A high-order, degree-adaptive hybridizable discontinuous Galerkin (HDG) method is presented for two-fluid incompressible Stokes flows, with boundaries and interfaces described using NURBS. The NURBS curves are embedded in a fixed Cartesian grid, yielding an unfitted HDG scheme capable of treating the exact geometry of the boundaries/interfaces, circumventing the need for fitted, high-order, curved meshes. The framework of the NURBS-enhanced finite element method (NEFEM) is employed for accurate quadrature along immersed NURBS and in elements cut by NURBS curves. A Nitsche's formulation is used to enforce Dirichlet conditions on embedded surfaces, yielding unknowns only on the mesh skeleton as in standard HDG, without introducing any additional degree of freedom on non-matching boundaries/interfaces. The resulting unfitted HDG-NEFEM method combines non-conforming meshes, exact NURBS geometry and high-order approximations to provide high-fidelity results on coarse meshes, independent of the geometric features of the domain. Numerical examples illustrate the optimal accuracy and robustness of the method, even in the presence of badly cut cells or faces, and its suitability to simulate microfluidic systems from CAD geometries.

翻译：提出了一种高阶、自由度自适应的混合化不连续伽辽金（HDG）方法，用于求解双流体不可压缩斯托克斯流问题，其中边界和界面采用NURBS描述。NURBS曲线嵌入固定笛卡尔网格中，形成一种非拟合HDG格式，能够处理边界/界面的精确几何，从而避免了对拟合、高阶、曲网格的需求。采用NURBS增强有限元方法（NEFEM）框架进行精确求积，适用于浸入式NURBS曲线以及被NURBS曲线切割的单元。利用Nitsche公式实现嵌入表面上的Dirichlet条件，仅保留标准HDG中的网格骨架未知量，无需在非匹配边界/界面上引入额外自由度。由此得到的非拟合HDG-NEFEM方法结合了非一致网格、精确NURBS几何和高阶逼近，能在粗网格上提供高保真结果，且不受域几何特征的影响。数值算例展示了该方法的最优精度和鲁棒性，即使存在严重切割的单元或面，也适用于从CAD几何出发模拟微流体系统。