A SHOCK SENSOR BASED SECOND ORDER BLENDED (BX) UPWIND RESIDUAL - - PowerPoint PPT Presentation

A SHOCK SENSOR BASED SECOND ORDER BLENDED (BX) UPWIND RESIDUAL DISTRIBUTION SCHEME FOR STEADY AND UNSTEADY COMPRESSIBLE FLOW Jiˇ r ´ ı Dobeˇ s

Department of Technical Mathematics, Faculty of Mechanical Engineering, Czech Technical University, Prague, Czech Republic

Herman Deconinck

Von Karman Institute for Fluid Dynamics, Belgium Hyperbolic Problems, July 17–21, 2006 Lyon, France 1

Outline Introduction and motivation RDS intro (steady & unsteady) Linear schemes Non-linear schemes Bx scheme for steady problems Numerical examples Bx scheme for unsteady problems Numerical examples Conclusions

Hyperbolic Problems, July 17–21, 2006 Lyon, France 2

The problem System of Euler equations in d spatial variables ut + ∇ · f = 0 u conserved set of variables, u(

x, t) : Rd+1 → Rq

uh approximation of the solution

• f vector of flux functions,

f(u) : Rq → Rq×d Ideal gas assumption Initial and boundary conditions Vector of Jacobian matrices

• λ = ∂

f

∂u

Hyperbolic Problems, July 17–21, 2006 Lyon, France 3

The problem System of Euler equations in d spatial variables ut + ∇ · f = 0 u conserved set of variables, u(

x, t) : Rd+1 → Rq

uh approximation of the solution

• f vector of flux functions,

f(u) : Rq → Rq×d Ideal gas assumption Initial and boundary conditions Vector of Jacobian matrices

• λ = ∂

f

∂u

Scalar advection equation in d spatial dimensions

ut + λ · ∇u = 0,

• λ ∈ Rd

Hyperbolic Problems, July 17–21, 2006 Lyon, France 3-a

RD schemes Introduction An alternative to the FV and FEM schemes Unstructured meshes, continuous representation of variables Very strong stability properties (in L∞) Second order accurate on arbitrary meshes, compact stencil Do not rely on 1D physics (no Riemann solver)

Hyperbolic Problems, July 17–21, 2006 Lyon, France 4

RD schemes Introduction An alternative to the FV and FEM schemes Unstructured meshes, continuous representation of variables Very strong stability properties (in L∞) Second order accurate on arbitrary meshes, compact stencil Do not rely on 1D physics (no Riemann solver) RD schemes (steady version) Introduced almost 25 years ago by P. Roe

• E. van der Weide: PhD thesis (1998)

Abgrall: JCP (2001), Abgrall & Mezine: JCP (2004)

Hyperbolic Problems, July 17–21, 2006 Lyon, France 4-a

RD schemes Introduction An alternative to the FV and FEM schemes Unstructured meshes, continuous representation of variables Very strong stability properties (in L∞) Second order accurate on arbitrary meshes, compact stencil Do not rely on 1D physics (no Riemann solver) RD schemes (steady version) Introduced almost 25 years ago by P. Roe

• E. van der Weide: PhD thesis (1998)

Abgrall: JCP (2001), Abgrall & Mezine: JCP (2004) RD schemes (unsteady version) FEM formulation (mass matrix): Ferrante: VKI report (1997) Space-time approach 2001 (VKI, U. Bordeaux) Abgrall & Mezine: JCP (2003), Ricchiuto & Cs ´ ık & Deconinck: JCP (2005)

Hyperbolic Problems, July 17–21, 2006 Lyon, France 4-b

RD schemes Introduction An alternative to the FV and FEM schemes Unstructured meshes, continuous representation of variables Very strong stability properties (in L∞) Second order accurate on arbitrary meshes, compact stencil Do not rely on 1D physics (no Riemann solver) RD schemes (steady version) Introduced almost 25 years ago by P. Roe

• E. van der Weide: PhD thesis (1998)

Abgrall: JCP (2001), Abgrall & Mezine: JCP (2004) RD schemes (unsteady version) FEM formulation (mass matrix): Ferrante: VKI report (1997) Space-time approach 2001 (VKI, U. Bordeaux) Abgrall & Mezine: JCP (2003), Ricchiuto & Cs ´ ık & Deconinck: JCP (2005) VKI CFD Lecture Series 2003, 2005

Hyperbolic Problems, July 17–21, 2006 Lyon, France 4-c

RD schemes Introduction ut + ∇ · f = 0 Scheme for steady problem (ut → 0) Compute element residual

