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April  2022, 21(4): 1109-1137. doi: 10.3934/cpaa.2022012

Asymptotic analysis for the electric field concentration with geometry of the core-shell structure

Zhiwen Zhao a,b,

a. 

Beijing Computational Science Research Center, Beijing 100193, China
b. 

School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China

Received  June 2021 Revised  November 2021 Published  April 2022 Early access  December 2021

Fund Project: Z. W. Zhao was partially supported by NSFC (11971061) and CPSF (2021M700358)

In the perfect conductivity problem arising from composites, the electric field may become arbitrarily large as $ \varepsilon $, the distance between the inclusions and the matrix boundary, tends to zero. In this paper, by making clear the singular role of the blow-up factor $ Q[\varphi] $ introduced in [27] for some special boundary data of even function type with $ k $-order growth, we prove the optimality of the blow-up rate in the presence of $ m $-convex inclusions close to touching the matrix boundary in all dimensions. Finally, we give closer analysis in terms of the singular behavior of the concentrated field for eccentric and concentric core-shell geometries with circular and spherical boundaries from the practical application angle.

Keywords: Asymptotic expansions, perfect conductivity problem, blow-up factor, optimal blow-up rate, core-shell structure.
Mathematics Subject Classification: Primary: 78A48; Secondary: 35B44; 35B65; 35J25.
Citation: Zhiwen Zhao. Asymptotic analysis for the electric field concentration with geometry of the core-shell structure. Communications on Pure and Applied Analysis, 2022, 21 (4) : 1109-1137. doi: 10.3934/cpaa.2022012
References:
[1]

H. Ammari, G. Ciraolo, H. Kang, H. Lee and K. Yun, Spectral analysis of the Neumann-Poincaré operator and characterization of the stress concentration in anti-plane elasticity, Arch. Ration. Mech. Anal., 208 (2013), 275-304.  doi: 10.1007/s00205-012-0590-8.

[2]

H. Ammari, H. Kang and M. Lim, Gradient estimates to the conductivity problem, Math. Ann. 332 (2005), 277-286. doi: 10.1007/s00208-004-0626-y.

[3]

H. Ammari, H. Kang, H. Lee, J. Lee and M. Lim, Optimal estimates for the electric field in two dimensions, J. Math. Pures Appl., 88 (2007), 307-324.  doi: 10.1016/j.matpur.2007.07.005.

[4]

I. Babuška, B. Andersson, P. Smith and K. Levin, Damage analysis of fiber composites. I. Statistical analysis on fiber scale, Comput. Methods Appl. Mech. Engrg., 172 (1999), 27-77.  doi: 10.1016/S0045-7825(98)00225-4.

[5]

B. Budiansky and G. F. Carrier, High shear stresses in stiff fiber composites, J. App. Mech., 51 (1984), 733-735. 

[6]

E. Bao, Y. Y. Li and B. Yin, Gradient estimates for the perfect conductivity problem, Arch. Ration. Mech. Anal., 193 (2009), 195-226.  doi: 10.1007/s00205-008-0159-8.

[7]

E. Bao, Y. Y. Li and B. Yin, Gradient estimates for the perfect and insulated conductivity problems with multiple inclusions, Commun. Partial Differ. Equ., 35 (2010), 1982-2006.  doi: 10.1080/03605300903564000.

[8]

E. Bonnetier and F. Triki, Pointwise bounds on the gradient and the spectrum of the Neumann-Poincaré operator: the case of 2 discs, Multi-scale and high-contrast PDE: from modeling, to mathematical analysis, to inversion, Contemp. Math., 577, Amer. Math. Soc., Providence, RI, 2012, pp. 81–91. doi: 10.1090/conm/577.

[9]

E. Bonnetier and F. Triki, On the spectrum of the Poincaré variational problem for two close-to-touching inclusions in 2D, Arch. Ration. Mech. Anal., 209 (2013), 541-567.  doi: 10.1007/s00205-013-0636-6.

[10]

E. Bonnetier and M. Vogelius, An elliptic regularity result for a composite medium with ``touching'' fibers of circular cross-section, SIAM J. Math. Anal., 31 (2000), 651-677.  doi: 10.1137/S0036141098333980.

[11]

V. M. Calo, Y. Efendiev and J. Galvis, Asymptotic expansions for high-contrast elliptic equations, Math. Models Methods Appl. Sci., 24 (2014), 465-494.  doi: 10.1142/S0218202513500565.

