Asymptotic expansions and effective boundary conditions: a short review for smooth and nonsmooth geometries with thin layers

Problems involving materials with thin layers arise in various application fields. We present here the use of asymptotic expansions for linear elliptic problems to derive and justify so-called ap-proximate or effective boundary conditions. We first recall the known results of the literature, and then discuss the optimality of the error estimates in the smooth case. For non-smooth geometries, the results of [18, 57] are commented and adapted to a model problem, and two improvements of the approximate model are proposed to increase its numerical performance.

[1]  A. Majda,et al.  Absorbing boundary conditions for the numerical simulation of waves , 1977 .

[2]  P. Grisvard.  Boundary value problems in non-smooth domains , 1980 .

[3]  Giuseppe Buttazzo,et al.  Reinforcement problems in the calculus of variations (*) (*)Financially supported by a national research project of the Italian Ministry of Education. , 1986 .

[4]  M. Dauge.  Elliptic boundary value problems on corner domains , 1988 .

[5]  S. Hoole.  Experimental validation of the impedance boundary condition and a review of its limitations , 1989 .

[6]  Christophe Hazard,et al.  Variational formulations for the determination of resonant states in scattering problems , 1992 .

[7]  Laurence Halpern,et al.  Absorbing boundary conditions for diffusion equations , 1995 .

[8]  Martin Costabel,et al.  A singularly perturbed mixed boundary value problem , 1996 .

[9]  Keddour Lemrabet,et al.  The Effect of a Thin Coating on the Scattering of a Time-Harmonic Wave for the Helmholtz Equation , 1996, SIAM J. Appl. Math..

[10]  Frédéric Valentin,et al.  Effective Boundary Conditions for Laminar Flows over Periodic Rough Boundaries , 1998 .

[11]  Habib Ammari,et al.  Effective impedance boundary conditions for an inhomogeneous thin layer on a curved metallic surface , 1998 .

[12]  Stefano Lenci,et al.  Mathematical Analysis of a Bonded Joint with a Soft Thin Adhesive , 1999 .

[13]  R. Phillips,et al.  Crust‐mantle decoupling by flexure of continental lithosphere , 2000 .

[14]  Willi Jäger,et al.  On the Roughness-Induced Effective Boundary Conditions for an Incompressible Viscous Flow , 2001 .

[15]  Dimitra I. Kaklamani,et al.  Electromagnetic scattering analysis of coated conductors with edges using the method of auxiliary sources (MAS) in conjunction with the standard impedance boundary condition (SIBC) , 2002 .

[16]  Igor Tsukerman,et al.  Method of overlapping patches for electromagnetic computation near imperfectly conducting cusps and edges , 2002 .

[17]  Dan Givoli,et al.  Finite Element Modeling of Thin Layers , 2004 .

[18]  José M. Galán,et al.  Nonreflecting Boundary Conditions for the Nonlinear , 2005 .

[19]  D. De Zutter,et al.  Skin effect modeling based on a differential surface admittance operator , 2005, IEEE Transactions on Microwave Theory and Techniques.

[21]  D. Gérard-Varet,et al.  Wall laws for fluid flows at a boundary with random roughness , 2006, math/0606768.

[22]  Yves Capdeboscq,et al.  Pointwise polarization tensor bounds, and applications to voltage perturbations caused by thin inhomogeneities , 2006, Asymptot. Anal..

[23]  Martin Costabel,et al.  Asymptotic expansion of the solution of an interface problem in a polygonal domain with thin layer , 2006, Asymptot. Anal..

[24]  Patrick Joly,et al.  Matching of Asymptotic Expansions for Wave Propagation in Media with Thin Slots I: The Asymptotic Expansion , 2006, Multiscale Model. Simul..

[25]  G. Vial,et al.  A multiscale correction method for local singular perturbations of the boundary , 2007 .

[26]  Houssem Haddar,et al.  GENERALIZED IMPEDANCE BOUNDARY CONDITIONS FOR SCATTERING PROBLEMS FROM STRONGLY ABSORBING OBSTACLES: THE CASE OF MAXWELL'S EQUATIONS , 2008 .

