isogeometric analysis and shape optimization in fluid...

Introduction Navier-Stokes Flow Shape Optimization Flow Acoustics Conclusions

Isogeometric Analysis and ShapeOptimization 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

Goals and Outline

The aim is to analyze and optimize flows using isogeometric analysis

Shape Optimization

Navier-Stokes Flow Model

Isogeometric Analysis

Flow Acoustics Model

Goals and Outline

Shape Optimization

Goals and Outline

Shape Optimization

Goals and Outline

Shape Optimization

Fluid Mechanics: Navier-Stokes Equations

Flow problems are governed by a boundary value problem

u velocityp pressureρ densityµ viscosityf force

[M. Van Dyke]

Re = ρULµ . 103

I viscous fluid

I slow flow

I small scale

2D steady-state incompressible Navier-Stokes flow

ρ(u · ∇)u+∇p− µ∇2u = ρf in Ω

∇ · u = 0 in Ω

u = u∗ on ΓD

(µ∇ui − p ei ) · n = 0 on ΓN

Challenge: solve this using isogeometric analysis[Bazilevs et al., 2006b; Bazilevs & Hughes, 2008; Akkerman et al., 2010; . . .]

[M. Van Dyke]

Re = ρULµ . 103

I viscous fluid

I slow flow

I small scale

ρ(u · ∇)u+∇p− µ∇2u = ρf in Ω

∇ · u = 0 in Ω

u = u∗ on ΓD

(µ∇ui − p ei ) · n = 0 on ΓN

[M. Van Dyke]

Re = ρULµ . 103

I viscous fluid

I slow flow

I small scale

ρ(u · ∇)u+∇p− µ∇2u = ρf in Ω

∇ · u = 0 in Ω

u = u∗ on ΓD

(µ∇ui − p ei ) · n = 0 on ΓN

[M. Van Dyke]

Re = ρULµ . 103

I viscous fluid

I slow flow

I small scale

ρ(u · ∇)u+∇p− µ∇2u = ρf in Ω

∇ · u = 0 in Ω

u = u∗ on ΓD

(µ∇ui − p ei ) · n = 0 on ΓN

Numerical Method

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

X(ξ, η) =Ng∑i=1

xiRgi (ξ, η)

u(ξ, η) =Nu∑i=1

uiPui (ξ, η)

p(ξ, η) =

Np∑i=1

piPpi (ξ, η)

[0, 1]2y

Rui ,R

Rgi Bivariate NURBS

/B-splineui, pi,

xi control point

/variable

Numerical Method

X(ξ, η) =

Ng∑i=1

xiRgi (ξ, η)

u(ξ, η) =Nu∑i=1

uiPui (ξ, η)

p(ξ, η) =

Np∑i=1

piPpi (ξ, η)

[0, 1]2y

Rui ,R

Rgi Bivariate NURBS

/B-splineui, pi,

xi control point

/variable

Numerical Method

X(ξ, η) =

Ng∑i=1

xiRgi (ξ, η)

u(ξ, η) =

Nu∑i=1

uiPui (ξ, η)

p(ξ, η) =

Np∑i=1

piPpi (ξ, η)

[0, 1]2y

Rui ,R

gi Bivariate NURBS/B-spline

ui, pi, xi control point/variable

Numerical Method

X(ξ, η) =

Ng∑i=1

xiRgi (ξ, η)

u(ξ, η) =

Nu∑i=1

uiPui (ξ, η)

p(ξ, η) =

Np∑i=1

piPpi (ξ, η)

[0, 1]2y

Rui ,R

Univariate B-spline:

Ni(ξ1), Mj(ξ2)

• knot vector• polynomial degree

Parameter domain [0, 1]2

Numerical Method

X(ξ, η) =

Ng∑i=1

xiRgi (ξ, η)

u(ξ, η) =

Nu∑i=1

uiPui (ξ, η)

p(ξ, η) =

Np∑i=1

piPpi (ξ, η)

