CAD‐consistent adaptive refinement using a NURBS‐based discontinuous Galerkin method

Abstract

This study concerns the development of a new method combining high-order CAD-consistent grids and adaptive refinement / coarsening strategies for efficient analysis of compressible flows. The proposed approach allows to use geometrical data from Computer-Aided Design (CAD) without any approximation. Thus, the simulations are based on the exact geometry, even for the coarsest discretizations. Combining this property with a local refinement method allows to start computations using very coarse grids and then rely on dynamic adaption to construct suitable computational domains. The resulting approach facilitates interactions between CAD and Computational Fluid Dynamics (CFD) solvers and focuses the computational effort on the capture of physical phenomena, since geometry is exactly taken into account. The proposed methodology is based on a Discontinuous Galerkin (DG) method for compressible Navier-Stokes equations, modified to use Non-Uniform Rational B-Spline (NURBS) representations. Local refinement and coarsening are introduced using intrinsic properties of NURBS associated to a local error indicator. A verification of the accuracy of the method is achieved and a set of applications are presented, ranging from viscous subsonic to inviscid trans-and supersonic flow problems.

Shape Sensitivity Analysis in Aerodynamics Using an Isogeometric Discontinuous Galerkin Method

“…The current study is conducted in this framework. As shown in previous works [9,10], the NURBS-based DG scheme allows to solve hyperbolic systems with a high accuracy and is fully consistent with CAD representations. Therefore, we intend to evaluate its potentiality for sensitivity analysis including shape parameters derived from CAD representations.…”

A NURBS-based discontinuous Galerkin method for conservation laws with high-order moving meshes

“…The rational Bernstein functions (17) are defined on a parametric domain, therefore, as in Isogeometric Analysis, the elements are mapped from the physical space to the parametric domain. In this context, a mapped ALE approach would require an additional map function to simulate the moving domain, resulting in a complex and cumbersome implementation.…”

“…All these approaches have demonstrated the interest of using high-order boundary representations in terms of accuracy for some selected test-cases governed by compressible or incompressible flow models. Besides, the DG framework may alleviate the mentioned difficulties of generating NURBS-based grids, thanks to the higher flexibility of the DG method in handling non-conformities and local refinement (17).…”

“…As a consequence, the present study aims at proposing a new framework for simulating compressible flows with arbitrary displacement of boundaries defined by NURBS. The proposed approach relies on a DG formulation based on rational Bézier elements (16), that can be easily constructed from a set of NURBS boundaries using local refinement (17), preserving the CAD geometry exactly.…”

A NURBS-based Discontinuous Galerkin Framework for Compressible Aerodynamics

“…This is achieved by using a least-squares projection. The whole adaptation procedure is detailed in [15]. When an adapted mesh is subject to non-linear deformations, the refinement operation might become irreversible.…”