Isogeometric Analysis and Shape Optimization in Fluid Mechanics - - PowerPoint PPT Presentation

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Isogeometric Analysis and Shape Optimization in Fluid Mechanics - - PowerPoint PPT Presentation

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Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Isogeometric Analysis and Shape Optimization in Fluid Mechanics Peter Nrtoft DTU Compute Joint work with Jens Gravesen, Allan R. Gersborg, Niels L. Pedersen,

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SLIDE 1

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions

Isogeometric Analysis and Shape Optimization in Fluid Mechanics

Peter Nørtoft DTU Compute

Joint work with Jens Gravesen, Allan R. Gersborg, Niels L. Pedersen, Morten Willatzen, Anton Evgrafov, Dang Manh Nguyen, and Tor Dokken

Scientific Computing Section Seminar, September 17, 2013

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SLIDE 2

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Goals and Outline

The aim is to analyze and optimize flows using isogeometric analysis

Shape Optimization drag

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SLIDE 3

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Goals and Outline

The aim is to analyze and optimize flows using isogeometric analysis

Shape Optimization drag Navier-Stokes Flow Model

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SLIDE 4

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Goals and Outline

The aim is to analyze and optimize flows using isogeometric analysis

Shape Optimization drag Navier-Stokes Flow Model Flow Acoustics Model

+

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SLIDE 5

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Goals and Outline

The aim is to analyze and optimize flows using isogeometric analysis

Shape Optimization drag Navier-Stokes Flow Model Isogeometric Analysis

  • ✒

Flow Acoustics Model

+

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SLIDE 6

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Fluid Mechanics: Navier-Stokes Equations

Flow problems are governed by a boundary value problem

ΓN ΓD Ω u velocity p pressure ρ density µ viscosity f force

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SLIDE 7

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Fluid Mechanics: Navier-Stokes Equations

Flow problems are governed by a boundary value problem

ΓN ΓD Ω u velocity p pressure ρ density µ viscosity f force

2D steady-state incompressible Navier-Stokes flow ρ(u · ∇)u + ∇p − µ∇2u = ρf in Ω ∇ · u = 0 in Ω u = u∗

  • n ΓD

( µ∇ui − p ei ) · n = 0

  • n ΓN
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SLIDE 8

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Fluid Mechanics: Navier-Stokes Equations

Flow problems are governed by a boundary value problem

ΓN ΓD Ω u velocity p pressure ρ density µ viscosity f force

[M. Van Dyke]

Re = ρ UL

µ

103

◮ viscous fluid ◮ slow flow ◮ small scale

2D steady-state incompressible Navier-Stokes flow ρ(u · ∇)u + ∇p − µ∇2u = ρf in Ω ∇ · u = 0 in Ω u = u∗

  • n ΓD

( µ∇ui − p ei ) · n = 0

  • n ΓN
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SLIDE 9

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Fluid Mechanics: Navier-Stokes Equations

Flow problems are governed by a boundary value problem

ΓN ΓD Ω u velocity p pressure ρ density µ viscosity f force

[M. Van Dyke]

Re = ρ UL

µ

103

◮ viscous fluid ◮ slow flow ◮ small scale

2D steady-state incompressible Navier-Stokes flow ρ(u · ∇)u + ∇p − µ∇2u = ρf in Ω ∇ · u = 0 in Ω u = u∗

  • n ΓD

( µ∇ui − p ei ) · n = 0

  • n ΓN

Challenge: solve this using isogeometric analysis

[Bazilevs et al., 2006b; Bazilevs & Hughes, 2008; Akkerman et al., 2010; . . .]

