American Institute of Mathematical Sciences

June  2017, 7(2): 139-169. doi: 10.3934/naco.2017010

Computational networks and systems-homogenization of self-adjoint differential operators in variational form on periodic networks and micro-architectured systems

 1 University of the Bundeswehr Munich, Faculty of Informatics, Werner-Heisenberg-Weg 39, 85577 Neubiberg, Germany 2 Middle East Technical University, Institute of Applied Mathematics, 06531 Ankara, Turkey

* Corresponding author: Erik Kropat

+ Honorary positions: Faculty of Economics, Business and Law, University of Siegen, Germany; Center for Research on Optimization and Control, University of Aveiro, Portugal; University of North Sumatra, Indonesia

Received  July 2016 Revised  April 2017 Published  June 2017

Micro-architectured systems and periodic network structures play an import role in multi-scale physics and material sciences. Mathematical modeling leads to challenging problems on the analytical and the numerical side. Previous studies focused on averaging techniques that can be used to reveal the corresponding macroscopic model describing the effective behavior. This study aims at a mathematical rigorous proof within the framework of homogenization theory. As a model example, the variational form of a self-adjoint operator on a large periodic network is considered. A notion of two-scale convergence for network functions based on a so-called two-scale transform is applied. It is shown that the sequence of solutions of the variational microscopic model on varying networked domains converges towards the solution of the macroscopic model. A similar result is achieved for the corresponding sequence of tangential gradients. The resulting homogenized variational model can be easily solved with standard PDE-solvers. In addition, the homogenized coefficients provide a characterization of the physical system on a global scale. In this way, a mathematically rigorous concept for the homogenization of self-adjoint operators on periodic manifolds is achieved. Numerical results illustrate the effectiveness of the presented approach.

