finite element methods for flow simulations daniel...

Finite Element Methods for Flow Simulations

Daniel Arndt

Heidelberg University

Interdisciplinary Center for Scientific Computing

The Heidelberg Laureate Forum atMATCH

21. September 2016

Introduction Discretization Numerical Results Further Topics

Overview

1 Introduction

2 Discretization

3 Numerical Results

4 Further Topics

21. September 2016 FEM for Flow Simulations 2 Daniel Arndt 2 / 21

Introduction Discretization Numerical Results Further Topics

Incompressible Navier-Stokes Equations

Momentum Equation - Inertial Frame of Reference

∂t u − ν∆u + (u · ∇)u +∇p = fu∇ · u = 0

Motion of viscuous, incompressible fluids

u velocity

p velocity

ν kinematic viscosity

Parabolic PDE of second order

Non-linear

Challenge: ν → 0

C. Fukushima and J. Westerweel,Technical University of Delft(Wikimedia Commons)

21. September 2016 FEM for Flow Simulations 3 Daniel Arndt 3 / 21

Introduction Discretization Numerical Results Further Topics

Navier-Stokes Equations - Millenium Problem1

Momentum Equation - Inertial Frame of Reference

∂t u − ν∆u + (u · ∇)u +∇p = fu (1)

∇ · u = 0 (2)

Open question: Existence of a solution satisfying

(1), (2)

u, p ∈ C∞(R3 × [0,∞))

f u ≡ 0∫Rn |u(x, t)|2 dx ≤ C ∀t ≥ 0

for a given u(x, 0) ∈ C∞(R3), ∇ · u(x, 0) = 0.

1Charles L Fefferman. “Existence and smoothness of the Navier-Stokes equation”.In: The millennium prize problems (2006), pp. 57–67

21. September 2016 FEM for Flow Simulations 4 Daniel Arndt 4 / 21

Introduction Discretization Numerical Results Further Topics

Weak Formulation

Consider a bounded domain Ω and the function spaces

V := [W 1,20 (Ω)]d Q := L2

∗(Ω)

Find (u, p) : (t0, T )→ V × Q such that

(∂t u, v) + cu(u; u, v) + ν(∇u,∇v)− (p,∇ · v) = (fu, v),

(∇ · u, q) = 0

for all (v , q) ∈ V × Q and t ∈ (t0, T ) a.e.

cu(w ; u, v) :=12

[((w · ∇)u, v)− ((w · ∇)v , u)

]Existence X Uniqueness ?

21. September 2016 FEM for Flow Simulations 5 Daniel Arndt 5 / 21

Introduction Discretization Numerical Results Further Topics

Overview

1 Introduction

2 Discretization

3 Numerical Results

4 Further Topics

21. September 2016 FEM for Flow Simulations 6 Daniel Arndt 6 / 21

Introduction Discretization Numerical Results Further Topics

Weak Formulation - Conforming Methods

Finite dimensional spaces

Vh := [W 1,20 (Ω)]d Qh ⊂ Q := L2

∗(Ω)

Find (uh, ph) : (t0, T )→ V h × Qh such that

(∂t uh, vh) + cu(uh; uh, vh) + ν(∇uh,∇vh)− (ph,∇ · vh) = (fu, vh),

(∇ · uh, qh) = 0

for all (vh, qh) ∈ Vh × Qh and t ∈ (t0, T ) a.e.

→ Nonlinear finite dimensional problemLinearize−→ Linear algebra

Existence X Uniqueness X

Problem: ∇ · Vh 6⊂ Qh =⇒ ∇ · uh 6= 0

21. September 2016 FEM for Flow Simulations 7 Daniel Arndt 7 / 21

Introduction Discretization Numerical Results Further Topics

Problems

Dominant convection Shear and boundary layers

Poor mass conservation Large Coriolis forces(non-inertial frame of reference)

21. September 2016 FEM for Flow Simulations 8 Daniel Arndt 8 / 21

Introduction Discretization Numerical Results Further Topics

Stabilization

(∂t u, v) + cu(u; u, v) + ν(∇u,∇v)− (p,∇ · v) = (fu, v),

(∇ · u, q) = 0

0 0.2 0.4 0.6 0.8 1

-1.5

-1

-0.5

0

0.5

1

1.5

=

0 0.2 0.4 0.6 0.8 1

-1.5

-1

-0.5

0

0.5

1

1.5

+

0 0.2 0.4 0.6 0.8 1

-1.5

-1

-0.5

0

0.5

1

1.5

Idea: Stabilize only small scales

Family of macro decompositions Mh

DM ⊂ [L∞(M)]d finite elemente ansatz space on M ∈Mh.

