44 (2006). Mean, ensemble-averaged, normalized concentration (e.g. of dye) Generalized Master Equation (GME): Continuous transitions of tracer transport Defined later

8 MESA LAB CTRW Background AFC Workshop Series @ MESALAB @ UCMerced 06/16/2014 Based on paper: Berkowitz et. al. Modeling Non-Fickian Transport in Geological Formations as a Continuous Time Random Walk. Review of Geophysics, 44 (2006). Probability/time for a walker to just arrive at site s at time t CTRW Transport Equations: Using Laplace Transform, GME = CTRW Probability/time for a displacement s with a difference of arrival times of t

9 MESA LAB CTRW Background AFC Workshop Series @ MESALAB @ UCMerced 06/16/2014 Based on paper: Berkowitz et. al. Modeling Non-Fickian Transport in Geological Formations as a Continuous Time Random Walk. Review of Geophysics, 44 (2006). Basic PDF. Determines nature of transport (Fickian or Non-Fickian) Correspondence between GME and LT of GME CTRW Transport Equations: Using Laplace Transform, GME = CTRW

10 MESA LAB CTRW MATLAB Basic Equations AFC Workshop Series @ MESALAB @ UCMerced 06/16/2014 Based on paper: Cortis et.al. The CTRW Matlab toolbox v3.1: a practical user’s guide (2013). Laplace Transform of CTRW Transport velocity Generalized Diffusion Coefficient “heart” of CTRW formulation

11 MESA LAB CTRW MATLAB Basic Equations AFC Workshop Series @ MESALAB @ UCMerced 06/16/2014 Based on paper: Cortis et.al. The CTRW Matlab toolbox v3.1: a practical user’s guide (2013).

12 MESA LAB 1D Forward Modeling: Generating Temporal Profiles AFC Workshop Series @ MESALAB @ UCMerced 06/16/2014 Based on paper: Cortis et.al. The CTRW Matlab toolbox v3.1: a practical user’s guide (2013). The Options Variable:

13 MESA LAB 1D Forward Modeling: Generating Temporal Profiles AFC Workshop Series @ MESALAB @ UCMerced 06/16/2014 Based on paper: Cortis et.al. The CTRW Matlab toolbox v3.1: a practical user’s guide (2013). The Options Variable:

14 MESA LAB 1D Forward Modeling: Generating Temporal Profiles AFC Workshop Series @ MESALAB @ UCMerced 06/16/2014 Based on paper: Cortis et.al. The CTRW Matlab toolbox v3.1: a practical user’s guide (2013). The Options Variable: ?

15 MESA LAB 1D Forward Modeling: Generating Temporal Profiles AFC Workshop Series @ MESALAB @ UCMerced 06/16/2014 Based on paper: Cortis et.al. The CTRW Matlab toolbox v3.1: a practical user’s guide (2013). The Options Variable:

16 MESA LAB 1D Forward Modeling: Generating Temporal Profiles AFC Workshop Series @ MESALAB @ UCMerced 06/16/2014 “A concentration measurement in the field or laboratory will be most closely related to either the resident or the flux concentration. The former describes all solute particles present at some point while the latter describes solute particles that have moved irrevocably past some point.” Zhang et al. Geophysical Research Letters. (33) 2006.

17 MESA LAB AFC Workshop Series @ MESALAB @ UCMerced 06/16/2014 Based on paper: Cortis et.al. The CTRW Matlab toolbox v3.1: a practical user’s guide (2013). Assume: Porous Column, use ADE model 1D Forward Modeling: Generating Temporal Profiles Ex: Breakthrough Curve/Concentration Profile (temporal profile) for classical ADE Equation

18 MESA LAB 1D Forward Modeling: Generating Temporal Profiles AFC Workshop Series @ MESALAB @ UCMerced 06/16/2014 Based on paper: Cortis et.al. The CTRW Matlab toolbox v3.1: a practical user’s guide (2013). Ex: Breakthrough Curve/Concentration Profile (temporal profile) for classical ADE Equation Matlab Commands:

19 MESA LAB 1D Forward Modeling: Generating Temporal Profiles AFC Workshop Series @ MESALAB @ UCMerced 06/16/2014 Based on paper: Cortis et.al. The CTRW Matlab toolbox v3.1: a practical user’s guide (2013). Ex: Breakthrough Curve/Concentration Profile (temporal profile) for classical ADE Equation Matlab Commands:

20 MESA LAB 1D Forward Modeling: Generating Temporal Profiles AFC Workshop Series @ MESALAB @ UCMerced 06/16/2014 Based on paper: Cortis et.al. The CTRW Matlab toolbox v3.1: a practical user’s guide (2013). Ex: Breakthrough Curve/Concentration Profile (temporal profile) for classical ADE Equation Resulting Temporal Profile:

21 MESA LAB Outline 1.Continuous Time Random Walk (CTRW) Background 2.CTRW MATLAB Basic Equations 3.1D Forward Modeling: Generating Temporal/ Spatial Profiles 4.1D Inverse Problem: Fitting CTRW Solutions to Data Sets 5.Conclusions and Future Work AFC Workshop Series @ MESALAB @ UCMerced 06/16/2014

22 MESA LAB Conclusions AFC Workshop Series @ MESALAB @ UCMerced 06/16/2014 Based on paper: Cortis et.al. The CTRW Matlab toolbox v3.1: a practical user’s guide (2013). 1.CTRW is good model for Fickian and Non-Fickian transport – “CTRW’s are rather good and general phenomenological models of diffusion, including processes of anomalous transport.” 2.MATLAB toolbox available with detailed instruction manual to generate temporal/spatial concentration profiles and model data using either TPM, ADE, or ETA model

23 MESA LAB Future Work AFC Workshop Series @ MESALAB @ UCMerced 06/16/2014 Based on paper: Cortis et.al. The CTRW Matlab toolbox v3.1: a practical user’s guide (2013).

24 MESA LAB Future Work AFC Workshop Series @ MESALAB @ UCMerced 06/16/2014 Based on paper: Cortis et.al. The CTRW Matlab toolbox v3.1: a practical user’s guide (2013).

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