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Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Numerical Method

Both geometry, velocity and pressure are parametrized by B-splines Ω

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SLIDE 11

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Numerical Method

Both geometry, velocity and pressure are parametrized by B-splines

X(ξ, η) =

Ng

  • i=1

¯ xiRg

i (ξ, η)

Ω X [0, 1]2

y x η ξ

Rg

i

Bivariate NURBS ¯ xi control point

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SLIDE 12

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Numerical Method

Both geometry, velocity and pressure are parametrized by B-splines

X(ξ, η) =

Ng

  • i=1

¯ xiRg

i (ξ, η)

u(ξ, η) =

Nu

  • i=1

¯ uiPu

i (ξ, η)

p(ξ, η) =

Np

  • i=1

¯ piPp

i (ξ, η)

Ω X [0, 1]2

y x η ξ

Ru

i , Rp i , Rg i

Bivariate NURBS/B-spline ¯ ui, ¯ pi, ¯ xi control point/variable

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Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Numerical Method

Both geometry, velocity and pressure are parametrized by B-splines

X(ξ, η) =

Ng

  • i=1

¯ xiRg

i (ξ, η)

u(ξ, η) =

Nu

  • i=1

¯ uiPu

i (ξ, η)

p(ξ, η) =

Np

  • i=1

¯ piPp

i (ξ, η)

Ω X [0, 1]2

y x η ξ

Ru

i , Rp i , Rg i

Bivariate NURBS/B-spline ¯ ui, ¯ pi, ¯ xi control point/variable

Univariate B-spline:

  • knot vector
  • polynomial degree

Parameter domain [0, 1]2

0.5 1 0.2 0.4 0.6 0.8 1 ξ1

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SLIDE 14

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Numerical Method

Both geometry, velocity and pressure are parametrized by B-splines

X(ξ, η) =

Ng

  • i=1

¯ xiRg

i (ξ, η)

u(ξ, η) =

Nu

  • i=1

¯ uiPu

i (ξ, η)

p(ξ, η) =

Np

  • i=1

¯ piPp

i (ξ, η)

Ω X [0, 1]2

y x η ξ

Ru

i , Rp i , Rg i

Bivariate NURBS/B-spline ¯ ui, ¯ pi, ¯ xi control point/variable

Univariate B-spline: Ni(ξ1)

  • knot vector
  • polynomial degree

Parameter domain [0, 1]2

0.5 1 0.2 0.4 0.6 0.8 1 ξ1

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SLIDE 15

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Numerical Method

Both geometry, velocity and pressure are parametrized by B-splines

X(ξ, η) =

Ng

  • i=1

¯ xiRg

i (ξ, η)

u(ξ, η) =

Nu

  • i=1

¯ uiPu

i (ξ, η)

p(ξ, η) =

Np

  • i=1

¯ piPp

i (ξ, η)

Ω X [0, 1]2

y x η ξ

Ru

i , Rp i , Rg i

Bivariate NURBS/B-spline ¯ ui, ¯ pi, ¯ xi control point/variable

Univariate B-spline: Ni(ξ1)

  • knot vector
  • polynomial degree

Parameter domain [0, 1]2

0.5 1 0.2 0.4 0.6 0.8 1 ξ1

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SLIDE 16

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Numerical Method

Both geometry, velocity and pressure are parametrized by B-splines

X(ξ, η) =

Ng

  • i=1

¯ xiRg

i (ξ, η)

u(ξ, η) =

Nu

  • i=1

¯ uiPu

i (ξ, η)

p(ξ, η) =

Np

  • i=1

¯ piPp

i (ξ, η)

Ω X [0, 1]2

y x η ξ

Ru

i , Rp i , Rg i

Bivariate NURBS/B-spline ¯ ui, ¯ pi, ¯ xi control point/variable

Univariate B-spline: Ni(ξ1)

  • knot vector
  • polynomial degree

Parameter domain [0, 1]2

0.5 1 0.5 1 0.5 1 ξ1 ξ2

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SLIDE 17

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Numerical Method

Both geometry, velocity and pressure are parametrized by B-splines

X(ξ, η) =

Ng

  • i=1

¯ xiRg

i (ξ, η)

u(ξ, η) =

Nu

  • i=1

¯ uiPu

i (ξ, η)

p(ξ, η) =

Np

  • i=1

¯ piPp

i (ξ, η)

