自适应边界元法求解双曲型异向介质透射问题
An adaptive boundary element method for the transmission problem with hyperbolic metamaterials
摘要：
本文提出了一种自适应边界元方法来计算双曲型异向介质中波相互作用的电磁响应。双曲型超材料的一个独特之处是其传播锥中存在强方向性波，当波开始指数衰减或增长时，这会导致积分方程的解在锥边界处发生急剧的跃迁。为了避免在整个边界上的全局加细网格，我们采用了两级后验误差估计和自适应网格细化过程来解决局部奇异性的积分方程的解决方案。这样的自适应过程允许积分方程求解器的自由度显著减小，同时实现解的期望精度。此外，为了解决快速过渡的基本解和其衍生物准确地跨越传播锥边界，自适应数值求积规则被应用于评估的刚性矩阵的积分。最后，为了建立边界上的积分方程，我们还导出了目标点接近边界时双曲介质中层势及其导数的极限。
英文摘要：
In this work we present an adaptive boundary element method for computing the electromagnetic response of wave interactions in hyperbolic metamaterials. One unique feature of hyperbolic metamaterial is the strongly directional wave in its propagating cone, which induces sharp transition for the solution of the integral equation across the cone boundary when wave starts to decay or grow exponentially. In order to avoid a global refined mesh over the whole boundary, we employ a two-level a posteriori error estimator and an adaptive mesh refinement procedure to resolve the singularity locally for the solution of the integral equation. Such an adaptive procedure allows for the reduction of the degree of freedom significantly for the integral equation solver while achieving desired accuracy for the solution. In addition, to resolve the fast transition of the fundamental solution and its derivatives accurately across the propagation cone boundary, adaptive numerical quadrature rules are applied to evaluate the integrals for the stiff matrices. Finally, in order to formulate the integral equations over the boundary, we also derive the limits of layer potentials and their derivatives in the hyperbolic media when the target points approach the boundary.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