ϕE ≈

• E

∇ · f dx

φ

Hyperbolic Problems, July 17–21, 2006 Lyon, France 5

RD schemes Introduction ut + ∇ · f = 0 Scheme for steady problem (ut → 0) Compute element residual

ϕE ≈

• E

∇ · f dx

Distribute it to nodes with distribution coefficient (matrix)

ϕE

i = βiϕE

φ φ

Hyperbolic Problems, July 17–21, 2006 Lyon, France 5-a

RD schemes Introduction ut + ∇ · f = 0 Scheme for steady problem (ut → 0) Compute element residual

ϕE ≈

• E

∇ · f dx

Distribute it to nodes with distribution coefficient (matrix)

ϕE

i = βiϕE

Update nodal solution

un+1

i

= un

i − αi

• E∈i

ϕE

i

The whole task is to define βi (or ϕE

i ).

φ φ φ φ φ φ φ φ

Hyperbolic Problems, July 17–21, 2006 Lyon, France 5-b

RD schemes Linear schemes Linear schemes: LDA scheme High accuracy (2nd order accurate) Enough dissipative even for some transonic flows This is not the case for unlimited FV method with linear reconstruction! N scheme Positive Quite accurate for 1st order schemes Very robust

Hyperbolic Problems, July 17–21, 2006 Lyon, France 6

RD schemes Linear schemes Linear schemes: LDA scheme High accuracy (2nd order accurate) Enough dissipative even for some transonic flows This is not the case for unlimited FV method with linear reconstruction! N scheme Positive Quite accurate for 1st order schemes Very robust Amazing iterative convergence Very robust Unfortunately, cannot be positive and 2nd order accurate

Hyperbolic Problems, July 17–21, 2006 Lyon, France 6-a

RD schemes Non-linear schemes N-modified scheme (PSI): Abgrall & Mezine: JCP (2004) 2nd order accurate for steady problems Positive for scalar problems Unsteady version avaliable B scheme: Cs ´ ık & Deconinck & Poedts: AIAA J. (2001) 2nd order accurate for steady problems Does not show oscillatory behavior Performs very similarly

Hyperbolic Problems, July 17–21, 2006 Lyon, France 7

RD schemes Non-linear schemes N-modified scheme (PSI): Abgrall & Mezine: JCP (2004) 2nd order accurate for steady problems Positive for scalar problems Unsteady version avaliable B scheme: Cs ´ ık & Deconinck & Poedts: AIAA J. (2001) 2nd order accurate for steady problems Does not show oscillatory behavior Performs very similarly Persisting problems: Accuracy in smooth parts of the solution Poor iterative convergence High nonlinearity of the scheme (bad for implicit method)

Hyperbolic Problems, July 17–21, 2006 Lyon, France 7-a

New Bx scheme (steady) Basic idea Use the LDA scheme everywhere except the shock waves Linear stability is sufficient in smooth parts of the flow Use the N scheme only in shock waves

Hyperbolic Problems, July 17–21, 2006 Lyon, France 8

New Bx scheme (steady) Basic idea Use the LDA scheme everywhere except the shock waves Linear stability is sufficient in smooth parts of the flow Use the N scheme only in shock waves Blending coefficient (LDA 0 ≤ θ ≤ 1 N) Bx = θ · N + (1 − θ) · LDA wish list: Order θ = O(h) in smooth parts of the solution Error of the N scheme will be multiplied by O(h) 2nd order of accuracy Very smooth Good convergence towards the steady state solution Implicit method (good Jacobian approximation)

Hyperbolic Problems, July 17–21, 2006 Lyon, France 8-a

New Bx scheme (steady) Shock sensor Shock sensor sc =

∇p · v δpv + ≈ 1 δpvµ(E)

• v ·
• T
• ∇p dx

+ ,

Compression sc > 0, expansion sc = 0 Smooth solution sc = O(1), shock sc ≫ O(1)

Hyperbolic Problems, July 17–21, 2006 Lyon, France 9

New Bx scheme (steady) Shock sensor Shock sensor sc =

∇p · v δpv + ≈ 1 δpvµ(E)

• v ·
• T
• ∇p dx

+ ,

Compression sc > 0, expansion sc = 0 Smooth solution sc = O(1), shock sc ≫ O(1) Blending coefficient

θ = min(1, sc2 · h)

Order θ = O(h) in smooth flow

Hyperbolic Problems, July 17–21, 2006 Lyon, France 9-a

New Bx scheme (steady) Chanel Maiso = 0.675 10 % circular bump chanel Maiso = 0.675 Mesh h = 1/30 Mach number isolines

