N ASPECTS OF SONIC-BOOM S OPTIMIZATION Navid Allahverdi · 2014-06-16 · Navid Allahverdi Joint...

NUMERICAL ASPECTS OF SONIC-BOOM SIMULATION ANDOPTIMIZATION

Navid AllahverdiJoint work with:

A. Pozo (BCAM)S. Vincent (BCAM)

E. Zuazua (BCAM & Ikerbasque)

BCAM – Basque Center for Applied Mathematics

MTM2011 – Annual Project MeetingJUNE 12-13, 2014

SimulationOptimization

physicsABE

Outline

Sonic boom propagation in the atmosphereDescribing the physics of sonic boom

Modeling via “Augmented Burgers Equation”

Sonic boom minimizationShaped Sonic Boom Demonstrator (SSBD)

Optimization with Burgers Equation

Navid Allahverdi Sonic Boom Optimization


physicsABE

Sonic Boom Propagation



physicsABE

The sonic boom model

Augmented Burgers Equation (R. O. Cleveland, 1995):

∂P∂σ

= P∂P∂τ

+1Γ∂2P∂τ2︸︷︷︸

absorption

+∑ν

Cν∂2

∂τ2

1 + θν∂∂τ

P︸︷︷︸molecularrelaxations

−1

2A∂A∂σ

P︸︷︷︸ray tubespreading

+1

2ρ0c0

∂(ρ0c0)∂σ

P,︸︷︷︸atmosphericstratification

where:θν

1 + θν∂∂τ

f (τ) =∫ τ

−∞e(s−τ)/θν f (s)ds.

R. O. Cleveland, Propagation of sonic booms through a real, stratified atmosphere, PhD Thesis,University of Texas at Austin (1995).

S. K. Rallabhandi, Advanced sonic boom prediction using augmented Burgers equation, Journal ofAircraft 48, 4 (2011), 1245-1253.



physicsABE

Operator splitting

First Order Operator Splitting:

∂P∂σ

= LP in (0, σ) P(0) = P0

L = ΣSs=1Ls (differential or algebraic)

∂P(s)

∂σ= Ls P(s) in (σn, σn+1)

P(s)(σn) = P(s−1)(σn+1), P(0)(σn+1) = P(σn), P(σn+1) = P(S)(σn+1)

Each subproblem can be discretized independently using a proper discretization,The splitting error is of O(∆σ), so a first order time integration will suffice,Each sub problem can be integrated using implicit or explicit time integrationmethods,Using different time steps is possible for solving each sub-problem (sub-stepping).



physicsABE

Splitting for ABE

–

∂P∂σ

= P∂P∂τ

→space:EO, time:Euler

+1Γ∂2P∂τ2 →space:CD, time:Crank-Nicolson

+∑ν

Cν∂2

∂τ2

1 + θν∂∂τ

P →space:CD, time:Crank-Nicolson

−1

2A∂A∂σ

P →P(σn+1) =P(σn)√

1 + ∆σ/σ

+1

2ρ0c0

∂(ρ0c0)∂σ

P →P(σn+1) = P(σn)

√ρ0c0|σn+1

ρ0c0|σn

Even basic operator splitting techniques work fine.



physicsABE

Numerical test

Sinusoidal wave propagation with one relaxation

Sinusoidal wave with finite amplitude propagating in a thermo-viscous mediumwith one relaxation process.

∂P∂σ

= P∂P∂τ

+1Γ∂2P∂τ2 +

C ∂2

∂τ2

1 + θ ∂∂τ

P in (−π,+π)

P(τ, σ = 0) = sin(ω0τ), ω0 = 1, 15, Γ = 400, C = 0.5, θ = 1.

Forθ

ω0� 1: Dissipative effects

Forθ

ω0' 1: Dissipative + Dispersive effect



physicsABE

Numerical test, cont’d

Sinusoidal wave propagation with one relaxation, ω0 = 1,θ

ω0= 1

−π −π/2 0 +π/2 +π

τ

−1.0

−0.5

0.0

0.5

1.0

σ = 0

−π −π/2 0 +π/2 +π

τ

−1.0

−0.5

0.0

0.5

1.0

σ = 1

−π −π/2 0 +π/2 +π

τ

−1.0

−0.5

0.0

0.5

1.0

σ = 2

−π −π/2 0 +π/2 +π

τ

−1.0

−0.5

0.0

0.5

1.0

σ = 3



physicsABE

Numerical test, cont’d

Sinusoidal wave propagation with one relaxation, ω0 = 15,θ

ω0� 1

−π −π/2 0 +π/2 +π

τ

−1.0

−0.5

0.0

0.5

1.0

σ = 0

−π −π/2 0 +π/2 +π

τ

−1.0

−0.5

0.0

0.5

1.0

σ = 1

−π −π/2 0 +π/2 +π

τ

−1.0

−0.5

0.0

0.5

1.0

σ = 2

−π −π/2 0 +π/2 +π

τ

−1.0

−0.5

0.0

0.5

1.0

σ = 3



physicsABE

Near-field Pressure

Whitham F-function

p = p0 + ∆p

∆p(x , r)γp∞M2

∞=

12π√

2βr

∫ x−βr

0

A′′e (ξ)√(x − βr)− ξ

dξ

F (τ) =1

2π

∫ τ

0

A′′e (x)√τ − x

dx

∆p(τ, r) = p∞γM2∞√

2βrF (τ)