πM : [L2(M)]d → DM orthogonal L2 projection,κM = Id − πM fluctuation operator

(∇ · uh, q) −→ τu,gd,M(∇ · uh,∇ · vh)M

cu(uh; uh, vh) −→ τu,M(κM((uM · ∇)uh), κM(uM · ∇vh))M

21. September 2016 FEM for Flow Simulations 9 Daniel Arndt 9 / 21

Introduction Discretization Numerical Results Further Topics

Stabilization

(∂t u, u) + cu(u; u, u) + ν(∇u,∇u)− (p,∇ · u) = (fu, u),

(∇ · u, p) = 0

0 0.2 0.4 0.6 0.8 1

-1.5

-1

-0.5

0

0.5

1

1.5

=

0 0.2 0.4 0.6 0.8 1

-1.5

-1

-0.5

0

0.5

1

1.5

+

0 0.2 0.4 0.6 0.8 1

-1.5

-1

-0.5

0

0.5

1

1.5

Idea: Stabilize only small scales

Family of macro decompositions Mh

DM ⊂ [L∞(M)]d finite elemente ansatz space on M ∈Mh.

πM : [L2(M)]d → DM orthogonal L2 projection,κM = Id − πM fluctuation operator

(∇ · uh, q) −→ τu,gd,M(∇ · uh,∇ · vh)M

cu(uh; uh, vh) −→ τu,M(κM((uM · ∇)uh), κM(uM · ∇vh))M

21. September 2016 FEM for Flow Simulations 9 Daniel Arndt 9 / 21

Introduction Discretization Numerical Results Further Topics

Stabilization

(∂t u, u) +cu(u; u, u) + ν(∇u,∇u)−(p,∇ · u) = (fu, u),

(∇ · u, p) = 0

0 0.2 0.4 0.6 0.8 1

-1.5

-1

-0.5

0

0.5

1

1.5

=

0 0.2 0.4 0.6 0.8 1

-1.5

-1

-0.5

0

0.5

1

1.5

+

0 0.2 0.4 0.6 0.8 1

-1.5

-1

-0.5

0

0.5

1

1.5

Idea: Stabilize only small scales

Family of macro decompositions Mh

DM ⊂ [L∞(M)]d finite elemente ansatz space on M ∈Mh.

πM : [L2(M)]d → DM orthogonal L2 projection,κM = Id − πM fluctuation operator

(∇ · uh, q) −→ τu,gd,M(∇ · uh,∇ · vh)M

cu(uh; uh, vh) −→ τu,M(κM((uM · ∇)uh), κM(uM · ∇vh))M

21. September 2016 FEM for Flow Simulations 9 Daniel Arndt 9 / 21

Introduction Discretization Numerical Results Further Topics

Stabilization

(∂t u, u) +cu(u; u, u) + ν(∇u,∇u)−(p,∇ · u) = (fu, u),

(∇ · u, p) = 0

0 0.2 0.4 0.6 0.8 1

-1.5

-1

-0.5

0

0.5

1

1.5

=

0 0.2 0.4 0.6 0.8 1

-1.5

-1

-0.5

0

0.5

1

1.5

+

0 0.2 0.4 0.6 0.8 1

-1.5

-1

-0.5

0

0.5

1

1.5

Idea: Stabilize only small scales

Family of macro decompositions Mh

DM ⊂ [L∞(M)]d finite elemente ansatz space on M ∈Mh.

πM : [L2(M)]d → DM orthogonal L2 projection,κM = Id − πM fluctuation operator

(∇ · uh, q) −→ τu,gd,M(∇ · uh,∇ · vh)M

cu(uh; uh, vh) −→ τu,M(κM((uM · ∇)uh), κM(uM · ∇vh))M

21. September 2016 FEM for Flow Simulations 9 Daniel Arndt 9 / 21

Introduction Discretization Numerical Results Further Topics

Stabilization

(∂t u, u) +cu(u; u, u) + ν(∇u,∇u)−(p,∇ · u) = (fu, u),

(∇ · u, p) = 0

0 0.2 0.4 0.6 0.8 1

-1.5

-1

-0.5

0

0.5

1

1.5

=

0 0.2 0.4 0.6 0.8 1

-1.5

-1

-0.5

0

0.5

1

1.5

+

0 0.2 0.4 0.6 0.8 1

-1.5

-1

-0.5

0

0.5

1

1.5

Idea: Stabilize only small scales

Family of macro decompositions Mh

DM ⊂ [L∞(M)]d finite elemente ansatz space on M ∈Mh.