Ω X [0, 1]2

y x η ξ

Ru

i , Rp i , Rg i

Bivariate NURBS/B-spline ¯ ui, ¯ pi, ¯ xi control point/variable

Univariate B-spline: Ni(ξ1), Mj(ξ2)

  • knot vector
  • polynomial degree

Parameter domain [0, 1]2

0.5 1 0.5 1 0.5 1 ξ1 ξ2

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SLIDE 18

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Numerical Method

Both geometry, velocity and pressure are parametrized by B-splines

X(ξ, η) =

Ng

  • i=1

¯ xiRg

i (ξ, η)

u(ξ, η) =

Nu

  • i=1

¯ uiPu

i (ξ, η)

p(ξ, η) =

Np

  • i=1

¯ piPp

i (ξ, η)

Ω X [0, 1]2

y x η ξ

Ru

i , Rp i , Rg i

Bivariate NURBS/B-spline ¯ ui, ¯ pi, ¯ xi control point/variable

Univariate B-spline: Ni(ξ1), Mj(ξ2)

  • knot vector
  • polynomial degree

Parameter domain [0, 1]2

0.5 1 0.5 1 0.5 1 ξ1 ξ2

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SLIDE 19

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Numerical Method

Both geometry, velocity and pressure are parametrized by B-splines

X(ξ, η) =

Ng

  • i=1

¯ xiRg

i (ξ, η)

u(ξ, η) =

Nu

  • i=1

¯ uiPu

i (ξ, η)

p(ξ, η) =

Np

  • i=1

¯ piPp

i (ξ, η)

Ω X [0, 1]2

y x η ξ

Ru

i , Rp i , Rg i

Bivariate NURBS/B-spline ¯ ui, ¯ pi, ¯ xi control point/variable

Univariate B-spline: Ni(ξ1), Mj(ξ2)

  • knot vector
  • polynomial degree

Parameter domain [0, 1]2 Bivariate Tensor Product B-spline:

  • 2 knot vectors
  • 2 polynomial degrees
  • Pi,j(ξ1, ξ2) = Ni(ξ1)Mj(ξ2)
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SLIDE 20

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Numerical Method

Both geometry, velocity and pressure are parametrized by B-splines

X(ξ, η) =

Ng

  • i=1

¯ xiRg

i (ξ, η)

u(ξ, η) =

Nu

  • i=1

¯ uiPu

i (ξ, η)

p(ξ, η) =

Np

  • i=1

¯ piPp

i (ξ, η)

Ω X [0, 1]2

y x η ξ

Ru

i , Rp i , Rg i

Bivariate NURBS/B-spline ¯ ui, ¯ pi, ¯ xi control point/variable

Univariate B-spline: Ni(ξ1), Mj(ξ2)

  • knot vector
  • polynomial degree

Parameter domain [0, 1]2 Bivariate Tensor Product B-spline:

  • 2 knot vectors
  • 2 polynomial degrees
  • Pi,j(ξ1, ξ2) = Ni(ξ1)Mj(ξ2)

Physical domain Ω

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SLIDE 21

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Numerical Method

Both geometry, velocity and pressure are parametrized by B-splines

X(ξ, η) =

Ng

  • i=1

¯ xiRg

i (ξ, η)

u(ξ, η) =

Nu

  • i=1

¯ uiPu

i (ξ, η)

p(ξ, η) =

Np

  • i=1

¯ piPp

i (ξ, η)

Ω X [0, 1]2

y x η ξ

Ru

i , Rp i , Rg i

Bivariate NURBS/B-spline ¯ ui, ¯ pi, ¯ xi control point/variable

Univariate B-spline: Ni(ξ1), Mj(ξ2)

  • knot vector
  • polynomial degree

Parameter domain [0, 1]2 Bivariate Tensor Product B-spline:

  • 2 knot vectors
  • 2 polynomial degrees
  • Pi,j(ξ1, ξ2) = Ni(ξ1)Mj(ξ2)