[12]

G. Ciraolo and A. Sciammetta, Gradient estimates for the perfect conductivity problem in anisotropic media, J. Math. Pures Appl., 127 (2019), 268-298.  doi: 10.1016/j.matpur.2018.09.006.

[13]

G. Ciraolo and A. Sciammetta, Stress concentration for closely located inclusions in nonlinear perfect conductivity problems, J. Differ. Equ., 266 (2019), 6149-6178.  doi: 10.1016/j.jde.2018.10.041.

[14]

H. J. Dong and H. G. Li, Optimal estimates for the conductivity problem by Green's function method, Arch. Ration. Mech. Anal., 231 (2019), 1427-1453.  doi: 10.1007/s00205-018-1301-x.

[15]

Y. Gorb and A. Novikov, Blow-up of solutions to a $p$-Laplace equation, Multiscale Model. Simul., 10 (2012), 727-743.  doi: 10.1137/110857167.

[16]

Y. Gorb, Singular behavior of electric field of high-contrast concentrated composites, Multiscale Model. Simul., 13 (2015), 1312-1326.  doi: 10.1137/140982076.

[17]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, 1998.

[18]

J. B. Keller, Stresses in narrow regions, Trans. ASME J. Appl. Mech., 60 (1993), 1054-1056. 

[19]

H. Kang, M. Lim and K. Yun, Asymptotics and computation of the solution to the conductivity equation in the presence of adjacent inclusions with extreme conductivities, J. Math. Pures Appl., 9 (2013), 234-249.  doi: 10.1016/j.matpur.2012.06.013.

[20]

H. Kang, M. Lim and K. Yun, Characterization of the electric field concentration between two adjacent spherical perfect conductors, SIAM J. Appl. Math., 74 (2014), 125-146.  doi: 10.1137/130922434.

[21]

H. Kang, H. Lee and K. Yun, Optimal estimates and asymptotics for the stress concentration between closely located stiff inclusions, Math. Ann., 363 (2015), 1281-1306.  doi: 10.1007/s00208-015-1203-2.

[22]

J. Kim and M. Lim, Electric field concentration in the presence of an inclusion with eccentric core-shell geometry, Math. Ann., 373 (2019), 517-551.  doi: 10.1007/s00208-018-1688-6.

[23]

J. Lekner, Electrostatics of two charged conducting spheres, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 468 (2012), 2829-2848.  doi: 10.1098/rspa.2012.0133.

[24]

H. G. Li, Y. Y. Li, E. S. Bao and B. Yin, Derivative estimates of solutions of elliptic systems in narrow regions, Quart. Appl. Math. 72 (2014), 589–596. doi: 10.1090/S0033-569X-2014-01339-0.

[25]

H. G. Li, Y. Y. Li and Z. L. Yang, Asymptotics of the gradient of solutions to the perfect conductivity problem, Multiscale Model. Simul., 17 (2019), 899-925.  doi: 10.1137/18M1214329.

[26]

H. G. Li, F. Wang and L. J. Xu, Characterization of electric fields between two spherical perfect conductors with general radii in 3D, J. Differ. Equ., 267 (2019), 6644-6690.  doi: 10.1016/j.jde.2019.07.007.

[27]

H. G. Li and L. J. Xu, Optimal estimates for the perfect conductivity problem with inclusions close to the boundary, SIAM J. Math. Anal., 49 (2017), 3125-3142.  doi: 10.1137/16M1067858.

[28]

H. G. Li, Asymptotics for the electric field concentration in the perfect conductivity problem, SIAM J. Math. Anal., 52 (2020), 3350-3375.  doi: 10.1137/19M1282623.

[29]

Y. Y. Li and L. Nirenberg, Estimates for elliptic system from composite material, Commun. Pure Appl. Math., 56 (2003), 892-925.  doi: 10.1002/cpa.10079.

[30]

Y. Y. Li and M. Vogelius, Gradient estimates for solutions to divergence form elliptic equations with discontinuous coefficients, Arch. Rational Mech. Anal., 153 (2000), 91-151.  doi: 10.1007/s002050000082.

[31]

M. Lim and K. Yun, Blow-up of electric fields between closely spaced spherical perfect conductors, Commun. Partial Differ. Equ., 34 (2009), 1287-1315.  doi: 10.1080/03605300903079579.

