WS, Antonio Huerta: NURBS-Embedded HDG for Incompressible Flows

Speaker
Antonio Huerta,Polytechnic University of Catalonia

Abstract
A high-order, degree-adaptive Hybridizable Discontinuous Galerkin (HDG) method is presented for incompressible flows, governed by either the Stokes or Navier–Stokes equations, with boundaries and interfaces exactly described by NURBS. The geometries are embedded in a fixed Cartesian mesh, resulting in an unfitted HDG scheme that captures complex geometrical features without the need for curved, body-fitted high-order element meshes.
To enable accurate integration over these embedded geometries, the method employs the NURBS-Enhanced Finite Element Method (NEFEM) for quadrature in elements intersected by NURBS curves [1,2]. Dirichlet and Neumann boundary conditions are handled seamlessly at any polynomial order. Importantly, the method retains the HDG structure, with global unknowns defined only on the mesh skeleton, and no additional degrees of freedom on non-conforming boundaries or interfaces.
This NURBS-embedded HDG framework (unfitted HDG-NEFEM method) combines non-conforming Cartesian grids, exact CAD-based geometry, and high-order accuracy, enabling robust and precise simulations even on coarse meshes. While the method was initially developed for two-fluid Stokes flows [3], it extends naturally to incompressible Navier–Stokes problems, maintaining optimal convergence and robustness, even in the presence of severely cut elements.
Numerical results confirm the method’s accuracy, stability, and effectiveness in capturing high-fidelity flow features, including applications to microfluidic systems directly from CAD models.

[1] R. Sevilla, S. Fernández-Méndez, A. Huerta, “NURBS-enhanced finite element method (NEFEM),” International Journal for Numerical Methods in Engineering, 76(1), pp. 56–83 (2008).
[2] R. Sevilla, A. Huerta, “HDG-NEFEM with Degree Adaptivity for Stokes Flows,” Journal of Scientific Computing, 77(3), pp. 1953–1980 (2018).
[3] S. Piccardo, M. Giacomini, A. Huerta, “An unfitted high-order HDG method for two-fluid Stokes flow with exact NURBS geometries,” Journal of Computational Physics, 512, 113143 (2024).