Section: New Results

Isogeometric Discontinuous Galerkin method for compressible flows

Participants : Régis Duvigneau, Stefano Pezzano, Maxime Stauffert.

The co-existence of different geometrical representations in the design loop (CAD-based and mesh-based) is a real bottleneck for the application of design optimization procedures in industry, yielding a major waste of human time to convert geometrical data. Isogeometric analysis methods, which consists in using CAD bases like NURBS in a Finite-Element framework, were proposed a decade ago to facilitate interactions between geometry and simulation domains.

We investigate the extension of such methods to Discontinuous Galerkin (DG) formulations, which are better suited to hyperbolic or convection-dominated problems. Specifically, we develop a DG method for compressible Euler and Navier-Stokes equations, based on rational parametric elements, that preserves exactly the geometry of boundaries defined by NURBS, while the same rational approximation space is adopted for the solution [37]. The following research axes are considered in this context:

• Adaptive refinement

  Properties of NURBS functions are used to define an adaptive refinement strategy, which refines locally the discretization according to an error indicator, while describing exactly CAD geometries whatever the refinement level. The resulting approach exhibits an optimal convergence rate and capture efficiently local flow features, like shocks or vortices, avoiding refinement due to geometry approximation [36], [47].

• Arbitrary Eulerian-Lagrangian formulation

  To enable the simulation of flows around moving bodies, an Arbitrary Eulerian-Lagrangian (ALE) formulation is proposed in the context of the isogeometric DG method. It relies on a NURBS-based boundary velocity, integrated along time over moving NURBS elements. The gain of using exact-geometry representations is clearly quantified [39].

• Isogeometric shape optimization

  On the basis of the isogeometric DG method, we develop a shape optimization procedure with sensitivity analysis, entirely based on NURBS representations [40]. The mesh, the shape parameters as well as the flow solutions are represented by NURBS, which avoids any geometrical conversion and allows to exploit NURBS properties, like regularity, hierarchy, etc.