An unfitted high-order HDG method for two-fluid Stokes flow with exact NURBS geometries

TL;DR

The paper presents an unfitted, high-order HDG method integrated with NEFEM to treat exact NURBS geometries for two-fluid Stokes flows. By embedding NURBS into a fixed Cartesian grid, it achieves high accuracy on non-matching meshes without additional DOFs on interfaces, while using Nitsche-based boundary treatment and interface coupling to preserve mass and symmetry. A degree-adaptive postprocessing scheme, together with an element-extension strategy for badly cut cells, yields robust, optimal convergence and precise mass conservation across challenging CAD geometries and microfluidic-like configurations. The approach demonstrates strong potential for CAD-derived microfluidic and multi-fluid simulations, scalable to complex 2D geometries and extendable to moving interfaces and 3D settings.

Abstract

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.

Figures (25)

• Figure 1: Schematic representation of the geometry and its discretization. Computational domain $\Omega_{\text{o}}$ partitioned into $6\times6$ square elements. The external physical boundary $\partial\Omega$ and the interface $\Upsilon$ not aligned with the mesh skeleton. Standard HDG elements in red, immersed boundary element (cut by $\partial\Omega$) in orange, interface element (cut by $\Upsilon$) in green.
• Figure 2: Schematic representation of the unknowns for the different element types. (a) Standard HDG element. (b) Immersed boundary element. (c) Interface element. Blue bullets represent local unknowns, red bullets hybrid unknowns, green crosses the doubled local unknowns employed in interface elements, and yellow crosses the doubled hybrid unknowns employed in interface cut faces.
• Figure 3: Schematic representation of an unfitted NURBS interface $\Upsilon$ in the computational domain $\Omega_{\text{o}}$.
• Figure 4: Element $\Omega_e$ cut in two regions by an interface $\Upsilon_e$. (a) Identification of $\Omega_e^1$ and $\Omega_e^2$. (b) Triangulation for quadrature. The interface is composed of two NURBS curves connected at the green bullet. The red bullets denote the intersections between $\Upsilon$ and $\partial\Omega_e$ and the blue ones are the vertices of $\Omega_e$ and of the triangulation for qudrature.
• Figure 5: Element extension strategy. Left: initial configuration with badly cut cell (red) and well-cut cell (green). Right: extended element. Blue bullets are the element-based unknowns and red bullets the face-based unknowns of the resulting extended element.

Theorems & Definitions (10)

• Remark 1: Surface tension
• Remark 2: One-fluid problem
• Remark 3: Uniqueness of pressure
• Remark 4: Multi-fluid interface problem
• Remark 5: Stabilization for high viscosity constrast
• Remark 6: HDG with Cauchy stress formulation
• Remark 7: Extension to three dimensional problems
• Remark 8: Periodic boundary conditions in HDG
• Remark 9: NURBS sampling
• Remark 10: Multiple unfitted regions