[27]  Xuefeng Wang,et al.  Asymptotic analysis of a Dirichlet problem for the heat equation on a coated body , 2008 .

[28]  J. Marigo,et al.  Shallow layer correction for Spectral Element like methods , 2008 .

[29]  Nathan Ida,et al.  Surface Impedance Boundary Conditions: A Comprehensive Approach , 2009 .

[30]  Delphine Brancherie,et al.  Effect of micro-defects on structure failure Coupling asymptotic analysis and strong discontinuity , 2010 .

[31]  Bérangère Delourme.  Modèles et asymptotiques des interfaces fines et périodiques en électromagnétisme , 2010 .

[32]  P. Novák,et al.  A review of the features and analyses of the solid electrolyte interphase in Li-ion batteries , 2010 .

[33]  The influence of crustal rheology on plate subduction based on numerical modeling results , 2010 .

[34]  C. Poignard.  Boundary layer correctors and generalized polarization tensor for periodic rough thin layers. A review for the conductivity problem , 2012 .

[35]  Houssem Haddar,et al.  Approximate models for wave propagation across thin periodic interfaces , 2012 .

[36]  Frédéric Hecht,et al.  New development in freefem++ , 2012, J. Num. Math..

[37]  A. A. Moussa,et al.  Asymptotic study of thin elastic layer , 2013 .

[38]  Patrick Joly,et al.  EFFECTIVE TRANSMISSION CONDITIONS FOR THIN-LAYER TRANSMISSION PROBLEMS IN ELASTODYNAMICS. THE CASE OF A PLANAR LAYER MODEL , 2013 .

[39]  Kersten Schmidt,et al.  A Unified Analysis of Transmission Conditions for Thin Conducting Sheets in the Time-Harmonic Eddy Current Model , 2013, SIAM J. Appl. Math..

[40]  Clair Poignard,et al.  Asymptotic expansion of steady-state potential in a high contrast medium with a thin resistive layer , 2013, Appl. Math. Comput..

[41]  Victor Péron,et al.  Corner asymptotics of the magnetic potential in the eddy‐current model , 2014, ArXiv.

[42]  M. M. S. Fakhrabadi,et al.  Investigation of interphase effects on mechanical behaviors of carbon nanocone-based composites , 2014 .

[43]  A. Bendali,et al.  Scattering by a highly oscillating surface , 2015 .

[44]  Houssem Haddar,et al.  Axisymmetric eddy current inspection of highly conducting thin layers via asymptotic models , 2015 .

[45]  Sabrina Eberhart.  Mathematical And Numerical Aspects Of Wave Propagation , 2016 .

[46]  Jin-quan Xu,et al.  Interface models for thin interfacial layers , 2016 .

[47]  Z. Yao,et al.  Boundary conditions for the Stokes fluid in a bounded domain with a thin layer , 2016 .

[48]  C. Ruyer-Quil,et al.  A three-equation model for thin films down an inclined plane , 2016, Journal of Fluid Mechanics.

[49]  Giuseppe Geymonat,et al.  Asymptotic Analysis of a Linear Isotropic Elastic Composite Reinforced by a Thin Layer of Periodically Distributed Isotropic Parallel Stiff Fibres , 2016 .

[50]  V. Ramachandran,et al.  Influence of interphase material and clay particle shape on the effective properties of epoxy-clay nanocomposites , 2016 .

[51]  F. Caubet,et al.  New Transmission Condition Accounting For Diffusion Anisotropy In Thin Layers Applied To Diffusion MRI , 2017 .

[52]  Helmut Harbrecht,et al.  Numerical solution of the homogeneous Neumann boundary value problem on domains with a thin layer of random thickness , 2017, J. Comput. Phys..

[53]  Alexis Auvray,et al.  Improved impedance conditions for a thin layer problem in a nonsmooth domain , 2019, Mathematical Methods in the Applied Sciences.