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.

我们提出了一种新的方法来为界面层问题导出不完善的界面模型。测试用例是潜在的问题，例如导热性，反平面弹性等。该方法将经典渐近分析与复值函数理论中的概念结合在一起。与现有渐近方法相比，它的主要优点是可以直接得出包含任意阶数的表面微分算子的跳跃条件，从而形成了一个模型层次结构，该模型相对于层厚度及其曲率保持了任意阶数的准确性。与低阶模型不同，导出的高阶模型可以准确地表示比相邻的块状材料明显软或更硬的层，并显示变化的曲率，或相对于相邻块区域的特征长度尺度具有有限的厚度。将通过我们的方法学获得的接口模型与现有的不同顺序模型进行比较，针对已知的接口机制对它们的限制行为进行了验证，并为基准示例说明了高阶变量的改进精度。虽然在这里我们将自己局限于二维标量问题，但二维矢量问题的扩展却很简单。我们还将讨论将我们的方法扩展到三维标量和矢量问题的途径。并举例说明了高阶变量的改进精度。虽然在这里我们将自己局限于二维标量问题，但二维矢量问题的扩展却很简单。我们还将讨论将我们的方法扩展到三维标量和矢量问题的途径。并举例说明了高阶变量的改进精度。虽然在这里我们将自己局限于二维标量问题，但二维矢量问题的扩展却很简单。我们还将讨论将我们的方法扩展到三维标量和矢量问题的途径。