Physical domain Ω

M(U)U = F

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SLIDE 22

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Error Convergence

Test of error convergence: flow with analytical solution

f1 = f1(x,y) f2 = f2(x,y) u|Γ =

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Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Error Convergence

Test of error convergence: flow with analytical solution

−3 −2 −1 1 2 3 −1.5 −1 −0.5 0.5 1 1.5 x y −3 −2 −1 1 2 3 −1.5 −1 −0.5 0.5 1 1.5 x y

f1 = f1(x,y) f2 = f2(x,y) u|Γ = u⋆

1

= −U sin(π˜ r2)y u⋆

2

= U/4 sin(π˜ r2)x p⋆ = 4/π2 + cos(π˜ r) ˜ r =

√

(x/2)2+ y2

u⋆ p⋆

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SLIDE 24

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Error Convergence

Test of error convergence: flow with analytical solution

−3 −2 −1 1 2 3 −1.5 −1 −0.5 0.5 1 1.5 x y −3 −2 −1 1 2 3 −1.5 −1 −0.5 0.5 1 1.5 x y

f1 = f1(x,y) f2 = f2(x,y) u|Γ = u⋆

1

= −U sin(π˜ r2)y u⋆

2

= U/4 sin(π˜ r2)x p⋆ = 4/π2 + cos(π˜ r) ˜ r =

√

(x/2)2+ y2

u⋆ p⋆

Re = 200

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Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Error Convergence

Test of error convergence: flow with analytical solution

−3 −2 −1 1 2 3 −1.5 −1 −0.5 0.5 1 1.5 x y −3 −2 −1 1 2 3 −1.5 −1 −0.5 0.5 1 1.5 x y

f1 = f1(x,y) f2 = f2(x,y) u|Γ = u⋆

1

= −U sin(π˜ r2)y u⋆

2

= U/4 sin(π˜ r2)x p⋆ = 4/π2 + cos(π˜ r) ˜ r =

√

(x/2)2+ y2

u⋆ p⋆

Re = 200 ǫ2

u

=

  • Ω

u(x,y)−u⋆(x,y)2 dx dy

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SLIDE 26

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Error Convergence

Discretizations with higher regularity perform better

10

−5

10 εu

DOF 102 103 104 105

u41

141 1 v41 141 1 p40 140 1

u40

240 2 v40 240 2 p40 140 1

u41

141 1 v41 141 1 p30 130 1

u40

240 2 v40 240 2 p30 130 1

u41

141 1 v41 141 1 p20 120 1

u40

240 2 v40 240 2 p20 120 1

u40

140 1 v40 140 1 p20 120 1

u41

141 2 v41 241 1 p30 130 1

u41

131 1 v31 141 1 p30 130 1

a b c d e f g h i

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SLIDE 27

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Error Convergence

Discretizations with higher regularity perform better

10

−5

10 εu

DOF 102 103 104 105 1 2

u41

141 1 v41 141 1 p40 140 1

u40

240 2 v40 240 2 p40 140 1

u41

141 1 v41 141 1 p30 130 1

u40

240 2 v40 240 2 p30 130 1

u41

141 1 v41 141 1 p20 120 1

u40

240 2 v40 240 2 p20 120 1

u40

140 1 v40 140 1 p20 120 1

u41

141 2 v41 241 1 p30 130 1

u41

131 1 v31 141 1 p30 130 1

a b c d e f g h i 1 2 1 2 1 2 1 2

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SLIDE 28

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Error Convergence

Discretizations with higher regularity perform better

10

−5

10 εu

DOF 102 103 104 105 1 2

Smoothness Mesh Density Strategy

1

High High Strategy

2

Low Low

u41

141 1 v41 141 1 p40 140 1

u40

240 2 v40 240 2 p40 140 1

u41

141 1 v41 141 1 p30 130 1

u40

240 2 v40 240 2 p30 130 1

u41

141 1 v41 141 1 p20 120 1

u40

240 2 v40 240 2 p20 120 1

u40

140 1 v40 140 1 p20 120 1

u41

141 2 v41 241 1 p30 130 1

u41

131 1 v31 141 1 p30 130 1

a b c d e f g h i 1 2 1 2 1 2 1 2

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SLIDE 29

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Design of Minimal Drag Body

We design a body with minimal drag

[Pironneau, 1973; 1974; Mohammadi & Pironneau, 2010]