πM : [L2(M)]d → DM orthogonal L2 projection,κM = Id − πM fluctuation operator

(∇ · uh, q) −→ τu,gd,M(∇ · uh,∇ · vh)M

cu(uh; uh, vh) −→ τu,M(κM((uM · ∇)uh), κM(uM · ∇vh))M

21. September 2016 FEM for Flow Simulations 9 Daniel Arndt 9 / 21

Introduction Discretization Numerical Results Further Topics

Analytical Results

U := (u, p), Uh := (uh, ph)

|||U |||2Stab := ν‖∇u‖20 +

∑M∈Mh

(γM‖∇ · u‖2

0,M + τM‖κM((uM · ∇)u)‖20,M

)Aims:

suitable choice of stabilization parameters

quasi-optimal error estimates

‖U − Uh‖2l∞(t0,T ;L2(Ω)) + |||U − Uh|||2l2(t0,T ;Stab)

≤ CeCG(T−t0)

(infWh

‖U −Wh‖2

l∞(t0,T ;L2(Ω)) + |||U −Wh|||2l2(t0,T ;Stab)

)semi-robust error estimates C(ν), CG(ν)

21. September 2016 FEM for Flow Simulations 10 Daniel Arndt 10 / 21

Introduction Discretization Numerical Results Further Topics

Overview

1 Introduction

2 Discretization

3 Numerical Results

4 Further Topics

21. September 2016 FEM for Flow Simulations 11 Daniel Arndt 11 / 21

Introduction Discretization Numerical Results Further Topics

Analytical Test Case - Setting

u(x, t) =

(− cos

(π2 x)· sin(π2 y) · sin(π · t)

sin(π2 x)· cos(π2 y) · sin(π · t)

)p(x, t) = −π · sin

(π2

x)· sin

(π2

y)· sin(π · t)

‖u‖k+1

‖p‖k= 1 Ω = [−1, 1]2

T = 4 · 10−1 ∆t = 4 · 10−5

-1.0 -0.5 0.0 0.5 1.0

-1.0

-0.5

0.0

0.5

1.0

21. September 2016 FEM for Flow Simulations 12 Daniel Arndt 12 / 21

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Analytical Test Case - Parameter Choice - ν = 10−4

10−4

10−2

100

102

104

10−4

10−3

10−2

10−1

100

101

102

Q2Q1

L2 velocity

H1 velocity

Linfty velocity

L2 div velocity

L2 pressure

H1 pressure

Linfty pressure

21. September 2016 FEM for Flow Simulations 13 Daniel Arndt 13 / 21

Introduction Discretization Numerical Results Further Topics

Analytical Test Case - Parameter Choice

h

5 ·10-2

10-1

2 ·10-1 5 ·10

-1

H1(u

) e

rro

r

10-6

10-4

10-2

100

102

104

h5 ·10

-210

-12 ·10

-1 5 ·10-1

L2(p

) e

rro

r

10-4

10-3

10-2

10-1

γ = 1 ν = 1

γ = 0 ν = 1

γ = 1 ν = 10−3

γ = 0 ν = 10−3

γ = 1 ν = 10−6

γ = 0 ν = 10−6

h2

21. September 2016 FEM for Flow Simulations 14 Daniel Arndt 14 / 21

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Blasius Flow, Flow over Horizontal Plate

Prandtl’s Boundary Layer Equation

u = u∞f ′(η)

2f ′′′(η) + f (η)f ′′(η) = 0

f (0) = f ′(0) = 0, f ′(∞) = 1

where η = y√

u∞/(νx) is a non-dimensional variable.

21. September 2016 FEM for Flow Simulations 15 Daniel Arndt 15 / 21

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Blasius Flow

η=y (u∞

/( ν x))1/2

0 2 4 6 8

u/u

∞

0

0.2

0.4

0.6

0.8

1

1.2

Blasius profilex=0.1x=0.5x=0.9

ν = 10−3, γ = 1, τM = 0, h = 2−5

21. September 2016 FEM for Flow Simulations 16 Daniel Arndt 16 / 21

Introduction Discretization Numerical Results Further Topics

Blasius Flow - ν = 10−3

Adaptively refined mesh, τM = 0 Globally refined mesh, τM = 1

21. September 2016 FEM for Flow Simulations 17 Daniel Arndt 17 / 21

Introduction Discretization Numerical Results Further Topics

Overview

1 Introduction

2 Discretization

3 Numerical Results

4 Further Topics

21. September 2016 FEM for Flow Simulations 18 Daniel Arndt 18 / 21

Introduction Discretization Numerical Results Further Topics

Further Topics

Not covered today

Time discretization

Linear algebra

Fast solverMultigridMatrix-Free methods

Implementation⇒ deal.II

Turbulence modeling

MHD, nonisothermal flow, . . .

k

100

101

102

E

10-4

10-3

10-2

10-1

100

τM = h/(2‖uh‖∞,M )τM = 1/(2‖uh‖∞,M )τM = 1

k−5/3

21. September 2016 FEM for Flow Simulations 19 Daniel Arndt 19 / 21

Thanks for your attention!

Rayleigh-Bénard Convection

Ra = 105 Ra = 107 Ra = 109

T = 1000, Pr = 0.786, N = 10 · 163, γM = 0.1