[0, 1]2y

Rui ,R

Univariate B-spline: Ni(ξ1)

, Mj(ξ2)

Numerical Method

X(ξ, η) =

Ng∑i=1

xiRgi (ξ, η)

u(ξ, η) =

Nu∑i=1

uiPui (ξ, η)

p(ξ, η) =

Np∑i=1

piPpi (ξ, η)

[0, 1]2y

Rui ,R

, Mj(ξ2)

Numerical Method

X(ξ, η) =

Ng∑i=1

xiRgi (ξ, η)

u(ξ, η) =

Nu∑i=1

uiPui (ξ, η)

p(ξ, η) =

Np∑i=1

piPpi (ξ, η)

[0, 1]2y

Rui ,R

, Mj(ξ2)

Numerical Method

X(ξ, η) =

Ng∑i=1

xiRgi (ξ, η)

u(ξ, η) =

Nu∑i=1

uiPui (ξ, η)

p(ξ, η) =

Np∑i=1

piPpi (ξ, η)

[0, 1]2y

Rui ,R

Univariate B-spline: Ni(ξ1), Mj(ξ2)• knot vector• polynomial degree

Numerical Method

X(ξ, η) =

Ng∑i=1

xiRgi (ξ, η)

u(ξ, η) =

Nu∑i=1

uiPui (ξ, η)

p(ξ, η) =

Np∑i=1

piPpi (ξ, η)

[0, 1]2y

Rui ,R

Numerical Method

X(ξ, η) =

Ng∑i=1

xiRgi (ξ, η)

u(ξ, η) =

Nu∑i=1

uiPui (ξ, η)

p(ξ, η) =

Np∑i=1

piPpi (ξ, η)

[0, 1]2y

Rui ,R

Bivariate Tensor Product B-spline:• 2 knot vectors• 2 polynomial degrees• Pi,j(ξ1, ξ2) = Ni(ξ1)Mj(ξ2)

Numerical Method

X(ξ, η) =

Ng∑i=1

xiRgi (ξ, η)

u(ξ, η) =

Nu∑i=1

uiPui (ξ, η)

p(ξ, η) =

Np∑i=1

piPpi (ξ, η)

[0, 1]2y

Rui ,R

Physical domain Ω

Numerical Method

X(ξ, η) =

Ng∑i=1

xiRgi (ξ, η)

u(ξ, η) =

Nu∑i=1

uiPui (ξ, η)

p(ξ, η) =

Np∑i=1

piPpi (ξ, η)

[0, 1]2y

Rui ,R

Physical domain ΩM(U)U = F

Error Convergence

Test of error convergence: flow with analytical solution

−3 −2 −1 0 1 2 3−1.5

−0.5

−3 −2 −1 0 1 2 3−1.5

−0.5

f1 = f1(x,y)

f2 = f2(x,y)

u|Γ = 0

u?1 = −U sin(πr2)y

u?2 = U/4 sin(πr2)x

p? = 4/π2 + cos(πr)

r =√

(x/2)2+ y2

p?Re = 200

ε2u =∫∫Ω

‖u(x,y)−u?(x,y)‖2 dxdy

Error Convergence

−3 −2 −1 0 1 2 3−1.5

−0.5

−3 −2 −1 0 1 2 3−1.5

−0.5

f1 = f1(x,y)

f2 = f2(x,y)

u|Γ = 0

p? = 4/π2 + cos(πr)

r =√

(x/2)2+ y2

Re = 200

ε2u =∫∫Ω

Error Convergence

−3 −2 −1 0 1 2 3−1.5

−0.5

−3 −2 −1 0 1 2 3−1.5

−0.5

f1 = f1(x,y)

f2 = f2(x,y)

u|Γ = 0

p? = 4/π2 + cos(πr)

r =√

(x/2)2+ y2

p?Re = 200

ε2u =∫∫Ω

Error Convergence

−3 −2 −1 0 1 2 3−1.5

−0.5

−3 −2 −1 0 1 2 3−1.5

−0.5

f1 = f1(x,y)

