September  2021, 16(3): 427-458. doi: 10.3934/nhm.2021012

Combined effects of homogenization and singular perturbations: A bloch wave approach

Tata Institute of Fundamental Research, Centre for Applicable Mathematics, Bangalore, Karnataka, 560065, India

Received  December 2020 Published  September 2021 Early access  May 2021

In this work, we study Bloch wave homogenization of periodically heterogeneous media with fourth order singular perturbations. We recover different homogenization regimes depending on the relative strength of the singular perturbation and length scale of the periodic heterogeneity. The homogenized tensor is obtained in terms of the first Bloch eigenvalue. The higher Bloch modes do not contribute to the homogenization limit. The main difficulty is the presence of two parameters which requires us to obtain uniform bounds on the Bloch spectral data in various regimes of the parameter.

Citation: Vivek Tewary. Combined effects of homogenization and singular perturbations: A bloch wave approach. Networks and Heterogeneous Media, 2021, 16 (3) : 427-458. doi: 10.3934/nhm.2021012
References:
[1]

E. C. Aifantis, On the Microstructural Origin of Certain Inelastic Models, Journal of Engineering Materials and Technology, 106 (1984), 326-330.  doi: 10.1115/1.3225725.

[2]

G. AllaireM. Briane and M. Vanninathan, A comparison between two-scale asymptotic expansions and Bloch wave expansions for the homogenization of periodic structures, SeMA Journal. Boletin de la Sociedad Espanñola de Matemática Aplicada, 73 (2016), 237-259.  doi: 10.1007/s40324-016-0067-z.

[3]

G. AllaireT. Ghosh and M. Vanninathan, Homogenization of Stokes system using Bloch waves, Networks and Heterogeneous Media, 12 (2017), 525-550.  doi: 10.3934/nhm.2017022.

[4]

S. Bair, Chapter Ten - Shear localization, slip, and the limiting stress, in High Pressure Rheology for Quantitative Elastohydrodynamics second edition, Elsevier, 2019, 271-286. doi: 10.1016/B978-0-444-64156-4.00010-6.

[5]

H. Baumgärtel, Analytic Perturbation Theory for Matrices and Operators, Operator Theory: Advances and Applications, Vol. 15, Birkhäuser Verlag, Basel, 1985.

[6]

A. Bensoussan, J. -L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, AMS Chelsea Publishing, Providence, RI, 2011. doi: 10.1090/chel/374.

[7]

M. Birman and T. Suslina, Periodic second-order differential operators. Threshold properties and homogenization, St. Petersburg Mathematical Journal, 15 (2004), 639-714.  doi: 10.1090/S1061-0022-04-00827-1.

[8]

A. Braides and C. I. Zeppieri, Multiscale analysis of a prototypical model for the interaction between microstructure and surface energy, Interfaces and Free Boundaries, 11 (2009), 61-118.  doi: 10.4171/IFB/204.

[9]

L. M. Chasman, An isoperimetric inequality for fundamental tones of free plates, Communications in Mathematical Physics, 303 (2011), 421-449.  doi: 10.1007/s00220-010-1171-z.

[10]

C. ConcaD. GómezM. Lobo and E. Pérez, The Bloch approximation in periodically perforated media, Applied Mathematics and Optimization, 52 (2005), 93-127.  doi: 10.1007/s00245-005-0822-5.

[11]

C. ConcaR. Orive and M. Vanninathan, Bloch approximation in homogenization and applications, SIAM Journal on Mathematical Analysis, 33 (2002), 1166-1198.  doi: 10.1137/S0036141001382200.

[12]

C. ConcaR. Orive and M. Vanninathan, Bloch approximation in homogenization on bounded domains, Asymptotic Analysis, 41 (2005), 71-91. 

[13]

C. Conca and M. Vanninathan, Homogenization of periodic structures via Bloch decomposition, SIAM Journal on Applied Mathematics, 57 (1997), 1639-1659.  doi: 10.1137/S0036139995294743.

[14]

T. DohnalA. Lamacz and B. Schweizer, Bloch-wave homogenization on large time scales and dispersive effective wave equations, Multiscale Modeling & Simulation. A SIAM Interdisciplinary Journal, 12 (2014), 488-513.  doi: 10.1137/130935033.

[15]

T. DohnalA. Lamacz and B. Schweizer, Dispersive homogenized models and coefficient formulas for waves in general periodic media, Asymptotic Analysis, 93 (2015), 21-49.  doi: 10.3233/ASY-141280.

[16]

D. DupuyR. Orive and L. Smaranda, Bloch waves homogenization of a Dirichlet problem in a periodically perforated domain, Asymptotic Analysis, 61 (2009), 229-250.  doi: 10.3233/ASY-2008-0912.