Citation: Erik Kropat, Silja Meyer-Nieberg, Gerhard-Wilhelm Weber. Computational networks and systems-homogenization of self-adjoint differential operators in variational form on periodic networks and micro-architectured systems. Numerical Algebra, Control and Optimization, 2017, 7 (2) : 139-169. doi: 10.3934/naco.2017010
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), 823-836.  doi: 10.1137/0521046. [2] J.-L. Auriault and J. Lewandoska, Diffusion/adsorption/advection macrotransport in soils, European Journal of Mechanics, A/Solids, 15 (1996), 681-704. [3] J. Bear, Dynamics of Fluids in Porous Media, Dover Publications, New York, 1988. [4] G. Bouchitté and I. Fragalà, Homogenization of elastic thin structures: a measure-fattening approach, Journal Of Convex Analysis, 9 (2002), 1-24. [5] G. Bouchitté and I. Fragalà, Homogenization of thin structures by two-scale method with respect to measures, SIAM Journal on Mathematical Analysis, 32 (2002), 1198-1226.  doi: 10.1137/S0036141000370260. [6] E. Canon and M. Lenczner, Modelling of thin elastic plates with small piezoelectric inclusions and distributed electronic circuits. Models for inclusions that are small with respect to the thickness of the plate, Journal of Elasticity, 55 (1999), 111-141.  doi: 10.1023/A:1007609122248. [7] G. A. Chechkin, V. V. Jikov, D. Lukkassen and A. L. Piatnitski, On homogenization of networks and junctions, Asymptotic Analysis, 30 (2002), 61-80. [8] D. Cioranescu and J. Saint Jean Paulin, Homogenization of Reticulated Structures, Springer, New York, 1999. doi: 10.1007/978-1-4612-2158-6. [9] F. Civan, Porous Media Transport Phenomena, John Wiley & Sons, Hoboken, New Jersey, 2011. [10] A. Fortin, J. M. Urquiza and R. Bois, A mesh adaptation method for 1D-boundary layer problems, International Journal of Numerical Analysis and Modeling, Series B, 3 (2012), 408-428. [11] L. Gibson and M. Ashby, Cellular Solids. Structure and Properties, Pergamon Press, New York, 1999. [12] I. G. Graham, T. Y. Hou, O. Lakkis and Robert Scheichl, Numerical Analysis of Multiscale Problems, Lecture Notes in Computational Science and Engineering, 83, Springer, Berlin, Heidelberg, 2012. Interdisciplinary Applied Mathematics, Springer, New York, 2003. [13] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer, Berlin, Heidelberg, 2001. [14] 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), 2259-2283.  doi: 10.1016/j.jmps.2005.04.010. [15] U. Hornung, Homogenization and Porous Media, Interdisciplinary Applied Mathematics, Springer, Stuttgart, 1997. doi: 10.1007/978-1-4612-1920-0. [16] J. D. Joannopoulos, R. D. Meade and J. N. Winn, Photonic Crystals, Princeton University Press, Princeton, NJ, 2008. [17] G. E. Karniadakis, A. Beskok and N. Aluru, Microflows and Nanoflows. Fundamentals and Simulation, Interdisciplinary Applied Mathematics, Springer, New York, 2005. [18] P. Kogut and G. Leugering, Homogenization of constrained optimal control problems for one-dimensional elliptic equations on periodic graphs, ESAIM: Control, Optimisation and Calculus of Variations, 15 (2009), 471-498.  doi: 10.1051/cocv:2008037. [19] P. Kogut and G. Leugering, Asymptotic Analysis of Optimal Control Problems on Periodic Singular Graphs, in Optimal Control Problems for Partial Differential Equations on Reticulated Domains (eds. P. Kogut and G. Leugering), Birkhäuser Boston, (2011), 409–440. [20] E. Kropat, Über die Homogenisierung von Netzwerk-Differentialgleichungen, Wissenschaftlicher Verlag Berlin, Berlin, 2007. [21] E. Kropat and S. Meyer-Nieberg, Homogenization of singularly perturbed diffusion-advectionreaction equations on periodic networks, in Proceedings of the 15th IFAC Workshop on Control Applications of Optimization (CAO 2012), September 13-16,2012, Rimini, Italy, (2012), 83–88. [22] E. Kropat, S. Meyer-Nieberg and G.