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CFD- Discretization Methods
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Computational Fluid DynamicsII. Basic Discretization Methods
C.-D. Munz
Universität Stuttgart, Institut für Aerodynamik und Gasdynamik
Pfaffenwaldring 21, 70550 StuttgartTel. +49-711/685-63401 (Sekr.)
Fax +49-711/685-63438 e-mail [email protected]
ContentsI. Equations
II. Basic Discretization Methods1. Finite Difference Schemes2. Finite Volume Schemes3. Finite Element Schemes4. Grids – Discretization in Space
III. Numerical Gasdynamics and Finite VolumeSchemes
VI. Numerical Solution of Incompressible and WeaklyCompressibleFluid Flow
1. Finite Difference (FD) Schemes
∆x
u(x)∆x)u(xlim(x)u
0∆x
−+=′→
1. Step: Discretize computational domain (Grid)
2. Step: Choose finite differences and replacederivatives by finite differences
3. Step: Order the difference equations
4. Step: Solve the system of difference equations
Basic idea:
Replace derivatives by difference quotients
PDE,ODE System of algebraic equations
PDE: partial differential equation, ODE: ordinary differential equation
Finite DifferencesTaylor expansions
(1)
(2)
( ) ( ) ( ) ( ) ( ) K−+′′′∆−′′∆+′∆−=− i
3
i
2
ii1i xu6
xxu
2
xxuxxuxu
( ) ( ) ( ) ( ) ( ) K+′′′∆+′′∆+′∆+=+ i
3
i
2
ii1i xu6
xxu
2
xxuxxuxu
Difference quotient for derivative: (2) – (1)
( ) ( ) ( ) ( ) ( )5i
3
i1i1i xOxu6
x2xux2xuxu ∆+′′′∆+′∆=− −+
( ) ( ) ( ) ( )2i
1i1i xOxux2
xuxu ∆+′=∆− −+
( ) ( ) ( ) ( )xOxux
xuxui
i1i ∆+′=∆
−+
( ) ( ) ( ) ( )xOxux
xuxui
1ii ∆+′=∆− −
central difference 2nd order
right-sided difference 1st order
left-sided difference 1st order
Finite difference for 2nd derivative: (1) + (2) + 2y(xi)
( ) ( ) ( ) ( ) ( )2i2
1ii1i hOxuh
xux2uxu +′′=+−⇒ −+
( ) ( ) ( ) ( ) ( )( ) ( )2i
IV4
i2
1ii1i hOxu12
hxuhxux2uxu ++′′=+− −+
central difference quotient, 2nd order.
•Finite differences of higher order need more than 3 points
Taylor expansion gives consistency = guarantee of approximation – assumption: u smooth enough
Conclusion Finite Differences
– Approximate values are values at grid points– Simple coding also for complicated equations– Multidimensional extension is straightforward on Cartesian
grids– Extension to structured grids– Solution has to be smooth enough and can be approximated
by a smooth approximation
u
x
2. Finite Volume (FV) Schemes
( ) [ ]T,0Din 0ufut ×=⋅∇+
smooth piecewise Cboundary
kjfür CC ,CD
j
kjjj
∂
≠Φ=∩=U
iCDiscretization of space
Grid
Numerical scheme for conservation equations
( )( ) dt dS nt,xufuCuC1n
n j
t
t C
njj
1njj ∫ ∫
+
∂
+ ⋅−=r
[ ]1nnj t,tCover n Integratio +×
Evolution equations for integral mean values
Direct approximation of the integral conservation law
Basic Basic Basic Basic partspartspartsparts:1. Discretization in space (quite general grids) 2. Reconstruction of local values3. Appropriate approximation of the flux, numerical flux
Finite Volume Scheme in One Space Dimension
∫ ∫∫ ∫+ +
−
+ +
−
=+1n
n
1/2i
1/2i
1n
n
1/2i
1/2i
t
t
x
x
x
t
t
x
x
t 0dxdtt))f(u(x, t)dx (x,u
( ) 0ufu xt =+
Conservation equation t
1−ix ix 1+ix
1nt +
nt
x
t
2/1ix − 2/1ix +ixIntegration over [xi-1/2,xi+1/2]x[tn,tn+1]:
∫∫++
−
=+ −++
1n
n
1/2i
1/2i
t
t
1/2i1/2i
x
x
n1n 0t))dt,f(u(x-t)),f(u(x )dx tu(x,-)tu(x,
)g - (g ∆x
∆tuu n
1/2-in
1/2ini
1ni +
+ −=flux numerical called is g 1/2i+
Conclusions Finite Volume Schemes
• Approximation of integral values• Consistent with integral conservation• No continuity assumption– approximation of the integral
conservation law• Flux calculation is the basic building block based on
local wave propagation - Shock-capturing possible• Reconstruction to get local data• general grids• stable approximation of
underresolved phenomena
u
x
3. Finite Element (FE) Schemes
Approximation is a function – usually continuous
∑=
=N
1iiih (x)φ(t)c t)(x,u
basis functionsDOF
The DOFs are determined in such a way that the trialfunction becomes a good approximation
Discretization in function space
here, for time dependent problems
DOF = degrees of freedom (time dependent)
trial function
Often Used Basis FunctionsBasis functions are defined to be local, e.g.,
nodal basis: DOF are values at some points of the grid cell
hat functions
modal basis: Coefficients of polynomials
=(x)φ i
i1ii xxxfür)x(xh
11 <<−+ −
1iii xxxfür)x(xh
11 +<<−−
otherwise0
Approximate Solution (Piecewise Linear)
Calculation of DOFs
Method of RitzThe problem may be formulated as a variational problem, too. Solve the variational problem for the trial function
CollocationThe approximate solution solves the problemat some points exactly:
Number of points = number of DOFsMethod of weighted residuals: Trial function
is inserted, DOFs are determined to minimizethe residual with respect to
Galerkin method: OrthogonalityLeast squares method: Least sqare deviation
Calculation of DOFs
Method of RitzThe problem may be formulated as a variational problem, too. Solve the variational problem for the trial function
CollocationThe approximate solution solves the problemat some points exactly:
Number of points = number of DOFsMethod of weighted residuals: Trial function
is inserted, DOFs are determined to minimizethe residual with respect to
Galerkin method: OrthogonalityLeast squares method: Least sqare deviation
Conclusion Finite Elements
- Data as degrees of freedomof a trial function
- Difficult coding- General grids- Problems at strong gradients- Continuous solutions and
approximations
u
Very recent approach: Discontinuous Galerkin schemesfor future aerodynamic codes ?
