2016, 6(2): 183-219. doi: 10.3934/naco.2016008

Singularly perturbed diffusion-advection-reaction processes on extremely large three-dimensional curvilinear networks with a periodic microstructure -- efficient solution strategies based on homogenization theory

1. 

University of the Bundeswehr Munich, Faculty of Informatics, Werner-Heisenberg-Weg 39, 85577 Neubiberg, Germany, Germany

2. 

Middle East Technical University, Institute of Applied Mathematics, 06531 Ankara, Turkey

Received  October 2015 Revised  July 2016 Published  August 2016

Boundary value problems on large periodic networks arise in many applications such as soil mechanics in geophysics or the analysis of photonic crystals in nanotechnology. As a model example, singularly perturbed elliptic differential equations of second order are addressed. Typically, the length of periodicity is very small compared to the size of the covered region. The overall complexity of the networks raises serious problems on the computational side. The high density of the graph, the huge number of edges and vertices and highly oscillating coefficients necessitate solution schemes, where even a numerical approximation is no longer feasible. Realizing that such a system depends on two spatial scales - global scale (full domain) and local scale (microstructure) - a two-scale asymptotic analysis for network differential equations is applied. The limit process leads to a homogenized model on the full domain. The homogenized coefficients cover the micro-oscillations and the topology of the periodic network and characterize the effective behaviour. The approximate model's quality is guaranteed by error estimates. Furthermore, singularly perturbed microscopic models with a decreasing diffusion part and transport-dominant problems are discussed. The effectiveness of the two-scale limit analysis is demonstrated by numerical examples of diffusion-advection-reaction problems on large periodic grids.
Citation: Erik Kropat, Silja Meyer-Nieberg, Gerhard-Wilhelm Weber. Singularly perturbed diffusion-advection-reaction processes on extremely large three-dimensional curvilinear networks with a periodic microstructure -- efficient solution strategies based on homogenization theory. Numerical Algebra, Control & Optimization, 2016, 6 (2) : 183-219. doi: 10.3934/naco.2016008
References:
[1]

T. Arbogast, J. Douglas Jr. and U. Hornung, Derivation of the double porosity model of single phase flow via homogenization theory,, SIAM Journal on Mathematical Analysis, 21 (1990), 823. doi: 10.1137/0521046.

[2]

J. L. Auriault and J. Lewandoska, Diffusion/adsorption/advection macrotransport in soils,, European Journal of Mechanics, 15 (1996), 681.

[3]

J. Bear, Dynamics of Fluids in Porous Media,, Dover Publications, (1988).

[4]

A. Bensoussan, J. L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures,, American Mathematical Society, (2011). doi: 10.1090/chel/374.

[5]

G. Bouchitté and I. Fragalà, Homogenization of elastic thin structures: a measure-fattening approach,, J. Convex Anal., 9 (2002), 1.

[6]

C. Boutin, A. Rallu and S. Hans, Large scale modulation of high frequency acoustic waves in periodic porous media,, The Journal of the acoustical society of America, 132 (2012), 3622.

[7]

E. Canon and M. Lenczner, Modelling of thin elastic plates with small piezoelectric inclusions and sistributed electronic circuits. Models for inclusions that are small with respect to the thickness of the plate,, Journal of Elasticity, 55 (1999), 111. doi: 10.1023/A:1007609122248.

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F. Civan, Porous Media Transport Phenomena,, John Wiley & Sons, (2011).

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M. Cunha and L. Nunes, Groundwater Characterization, Management and Monitoring,, WITPress, (2010).

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L. Dormieux and E. Lemarchand, Homogenization approach of advection and diffusion in cracked porous material,, Journal of Engineering Mechanics, 127 (2001), 1267.

[11]

M. Espedal, A. Fasano and A. Mikelić, Filtration in porous media and industrial application,, Lecture Notes in Mathematics, 1734 (2000), 24. doi: 10.1007/BFb0103973.

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P. Exner, J. P. Keating, P. Kuchment, T. Sunada and A. Teplyaev, Analysis on Graphs and Its Applications - Proceedings of Symposia in Pure Mathematics,, American Mathematical Society, 77 (2008). doi: 10.1090/pspum/077.

[13]

S. Franz and H.-G. Roos, The capriciousness of numerical methods for singular perturbations,, SIAM review, 53 (2011), 157. doi: 10.1137/090757344.

[14]

A. Fortin, J. M. Urquiza and R. Bois, A mesh adaptation method for 1D-boundary layer problems,, International Journal of Numerical Analysis and Modeling, 3 (2012), 408.

[15]

S. Göktepe and C. Miehe, A micro-macro approach to rubber-like materials. Part III: The micro-sphere model of anisotropic Mullins-type damage,, Journal of the Mechanics and Physics of Solids, 53 (2005), 2259. doi: 10.1016/j.jmps.2005.04.010.