f2 = f2(x,y)

u|Γ = 0

p? = 4/π2 + cos(πr)

r =√

(x/2)2+ y2

p?Re = 200

ε2u =∫∫Ω

Error Convergence

Discretizations with higher regularity perform better

10−5

10−6

10−4

10−2

ε∇ u

10−4

10−2

ε∇ p

DOF102 103 104 105

Smoothness Mesh Density

Strategy 1 High High

Strategy 2 Low Low

10−5

10−6

10−4

10−2

ε∇ u

10−4

10−2

ε∇ p

Error Convergence

10−5

10−6

10−4

10−2

ε∇ u

10−4

10−2

ε∇ p

DOF102 103 104 105

Strategy 2 Low Low

10−5

10−6

10−4

10−2

ε∇ u

10−4

10−2

ε∇ p

Error Convergence

10−5

10−6

10−4

10−2

ε∇ u

10−4

10−2

ε∇ p

DOF102 103 104 105

Strategy 2 Low Low

10−5

10−6

10−4

10−2

ε∇ u

10−4

10−2

ε∇ p

Design of Minimal Drag Body

We design a body with minimal drag

Design boundary γ of body with

area A0 travelling at constant speed

U to minimize the drag D

Optimization Problem

minγ(xb)

c = D + εR

s. t. Area ≥ A0

L−(xb) ≤ L(xb) ≤ L+(xb)

MU = F

dragobjective

areaconstraint

lineardesignconstraints

governingequations

D =∫γ

(− pI + µ

(∇u + (∇u)T

))n ds · eu

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

u = Ue1

v = ∂u/∂n = 0

20r 40r

minγ(xb)

c = D + εR

s. t. Area ≥ A0

L−(xb) ≤ L(xb) ≤ L+(xb)

MU = F

dragobjective

areaconstraint

governingequations

D =∫γ

(− pI + µ

(∇u + (∇u)T

))n ds · eu

γu = 0

u = Ue1

v = ∂u/∂n = 0

20r 40r

minγ(xb)

c = D + εR

s. t. Area ≥ A0

L−(xb) ≤ L(xb) ≤ L+(xb)

MU = F

dragobjective

areaconstraint

governingequations

D =∫γ

(− pI + µ

(∇u + (∇u)T

))n ds · eu

γu = 0

u = Ue1

v = ∂u/∂n = 0

20r 40r

minγ(xb)

c = D + εR

s. t. Area ≥ A0

L−(xb) ≤ L(xb) ≤ L+(xb)

MU = F

dragobjective

areaconstraint

governingequations

D =∫γ

(− pI + µ

(∇u + (∇u)T

))n ds · eu

γu = 0

u = Ue1

v = ∂u/∂n = 0

20r 40r

minγ(xb)

c = D + εR

s. t. Area ≥ A0

L−(xb) ≤ L(xb) ≤ L+(xb)

MU = F

dragobjective

areaconstraint

governingequations

D =∫γ

(− pI + µ

(∇u + (∇u)T

))n ds · eu

γu = 0

u = Ue1

v = ∂u/∂n = 0

20r 40r

The optimal shape is longer and more slender for higher speeds

Optimal shapes for different speeds

Initial

U = 10

U = 40

U = 100

Flow before optimization (U=1)

Flow after optimization (U=100)