4. Grids – Discretization in Space• The use of the grid may be different for different discretizaiton
methods
Finite differences
grid points withapproximate values
Finite volumes
Grid cells withapproximate
integral means
Finite elements
Support of trialfunction
∑ φn
1iia
Cartesian Grids
• Simple generation• Good data management• Adaptation difficult• Limited to simple geometries• Curved boundary treatment on Cartesian
meshes possible, but difficult
equidistant non equidistant
Transformation of the equations from physical to logical space
( ) ( ):ξr)r
uxu → 2,1 with ˆ2
2
1
1 =∂∂
∂∂+
∂∂
∂∂=
∂∂
mux
xu
x
xu
mmm ξξξ
2m
2
1m
1
m xxx ξ∂∂
∂ξ∂+
ξ∂∂
∂ξ∂=
∂∂
Derivatives:
Example continuity equation:
( )
( ) ( )0
ˆˆˆˆˆ
0
2
1 2
2
1
1
2
1
=
∂∂
∂∂+
∂∂
∂∂+
∂∂
=∂
∂+∂∂
∑
∑
=
=
m
m
m
m
m
m m
m
u
x
u
xt
x
u
t
ξρξ
ξρξρ
ρρ
Boundary-fitted Structured GridsRequirement: Bijective transformation to a Cartesian grid
),()x,x(:T 2121 ξξ→
Function:
physical coordinates logical coordinates
Boundary-fitted Structured Grid (H-Grid)
Transformation to a Cartesian grid
η
ξ
y
x
a b c d a b c d
η
ξ
y
x
e a = a‘c b = b‘
d
f
fd e
a‘a cb b‘
Boundary-fitted Structured Grid (C-Grid)
η
ξ
y
x
d a = a‘b c = c‘
c‘c d
a‘a b
Boundary-fitted Structured Grid (O-Grid)
Unstructured Grids
• Best approach for complex geometries• Usual triangles and quadrilaterals in 2D or tetrahedrons
and hexahedrons in 3D
Dual Grid
Grid cell edges of the dual grid cell isdefined by the lines betweenbarycenters and midpoints of edges of the primal grid
e.g., implemented in Tau-Code of DLRAirbus code for unstructured grids
flow direction
fan
spinner
Noise calculation of the inlet of turbo engine by solvingthe linearized Euler equations
mean flow given by RANS simulation
Source terms are given by LES
Cooperation withF. Thiele, Berlin
Numerical Results
Block-Structured Chimera-TypeFinite Difference (FD) Mesh
(5,469,928 points / DOF)Mesh generation time: days
Fully Unstructured TetrahedralDiscontinuous Galerkin (DG) Mesh(123,304 elements – 2,466,080 DOF)
Mesh generation time: 55 s.
FD results ( 6.0 CPUh / ms ) DG results ( 2.2 CPUh / ms )
Numerical Methods on Unstructered Grids
• Methods for unstructered grids:– Finite volumes
– Finite elements• The data management needs much more
overhead: Information about neighbors, edges, ..• Adaptation technically simple
Fluid Flow around a Sphere
Flexibility of Discontinuous Galerkin Schemes
our calculations
Literatur - Grundlagen
• C.-D. Munz, T. Westermann: Numerische Behandlung von gewöhnlichen und partiellen Differenzial-gleichungen, Springer-Verlag 2009, auch als e-bookohne CD erhältlich
• C.Hirsch: Numerical Computation of Internal and External Flows, 2nd edition, Elsevier John Wiley and Sons 2007
• K. A. Hoffmann, S. T. Chiang: Computational FluidDynamics for Engineers, Vol. I