5 1 1.5 2

0.5 1 1.5 2 2.5 3 0.2 0.4 0.6 0.8 1 0.5 1 1.5 2 2.5 3 0.2 0.4 0.6 0.8 1

Hyperbolic Problems, July 17–21, 2006 Lyon, France 10

New Bx scheme (steady) Chanel Maiso = 0.675

x Mach

1.7 1.75 1.8 0.65 0.7 0.75 0.8 0.85 0.9 0.95 Bx N LDA B N modif FV Barth

x Mach

1.5 1.6 1.7 1.8 1.2 1.25 1.3 1.35 1.4 1.45 Bx N LDA B N modif FV Barth

Hyperbolic Problems, July 17–21, 2006 Lyon, France 11

New Bx scheme (steady) Chanel Maiso = 0.675

Iterations Log(res) 20 40 60 80

• 12
• 10
• 8
• 6
• 4
• 2

Bx N LDA FV Barth N modif B

Hyperbolic Problems, July 17–21, 2006 Lyon, France 12

New Bx scheme (steady) Ma∞ = 0.38 flow past a cylinder Sub-critical flow, Ma∞ = 0.38 Mach number isolines ∆Ma = 0.02 LDA Bx

Hyperbolic Problems, July 17–21, 2006 Lyon, France 13

New Bx scheme (steady) Ma∞ = 0.38 flow past a cylinder B N-modif

Remark: FV2 + Barth – similar to the B scheme result Hyperbolic Problems, July 17–21, 2006 Lyon, France 14

New Bx scheme (steady) Mach 20 bow shock Mach 20 bow shock One can apply convergence fix (do not decrease blending coeff.)

θn

i = max(θi, θn−1 i

)

N-modified scheme Bx scheme

Hyperbolic Problems, July 17–21, 2006 Lyon, France 15

New Bx scheme (steady) Mach 20 bow shock

x Mach

0.5 0.6 0.7 0.8 0.9 1 5 10 15 20 Bx Bx fix N N modif

Iterations log(||res||)

5000 10000 15000

• 4
• 3
• 2
• 1

1 2 3

Nmodified N Bx Bx fix

Hyperbolic Problems, July 17–21, 2006 Lyon, France 16

New Bx scheme (steady) Onera M6 wing Onera M6 wing Ma∞ = 0.8395, α = 3.06◦ Mesh 57041 nodes and 306843 elements Mach number isolines Note the λ-shock

X Y Z

Mesh FV2 Barth Bx scheme

Hyperbolic Problems, July 17–21, 2006 Lyon, France 17

New Bx scheme (steady) Onera M6 wing CPU time (4 x Pentium4 2.80, 3.2GHz – nonhomogenous cluster) CFL ≈ 100 An example – a large oportunity for speed-up!

CPU time [h] res

5 10 15

• 14
• 12
• 10
• 8
• 6
• 4
• 2

2 4 FV2 Barth Bx

Hyperbolic Problems, July 17–21, 2006 Lyon, France 18

Unsteady LDA scheme FEM formulation LDA scheme for unsteady scalar problems

∂u ∂t + λ · ∇u = 0

Multiply by test function ϕi and integrate

• Ω

ϕi ∂u ∂t d x +

• Ω

ϕi λ · ∇u d x = 0

Split integration over elements in the stencil Approximate the solution with pice-wise linear (tent-like) trial function

uh =

• j∈E

ujψj

Use linearization for the vector of Jacobian matrices

Hyperbolic Problems, July 17–21, 2006 Lyon, France 19

Unsteady LDA scheme FEM formulation Take constants in front of the integral

• E∈i

 

j∈E

∂uj ∂t

E

ϕiψj d x

• +
• µ(E)

λ · ∇u

• 1

µ(E)

• E

ϕi d x   = 0

Final form

ϕE

i =

• j∈E

∂uj ∂t mij + βiϕE, un+1,m+1

i

= un+1,m

i

− αi

• E∈i

ϕE

i

with

mij =

• E

ϕiψj d x, ϕE =

• E
• λ · ∇u d

x, βi = 1 µ(E)

• E

ϕi d x

Hyperbolic Problems, July 17–21, 2006 Lyon, France 20

Unsteady LDA scheme FEM formulation Take constants in front of the integral

• E∈i

 

j∈E

∂uj ∂t

E

ϕiψj d x

• +
• µ(E)

λ · ∇u

• 1

µ(E)

• E

ϕi d x   = 0

Final form

ϕE

i =

• j∈E

∂uj ∂t mij + βiϕE, un+1,m+1

i

= un+1,m

i

− αi

• E∈i

ϕE

i

with

mij =

• E

ϕiψj d x, ϕE =

• E
• λ · ∇u d

x, βi = 1 µ(E)

