BAIL 2006 Book of Abstracts - Institut für Numerische und ...
BAIL 2006 Book of Abstracts - Institut für Numerische und ... BAIL 2006 Book of Abstracts - Institut für Numerische und ...
R. HARTMANN: Discontinuous Galerkin methods for compressible flows: higher order accuracy, error estimation and adaptivity ✬ ✫ Discontinuous Galerkin methods for compressible flows: higher order accuracy, error estimation and adaptivity Ralf Hartmann Institute of Aerodynamics and Flow Technology, German Aerospace Center (DLR), Lilienthalplatz 7, 38108 Braunschweig, Germany The Discontinuous Galerkin (DG) method for the compressible Euler equations is extended to the symmetric interior penalty (SIP)DG method for the compressible Navier-Stokes equations, see Hartmann & Houston [2]. Shock-capturing is used to avoid overshoots near shocks, see Hartmann [1]. The nonlinear equations are solved using a fully implicit solver. We demonstrate the accuracy of higher order DG discretizations with respect to the approximation of aerodynamical force coefficients and to the approximation of viscous boundary layers, see Figure 1 and Hartmann & Houston [2]. Finally, we demonstrate the use of a posteriori error estimation and goal-oriented (also called weighted-residual-based or adjoint-based) adaptive mesh refinement for subsonic flows, Hartmann & Houston [3], and for supersonic compressible flows, see Figure 2 and Hartmann [1]. References [1] R. Hartmann. Adaptive discontinuous Galerkin methods with shock-capturing for the compressible Navier-Stokes equations. Int. J. Numer. Meth. Fluids, 2006. To appear. [2] R. Hartmann and P. Houston. Symmetric interior penalty DG methods for the compressible Navier–Stokes equations I: Method formulation. Int. J. Num. Anal. Model., 3(1):1–20, 2006. [3] R. Hartmann and P. Houston. Symmetric interior penalty DG methods for the compressible Navier–Stokes equations II: Goal–oriented a posteriori error estimation. Int. J. Num. Anal. Model., 3(2):141–162, 2006. Speaker: HARTMANN, R. 22 BAIL 2006 1 ✩ ✪
R. HARTMANN: Discontinuous Galerkin methods for compressible flows: higher order accuracy, error estimation and adaptivity ✬ ✫ c dp 0.024 0.022 0.02 0.018 0.016 0.014 0.012 0.01 0.008 0.006 reference cdp DG(3), global refinement DG(2), global refinement DG(1), global refinement 10000 100000 number of elements |c dp − 0.0222875| 0.01 0.001 0.0001 (a) (b) DG(1), global refinement DG(2), global refinement DG(3), global refinement 10000 100000 number of elements Figure 1: Subsonic laminar flow around the NACA0012 airfoil at M = 0.5, α = 0 and Re = 5000: Convergence of the pressure induced drag cdp under global refinement for DG(p), p = 1,2,3: (a) cdp versus number of elements; (b) Error in cdp (reference cdp − cdp) versus number of elements. For more detail cf. Hartmann & Houston [2]. 8 4 0 -4 -8 -4 0 4 8 8 4 0 -4 -8 -4 0 4 8 (a) (b) Figure 2: Supersonic laminar flow around the NACA0012 airfoil at M = 1.2, α = 0 and Re = 1000: (a) Residual-based refined mesh of 17670 elements with 282720 degrees of freedom and |Jcdp (u) − Jcdp (uh)| = 1.9 · 10 −3 ; (b) Adjoint-based refined mesh for cdp: mesh of 10038 elements with 160608 degrees of freedom and |Jcdp (u) − Jcdp (uh)| = 1.6 · 10 −4 . For more detail cf. Hartmann [1]. Speaker: HARTMANN, R. 23 BAIL 2006 2 ✩ ✪
- Page 1: BAIL 2006 International Conference
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- Page 17 and 18: Contents Greetings . . . . . . . .