D γ A0 U

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SLIDE 30

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Design of Minimal Drag Body

We design a body with minimal drag

Aim

Design boundary γ of body with area A0 travelling at constant speed U to minimize the drag D

[Pironneau, 1973; 1974; Mohammadi & Pironneau, 2010]

D γ A0 U

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SLIDE 31

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Design of Minimal Drag Body

We design a body with minimal drag

Aim

Design boundary γ of body with area A0 travelling at constant speed U to minimize the drag D

Optimization Problem

min

γ(¯ xb)

c = D + ǫR

  • s. t.

Area ≥ A0 L−(¯ xb) ≤ L(¯ xb) ≤ L+(¯ xb) MU = F

drag

  • bjective

area constraint linear design constraints governing equations D =

  • γ
  • − pI + µ
  • ∇u + (∇u)T

n ds · eu [Pironneau, 1973; 1974; Mohammadi & Pironneau, 2010]

D γ A0 U

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SLIDE 32

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Design of Minimal Drag Body

We design a body with minimal drag

Aim

Design boundary γ of body with area A0 travelling at constant speed U to minimize the drag D

Optimization Problem

min

γ(¯ xb)

c = D + ǫR

  • s. t.

Area ≥ A0 L−(¯ xb) ≤ L(¯ xb) ≤ L+(¯ xb) MU = F

drag

  • bjective

area constraint linear design constraints governing equations D =

  • γ
  • − pI + µ
  • ∇u + (∇u)T

n ds · eu [Pironneau, 1973; 1974; Mohammadi & Pironneau, 2010]

D γ A0 U

γ

u = 0

✲ ✛

r

slide-33
SLIDE 33

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Design of Minimal Drag Body

We design a body with minimal drag

Aim

Design boundary γ of body with area A0 travelling at constant speed U to minimize the drag D

Optimization Problem

min

γ(¯ xb)

c = D + ǫR

  • s. t.

Area ≥ A0 L−(¯ xb) ≤ L(¯ xb) ≤ L+(¯ xb) MU = F

drag

  • bjective

area constraint linear design constraints governing equations D =

  • γ
  • − pI + µ
  • ∇u + (∇u)T

n ds · eu [Pironneau, 1973; 1974; Mohammadi & Pironneau, 2010]

D γ A0 U

γ

u = 0

✲ ✛

r u = Ue1 v = ∂u/∂n = 0 ΓN 20r 20r 40r

✻ ❄ ✲ ✛ ✲ ✛

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SLIDE 34

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Design of Minimal Drag Body

The optimal shape is longer and more slender for higher speeds Flow before optimization (U=1)

slide-35
SLIDE 35

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Design of Minimal Drag Body

The optimal shape is longer and more slender for higher speeds Optimal shapes for different speeds Initial U = 1 U = 10 U = 40 U = 100 Flow before optimization (U=1)

slide-36
SLIDE 36

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Design of Minimal Drag Body

The optimal shape is longer and more slender for higher speeds Optimal shapes for different speeds Initial U = 1 U = 10 U = 40 U = 100 Flow before optimization (U=1)

slide-37
SLIDE 37

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Design of Minimal Drag Body

The optimal shape is longer and more slender for higher speeds Optimal shapes for different speeds Initial U = 1 U = 10 U = 40 U = 100 Flow before optimization (U=1)

slide-38
SLIDE 38

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Design of Minimal Drag Body

The optimal shape is longer and more slender for higher speeds Optimal shapes for different speeds Initial U = 1 U = 10 U = 40 U = 100 Flow before optimization (U=1)

slide-39
SLIDE 39

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Design of Minimal Drag Body

The optimal shape is longer and more slender for higher speeds Optimal shapes for different speeds Initial U = 1 U = 10 U = 40 U = 100 Flow before optimization (U=1)

slide-40
SLIDE 40

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Design of Minimal Drag Body

The optimal shape is longer and more slender for higher speeds Optimal shapes for different speeds Initial U = 1 U = 10 U = 40 U = 100 Flow before optimization (U=1) Flow after optimization (U=100) Note: low Reynolds numbers

slide-41
SLIDE 41

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Introduction and Governing Equations

