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

Figure
1. Mesh of a wing section

This
project has been done in collaboration with Prof.
Paul Houston.
We
derived highorder finite numerical methods (FEMs), based on
discontinuous Galerkin (DG) approximations, to solve the compressible
Euler and
NavierStokes 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: goaloriented 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 hpmesh 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:
Figure
2. Anisotropic hmesh refinement


Figure
3. hpmesh
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 firstorder 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
nonconforming meshes, allow for approximations of various orders, thus
facilitating hpadaptivity,
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.

Figure 4. Polynomial degrees in the x
direction


Figure 5. Polynomial degrees in the y direction


Goal Oriented Error
Estimators:
Figure
6. Convergence


Figure
7. Fichera corner singularity


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 hmesh 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 hpmesh 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 hpmesh 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 hpmesh 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 reentering corner.
 Fig.
8: The final mesh after 5 steps
of isotropic hpmesh refinement for
the Fichera corner domain with a singularity in the reentering corner.
References:
S. Giani and P. Houston,
HighOrder hpAdaptive 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 HigherOrder Variational Methods for Aerospace Applications, Springer, 2010
S. Giani and P. Houston (2012), Anisotropic hpadaptive discontinuous Galerkin finite element methods for compressible fluid flows.
 International Journal of Numerical Analysis and Modeling 9(4), 928949.

