# American Institute of Mathematical Sciences

September  2017, 7(3): 223-250. doi: 10.3934/naco.2017016

## Bridging the gap between variational homogenization results and two-scale asymptotic averaging techniques on periodic network structures

 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  August 2016 Revised  July 2017 Published  July 2017

In modern material sciences and multi-scale physics homogenization approaches provide a global characterization of physical systems that depend on the topology of the underlying microgeometry. Purely formal approaches such as averaging techniques can be applied for an identification of the averaged system. For models in variational form, two-scale convergence for network functions can be used to derive the homogenized model. The sequence of solutions of the variational microcsopic models and the corresponding sequence of tangential gradients converge toward limit functions that are characterized by the solution of the variational macroscopic model. Here, a further extension of this result is proved. The variational macroscopic model can be equivalently represented by a homogenized model on the superior domain and a certain number of reference cell problems. In this way, the results obtained by averaging strategies are supported by notions of convergence for network functions on varying domains.

Citation: Erik Kropat, Silja Meyer-Nieberg, Gerhard-Wilhelm Weber. Bridging the gap between variational homogenization results and two-scale asymptotic averaging techniques on periodic network structures. Numerical Algebra, Control & Optimization, 2017, 7 (3) : 223-250. doi: 10.3934/naco.2017016
##### References:
 [1] A. Abdulle, Y. Bai and G. Vilmart, An offline-online homogenization strategy to solve quasilinear two-scale problems at the cost of one-scale problems, International Journal for Numerical Methods in Engineering, 99 (2014), 469-486.  doi: 10.1002/nme.4682.  Google Scholar [2] 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.  Google Scholar [3] A. Braides, Γ-Convergence for Beginners, Oxford Lecture Series in Mathematics and Its Applications, Oxford University Press, Oxford, 1988. doi: 10.1093/acprof:oso/9780198507840.001.0001.  Google Scholar [4] S. Bochner, Beiträge zur theorie der fastperiodischen funktionen, Math. Annalen., 96 (1926), 119-147.  doi: 10.1007/BF01209156.  Google Scholar [5] S. Bochner and J. von Neumann, Almost periodic function in a group Ⅱ, Trans. Amer. Math. Soc., 37 (1935), 21-50.  doi: 10.2307/1989694.  Google Scholar [6] H. Bohr, Zur theorie der fastperiodischen funktionen I, Acta Mathematica, 45 (1925), 29-127.  doi: 10.1007/BF02395468.  Google Scholar [7] D. Cioranescu and J. Saint Jean Paulin, Homogenization of Reticulated Structures, Springer, New York, 1999. doi: 10.1007/978-1-4612-2158-6.  Google Scholar [8] G. Dal Maso, An Introduction to Γ-Convergence, Progress in Nonlinear Differential Equations and Their Applications, Birkhuser, Basel, 1993. doi: 10.1007/978-1-4612-0327-8.  Google Scholar [9] 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.   Google Scholar [10] I. G. Graham, T. Y. Hou, O. Lakkis and R. 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. doi: 10.1007/978-3-642-22061-6.  Google Scholar [11] 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.  Google Scholar [12] B. Hassani and, E. Hinton, Homogenization and Structural Topology Optimization: Theory, Practice and Software, Springer, London, 2011. doi: 10.1007/978-1-4471-0891-7.  Google Scholar [13] V. V. Jikov, S. M. Kozlov and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer, Berlin, Heidelberg, 1994. doi: 10.1007/978-3-642-84659-5.  Google Scholar [14] E. Kropat, Über die Homogenisierung von Netzwerk-Differentialgleichungen, Wissenschaftlicher Verlag Berlin, Berlin, 2007. Google Scholar [15] E. Kropat, Homogenization of optimal control problems on large curvilinear networks with a periodic microstructure -Results on $S$-homogenization and Γ-convergence, Numerical Algebra, Control and Optimization, 7 (2017), 51-76.  doi: 10.3934/naco.2017003.  Google Scholar [16] 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. Google Scholar [17] 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, 22 (2015), 293-324.   Google Scholar [18] 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.  Google Scholar [19] 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.   Google Scholar [20] M. Lenczner, Homogénéisation d'un circuit électrique, Comptes Rendus de l'Académie des Sciences -Series IIB -Mechanics-Physics-Chemistry-Astronomy, 324 (1997), 537-542.   Google Scholar [21] M. Lenczner, Multiscale model for atomic force microscope array mechanical behaviour, Applied Physics Letters, 90 (2007), 091908.   Google Scholar [22] 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.  Google Scholar [23] 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.  Google Scholar [24] 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.  Google Scholar [25] 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.  Google Scholar [26] 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, (eds. H. Ralph and Jr. Sprague), IEEE Computer Society, (2012), 1089–1096. Google Scholar [27] G. A. Pavliotis and A. Stuart, Multiscale methods, averaging and homogenization, Texts in Applied Mathematics Vol. 53, Springer, New York, 2008.  Google Scholar [28] 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.  Google Scholar [29] L. Tartar, The General Theory of Homogenization: A Personalized Introduction, Springer, Berlin, Heidelberg, 2010. doi: 10.1007/978-3-642-05195-1.  Google Scholar [30] M. Vogelius, A homogenization result for planar, polygonal networks, RAIRO Modélisation Mathématique et Analyse Numérique, 25 (1991), 483-514.  doi: 10.1051/m2an/1991250404831.  Google Scholar