-W. Weber, Two-scale asymptotic analysis of singularly perturbed elliptic differential equations on large periodic networks, Dynamics of Continuous, Discrete and Impulsive Systems -Series B: Applications & Algorithms, 22 (2015), 293-324. [23] E. Kropat, S. Meyer-Nieberg and G.-W. 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 and Optimization, 9 (2016), 183-219.  doi: 10.3934/naco.2016008. [24] E. Kropat, S. Meyer-Nieberg and G.-W. Weber, A topology optimization approach for micro-architectured systems on singularly perturbed periodic manifolds? Two-scale asymptotic analysis and the influence of the network topology, Dynamics of Continuous, Discrete and Impulsive Systems -Series B: Applications & Algorithms, 23 (2016), 155-193. [25] P. Kuchment, Quantum graphs I: Some basic structures, Waves Random Media, 14 (2004), 107-128.  doi: 10.1088/0959-7174/14/1/014. [26] P. Kuchment, Quantum graphs: An Introduction and a Brief Survey, In Analysis on Graphs and Its Applications -Proceedings of Symposia in Pure Mathematics 77 (eds. P. Exner, J. P. Keating, P. Kuchment, T. Sunada and A. Teplyaev), American Mathematical Society, (2008), 291–312. doi: 10.1090/pspum/077/2459876. [27] P. Kuchment and L. Kunyansky, Differential operators on graphs and photonic crystals, Advances in Computational Mathematics, 16 (2008), 263-290.  doi: 10.1023/A:1014481629504. [28] M. Lenczner, Homogénéisation d'un circuit électrique, Comptes Rendus de l'Académie des Sciences -Series IIB -Mechanics-Physics-Chemistry-Astronomy, 324 (1997), 537-542. [29] M. Lenczner, Multiscale model for atomic force microscope array mechanical behaviour, Applied Physics Letters, 90 (2007), 091908. [30] M. Lenczner and D. Mercier, Homogenization of periodic electrical networks including voltage to current amplifiers, Multiscale Modeling and Simulation, 2 (2004), 359-397.  doi: 10.1137/S1540345903423919. [31] M. Lenczner and G. Senouci-Bereksi, Homogenization of electrical networks including voltage-to-voltage amplifiers, Mathematical Models and Methods in Applied Sciences, 9 (1999), 899-932.  doi: 10.1142/S0218202599000415. [32] V. G. Mazja, S. A. Nasarow and B. A. Plamenewski, Asymptotische Theorie elliptischer Randwertaufgaben in singulär gestörten Gebieten II, Wiley, Berlin, 1991. [33] 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), 2231-2258.  doi: 10.1016/j.jmps.2005.04.006. [34] 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), 2617-2660.  doi: 10.1016/j.jmps.2004.03.011. [35] E. Nuhn and E. Kropat and W. Reinhardt and S. Pckl, Preparation of Complex Landslide Simulation Results with Clustering Approaches for Decision Support and Early Warning, in Proceedings of the 45th Annual Hawaii International Conference on System Sciences (HICSS-45), January 4-7,2012, Grand Wailea, Maui, Hawaii, , Ralph H. Sprague, Jr. (eds.), IEEE Computer Society, (2012), 1089–1096. [36] S. E. Pastukhova, Homogenization in Elasticity Problems on Combined Structures, Journal of Mathematical Sciences, 132 (2006), 313-330. [37] C. Pechstein, Finite and Boundary Element Tearing and Interconnecting Solvers for Multiscale Problems, Lecture Notes in Computational Science and Engineering, 90, Springer, Berlin, Heidelberg, 2013. doi: 10.1007/978-3-642-23588-7. [38] M. Sahimi, Heterogeneous Materials. Nonlinear and Breakdown Properties and Atomistic Modeling, [39] M. Vogelius, A homogenization result for planar, polygonal networks, RAIRO Modélisation Mathématique et Analyse Numérique, 25 (1991), 483-514.  doi: 10.1051/m2an/1991250404831. [40] V. V. Zhikov, On an extension of the method of two-scale convergence and its applications}, Sbornik: Mathematics, 191 (2000), 973-1014.  doi: 10.1070/SM2000v191n07ABEH000491.

