STEFANO GIANI
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High–Order/hp–Adaptive Discontinuous Galerkin Finite Element Methods
for Compressible Fluid Flows

 

mesh
Figure 1. Mesh of a wing section

This project has been done in collaboration with Prof. Paul Houston.

We derived high-order finite numerical methods (FEMs), based on discontinuous Galerkin (DG) approximations, to solve the compressible Euler and Navier-Stokes equations in both the 2D and 3D cases. In particular, we exploit a lot of new technologies to improve the reliability and the accuracy of our simulations, like: goal-oriented error estimators, which determine what areas of a mesh should be refined in order to improve the accuracy in the approximations of quantities of interested, and hp-mesh adaption techniques to reduce the size of the elements and to adjust their polynomial degrees in order to improve the computations. We show that our method is capable of achieving exponential convergence of the error to zero for quantities of interest, such as lift and drag coefficiens. This is particularly important considering that large aerodynamics simulations of viscous high Reynolds number flows around complex aircraft configurations are still very expensive, despite the progress made in Computational Fluid Dynamics (CFD), both in terms of user time and computational resources.

Discontinuous Galerkin Methods:

aniso h Figure 2. Anisotropic h-mesh refinement
 
hp mesh Figure 3. hp-mesh refinement

Discontinuous Galerkin (DG) methods have been introduced in 1973 by Reed and Hill for the numerical approximation of hyperbolic problems, and since that time there has been an active development of DG methods for hyperbolic and nearly hyperbolic problems. DG methods have received particular considerable interest for problems with a dominant first-order part, e.g. in electrodynamics and fluid mechanics. The reasons for this increase of interest in DG methods are numerous, but essentially lie in the fact that allowing for discontinuities in the finite element approximation gives tremendous flexibility in terms of mesh design and choice of shape functions. For example, DG methods easily handle non-conforming meshes, allow for approximations of various orders, thus facilitating hp-adaptivity, permit to handle in a natural way possible discontinuities in the coefficients of the physical model and to approximate weakly the boundary conditions. The compact form of the DG method makes it well suited for parallel computer platforms.

Polynomial in x direction Figure 4. Polynomial degrees in the x direction
     
Polynomial in y direction Figure 5. Polynomial degrees in the y direction

 

Goal Oriented Error Estimators:

Convergence
Figure 6. Convergence
 
Fichera Corner
Figure 7. Fichera corner singularity
 
Fichera Corner hp
Figure 8. Polynomial distribution on the refined mesh

By employing a duality argument we derive so–called weighted or Type I a posteriori estimates which bound the error between the true value of the prescribed functional, and the actual computed value. In these error estimates, the element residuals of the computed numerical solution are multiplied by local weights involving the solution of a certain dual or adjoint problem. On the basis of the resulting a posteriori error bound, we design and implement an adaptive finite element algorithm to ensure reliable and efficient control of the error in the computed functional with respect to a user–defined tolerance. For the hp–version of the discontinuous Galerkin finite element method, the decision as to whether to h–refine or p–refine an element is based on estimating the local analyticity of the primal and dual solutions via truncated Legendre series expansions. The performance of the resulting hp–refinement algorithm is demonstrated through a series of numerical experiments. 

 

Figures:

  • Fig. 1: The unstructured initial mesh of the section of a wing
  • Fig. 2: The final mesh after 9 steps of anisotropic h-mesh refinement. The refinement has been done to minimize error in the lift and drag coefficients, as can be seen, the refinement is mostly along the wing and in the wake.
  • Fig. 3: The final mesh after 9 steps of isotropic hp-mesh refinement. The colors indicates the polynomial degrees in each element. The refinement has been done to minimize error in the lift and drag coefficients.
  • Fig. 4: The final mesh after 9 steps of anisotropic hp-mesh refinement. The colors indicates the polynomial degrees in each element along the x direction. The refinement has been done to minimize error in the lift and drag coefficients, as can be seen, the polynomial degrees are almost constant along the wing.
  • Fig. 5: The final mesh after 9 steps of anisotropic hp-mesh refinement. The colors indicates the polynomial degrees in each element along the y direction. The refinement has been done to minimize error in the lift and drag coefficients, as can be seen, the polynomial degrees increase rapidly approaching the surface of the wing.
  • Fig. 6: The convergence for each different refinement scheme.
  • Fig. 7: A Fichera corner domain with a singularity in the re-entering corner.
  • Fig. 8: The final mesh after 5 steps of isotropic hp-mesh refinement for the Fichera corner domain with a singularity in the re-entering corner.

 

References:

S. Giani and P. Houston, High-Order hp-Adaptive Discontinuous Galerkin Finite Element Methods for Compressible Fluid Flows. - In N. Kroll, H. Bieler, H. Deconinck, V. Couallier, H. van der Ven and K. Sorensen, editors, ADIGMA - A European Initiative on the Development of Adaptive Higher-Order Variational Methods for Aerospace Applications, Springer, 2010

S. Giani and P. Houston (2012), Anisotropic hp-adaptive discontinuous Galerkin finite element methods for compressible fluid flows. - International Journal of Numerical Analysis and Modeling 9(4), 928-949.

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