We present a novel isogeometric method, namely the Immersed Boundary-Conformal Method (IBCM), that features a layer of discretization conformal to the boundary while employing a simple background mesh for the remaining domain. In this manner, we leverage the geometric flexibility of the immersed boundary method with the advantages of a conformal discretization, such as intuitive control of mesh resolution around the boundary, higher accuracy per degree of freedom, automatic satisfaction of interface kinematic conditions, and the ability to strongly impose Dirichlet boundary conditions. In the proposed method, starting with a boundary representation of a geometric model, we extrude it to obtain a corresponding conformal layer. Next, a given background B-spline mesh is cut with the conformal layer, leading to two disconnected regions: an exterior region and an interior region. Depending on the problem of interest, one of the two regions is selected to be coupled with the conformal layer through Nitsche’s method. Such a construction involves Boolean operations such as difference and union, which therefore require proper stabilization to deal with arbitrarily cut elements. In this regard, we follow our precedent work called the minimal stabilization method (Antolin et al in SIAM J Sci Comput 43(1):A330–A354, 2021). In the end, we solve several 2D benchmark problems to demonstrate improved accuracy and expected convergence with IBCM. Two applications that involve complex geometries are also studied to show the potential of IBCM, including a spanner model and a fiber-reinforced composite model. Moreover, we demonstrate the effectiveness of IBCM in an application that exhibits boundary-layer phenomena.

我们提出了一种新的等几何方法，即浸入边界保形方法(IBCM)，它具有与边界共形的离散化层，同时对剩余域采用简单的背景网格。通过这种方式，我们利用浸入边界方法的几何灵活性和保形离散化的优点，例如边界周围网格分辨率的直观控制、每个自由度的更高精度、界面运动学条件的自动满足以及能力强强加 Dirichlet 边界条件。在所提出的方法中，从几何模型的边界表示开始，我们挤压它以获得相应的保形层。接下来，给定的背景 B 样条网格被共形层切割，导致两个不连续的区域：外部区域和内部区域。根据感兴趣的问题，通过 Nitsche 方法选择两个区域之一与共形层耦合。这种构造涉及诸如差和并之类的布尔运算，因此需要适当的稳定性来处理任意切割的元素。在这方面，我们遵循称为最小稳定方法的先例工作（Antolin 等人在 SIAM J Sci Comput 43(1):A330–A354, 2021 中）。最后，我们解决了几个 2D 基准问题，以证明使用 IBCM 提高了准确性和预期收敛性。还研究了涉及复杂几何形状的两个应用程序，以展示 IBCM 的潜力，包括扳手模型和纤维增强复合材料模型。此外，我们证明了 IBCM 在表现出边界层现象的应用程序中的有效性。