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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), 823-836.  doi: 10.1137/0521046. [2] J.-L. Auriault and J. Lewandoska, Diffusion/adsorption/advection macrotransport in soils, European Journal of Mechanics, A/Solids, 15 (1996), 681-704. [3] J. Bear, Dynamics of Fluids in Porous Media, Dover Publications, New York, 1988. [4] G. Bouchitté and I. Fragalà, Homogenization of elastic thin structures: a measure-fattening approach, Journal Of Convex Analysis, 9 (2002), 1-24. [5] G. Bouchitté and I. Fragalà, Homogenization of thin structures by two-scale method with respect to measures, SIAM Journal on Mathematical Analysis, 32 (2002), 1198-1226.  doi: 10.1137/S0036141000370260. [6] E. Canon and M. Lenczner, Modelling of thin elastic plates with small piezoelectric inclusions and distributed electronic circuits. Models for inclusions that are small with respect to the thickness of the plate, Journal of Elasticity, 55 (1999), 111-141.  doi: 10.1023/A:1007609122248. [7] G. A. Chechkin, V. V. Jikov, D. Lukkassen and A. L. Piatnitski, On homogenization of networks and junctions, Asymptotic Analysis, 30 (2002), 61-80. [8] D. Cioranescu and J. Saint Jean Paulin, Homogenization of Reticulated Structures, Springer, New York, 1999. doi: 10.1007/978-1-4612-2158-6. [9] F. Civan, Porous Media Transport Phenomena, John Wiley & Sons, Hoboken, New Jersey, 2011. [10] A. Fortin, J. M. Urquiza and R. Bois, A mesh adaptation method for 1D-boundary layer problems, International Journal of Numerical Analysis and Modeling, Series B, 3 (2012), 408-428. [11] L. Gibson and M. Ashby, Cellular Solids. Structure and Properties, Pergamon Press, New York, 1999. [12] I. G. Graham, T. Y. Hou, O. Lakkis and Robert Scheichl, Numerical Analysis of Multiscale Problems, Lecture Notes in Computational Science and Engineering, 83, Springer, Berlin, Heidelberg, 2012. Interdisciplinary Applied Mathematics, Springer, New York, 2003. [13] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer, Berlin, Heidelberg, 2001. [14] 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), 2259-2283.  doi: 10.1016/j.jmps.2005.04.010. [15] U. Hornung, Homogenization and Porous Media, Interdisciplinary Applied Mathematics, Springer, Stuttgart, 1997. doi: 10.1007/978-1-4612-1920-0. [16] J. D. Joannopoulos, R. D. Meade and J. N. Winn, Photonic Crystals, Princeton University Press, Princeton, NJ, 2008. [17] G. E. Karniadakis, A. Beskok and N. Aluru, Microflows and Nanoflows. Fundamentals and Simulation, Interdisciplinary Applied Mathematics, Springer, New York, 2005. [18] P. Kogut and G. Leugering, Homogenization of constrained optimal control problems for one-dimensional elliptic equations on periodic graphs, ESAIM: Control, Optimisation and Calculus of Variations, 15 (2009), 471-498.  doi: 10.1051/cocv:2008037. [19] P. Kogut and G. Leugering, Asymptotic Analysis of Optimal Control Problems on Periodic Singular Graphs, in Optimal Control Problems for Partial Differential Equations on Reticulated Domains (eds. P. Kogut and G. Leugering), Birkhäuser Boston, (2011), 409–440. [20] E. Kropat, Über die Homogenisierung von Netzwerk-Differentialgleichungen, Wissenschaftlicher Verlag Berlin, Berlin, 2007. [21] E. Kropat and S. Meyer-Nieberg, Homogenization of singularly perturbed diffusion-advectionreaction equations on periodic networks, in Proceedings of the 15th IFAC Workshop on Control Applications of Optimization (CAO 2012), September 13-16,2012, Rimini, Italy, (2012), 83–88. [22] E. Kropat, S. Meyer-Nieberg and G.-W. Weber, Two-scale asymptotic analysis of singularly perturbed elliptic differential equations on large periodic networks, Dynamics of Continuous, Discrete and Impulsive Systems -Series B: Applications & Algorithms, 22 (2015), 293-324. [23] E. Kropat, S. Meyer-Nieberg and G.-W. 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 and Optimization, 9 (2016), 183-219.  doi: 10.3934/naco.2016008. [24] E. Kropat, S. Meyer-Nieberg and G.-W. Weber, A topology optimization approach for micro-architectured systems on singularly perturbed periodic manifolds? Two-scale asymptotic analysis and the influence of the network topology, Dynamics of Continuous, Discrete and Impulsive Systems -Series B: Applications & Algorithms, 23 (2016), 155-193. [25] P. Kuchment, Quantum graphs I: Some basic structures, Waves Random Media, 14 (2004), 107-128.  