We present a new methodology to derive imperfect interface models for the problems with interphase layers. The test case is potential problems, e.g., thermal conductivity, antiplane elasticity, etc. The methodology combines classical asymptotic analysis with concepts from the theory of complex-valued functions. Its major advantage over existing asymptotic approaches is the straightforward derivation of jump conditions that involve surface differential operators of arbitrary order, resulting in a hierarchy of models that maintain arbitrary-order accuracy with respect to the layer thickness and its curvature. Unlike low-order models, the derived higher-order models can accurately represent layers that are significantly softer or stiffer than the adjacent bulk materials, exhibit varying curvature, or are of finite thickness with respect to the characteristic length scale of the adjacent bulk regions. The interface models obtained via our methodology are compared with existing models of different orders, their limiting behavior is validated with respect to known interface regimes, and the improved accuracy of higher-order variants is illustrated for a benchmark example. While here we limited ourselves to scalar problems in two dimensions, the extension to vector problems in two-dimensions is straightforward. We also discuss the pathway to extend our methodology to scalar and vector problems in three-dimensions.
Bibliographical noteFunding Information:
The first author (S.B.) gratefully acknowledges the support of the International Student Work Opportunity Program (ISWOP) at the University of Minnesota. The second author (S.G.M.) gratefully acknowledges the support from the Theodore W. Bennett Chair at the University of Minnesota. The first and second authors (S.B., S.G.M.) gratefully acknowledge support of the Isaac Newton Institute for Mathematical Sciences (INI) at Cambridge University, which enabled their participation in the INI programme on “Complex analysis: techniques, applications and computations” ( EPSRC grant EP/R014604/1 ). In this context, the second author (S.G.M.) also gratefully acknowledges the support of the Simons Foundation through a Simons INI Fellowship. The third and fourth authors (T.H.N., D.S.) gratefully acknowledge funding from the German Research Foundation through the DFG Emmy Noether Award SCH 1249/2-1 .
- Asymptotic analysis
- Higher-order imperfect interface models
- Interphase layer
- Potential problems
- Theory of complex-valued functions