show all references

##### References:
 [1] A. Abdulle, Y. Bai and G. Vilmart, An offline-online homogenization strategy to solve quasilinear two-scale problems at the cost of one-scale problems, International Journal for Numerical Methods in Engineering, 99 (2014), 469-486.  doi: 10.1002/nme.4682.  Google Scholar [2] 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.  Google Scholar [3] A. Braides, Γ-Convergence for Beginners, Oxford Lecture Series in Mathematics and Its Applications, Oxford University Press, Oxford, 1988. doi: 10.1093/acprof:oso/9780198507840.001.0001.  Google Scholar [4] S. Bochner, Beiträge zur theorie der fastperiodischen funktionen, Math. Annalen., 96 (1926), 119-147.  doi: 10.1007/BF01209156.  Google Scholar [5] S. Bochner and J. von Neumann, Almost periodic function in a group Ⅱ, Trans. Amer. Math. Soc., 37 (1935), 21-50.  doi: 10.2307/1989694.  Google Scholar [6] H. Bohr, Zur theorie der fastperiodischen funktionen I, Acta Mathematica, 45 (1925), 29-127.  doi: 10.1007/BF02395468.  Google Scholar [7] D. Cioranescu and J. Saint Jean Paulin, Homogenization of Reticulated Structures, Springer, New York, 1999. doi: 10.1007/978-1-4612-2158-6.  Google Scholar [8] G. Dal Maso, An Introduction to Γ-Convergence, Progress in Nonlinear Differential Equations and Their Applications, Birkhuser, Basel, 1993. doi: 10.1007/978-1-4612-0327-8.  Google Scholar [9] 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.   Google Scholar [10] I. G. Graham, T. Y. Hou, O. Lakkis and R. 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. doi: 10.1007/978-3-642-22061-6.  Google Scholar [11] 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.  Google Scholar [12] B. Hassani and, E. Hinton, Homogenization and Structural Topology Optimization: Theory, Practice and Software, Springer, London, 2011. doi: 10.1007/978-1-4471-0891-7.  Google Scholar [13] V. V. Jikov, S. M. Kozlov and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer, Berlin, Heidelberg, 1994. doi: 10.1007/978-3-642-84659-5.  Google Scholar [14] E. Kropat, Über die Homogenisierung von Netzwerk-Differentialgleichungen, Wissenschaftlicher Verlag Berlin, Berlin, 2007. Google Scholar [15] E. Kropat, Homogenization of optimal control problems on large curvilinear networks with a periodic microstructure -Results on $S$-homogenization and Γ-convergence, Numerical Algebra, Control and Optimization, 7 (2017), 51-76.  doi: 10.3934/naco.2017003.  Google Scholar [16] 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. Google Scholar [17] 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, 22 (2015), 293-324.   Google Scholar [18] 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.  Google Scholar [19] 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.   Google Scholar [20] M. Lenczner, Homogénéisation d'un circuit électrique, Comptes Rendus de l'Académie des Sciences -Series IIB -Mechanics-Physics-Chemistry-Astronomy, 324 (1997), 537-542.   Google Scholar [21] M. Lenczner, Multiscale model for atomic force microscope array mechanical behaviour, Applied Physics Letters, 90 (2007), 091908.   Google Scholar [22] 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.  Google Scholar [23] 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.  Google Scholar [24] 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.  Google Scholar [25] 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.  Google Scholar [26] 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, (eds. H. Ralph and Jr. Sprague), IEEE Computer Society, (2012), 1089–1096. Google Scholar [27] G. A. Pavliotis and A. Stuart, Multiscale methods, averaging and homogenization, Texts in Applied Mathematics Vol. 53, Springer, New York, 2008.  Google Scholar [28] 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.  Google Scholar [29] L. Tartar, The General Theory of Homogenization: A Personalized Introduction, Springer, Berlin, Heidelberg, 2010. doi: 10.1007/978-3-642-05195-1.  Google Scholar [30] M. Vogelius, A homogenization result for planar, polygonal networks, RAIRO Modélisation Mathématique et Analyse Numérique, 25 (1991), 483-514.  doi: 10.1051/m2an/1991250404831.  Google Scholar
The homogenization process: The sequence of solutions of the microscopic model as well as the corresponding sequence of tangential gradients are weakly two-scale convergent. The limits of these sequences can be represented by the solution of the homogenized model
Two-scale transform: The function $x:\Omega \times {\mathscr{Y}} \rightarrow {\cal{N}}^\Omega_\varepsilon$ is surjective, but not injective
Two-scale transform: Mapping from ${\cal{N}}^\Omega_\varepsilon$ to the product $\Omega \times {\mathscr{Y}}$
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