We model geometric effects on sound propagation through flow in 2D ducts

Pipe Geometry

slide-42
SLIDE 42

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Introduction and Governing Equations

We model geometric effects on sound propagation through flow in 2D ducts

Pipe Geometry Flow

slide-43
SLIDE 43

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Introduction and Governing Equations

We model geometric effects on sound propagation through flow in 2D ducts

Pipe Geometry Flow Acoustics

slide-44
SLIDE 44

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Introduction and Governing Equations

We model geometric effects on sound propagation through flow in 2D ducts

Pipe Geometry Flow Acoustics p pressure u velocity ρ density

slide-45
SLIDE 45

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Introduction and Governing Equations

We model geometric effects on sound propagation through flow in 2D ducts

Pipe Geometry Flow Acoustics p pressure u velocity ρ density ρ∂u ∂t + ρ(u · ∇)u + ∇p − µ∇2u = 0 ∂ρ ∂t + ∇ · (ρu) = 0

slide-46
SLIDE 46

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Introduction and Governing Equations

We model geometric effects on sound propagation through flow in 2D ducts

Pipe Geometry Flow Acoustics p pressure u velocity ρ density ρ∂u ∂t + ρ(u · ∇)u + ∇p − µ∇2u = 0 ∂ρ ∂t + ∇ · (ρu) = 0 p = p0 + p′ u = u0 + u′ ρ = ρ0 + ρ′ Background Flow: p0, u0, ρ0 Acoustic Disturbance: p′, u′, ρ′

slide-47
SLIDE 47

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Introduction and Governing Equations

We model geometric effects on sound propagation through flow in 2D ducts

Pipe Geometry Flow Acoustics p pressure u velocity ρ density ρ∂u ∂t + ρ(u · ∇)u + ∇p − µ∇2u = 0 ∂ρ ∂t + ∇ · (ρu) = 0 p = p0 + p′ u = u0 + u′ ρ = ρ0 + ρ′ Background Flow: p0, u0, ρ0 Acoustic Disturbance: p′, u′, ρ′

1 2

⇓

slide-48
SLIDE 48

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Introduction and Governing Equations

We model geometric effects on sound propagation through flow in 2D ducts

Pipe Geometry Flow Acoustics p pressure u velocity ρ density ρ∂u ∂t + ρ(u · ∇)u + ∇p − µ∇2u = 0 ∂ρ ∂t + ∇ · (ρu) = 0 p = p0 + p′ u = u0 + u′ ρ = ρ0 + ρ′ Background Flow: p0, u0, ρ0 Acoustic Disturbance: p′, u′, ρ′

1 2

⇓

γ− γ+ Γw Γs Ω

slide-49
SLIDE 49

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Results: Flow-Acoustic Coupling

We examine how the flow affects the sound in different ducts ✲ ✲ ✲ ✲ ✲ Pipe A ✲ ✲ ✲ ✲ ✲ Pipe B ✲ ✲ ✲ ✲ ✲ Pipe C Fluid Radius Speed Frequency Air 1 cm 1 m/s ∼25 kHz

slide-50
SLIDE 50

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Results: Flow-Acoustic Coupling

We examine how the flow affects the sound in different ducts ✲ ✲ ✲ ✲ ✲ Pipe A

|u| Flow

✲ ✲ ✲ ✲ ✲ Pipe B

|u| Flow

✲ ✲ ✲ ✲ ✲ Pipe C

|u| Flow

Fluid Radius Speed Frequency Air 1 cm 1 m/s ∼25 kHz

slide-51
SLIDE 51

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Results: Flow-Acoustic Coupling