[32]

K. Yun, Estimates for electric fields blown up between closely adjacent conductors with arbitrary shape, SIAM J. Appl. Math., 67 (2007), 714-730.  doi: 10.1137/060648817.

[33]

K. Yun, Optimal bound on high stresses occurring between stiff fibers with arbitrary shaped cross-sections, J. Math. Anal. Appl., 350 (2009), 306-312.  doi: 10.1016/j.jmaa.2008.09.057.

[34]

Z. W. Zhao and X. Hao, Asymptotics for the concentrated field between closely located hard inclusions in all dimensions, Commun. Pure Appl. Anal., 20 (2021), 2379-2398.  doi: 10.3934/cpaa.2021086.

show all references

References:
[1]

H. Ammari, G. Ciraolo, H. Kang, H. Lee and K. Yun, Spectral analysis of the Neumann-Poincaré operator and characterization of the stress concentration in anti-plane elasticity, Arch. Ration. Mech. Anal., 208 (2013), 275-304.  doi: 10.1007/s00205-012-0590-8.

[2]

H. Ammari, H. Kang and M. Lim, Gradient estimates to the conductivity problem, Math. Ann. 332 (2005), 277-286. doi: 10.1007/s00208-004-0626-y.

[3]

H. Ammari, H. Kang, H. Lee, J. Lee and M. Lim, Optimal estimates for the electric field in two dimensions, J. Math. Pures Appl., 88 (2007), 307-324.  doi: 10.1016/j.matpur.2007.07.005.

[4]

I. Babuška, B. Andersson, P. Smith and K. Levin, Damage analysis of fiber composites. I. Statistical analysis on fiber scale, Comput. Methods Appl. Mech. Engrg., 172 (1999), 27-77.  doi: 10.1016/S0045-7825(98)00225-4.

[5]

B. Budiansky and G. F. Carrier, High shear stresses in stiff fiber composites, J. App. Mech., 51 (1984), 733-735. 

[6]

E. Bao, Y. Y. Li and B. Yin, Gradient estimates for the perfect conductivity problem, Arch. Ration. Mech. Anal., 193 (2009), 195-226.  doi: 10.1007/s00205-008-0159-8.

[7]

E. Bao, Y. Y. Li and B. Yin, Gradient estimates for the perfect and insulated conductivity problems with multiple inclusions, Commun. Partial Differ. Equ., 35 (2010), 1982-2006.  doi: 10.1080/03605300903564000.

[8]

E. Bonnetier and F. Triki, Pointwise bounds on the gradient and the spectrum of the Neumann-Poincaré operator: the case of 2 discs, Multi-scale and high-contrast PDE: from modeling, to mathematical analysis, to inversion, Contemp. Math., 577, Amer. Math. Soc., Providence, RI, 2012, pp. 81–91. doi: 10.1090/conm/577.

[9]

E. Bonnetier and F. Triki, On the spectrum of the Poincaré variational problem for two close-to-touching inclusions in 2D, Arch. Ration. Mech. Anal., 209 (2013), 541-567.  doi: 10.1007/s00205-013-0636-6.

[10]

E. Bonnetier and M. Vogelius, An elliptic regularity result for a composite medium with ``touching'' fibers of circular cross-section, SIAM J. Math. Anal., 31 (2000), 651-677.  doi: 10.1137/S0036141098333980.

[11]

V. M. Calo, Y. Efendiev and J. Galvis, Asymptotic expansions for high-contrast elliptic equations, Math. Models Methods Appl. Sci., 24 (2014), 465-494.  doi: 10.1142/S0218202513500565.

[12]

G. Ciraolo and A. Sciammetta, Gradient estimates for the perfect conductivity problem in anisotropic media, J. Math. Pures Appl., 127 (2019), 268-298.  doi: 10.1016/j.matpur.2018.09.006.

[13]

G. Ciraolo and A. Sciammetta, Stress concentration for closely located inclusions in nonlinear perfect conductivity problems, J. Differ. Equ., 266 (2019), 6149-6178.  doi: 10.1016/j.jde.2018.10.041.

[14]

H. J. Dong and H. G. Li, Optimal estimates for the conductivity problem by Green's function method, Arch. Ration. Mech. Anal., 231 (2019), 1427-1453.  doi: 10.1007/s00205-018-1301-x.