• E

ϕi d x

Time derivative: 3BDF scheme We solve it in dual time → element contributions to nodes

Hyperbolic Problems, July 17–21, 2006 Lyon, France 20-a

Unsteady LDA scheme FEM formulation Take constants in front of the integral

• E∈i

 

j∈E

∂uj ∂t

E

ϕiψj d x

• +
• µ(E)

λ · ∇u

• 1

µ(E)

• E

ϕi d x   = 0

Final form

ϕE

i =

• j∈E

∂uj ∂t mij + βiϕE, un+1,m+1

i

= un+1,m

i

− αi

• E∈i

ϕE

i

with

mij =

• E

ϕiψj d x, ϕE =

• E
• λ · ∇u d

x, βi = 1 µ(E)

• E

ϕi d x

Time derivative: 3BDF scheme We solve it in dual time → element contributions to nodes Extension for Euler equations: Distribution coefficient → matrix Conservative linearization Deconinck, Roe, Struijs: C&F 1993

Hyperbolic Problems, July 17–21, 2006 Lyon, France 20-b

Unsteady Bx scheme N scheme Standard N scheme (Weide, PhD thesis, 1998) Lumped mass matrix 3BDF time discretization Formulated in dual time

Hyperbolic Problems, July 17–21, 2006 Lyon, France 21

Unsteady Bx scheme N scheme Standard N scheme (Weide, PhD thesis, 1998) Lumped mass matrix 3BDF time discretization Formulated in dual time Shock sensor sc =

1 δpv D p D t + = 1 δpv ∂p ∂t + ∇p · v + ≈ ≈ 1 δpvµ(E)

• E

∂p ∂t d x + v ·

• E
• ∇p d

x +

Compression sc > 0, expansion sc = 0 Reverts to the steady definition for the stationary solution

Hyperbolic Problems, July 17–21, 2006 Lyon, France 21-a

Unsteady Bx scheme Vortex convection Coarse mesh: 40 points per period Evolution after one period

X Y

• 0.2

0.2

• 0.2

0.2

Bx

X Y

• 0.2

0.2

• 0.2

0.2

LDA

Hyperbolic Problems, July 17–21, 2006 Lyon, France 22

Unsteady Bx scheme Vortex convection

X Y

• 0.2

0.2

• 0.2

0.2

N-modif

X Y

• 0.2

0.2

• 0.2

0.2

FV Barth

Hyperbolic Problems, July 17–21, 2006 Lyon, France 23

Unsteady Bx scheme Vortex convection

x Pressure

0.2 0.4 0.6 0.8 1 93 94 95 96 97 98 99 100 LDA Ferrante LDA Caraeni N-modified Bx Exact

x Pressure

0.2 0.4 0.6 0.8 1 93 94 95 96 97 98 99 100 FV nolim FV WENO FV Barth Exact

Minimal pressure in the vortex core: Scheme: LDAmm Bx N-modif FV nolim FV Barth Error 9.23% 11.59% 45.04% 16.75% 81%

Hyperbolic Problems, July 17–21, 2006 Lyon, France 24

Unsteady Bx scheme 2D Riemann problem Unstructured mesh, h = 1/400

x y 0.25 0.5 0.75 0.25 0.5 0.75 x y 0.25 0.5 0.75 0.25 0.5 0.75

Bx N-modif

Hyperbolic Problems, July 17–21, 2006 Lyon, France 25

Conclusions A nonlinear blended Bx scheme developed Blend of low order N scheme and high order LDA scheme Simple blending coefficient Based on a shock sensor Smooth → good iterative convergence properties

O(h) in smooth regions → second order accuracy

Unsteady version LDA scheme with the mass matrix Formulated in dual time Does not follow standard mathematical arguments (no maximum principle, etc. . . ) Works very well Extension to moving meshes → ICCFD Gent 2006 2D, 3D transonic aeroelasic computations CFL ≈ 500

Hyperbolic Problems, July 17–21, 2006 Lyon, France 26

Conclusions A nonlinear blended Bx scheme developed Blend of low order N scheme and high order LDA scheme Simple blending coefficient Based on a shock sensor Smooth → good iterative convergence properties

O(h) in smooth regions → second order accuracy

Unsteady version LDA scheme with the mass matrix Formulated in dual time Does not follow standard mathematical arguments (no maximum principle, etc. . . ) Works very well Extension to moving meshes → ICCFD Gent 2006 2D, 3D transonic aeroelasic computations CFL ≈ 500 Thank you for your attention

Hyperbolic Problems, July 17–21, 2006 Lyon, France 26-a