论文标题
通过PDE受限的优化方法，用于3D-1D耦合问题的扩展有限元素
Extended Finite Elements for 3D-1D coupled problems via a PDE-constrained optimization approach
论文作者
论文摘要
在这项工作中，我们提出了在三维和一维椭圆问题之间耦合的上下文中的扩展有限元方法（XFEM）的应用。特别是，我们考虑了3D-1D耦合问题的情况，这是由于几何模型减少了一个完全三维问题的情况，其特征是嵌入在更宽的域中的细管夹杂物。在3D-1D耦合框架中，广泛采用了非符合网格的使用。但是，由于夹杂物通常是3D问题的单数水槽或源，因此在嵌入的1D域附近的网格适应可能是提高解决方案准确性和恢复最佳收敛速率的必要条件。 XFEM表示网格适应性的替代方法，我们在这里建议它增强基于优化的3D-1D耦合方法的近似功能。设计了一种有效的正交策略来整合富集功能，并提出了对单个和多个段的数值测试，以证明该方法的有效性。
In this work, we propose the application of the eXtended Finite Element Method (XFEM) in the context of the coupling between three-dimensional and one-dimensional elliptic problems. In particular, we consider the case in which the 3D-1D coupled problem arises from the geometrical model reduction of a fully three-dimensional problem, characterized by thin tubular inclusions embedded in a much wider domain. In the 3D-1D coupling framework, the use of non conforming meshes is widely adopted. However, since the inclusions typically behave as singular sinks or sources for the 3D problem, mesh adaptation near the embedded 1D domains may be necessary to enhance solution accuracy and recover optimal convergence rates. An alternative to mesh adaptation is represented by the XFEM, which we here propose to enhance the approximation capabilities of an optimization-based 3D-1D coupling approach. An effective quadrature strategy is devised to integrate the enrichment functions and numerical tests on single and multiple segments are proposed to demonstrate the effectiveness of the approach.