[15]

Y. Gorb and A. Novikov, Blow-up of solutions to a $p$-Laplace equation, Multiscale Model. Simul., 10 (2012), 727-743.  doi: 10.1137/110857167.

[16]

Y. Gorb, Singular behavior of electric field of high-contrast concentrated composites, Multiscale Model. Simul., 13 (2015), 1312-1326.  doi: 10.1137/140982076.

[17]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, 1998.

[18]

J. B. Keller, Stresses in narrow regions, Trans. ASME J. Appl. Mech., 60 (1993), 1054-1056. 

[19]

H. Kang, M. Lim and K. Yun, Asymptotics and computation of the solution to the conductivity equation in the presence of adjacent inclusions with extreme conductivities, J. Math. Pures Appl., 9 (2013), 234-249.  doi: 10.1016/j.matpur.2012.06.013.

[20]

H. Kang, M. Lim and K. Yun, Characterization of the electric field concentration between two adjacent spherical perfect conductors, SIAM J. Appl. Math., 74 (2014), 125-146.  doi: 10.1137/130922434.

[21]

H. Kang, H. Lee and K. Yun, Optimal estimates and asymptotics for the stress concentration between closely located stiff inclusions, Math. Ann., 363 (2015), 1281-1306.  doi: 10.1007/s00208-015-1203-2.

[22]

J. Kim and M. Lim, Electric field concentration in the presence of an inclusion with eccentric core-shell geometry, Math. Ann., 373 (2019), 517-551.  doi: 10.1007/s00208-018-1688-6.

[23]

J. Lekner, Electrostatics of two charged conducting spheres, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 468 (2012), 2829-2848.  doi: 10.1098/rspa.2012.0133.

[24]

H. G. Li, Y. Y. Li, E. S. Bao and B. Yin, Derivative estimates of solutions of elliptic systems in narrow regions, Quart. Appl. Math. 72 (2014), 589–596. doi: 10.1090/S0033-569X-2014-01339-0.

[25]

H. G. Li, Y. Y. Li and Z. L. Yang, Asymptotics of the gradient of solutions to the perfect conductivity problem, Multiscale Model. Simul., 17 (2019), 899-925.  doi: 10.1137/18M1214329.

[26]

H. G. Li, F. Wang and L. J. Xu, Characterization of electric fields between two spherical perfect conductors with general radii in 3D, J. Differ. Equ., 267 (2019), 6644-6690.  doi: 10.1016/j.jde.2019.07.007.

[27]

H. G. Li and L. J. Xu, Optimal estimates for the perfect conductivity problem with inclusions close to the boundary, SIAM J. Math. Anal., 49 (2017), 3125-3142.  doi: 10.1137/16M1067858.

[28]

H. G. Li, Asymptotics for the electric field concentration in the perfect conductivity problem, SIAM J. Math. Anal., 52 (2020), 3350-3375.  doi: 10.1137/19M1282623.

[29]

Y. Y. Li and L. Nirenberg, Estimates for elliptic system from composite material, Commun. Pure Appl. Math., 56 (2003), 892-925.  doi: 10.1002/cpa.10079.

[30]

Y. Y. Li and M. Vogelius, Gradient estimates for solutions to divergence form elliptic equations with discontinuous coefficients, Arch. Rational Mech. Anal., 153 (2000), 91-151.  doi: 10.1007/s002050000082.

[31]

M. Lim and K. Yun, Blow-up of electric fields between closely spaced spherical perfect conductors, Commun. Partial Differ. Equ., 34 (2009), 1287-1315.  doi: 10.1080/03605300903079579.

[32]

K. Yun, Estimates for electric fields blown up between closely adjacent conductors with arbitrary shape, SIAM J. Appl. Math., 67 (2007), 714-730.  doi: 10.1137/060648817.

[33]

K. Yun, Optimal bound on high stresses occurring between stiff fibers with arbitrary shaped cross-sections, J. Math. Anal. Appl., 350 (2009), 306-312.  doi: 10.1016/j.jmaa.2008.09.057.

[34]

Z. W. Zhao and X. Hao, Asymptotics for the concentrated field between closely located hard inclusions in all dimensions, Commun. Pure Appl. Anal., 20 (2021), 2379-2398.  doi: 10.3934/cpaa.2021086.

Figure 1.  Eccentric circles
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