Note: low Reynolds numbers

Initial

U = 10

U = 40

U = 100

Initial

U = 10

U = 40

U = 100

Initial

U = 10

U = 40

U = 100

Initial

U = 10

U = 40

U = 100

Initial

U = 10

U = 40

U = 100

Initial

U = 10

U = 40

U = 100

Introduction and Governing Equations

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

Pipe Geometry

Flow Acoustics

p pressureu velocityρ density

ρ∂u

∂t+ ρ(u · ∇)u+∇p− µ∇2u = 0

∂t+∇ · (ρu) = 0

p = p0 + p′

u = u0 + u′

ρ = ρ0 + ρ′

Background Flow: p0, u0, ρ0

Acoustic Disturbance: p′, u′, ρ′

γ− γ+

Pipe GeometryFlow

Acoustics

ρ∂u

∂t+ ρ(u · ∇)u+∇p− µ∇2u = 0

∂t+∇ · (ρu) = 0

p = p0 + p′

u = u0 + u′

ρ = ρ0 + ρ′

γ− γ+

Pipe GeometryFlow Acoustics

ρ∂u

∂t+ ρ(u · ∇)u+∇p− µ∇2u = 0

∂t+∇ · (ρu) = 0

p = p0 + p′

u = u0 + u′

ρ = ρ0 + ρ′

γ− γ+

ρ∂u

∂t+ ρ(u · ∇)u+∇p− µ∇2u = 0

∂t+∇ · (ρu) = 0

p = p0 + p′

u = u0 + u′

ρ = ρ0 + ρ′

γ− γ+

ρ∂u

∂t+ ρ(u · ∇)u+∇p− µ∇2u = 0

∂t+∇ · (ρu) = 0

p = p0 + p′

u = u0 + u′

ρ = ρ0 + ρ′

γ− γ+

ρ∂u

∂t+ ρ(u · ∇)u+∇p− µ∇2u = 0

∂t+∇ · (ρu) = 0

p = p0 + p′

u = u0 + u′

ρ = ρ0 + ρ′

γ− γ+

ρ∂u

∂t+ ρ(u · ∇)u+∇p− µ∇2u = 0

∂t+∇ · (ρu) = 0

p = p0 + p′

u = u0 + u′

ρ = ρ0 + ρ′

γ− γ+

ρ∂u

∂t+ ρ(u · ∇)u+∇p− µ∇2u = 0

∂t+∇ · (ρu) = 0

p = p0 + p′

u = u0 + u′

ρ = ρ0 + ρ′

γ− γ+

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

FluidRadiusSpeedFrequency

Air1 cm1 m/s∼25 kHz

-----Pipe A

|u|Flow

|p|Sound

-----Pipe B

|u|Flow

|p|Sound

-----Pipe C

|u|Flow

|p|Sound

-----Pipe A

|u|Flow

|p|Sound

-----Pipe B

|u|Flow

|p|Sound

-----Pipe C

|u|Flow

|p|Sound

The corrugated duct exhibits stronger flow-acoustic coupling

Acoustic Pressure p [Pa]

FlowUpstream Downstream⇒

Measure of Flow-Acoustic Coupling

〈δp〉 =

∫∫Ω |p(x)− p(−x)| dA∫∫

Ω |p(x)|dA

Sound Frequency Sensitivity

20 21 22 23 24 25 26 27 28 29 300

Pipe A

Pipe B

Pipe C

〈δp〉

f [kHz]

2520 30

Flow Speed Sensitivity

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

Pipe A

Pipe B

Pipe C

〈δp〉

U0 [m s−1]

〈δp〉 =

∫∫Ω |p(x)− p(−x)| dA∫∫

Ω |p(x)|dA

20 21 22 23 24 25 26 27 28 29 300

Pipe A

Pipe B

Pipe C

〈δp〉

f [kHz]

2520 30

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

Pipe A

Pipe B

Pipe C

〈δp〉

U0 [m s−1]

〈δp〉 =

∫∫Ω |p(x)− p(−x)| dA∫∫

Ω |p(x)|dA

20 21 22 23 24 25 26 27 28 29 300

Pipe A

Pipe B

Pipe C

〈δp〉

f [kHz]

2520 30

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

Pipe A

Pipe B

Pipe C

〈δp〉

U0 [m s−1]

〈δp〉 =

∫∫Ω |p(x)− p(−x)| dA∫∫

Ω |p(x)|dA

20 21 22 23 24 25 26 27 28 29 300

Pipe A

Pipe B

Pipe C

〈δp〉

f [kHz]