We examine how the flow affects the sound in different ducts ✲ ✲ ✲ ✲ ✲ Pipe A

|u| Flow |˜ p| Sound

✲ ✲ ✲ ✲ ✲ Pipe B

|u| Flow |˜ p| Sound

✲ ✲ ✲ ✲ ✲ Pipe C

|u| Flow |˜ p| Sound

Fluid Radius Speed Frequency Air 1 cm 1 m/s ∼25 kHz

slide-52
SLIDE 52

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Results: Flow-Acoustic Coupling

The corrugated duct exhibits stronger flow-acoustic coupling Acoustic Pressure ˜ p [Pa] A B C Flow

Upstream Downstream

⇒

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SLIDE 53

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Results: Flow-Acoustic Coupling

The corrugated duct exhibits stronger flow-acoustic coupling Acoustic Pressure ˜ p [Pa] A B C Flow

Upstream Downstream

⇒

Measure of Flow-Acoustic Coupling

δ˜ p =

  • Ω |˜

p(x) − ˜ p(−x)| dA

  • Ω |˜

p(x)| dA

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SLIDE 54

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Results: Flow-Acoustic Coupling

The corrugated duct exhibits stronger flow-acoustic coupling Acoustic Pressure ˜ p [Pa] A B C Flow

Upstream Downstream

⇒

Measure of Flow-Acoustic Coupling

δ˜ p =

  • Ω |˜

p(x) − ˜ p(−x)| dA

  • Ω |˜

p(x)| dA Sound Frequency Sensitivity

20 0.2 0.4 0.6 0.8 Pipe A Pipe B Pipe C

  • δ˜

p f [kHz]

25 20 30

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SLIDE 55

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Results: Flow-Acoustic Coupling

The corrugated duct exhibits stronger flow-acoustic coupling Acoustic Pressure ˜ p [Pa] A B C Flow

Upstream Downstream

⇒

Measure of Flow-Acoustic Coupling

δ˜ p =

  • Ω |˜

p(x) − ˜ p(−x)| dA

  • Ω |˜

p(x)| dA Sound Frequency Sensitivity

20 0.2 0.4 0.6 0.8 Pipe A Pipe B Pipe C

  • δ˜

p f [kHz]

25 20 30

Flow Speed Sensitivity

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Pipe A Pipe B Pipe C

δ˜ p U0 [m s−1]