doi: 10.1088/0959-7174/14/1/014. [26] P. Kuchment, Quantum graphs: An Introduction and a Brief Survey, In Analysis on Graphs and Its Applications -Proceedings of Symposia in Pure Mathematics 77 (eds. P. Exner, J. P. Keating, P. Kuchment, T. Sunada and A. Teplyaev), American Mathematical Society, (2008), 291–312. doi: 10.1090/pspum/077/2459876. [27] P. Kuchment and L. Kunyansky, Differential operators on graphs and photonic crystals, Advances in Computational Mathematics, 16 (2008), 263-290.  doi: 10.1023/A:1014481629504. [28] M. Lenczner, Homogénéisation d'un circuit électrique, Comptes Rendus de l'Académie des Sciences -Series IIB -Mechanics-Physics-Chemistry-Astronomy, 324 (1997), 537-542. [29] M. Lenczner, Multiscale model for atomic force microscope array mechanical behaviour, Applied Physics Letters, 90 (2007), 091908. [30] M. Lenczner and D. Mercier, Homogenization of periodic electrical networks including voltage to current amplifiers, Multiscale Modeling and Simulation, 2 (2004), 359-397.  doi: 10.1137/S1540345903423919. [31] M. Lenczner and G. Senouci-Bereksi, Homogenization of electrical networks including voltage-to-voltage amplifiers, Mathematical Models and Methods in Applied Sciences, 9 (1999), 899-932.  doi: 10.1142/S0218202599000415. [32] V. G. Mazja, S. A. Nasarow and B. A. Plamenewski, Asymptotische Theorie elliptischer Randwertaufgaben in singulär gestörten Gebieten II, Wiley, Berlin, 1991. [33] 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), 2231-2258.  doi: 10.1016/j.jmps.2005.04.006. [34] 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), 2617-2660.  doi: 10.1016/j.jmps.2004.03.011. [35] E. Nuhn and E. Kropat and W. Reinhardt and S. Pckl, Preparation of Complex Landslide Simulation Results with Clustering Approaches for Decision Support and Early Warning, in Proceedings of the 45th Annual Hawaii International Conference on System Sciences (HICSS-45), January 4-7,2012, Grand Wailea, Maui, Hawaii, , Ralph H. Sprague, Jr. (eds.), IEEE Computer Society, (2012), 1089–1096. [36] S. E. Pastukhova, Homogenization in Elasticity Problems on Combined Structures, Journal of Mathematical Sciences, 132 (2006), 313-330. [37] C. Pechstein, Finite and Boundary Element Tearing and Interconnecting Solvers for Multiscale Problems, Lecture Notes in Computational Science and Engineering, 90, Springer, Berlin, Heidelberg, 2013. doi: 10.1007/978-3-642-23588-7. [38] M. Sahimi, Heterogeneous Materials. Nonlinear and Breakdown Properties and Atomistic Modeling, [39] M. Vogelius, A homogenization result for planar, polygonal networks, RAIRO Modélisation Mathématique et Analyse Numérique, 25 (1991), 483-514.  doi: 10.1051/m2an/1991250404831. [40] V. V. Zhikov, On an extension of the method of two-scale convergence and its applications}, Sbornik: Mathematics, 191 (2000), 973-1014.  doi: 10.1070/SM2000v191n07ABEH000491.
Homogenization theory: The limit process
Periodic networks: The network $\mathcal{N}^\Omega_\varepsilon$ is the segment of the infinite network $\mathcal{N}_\varepsilon$ contained in the polyhedral domain $\Omega$, that is obtained by copying and scaling from the reference graph $\mathbb{G}$ in the unit cell $[0, 1)^d$
Covering with cells: The $\varepsilon$-cells and the corresponding graphs
Feasible networks: The network $\mathcal{N}^\Omega_1$ (left figure) and the network $\mathcal{N}^\Omega_{\frac{{1}}{{2}}}$ are feasible networks where the corresponding $\varepsilon$-cells cover the domain $\Omega$
Two-scale transform: The function $x:\Omega \times \mathbb{G} \rightarrow \mathcal{N}^\Omega_\varepsilon$ is surjective, but not injective
Two-scale transform: Mapping from $\mathcal{N}^\Omega_\varepsilon$ to the product $\Omega \times \mathbb{G}$
Illustrative example: The reference graph $\mathbb{G}$
Illustrative example: Diffusion coefficient and source term
Illustrative example: The periodic network $\mathcal{N}^\Omega_1$
Illustrative example: Solutions of the microscopic model for different lengths of periodicity $\varepsilon$
Illustrative example: The approximate homogenized model and the solution for $\varepsilon=0.2$
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