[17]

F. Ferraresso and P. D. Lamberti, On a Babuška paradox for polyharmonic operators: Spectral stability and boundary homogenization for intermediate problems, Integral Equations and Operator Theory, 91 (2019), Paper No. 55, 42 pp. doi: 10.1007/s00020-019-2552-0.

[18]

G. Franfort and S. Müller, Combined effects of homogenization and singular perturbations in elasticity, Journal für die reine und angewandte Mathematik (Crelles Journal), 454 (1994), 1-35.  doi: 10.1515/crll.1994.454.1.

[19]

S. S. Ganesh and M. Vanninathan, Bloch wave homogenization of scalar elliptic operators, Asymptotic Analysis, 39 (2004), 15-44. 

[20]

F. Gazzola, H. -C. Grunau and G. Sweers, Eigenvalue problems, in Polyharmonic Boundary Value Problems: Positivity Preserving and Nonlinear Higher Order Elliptic Equations in Bounded Domains, Lecture Notes in Mathematics, Vol. 1991, Springer-Verlag, Berlin, 2010, 61-98. doi: 10.1007/978-3-642-12245-3.

[21]

T. Kato, Perturbation Theory for Linear Operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995. doi: 10.1007/978-3-642-66282-9.

[22]

A. Lamacz-Keymling and I. Yousept, High-order homogenization in optimal control by the Bloch wave method, preprint, 2020, arXiv: 2010.04469v1.

[23]

Y. K. Lee, Conditions for shear banding and material instability in finite elastoplastic deformation, International Journal of Plasticity, 5 (1989), 197-226.  doi: 10.1016/0749-6419(89)90013-2.

[24]

F. Murat and L. Tartar, H-convergence, in Topics in the Mathematical Modelling of Composite Materials, Birkhäuser Boston, Boston, MA, 1997, 21-43. doi: 10.1007/978-1-4612-2032-9.

[25] R. Narasimhan, Several Complex Variables, Chicago Lectures in Mathematics, University of Chicago Press, 1971. 
[26]

W. Niu and Z. Shen, Combined Effects of Homogenization and Singular Perturbations: Quantitative Estimates, preprint, 2020, arXiv: 2005.12776v1.

[27]

W. Niu and Y. Yuan, Convergence rate in homogenization of elliptic systems with singular perturbations, Journal of Mathematical Physics, 60 (2019), 111509, 7 pp. doi: 10.1063/1.5124140.

[28]

J. OrtegaJ. S. Martín and L. Smaranda, Bloch wave homogenization of a non-homogeneous Neumann problem, Zeitschrift für angewandte Mathematik und Physik, 58 (2007), 969-993.  doi: 10.1007/s00033-007-6142-7.

[29]

S. E. Pastukhova, Homogenization estimates for singularly perturbed operators, Journal of Mathematical Sciences, 251 (2020), 724-747.  doi: 10.1007/s10958-020-05125-0.

[30]

L. Provenzano, A note on the Neumann eigenvalues of the biharmonic operator, Mathematical Methods in the Applied Sciences, 41 (2018), 1005-1012.  doi: 10.1002/mma.4063.

[31]

M. Reed and B. Simon, Methods of Modern Mathematical Physics. IV. Analysis of Operators, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978.

[32]

Y. C. Shu, Heterogeneous thin films of Martensitic materials, Archive for Rational Mechanics and Analysis, 153 (2000), 39-90.  doi: 10.1007/s002050000088.

[33]

S. Sivaji Ganesh and V. Tewary, Bloch approach to almost periodic homogenization and approximations of effective coefficients, Discrete and Continuous Dynamical Systems Series B, 2021, 1-36. doi: 10.3934/dcdsb. 2021119.

[34]

S. Sivaji Ganesh and V. Tewary, Bloch wave homogenisation of quasiperiodic media, European Journal of Applied Mathematics, 2020, 1-21. doi: 10.1017/S0956792520000352.

[35]

S. Sivaji Ganesh and M. Vanninathan, Bloch wave homogenization of linear elasticity system, ESAIM Control Optim. Calc. Var., 11 (2005), 542-573.  doi: 10.1051/cocv:2005018.

[36]

T. Suslina, Homogenization of the Dirichlet problem for higher-order elliptic equations with periodic coefficients, St. Petersburg Mathematical Journal, 29 (2018), 325-362.  doi: 10.1090/spmj/1496.

[37]

N. Veniaminov, Homogenization of periodic differential operators of high order, St. Petersburg Mathematical Journal, 22 (2011), 751-775.  doi: 10.1090/S1061-0022-2011-01166-5.

