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：本文提出了一种用于两相不可压缩Stokes流的高阶、可自适应阶次的混合间断伽辽金方法，其边界和界面采用NURBS进行描述。NURBS曲线嵌入在固定的笛卡尔网格中，形成一种非拟合HDG格式，能够精确处理边界/界面的几何形状，从而避免使用拟合的高阶曲边网格。采用NURBS增强有限元方法的框架，对浸入式NURBS曲线及被NURBS曲线切割的单元进行精确积分。通过Nitsche格式在嵌入曲面上施加Dirichlet边界条件，使得与标准HDG方法一样仅在网格骨架上存在未知量，无需在非匹配边界/界面上引入任何额外的自由度。最终得到的非拟合HDG-NEFEM方法结合了非匹配网格、精确NURBS几何与高阶近似，能够在粗网格上提供高保真度的计算结果，且独立于求解域的几何特征。数值算例验证了该方法即使在存在严重切割单元或面的情况下，仍具有最优的精度和鲁棒性，并展示了其适用于基于CAD几何的微流控系统模拟。