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SLIDE 56

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Outlook

Ongoing work

Flow Discretizations using Locally Refinable B-splines

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SLIDE 57

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Outlook

Ongoing work

Flow Discretizations using Locally Refinable B-splines

Region of Interest Computational Domain

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SLIDE 58

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Outlook

Ongoing work

Flow Discretizations using Locally Refinable B-splines

Region of Interest Global Refinement

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SLIDE 59

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Outlook

Ongoing work

Flow Discretizations using Locally Refinable B-splines

Region of Interest Local Refinement

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SLIDE 60

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Outlook

Ongoing work

Flow Discretizations using Locally Refinable B-splines

Region of Interest Local Refinement 1

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SLIDE 61

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Outlook

Ongoing work

Flow Discretizations using Locally Refinable B-splines

Region of Interest Local Refinement 1

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SLIDE 62

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Outlook

Ongoing work

Flow Discretizations using Locally Refinable B-splines

Region of Interest Local Refinement 1

slide-63
SLIDE 63

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Outlook

Ongoing work

Flow Discretizations using Locally Refinable B-splines

Region of Interest Local Refinement 1

slide-64
SLIDE 64

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Outlook

Ongoing work

Flow Discretizations using Locally Refinable B-splines

Region of Interest Local Refinement 1

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SLIDE 65

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Outlook

Ongoing work

Flow Discretizations using Locally Refinable B-splines

Region of Interest Local Refinement 1

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SLIDE 66

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Outlook

Ongoing work

Flow Discretizations using Locally Refinable B-splines

Region of Interest Local Refinement 1 2

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SLIDE 67

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Outlook

Ongoing work

Flow Discretizations using Locally Refinable B-splines

Region of Interest Local Refinement 1 2

slide-68
SLIDE 68

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Outlook

Ongoing work

Flow Discretizations using Locally Refinable B-splines

Region of Interest Local Refinement 1 2

slide-69
SLIDE 69

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Outlook

Ongoing work

Flow Discretizations using Locally Refinable B-splines

Region of Interest Local Refinement 1 2

slide-70
SLIDE 70

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Outlook

Ongoing work

Flow Discretizations using Locally Refinable B-splines

Region of Interest Local Refinement 1 2

slide-71
SLIDE 71

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Outlook

Ongoing work

Flow Discretizations using Locally Refinable B-splines

Region of Interest Local Refinement 1 2

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SLIDE 72

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Outlook

Ongoing work

Flow Discretizations using Locally Refinable B-splines

Region of Interest Local Refinement 1 2

Optimization of Parametrizations for Isogeometric Analysis L(U) = 0 Ω Γ

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SLIDE 73

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Outlook

Ongoing work

Flow Discretizations using Locally Refinable B-splines

Region of Interest Local Refinement 1 2

Optimization of Parametrizations for Isogeometric Analysis L(U) = 0 Ω Γ

?

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SLIDE 74

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Outlook

Ongoing work

Flow Discretizations using Locally Refinable B-splines

Region of Interest Local Refinement 1 2

Optimization of Parametrizations for Isogeometric Analysis L(U) = 0 Ω Γ

?

slide-75
SLIDE 75

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Outlook

Ongoing work

Flow Discretizations using Locally Refinable B-splines

Region of Interest Local Refinement 1 2

Optimization of Parametrizations for Isogeometric Analysis L(U) = 0 Ω Γ

? ?

slide-76
SLIDE 76

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Outlook

Ongoing work

Flow Discretizations using Locally Refinable B-splines

Region of Interest Local Refinement 1 2

Optimization of Parametrizations for Isogeometric Analysis L(U) = 0 Ω Γ

? ?

minimize c(Ω, U) such that det(J) > 0 where ∂Ω = Γ L(U) = 0

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SLIDE 77

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Summary

Summary

Isogeometric Method Isogeometric Analysis of Flows Isogeometric Shape Optimization of Flows Isogeometric Analysis of Flow Acoustics

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SLIDE 78

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Summary

Summary

Isogeometric Method IGA = FEM (high regularity) + CAD (exact geometry) Isogeometric Analysis of Flows Isogeometric Shape Optimization of Flows Isogeometric Analysis of Flow Acoustics

  • ✒
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SLIDE 79

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Summary

Summary

Isogeometric Method IGA = FEM (high regularity) + CAD (exact geometry) Isogeometric Analysis of Flows Facilitates High-regularity discretizations of flow variables Isogeometric Shape Optimization of Flows Isogeometric Analysis of Flow Acoustics

  • ✒
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SLIDE 80

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Summary

Summary

Isogeometric Method IGA = FEM (high regularity) + CAD (exact geometry) Isogeometric Analysis of Flows Facilitates High-regularity discretizations of flow variables Isogeometric Shape Optimization of Flows Unites design and analysis models through B-splines Isogeometric Analysis of Flow Acoustics

  • ✒
slide-81
SLIDE 81

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions Summary

Summary

Isogeometric Method IGA = FEM (high regularity) + CAD (exact geometry) Isogeometric Analysis of Flows Facilitates High-regularity discretizations of flow variables Isogeometric Shape Optimization of Flows Unites design and analysis models through B-splines Isogeometric Analysis of Flow Acoustics Identifies geometric enhancement of flow-acoustic coupling

  • ✒
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SLIDE 82

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions

The 3 minute version: http://www.dr.dk/da

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