[38]

C. H. Wilcox, Theory of Bloch waves, J. Analyse Math., 33 (1978), 146-167.  doi: 10.1007/BF02790171.

[39]

C. I. Zeppieri, Stochastic homogenisation of singularly perturbed integral functionals, Annali di Matematica Pura ed Applicata (4), 195 (2016), 2183-2208.  doi: 10.1007/s10231-016-0558-7.

show all references

References:
[1]

E. C. Aifantis, On the Microstructural Origin of Certain Inelastic Models, Journal of Engineering Materials and Technology, 106 (1984), 326-330.  doi: 10.1115/1.3225725.

[2]

G. AllaireM. Briane and M. Vanninathan, A comparison between two-scale asymptotic expansions and Bloch wave expansions for the homogenization of periodic structures, SeMA Journal. Boletin de la Sociedad Espanñola de Matemática Aplicada, 73 (2016), 237-259.  doi: 10.1007/s40324-016-0067-z.

[3]

G. AllaireT. Ghosh and M. Vanninathan, Homogenization of Stokes system using Bloch waves, Networks and Heterogeneous Media, 12 (2017), 525-550.  doi: 10.3934/nhm.2017022.

[4]

S. Bair, Chapter Ten - Shear localization, slip, and the limiting stress, in High Pressure Rheology for Quantitative Elastohydrodynamics second edition, Elsevier, 2019, 271-286. doi: 10.1016/B978-0-444-64156-4.00010-6.

[5]

H. Baumgärtel, Analytic Perturbation Theory for Matrices and Operators, Operator Theory: Advances and Applications, Vol. 15, Birkhäuser Verlag, Basel, 1985.

[6]

A. Bensoussan, J. -L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, AMS Chelsea Publishing, Providence, RI, 2011. doi: 10.1090/chel/374.

[7]

M. Birman and T. Suslina, Periodic second-order differential operators. Threshold properties and homogenization, St. Petersburg Mathematical Journal, 15 (2004), 639-714.  doi: 10.1090/S1061-0022-04-00827-1.

[8]

A. Braides and C. I. Zeppieri, Multiscale analysis of a prototypical model for the interaction between microstructure and surface energy, Interfaces and Free Boundaries, 11 (2009), 61-118.  doi: 10.4171/IFB/204.

[9]

L. M. Chasman, An isoperimetric inequality for fundamental tones of free plates, Communications in Mathematical Physics, 303 (2011), 421-449.  doi: 10.1007/s00220-010-1171-z.

[10]

C. ConcaD. GómezM. Lobo and E. Pérez, The Bloch approximation in periodically perforated media, Applied Mathematics and Optimization, 52 (2005), 93-127.  doi: 10.1007/s00245-005-0822-5.

[11]

C. ConcaR. Orive and M. Vanninathan, Bloch approximation in homogenization and applications, SIAM Journal on Mathematical Analysis, 33 (2002), 1166-1198.  doi: 10.1137/S0036141001382200.

[12]

C. ConcaR. Orive and M. Vanninathan, Bloch approximation in homogenization on bounded domains, Asymptotic Analysis, 41 (2005), 71-91. 

[13]

C. Conca and M. Vanninathan, Homogenization of periodic structures via Bloch decomposition, SIAM Journal on Applied Mathematics, 57 (1997), 1639-1659.  doi: 10.1137/S0036139995294743.

[14]

T. DohnalA. Lamacz and B. Schweizer, Bloch-wave homogenization on large time scales and dispersive effective wave equations, Multiscale Modeling & Simulation. A SIAM Interdisciplinary Journal, 12 (2014), 488-513.  doi: 10.1137/130935033.

[15]

T. DohnalA. Lamacz and B. Schweizer, Dispersive homogenized models and coefficient formulas for waves in general periodic media, Asymptotic Analysis, 93 (2015), 21-49.  doi: 10.3233/ASY-141280.

[16]

D. DupuyR. Orive and L. Smaranda, Bloch waves homogenization of a Dirichlet problem in a periodically perforated domain, Asymptotic Analysis, 61 (2009), 229-250.  doi: 10.3233/ASY-2008-0912.

[17]

F. Ferraresso and P. D. Lamberti, On a Babuška paradox for polyharmonic operators: Spectral stability and boundary homogenization for intermediate problems, Integral Equations and Operator Theory, 91 (2019), Paper No. 55, 42 pp. doi: 10.1007/s00020-019-2552-0.