[16]

U. Hornung, Homogenization and Porous Media,, Springer, (1996). doi: 10.1007/978-1-4612-1920-0.

[17]

J. D. Joannopoulos, R. D. Meade and J. N. Winn, Photonic Crystals,, Princeton University Press, (2008).

[18]

P. Kogut and G. Leugering, Asymptotic analysis of optimal control problems on periodic singular graphs,, Systems & Control: Foundations & Applications Birkhäuser Boston, (2011). doi: 10.1007/978-0-8176-8149-4.

[19]

P. Kuchment, Quantum graphs I: Some basic structures,, Waves Random Media, 14 (2004), 107. doi: 10.1088/0959-7174/14/1/014.

[20]

P. Kuchment, Quantum graphs: An introduction and a brief survey,, in Analysis on Graphs and Its Applications - Proceedings of Symposia in Pure Mathematics(eds. P. Exner, 77 (2008), 291. doi: 10.1090/pspum/077/2459876.

[21]

P. Kuchment and L. Kunyansky, Differential operators on graphs and photonic crystals,, Advances in Computational Mathematics, 16 (2002), 263. doi: 10.1023/A:1014481629504.

[22]

E. Kropat, Über die Homogenisierung von Netzwerk-Differentialgleichungen,, Wissenschaftlicher Verlag Berlin, (2007).

[23]

M. Lenczner, Multiscale model for atomic force microscope array mechanical behaviour,, Applied Physics Letters, 90 (2007).

[24]

M. Lenczner and D. Mercier, Homogenization of periodic electrical networks including voltage to current amplifiers,, Multiscale Modeling and Simulation, 2 (2004), 359. doi: 10.1137/S1540345903423919.

[25]

M. Lenczner and G. Senouci-Bereksi, Homogenization of electrical networks including voltage-to-voltage amplifiers,, Mathematical Models and Methods in Applied Sciences, 9 (1999), 899. doi: 10.1142/S0218202599000415.

[26]

E. Marušić-Palokaa and S. Marušić, Computation of the permeability tensor for the fluid flow through a periodic net of thin channels,, Applicable Analysis, 64 (1997), 27. doi: 10.1080/00036819708840521.

[27]

V. G. Mazja, S. A. Nasarow and B. A. Plamenewski, Asymptotische Theorie elliptischer Randwertaufgaben in Singulär gestörten Gebieten II,, Wiley, (1991).

[28]

C. Miehe and S. Göktepe, A micro-macro approach to rubber-like materials. Part II: The micro-sphere model of finite rubber viscoelasticity,, Journal of the Mechanics and Physics of Solids, 53 (2005), 2231. doi: 10.1016/j.jmps.2005.04.006.

[29]

C. Miehe, S. Göktepe and F. Lulei, A micro-macro approach to rubber-like materials - Part I: the non-affine micro-sphere model of rubber elasticity,, Journal of the Mechanics and Physics of Solids, 52 (2004), 2617. doi: 10.1016/j.jmps.2004.03.011.

[30]

C. O. Ng and C. C. Mei, Homogenization theory applied to soil vapor extraction in aggregated soils,, Physics of Fluids, 8 (1996), 2298. doi: 10.1063/1.869017.

[31]

M. Vogelius, A homogenization result for planar, polygonal networks,, RAIRO Modélisation Mathématique et Analyse Numérique, 25 (1991), 483.

[32]

F. Yerlikaya-Özkurta, A. Askanb and G.-W. Weber, An alternative approach to the ground motion prediction problem by a non-parametric adaptive regression method,, Engineering Optimization, 46 (2014), 1651.

show all references

References:
[1]

T. Arbogast, J. Douglas Jr. and U. Hornung, Derivation of the double porosity model of single phase flow via homogenization theory,, SIAM Journal on Mathematical Analysis, 21 (1990), 823. doi: 10.1137/0521046.

[2]

J. L. Auriault and J. Lewandoska, Diffusion/adsorption/advection macrotransport in soils,, European Journal of Mechanics, 15 (1996), 681.

[3]

J. Bear, Dynamics of Fluids in Porous Media,, Dover Publications, (1988).

[4]

A. Bensoussan, J. L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures,, American Mathematical Society, (2011). doi: 10.1090/chel/374.

[5]

G. Bouchitté and I. Fragalà, Homogenization of elastic thin structures: a measure-fattening approach,, J. Convex Anal., 9 (2002), 1.

[6]

C. Boutin, A. Rallu and S. Hans, Large scale modulation of high frequency acoustic waves in periodic porous media,, The Journal of the acoustical society of America, 132 (2012), 3622.

[7]

E. Canon and M. Lenczner, Modelling of thin elastic plates with small piezoelectric inclusions and sistributed electronic circuits. Models for inclusions that are small with respect to the thickness of the plate,, Journal of Elasticity, 55 (1999), 111. doi: 10.1023/A:1007609122248.