2520 30

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

Pipe A

Pipe B

Pipe C

〈δp〉

U0 [m s−1]

Outlook

Ongoing work

Flow Discretizations using Locally Refinable B-splines

Optimization of Parametrizations for Isogeometric Analysis

L(U) = 0

minimize c(Ω, U)

such that det(J) > 0

where ∂Ω = Γ

L(U) = 0

Outlook

Ongoing work

Region of Interest

Computational Domain

L(U) = 0

minimize c(Ω, U)

where ∂Ω = Γ

L(U) = 0

Outlook

Ongoing work

Region of Interest

Global Refinement

L(U) = 0

minimize c(Ω, U)

where ∂Ω = Γ

L(U) = 0

Outlook

Ongoing work

Region of Interest

Local Refinement

L(U) = 0

minimize c(Ω, U)

where ∂Ω = Γ

L(U) = 0

Outlook

Ongoing work

Region of Interest

Local Refinement1

L(U) = 0

minimize c(Ω, U)

where ∂Ω = Γ

L(U) = 0

Outlook

Ongoing work

Region of Interest

Local Refinement1

L(U) = 0

minimize c(Ω, U)

where ∂Ω = Γ

L(U) = 0

Outlook

Ongoing work

Region of Interest

Local Refinement1

L(U) = 0

minimize c(Ω, U)

where ∂Ω = Γ

L(U) = 0

Outlook

Ongoing work

Region of Interest

Local Refinement1

L(U) = 0

minimize c(Ω, U)

where ∂Ω = Γ

L(U) = 0

Outlook

Ongoing work

Region of Interest

Local Refinement1

L(U) = 0

minimize c(Ω, U)

where ∂Ω = Γ

L(U) = 0

Outlook

Ongoing work

Region of Interest

Local Refinement1

L(U) = 0

minimize c(Ω, U)

where ∂Ω = Γ

L(U) = 0

Outlook

Ongoing work

Region of Interest

Local Refinement1 2

L(U) = 0

minimize c(Ω, U)

where ∂Ω = Γ

L(U) = 0

Outlook

Ongoing work

Region of Interest

Local Refinement1 2

L(U) = 0

minimize c(Ω, U)

where ∂Ω = Γ

L(U) = 0

Outlook

Ongoing work

Region of Interest

Local Refinement1 2

L(U) = 0

minimize c(Ω, U)

where ∂Ω = Γ

L(U) = 0

Outlook

Ongoing work

Region of Interest

Local Refinement1 2

L(U) = 0

minimize c(Ω, U)

where ∂Ω = Γ

L(U) = 0

Outlook

Ongoing work

Region of Interest

Local Refinement1 2

L(U) = 0

minimize c(Ω, U)

where ∂Ω = Γ

L(U) = 0

Outlook

Ongoing work

Region of Interest

Local Refinement1 2

L(U) = 0

minimize c(Ω, U)

where ∂Ω = Γ

L(U) = 0

Outlook

Ongoing work

Region of Interest

Local Refinement1 2

L(U) = 0

minimize c(Ω, U)

where ∂Ω = Γ

L(U) = 0

Outlook

Ongoing work

Region of Interest

Local Refinement1 2

L(U) = 0

minimize c(Ω, U)

where ∂Ω = Γ

L(U) = 0

Outlook

Ongoing work

Region of Interest

Local Refinement1 2

L(U) = 0

minimize c(Ω, U)

where ∂Ω = Γ

L(U) = 0

Outlook

Ongoing work

Region of Interest

Local Refinement1 2

L(U) = 0

minimize c(Ω, U)

where ∂Ω = Γ

L(U) = 0

Outlook

Ongoing work

Region of Interest

Local Refinement1 2

L(U) = 0

minimize c(Ω, U)

where ∂Ω = Γ

L(U) = 0

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

Summary

Isogeometric Method

Summary

Isogeometric Method

Summary

Isogeometric Method

Summary

Isogeometric Method

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