physicsABE

Boom propagation test

Flight DataParameter ValueAltitude 18288 mMach 2.7Length 91.5 mWeight 277,155 kg

F-function ParametersParameter ValueH 0.19

√m

yf 9.15 mC 0.03

√m

B 0D 0.04

√m

λ 82.58 myr 153.56 m



physicsABE

Boom propagation test, cont’d



physicsABE

Boom propagation test, cont’d

Verification

200 0 200 400 600 800t [ms]

80

60

40

20

0

20

40

60P [

Pa]

ground_signature.txt

ground_signature_sigma_10x.txt

Minelli-2014



SSBDInverse Design

Shaped Sonic Boom Demonstrator



SSBDInverse Design

Next generation of super-sonic airplanes

http://www.aeronautics.nasa.gov/



SSBDInverse Design

Inverse Design

Continuous SystemGiven u∗ and T find u0 such that minimizes:

J(u0) =12

∫R

(u(x ,T )− u∗(x))2 dx , (1)

where u is the solution of the viscous Burgers equation:∂t u + ∂x

(u2

2

)= ν∂xx u, x ∈ R, t > 0,

u(x , 0) = u0(x), x ∈ R.(2)

Ex. data assimilation, sonic boom propagation.solving min. problem = solving inverse (backward) problem (T −→ 0)backward problem for heat equation is ill-posed.backward problem for hyperbolic equation may have many solutions.



SSBDInverse Design

Inverse Design

Discretized systemIn the discretized system, we minimize:

J∆(u0j ) =

∆x2

∑Z

(uNj − u∗j )2 (3)

where unj is the solution of the discretized viscous Burgers equation:

un+1j = un

j −∆t∆x (gn

j+1/2 − gnj−1/2) + ν∆t

∆x2 (unj−1 − 2un

j + unj+1), n = 0, . . . ,N − 1,

u0j = 1

∆x

∫ xj+1/2xj−1/2

u0(x)dx .

(4)

Engquist-Osher (EO): gEO(v ,w) =v(v + |v |)

4+

w(w − |w |)4

, (5)

Lax-Friedrichs modified (LFM): gLFM (v ,w) =v2 + w2

4−

∆x∆t

(w − v4

). (6)



SSBDInverse Design

Problem Description

Target functionLet’s choose T = 50, and following target function:

u∗(x) =

3

2000

(− e−(5

√20+x)2 + e−(2

√20+x)2

+√π x(

erf(5√

20− x) + erf(2√

20 + x))), |x − 5| ≤ 25,

0, elsewhere.

(7)

Discretization

x ∈ (a, b)

spatial discretization: ∆x =(b − a)

M, M = 400, 600, 800, 1000, finite volume cells

temporal discretization to satisfy CFL:

∆t∆x

maxj|u0

j |+ ν∆t

∆x2 ≤12

(8)

physical viscosity is chosen as ν = 0.0001.



SSBDInverse Design

Engquist-Osher

−30 −20 −10 0 10 20 30 40 50

−0.05

0.00

0.05

0.10

0.15

0.20Engquist Osher Scheme, T=50, N=800 Cells

u0

uT

u ∗



SSBDInverse Design

Engquist-Osher Flux, 800 Cells



SSBDInverse Design

Modified Lax-Friedrichs

−30 −20 −10 0 10 20 30 40 50

−0.05

0.00

0.05

0.10

0.15

0.20Modified Lax-Friedrichs, T=50, N=800 Cells

u0

uT

u ∗



SSBDInverse Design

Modified Lax Friedrichs Flux, 800 Cells



SSBDInverse Design

Backward and forward solvers

solve the optimization (backward) problem with MLF and obtain u0

solve the forward problem with EO from u0



SSBDInverse Design

Future work

Work under progress:Adjoint formulation for augmented Burgers equation (ABE),

Inverse design, and optimization of the F-function (= effective area) for lowering boomsignature on the ground.

Next steps:Coupling 3D Euler’s equation (for near field) with ABE,

Using reduced order models for the near field.


N ASPECTS OF SONIC-BOOM S OPTIMIZATION Navid Allahverdi · 2014-06-16 · Navid Allahverdi Joint...

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Transcript of N ASPECTS OF SONIC-BOOM S OPTIMIZATION Navid Allahverdi · 2014-06-16 · Navid Allahverdi Joint...