In this paper, we propose an elegant methodology to treat sharp interfaces that are implicitly defined which does not require (a) enrichment functions, (b) additional linear and bilinear terms such as the inter-element penalty terms as in Nitsche's method, or use of multipliers like Lagrange multiplier, in the weak form for enforcing the jump conditions across the interface, and (c) modification to the standard virtual element method solution space. The background mesh consists of structured quadrilateral elements with each element consisting of eight nodes, namely, the four vertices and the mid-points of four edges. A simple and efficient idea to generate an interface-fitted mesh is discussed where the number of nodes remains invariant, esp., for moving boundary problems. A linear virtual element method approximation is assumed on the fitted mesh. The efficiency and accuracy of the presented technique is demonstrated by solving and verifying the rate of convergence in both norm and semi-norm, for the benchmark problems with interfaces of various geometries and moving interfaces.

中文翻译：

---

使用虚拟元素方法解决嵌入式接口问题的笛卡尔网格方法

在本文中，我们提出了一种优雅的方法来处理隐式定义的尖锐接口，该方法不需要（a）丰富函数，（b）额外的线性和双线性项，例如尼采方法中的元素间惩罚项，或使用像拉格朗日乘子这样的乘子，以弱形式强制跨接口的跳跃条件，以及（c）对标准虚拟元素方法解空间的修改。背景网格由结构化的四边形单元组成，每个单元由八个节点组成，即四个顶点和四个边的中点。讨论了生成界面拟合网格的简单而有效的想法，其中节点数量保持不变，尤其是对于移动边界问题。假设拟合网格采用线性虚拟元素法近似。通过求解和验证范数和半范数的收敛速度，证明了所提出技术的效率和准确性，针对具有各种几何形状的界面和移动界面的基准问题。