Inverse problem in anomalous diffusion with uncertainty propagation

• Autorzy: Silva L.G. , Knupp D.C. , Bevilacqua L. , Galeão A.C.N.R. , Silva Neto A.J.
• Języki publikacji: EN

Abstrakty

EN

Recently, Bevilacqua, Galeão and co-workers have developed a new analytical formulation for the simulation of diffusion with retention phenomena. This new formulation aims at the reduction of all diffusion processes with retention to a unifying model that can adequately simulate the retention effect. This model may have relevant applications in a number of different areas such as population spreading with partial hold up of the population to guarantee territorial domain chemical reactions inducing adsorption processes and multiphase flow through porous media. In this new formulation a discrete approach is firstly formulated taking into account a control parameter which represents the fraction of particles that are able to diffuse. The resulting governing equation for the modelling of diffusion with retention in a continuum medium requires a fourth-order differential term. Specific experimental techniques, together with an appropriate inverse analysis, need to be determined to characterize complementary parameters. The present work investigates an inverse problem which does not allow for simultaneous estimation of all model parameter. In addition a two-step characterization procedure proposed: in the first step the diffusion coefficient is estimated and in the second one the complementary parameters are estimated. In this paper, it is assumed that the first step is already completed and the diffusion coefficient is known with a certain degree of reliability. Therefore, this work is aimed at investigating the confidence intervals of the complementary parameters estimates considering both the uncertainties due to measurement errors in the experimental data and due to the uncertainty propagation of the estimated value of the diffusion coefficient. The inverse problem solution is carried out through the maximum likelihood approach, with the minimization problem solved with the Levenberg-Marquardt method, and the estimation of the confidence intervals is carried out through the Monte Carlo analysis.

Słowa kluczowe

EN

diffusion; inverse problems; uncertainty propagation; Monte Carlo method

PL

dyfuzja; zagadnienia odwrotne; propagacja niepewności; metoda Monte Carlo

• Wydawca: Instytut Podstawowych Problemów Techniki PAN
• Czasopismo: Computer Assisted Methods in Engineering and Science
• Rocznik: 2014
• Tom: Vol. 21, no. 3-4
• Strony: 245--255
• Opis fizyczny: Bibliogr. 18 poz., rys., tab., wykr.

Twórcy

autor

Silva L.G.

• Patrícia Oliva Soares Laboratory for Experimentation and Numerical Simulation in Heat and Mass Transfer (LEMA) Instituto Politécnico, Universidade do Estado do Rio de Janeiro IPRJ/UERJ, Nova Friburgo, RJ, Brazil

autor

Knupp D.C.

• Patrícia Oliva Soares Laboratory for Experimentation and Numerical Simulation in Heat and Mass Transfer (LEMA) Instituto Politécnico, Universidade do Estado do Rio de Janeiro IPRJ/UERJ, Nova Friburgo, RJ, Brazil

autor

Bevilacqua L.

• Universidade Federal do Rio de Janeiro, UFRJ/COPPE, Rio de Janeiro, RJ, Brazil

autor

Galeão A.C.N.R.

• Laboratório Nacional de Computação Científica, LNCC, Petrópolis, RJ, Brazil

autor

Silva Neto A.J.

• Patrícia Oliva Soares Laboratory for Experimentation and Numerical Simulation in Heat and Mass Transfer (LEMA) Instituto Politécnico, Universidade do Estado do Rio de Janeiro IPRJ/UERJ, Nova Friburgo, RJ, Brazil

Bibliografia

• [1] H. Atsumi. Hydrogen bulk retention in graphite and kinetics of diffusion. Journal of Nuclear Materials, 307/311: 1466–1470, 2002.
• [2] L. Bevilacqua, A.C.N.R. Galeão, F.P. Costa. A new analytical formulation of retention effects on particle diffusion process. Annals of the Brazilian Academy of Sciences, 83: 1443–1464, 2011.
• [3] B.F. Blackwell, K.J. Dowding, R.J. Cochran. Development and implementation of sensitivity coefficient equations for heat conduction problems. Numerical Heat Transfer — Part B, 36: 15–32, 1999.
• [4] S. Brandani, M. Jama, D. Ruthven. Diffusion, self-diffusion and counter-diffusion of benzene and p-xylene in silicalite. Micropor Mesopor Mat, 35/36: 283–300, 2000.
• [5] M.V. D’Angelo, E. Fontana, R. Chertcoff, M. Rosen. Retention phenomena in non-Newtonian fluid flows. Physica A: Statistical Mechanics and its Applications, 327: 44–48, 2003.
• [6] E. Deleersnijder, J.-M. Beckers, E.M. Delhez. The residence time of setting in the surface mixed layer. Environ Fluid Mech, 6: 25–42, 2006.
• [7] C. Derec, M. Smerlak, J. Servais, J.-C. Bacri. Anomalous diffusion in microchannel under magnetic field. Physics Procedia, 9: 109–112, 2010.
• [8] P.F. Green, Translational dynamics of macromolecules in metals. In: P. Neogi [Ed.], Diffusion in Polymers, Marcel Dekker Inc., 1996.
• [9] G.J. Hahn, S.S. Shapiro. Statistical models in engineering, Wiley, 1967.
• [10] S. Joannès, L. Mazé, A.R. Bunsell. A concentration-dependent diffusion coefficient model for water sorption in composite. Composite Structures, 108: 111–118, 2014.
• [11] H. Liu, K.E. Thompson. Numerical modeling of reactive polymer flow in porous media. Comput. Chem. Eng., 26: 1595–1610, 2002.
• [12] D.W. Marquardt. An algorithm for least-squares estimation of non-linear parameters. J. Soc. Industr. Appl. Math., 11: 431–441, 1963.
• [13] R. Metzler, J. Klafter. The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep., 339, 1—77, 2000.
• [14] N. Muhammad. Hydraulic, diffusion, and retention characteristics of inorganic chemicals in bentonite. Ph.D. Thesis, Department of Civil and Environmental Engineering, College of Engineering, University of South Florida, p. 251, 2004.
• [15] L.G. Silva, D.C. Knupp, L. Bevilacqua, A.C.N.R. Galeão, J.G. Simas, J.F. Vasconcellos, A.J. Silva Neto. Investigation of a new model for anomalous diffusion phenomena by means of an inverse analysis. Proceedings of the 4th Inverse Problems, Design and Optimization Symposium, Albi, France, 2013.
• [16] L.G. Silva, D.C. Knupp, L. Bevilacqua, A.C.N.R. Galeão, A.J. Silva Neto. Inverse problem in anomalous diffusion with uncertainty propagation. Proceedings of the 8th International Conference on Inverse Problems in Engineering, Cracow, Poland, 2014.
• [17] N.H. Thomsom, H.R.B. Orlande. Computation of sensitivity coefficients and estimation of thermophysical properties with the line heat source method. Proceedings of the III European Conference on Computational Mechanics. Solids, Structures and Coupled Problems in Engineering, Lisbon, Portugal, June 5–8, 2006.
• [18] J. Wu, K.M. Berland. Propagators and time-dependent diffusion coefficients for anomalous diffusion. Biophysical Journal, 95: 2049–2052, 2008.