[18]

G. Franfort and S. Müller, Combined effects of homogenization and singular perturbations in elasticity, Journal für die reine und angewandte Mathematik (Crelles Journal), 454 (1994), 1-35.  doi: 10.1515/crll.1994.454.1.

[19]

S. S. Ganesh and M. Vanninathan, Bloch wave homogenization of scalar elliptic operators, Asymptotic Analysis, 39 (2004), 15-44. 

[20]

F. Gazzola, H. -C. Grunau and G. Sweers, Eigenvalue problems, in Polyharmonic Boundary Value Problems: Positivity Preserving and Nonlinear Higher Order Elliptic Equations in Bounded Domains, Lecture Notes in Mathematics, Vol. 1991, Springer-Verlag, Berlin, 2010, 61-98. doi: 10.1007/978-3-642-12245-3.

[21]

T. Kato, Perturbation Theory for Linear Operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995. doi: 10.1007/978-3-642-66282-9.

[22]

A. Lamacz-Keymling and I. Yousept, High-order homogenization in optimal control by the Bloch wave method, preprint, 2020, arXiv: 2010.04469v1.

[23]

Y. K. Lee, Conditions for shear banding and material instability in finite elastoplastic deformation, International Journal of Plasticity, 5 (1989), 197-226.  doi: 10.1016/0749-6419(89)90013-2.

[24]

F. Murat and L. Tartar, H-convergence, in Topics in the Mathematical Modelling of Composite Materials, Birkhäuser Boston, Boston, MA, 1997, 21-43. doi: 10.1007/978-1-4612-2032-9.

[25] R. Narasimhan, Several Complex Variables, Chicago Lectures in Mathematics, University of Chicago Press, 1971. 
[26]

W. Niu and Z. Shen, Combined Effects of Homogenization and Singular Perturbations: Quantitative Estimates, preprint, 2020, arXiv: 2005.12776v1.

[27]

W. Niu and Y. Yuan, Convergence rate in homogenization of elliptic systems with singular perturbations, Journal of Mathematical Physics, 60 (2019), 111509, 7 pp. doi: 10.1063/1.5124140.

[28]

J. OrtegaJ. S. Martín and L. Smaranda, Bloch wave homogenization of a non-homogeneous Neumann problem, Zeitschrift für angewandte Mathematik und Physik, 58 (2007), 969-993.  doi: 10.1007/s00033-007-6142-7.

[29]

S. E. Pastukhova, Homogenization estimates for singularly perturbed operators, Journal of Mathematical Sciences, 251 (2020), 724-747.  doi: 10.1007/s10958-020-05125-0.

[30]

L. Provenzano, A note on the Neumann eigenvalues of the biharmonic operator, Mathematical Methods in the Applied Sciences, 41 (2018), 1005-1012.  doi: 10.1002/mma.4063.

[31]

M. Reed and B. Simon, Methods of Modern Mathematical Physics. IV. Analysis of Operators, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978.

[32]

Y. C. Shu, Heterogeneous thin films of Martensitic materials, Archive for Rational Mechanics and Analysis, 153 (2000), 39-90.  doi: 10.1007/s002050000088.

[33]

S. Sivaji Ganesh and V. Tewary, Bloch approach to almost periodic homogenization and approximations of effective coefficients, Discrete and Continuous Dynamical Systems Series B, 2021, 1-36. doi: 10.3934/dcdsb. 2021119.

[34]

S. Sivaji Ganesh and V. Tewary, Bloch wave homogenisation of quasiperiodic media, European Journal of Applied Mathematics, 2020, 1-21. doi: 10.1017/S0956792520000352.

[35]

S. Sivaji Ganesh and M. Vanninathan, Bloch wave homogenization of linear elasticity system, ESAIM Control Optim. Calc. Var., 11 (2005), 542-573.  doi: 10.1051/cocv:2005018.

[36]

T. Suslina, Homogenization of the Dirichlet problem for higher-order elliptic equations with periodic coefficients, St. Petersburg Mathematical Journal, 29 (2018), 325-362.  doi: 10.1090/spmj/1496.

[37]

N. Veniaminov, Homogenization of periodic differential operators of high order, St. Petersburg Mathematical Journal, 22 (2011), 751-775.  doi: 10.1090/S1061-0022-2011-01166-5.

[38]

C. H. Wilcox, Theory of Bloch waves, J. Analyse Math., 33 (1978), 146-167.  doi: 10.1007/BF02790171.

[39]

C. I. Zeppieri, Stochastic homogenisation of singularly perturbed integral functionals, Annali di Matematica Pura ed Applicata (4), 195 (2016), 2183-2208.  doi: 10.1007/s10231-016-0558-7.

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