[8]

F. Civan, Porous Media Transport Phenomena,, John Wiley & Sons, (2011).

[9]

M. Cunha and L. Nunes, Groundwater Characterization, Management and Monitoring,, WITPress, (2010).

[10]

L. Dormieux and E. Lemarchand, Homogenization approach of advection and diffusion in cracked porous material,, Journal of Engineering Mechanics, 127 (2001), 1267.

[11]

M. Espedal, A. Fasano and A. Mikelić, Filtration in porous media and industrial application,, Lecture Notes in Mathematics, 1734 (2000), 24. doi: 10.1007/BFb0103973.

[12]

P. Exner, J. P. Keating, P. Kuchment, T. Sunada and A. Teplyaev, Analysis on Graphs and Its Applications - Proceedings of Symposia in Pure Mathematics,, American Mathematical Society, 77 (2008). doi: 10.1090/pspum/077.

[13]

S. Franz and H.-G. Roos, The capriciousness of numerical methods for singular perturbations,, SIAM review, 53 (2011), 157. doi: 10.1137/090757344.

[14]

A. Fortin, J. M. Urquiza and R. Bois, A mesh adaptation method for 1D-boundary layer problems,, International Journal of Numerical Analysis and Modeling, 3 (2012), 408.

[15]

S. Göktepe and C. Miehe, A micro-macro approach to rubber-like materials. Part III: The micro-sphere model of anisotropic Mullins-type damage,, Journal of the Mechanics and Physics of Solids, 53 (2005), 2259. doi: 10.1016/j.jmps.2005.04.010.

[16]

U. Hornung, Homogenization and Porous Media,, Springer, (1996). doi: 10.1007/978-1-4612-1920-0.

[17]

J. D. Joannopoulos, R. D. Meade and J. N. Winn, Photonic Crystals,, Princeton University Press, (2008).

[18]

P. Kogut and G. Leugering, Asymptotic analysis of optimal control problems on periodic singular graphs,, Systems & Control: Foundations & Applications Birkhäuser Boston, (2011). doi: 10.1007/978-0-8176-8149-4.

[19]

P. Kuchment, Quantum graphs I: Some basic structures,, Waves Random Media, 14 (2004), 107. doi: 10.1088/0959-7174/14/1/014.

[20]

P. Kuchment, Quantum graphs: An introduction and a brief survey,, in Analysis on Graphs and Its Applications - Proceedings of Symposia in Pure Mathematics(eds. P. Exner, 77 (2008), 291. doi: 10.1090/pspum/077/2459876.

[21]

P. Kuchment and L. Kunyansky, Differential operators on graphs and photonic crystals,, Advances in Computational Mathematics, 16 (2002), 263. doi: 10.1023/A:1014481629504.

[22]

E. Kropat, Über die Homogenisierung von Netzwerk-Differentialgleichungen,, Wissenschaftlicher Verlag Berlin, (2007).

[23]

M. Lenczner, Multiscale model for atomic force microscope array mechanical behaviour,, Applied Physics Letters, 90 (2007).

[24]

M. Lenczner and D. Mercier, Homogenization of periodic electrical networks including voltage to current amplifiers,, Multiscale Modeling and Simulation, 2 (2004), 359. doi: 10.1137/S1540345903423919.

[25]

M. Lenczner and G. Senouci-Bereksi, Homogenization of electrical networks including voltage-to-voltage amplifiers,, Mathematical Models and Methods in Applied Sciences, 9 (1999), 899. doi: 10.1142/S0218202599000415.

[26]

E. Marušić-Palokaa and S. Marušić, Computation of the permeability tensor for the fluid flow through a periodic net of thin channels,, Applicable Analysis, 64 (1997), 27. doi: 10.1080/00036819708840521.

[27]

V. G. Mazja, S. A. Nasarow and B. A. Plamenewski, Asymptotische Theorie elliptischer Randwertaufgaben in Singulär gestörten Gebieten II,, Wiley, (1991).

[28]

C. Miehe and S. Göktepe, A micro-macro approach to rubber-like materials. Part II: The micro-sphere model of finite rubber viscoelasticity,, Journal of the Mechanics and Physics of Solids, 53 (2005), 2231. doi: 10.1016/j.jmps.2005.04.006.

[29]

C. Miehe, S. Göktepe and F. Lulei, A micro-macro approach to rubber-like materials - Part I: the non-affine micro-sphere model of rubber elasticity,, Journal of the Mechanics and Physics of Solids, 52 (2004), 2617. doi: 10.1016/j.jmps.2004.03.011.

[30]

C. O. Ng and C. C. Mei, Homogenization theory applied to soil vapor extraction in aggregated soils,, Physics of Fluids, 8 (1996), 2298. doi: 10.1063/1.869017.

[31]

M. Vogelius, A homogenization result for planar, polygonal networks,, RAIRO Modélisation Mathématique et Analyse Numérique, 25 (1991), 483.

[32]

F. Yerlikaya-Özkurta, A. Askanb and G.-W. Weber, An alternative approach to the ground motion prediction problem by a non-parametric adaptive regression method,, Engineering Optimization, 46 (2014), 1651.

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