- Page 19 and 20: CONTENTS S. HEMAVATHI, S. VALARMATH
- Page 21 and 22: CONTENTS G. LUBE: A stabilized fini
- Page 23: Plenary Presentations
- Page 26 and 27: P. HOUSTON: Discontinuous Galerkin
- Page 28 and 29: M. STYNES: Convection-diffusion pro
- Page 30 and 31: V. GRAVEMEIER, S. LENZ, W.A. WALL:
- Page 33: Minisymposia
- Page 36 and 37: R.K. DUNNE, E. O’RIORDAN, M.M. TU
- Page 38 and 39: G.I. SHISHKIN: A posteriori adapted
- Page 40 and 41: W. LAYTON, I. STANCULESCU: Numerica
- Page 42 and 43: H. WANG: A Component-Based Eulerian
- Page 46 and 47: V. HEUVELINE: On a new refinement s
- Page 48 and 49: J.A. MACKENZIE, A. NICOLA: A Discon
- Page 50 and 51: R. SCHNEIDER, P. JIMACK: Anisotropi
- Page 53 and 54: Speaker: Debopam Das, Tapan Sengupt
- Page 55 and 56: M.H. BUSCHMANN, M. GAD-EL-HAK: Turb
- Page 57 and 58: A. NAYAK, D. DAS: Three-dimesnional
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- Page 61 and 62: T.K. SENGUPTA, A. KAMESWARA RAO: Sp
- Page 63 and 64: L. SAVIĆ, H. STEINRÜCK: Asymptoti
- Page 65 and 66: G.I. Shiskin, P. Hemker Robust Meth
- Page 67 and 68: D. BRANLEY, A.F. HEGARTY, H. PURTIL
- Page 69 and 70: T. LINSS, N. MADDEN: Layer-adapted
- Page 71 and 72: L.P. SHISHKINA, G.I. SHISHKIN: A Di
- Page 73 and 74: I.V. TSELISHCHEVA, G.I. SHISHKIN: D
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R. HARTMANN: Discontinuous Galerkin methods for compressible flows: higher order<br />
accuracy, error estimation and adaptivity<br />
✬<br />
✫<br />
c dp<br />
0.024<br />
0.022<br />
0.02<br />
0.018<br />
0.016<br />
0.014<br />
0.012<br />
0.01<br />
0.008<br />
0.006<br />
reference cdp<br />
DG(3), global refinement<br />
DG(2), global refinement<br />
DG(1), global refinement<br />
10000 100000<br />
number <strong>of</strong> elements<br />
|c dp − 0.0222875|<br />
0.01<br />
0.001<br />
0.0001<br />
(a) (b)<br />
DG(1), global refinement<br />
DG(2), global refinement<br />
DG(3), global refinement<br />
10000 100000<br />
number <strong>of</strong> elements<br />
Figure 1: Subsonic laminar flow aro<strong>und</strong> the NACA0012 airfoil at M = 0.5, α = 0 and Re = 5000:<br />
Convergence <strong>of</strong> the pressure induced drag cdp <strong>und</strong>er global refinement for DG(p), p = 1,2,3: (a)<br />
cdp versus number <strong>of</strong> elements; (b) Error in cdp (reference cdp − cdp) versus number <strong>of</strong> elements.<br />
For more detail cf. Hartmann & Houston [2].<br />
8<br />
4<br />
0<br />
-4<br />
-8<br />
-4 0 4 8<br />
8<br />
4<br />
0<br />
-4<br />
-8<br />
-4 0 4 8<br />
(a) (b)<br />
Figure 2: Supersonic laminar flow aro<strong>und</strong> the NACA0012 airfoil at M = 1.2, α = 0 and<br />
Re = 1000: (a) Residual-based refined mesh <strong>of</strong> 17670 elements with 282720 degrees <strong>of</strong> freedom<br />
and |Jcdp (u) − Jcdp (uh)| = 1.9 · 10 −3 ; (b) Adjoint-based refined mesh for cdp: mesh <strong>of</strong> 10038<br />
elements with 160608 degrees <strong>of</strong> freedom and |Jcdp (u) − Jcdp (uh)| = 1.6 · 10 −4 . For more detail<br />
cf. Hartmann [1].<br />
Speaker: HARTMANN, R. 23 <strong>BAIL</strong> <strong>2006</strong><br />
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