Augmented Burgers Equation and Supersonic Aircraft Design

S. Vincent

June 2014

S. Vincent

Outline

The Sonic-Boom issue

Burgers-like Equation

Optimization with IPOpt

Augmented Burgers Equation

Credit: NASA, US Navy

The Sonic-Boom issue S. Vincent

Sonic Boom Propagation

Sonic-Boom Propagation1

▸ From the aircraft to the near-field:Eulers equation

▸ From the near-field to the ground:Augmented Burgers Equation(ABE)

▸ Forward Problem: ABE Solver

▸ Backward Problem: Optimization

1JUAN J. ALONSO AND MICHAEL R. COLONNO, Multidisciplinary Optimization with Applicationsto Sonic-Boom Minimization, Annu. Rev. Fluid. Mech. 44 (2012)

The Sonic-Boom issue S. Vincent

Augmented Burgers Equation

∂σP = P∂τP + 1

Γ∂2τP +∑

ν

Cν∂2τ

1 + θν∂τP − ∂σG

2GP + ∂σ(ρ0c0)

2ρ0c0P

▸ P∂τP : Non-linearity

▸1Γ∂2τP : Viscosity

▸ ∑νCν∂

2τ

1+θν∂τP : Molecular Relaxation

▸ − ∂σG2G

P : Ray Tube Spreading

▸∂σ(ρ0c0)

2ρ0c0P : Atmosphere Stratification

Parameters▸ P : dimensionless pressure▸ σ : dimensionless distance▸ τ : dimensionless time

Burgers-like Equation : ∂tu = u∂xu + ν∂2xu

1SRIRAM K. RALLABHANDI, Sonic Boom Adjoint Methodology and its Applications, Journal ofAircraft 48, 4 (2011)

The Sonic-Boom issue S. Vincent

Augmented Burgers Equation

∂σP = P∂τP + 1

Γ∂2τP +∑

ν

Cν∂2τ

1 + θν∂τP − ∂σG

2GP + ∂σ(ρ0c0)

2ρ0c0P

▸ P∂τP : Non-linearity

▸1Γ∂2τP : Viscosity

▸ ∑νCν∂

2τ

1+θν∂τP : Molecular Relaxation

▸ − ∂σG2G

P : Ray Tube Spreading

▸∂σ(ρ0c0)

2ρ0c0P : Atmosphere Stratification

Parameters▸ P : dimensionless pressure▸ σ : dimensionless distance▸ τ : dimensionless time

Burgers-like Equation : ∂tu = u∂xu + ν∂2xu

1SRIRAM K. RALLABHANDI, Sonic Boom Adjoint Methodology and its Applications, Journal ofAircraft 48, 4 (2011)

The Sonic-Boom issue S. Vincent

Augmented Burgers Equation

∂σP = P∂τP + 1

Γ∂2τP +∑

ν

Cν∂2τ

1 + θν∂τP − ∂σG

2GP + ∂σ(ρ0c0)

2ρ0c0P

▸ P∂τP : Non-linearity

▸1Γ∂2τP : Viscosity

▸ ∑νCν∂

2τ

1+θν∂τP : Molecular Relaxation

▸ − ∂σG2G

P : Ray Tube Spreading

▸∂σ(ρ0c0)

2ρ0c0P : Atmosphere Stratification

Parameters▸ P : dimensionless pressure▸ σ : dimensionless distance▸ τ : dimensionless time

Burgers-like Equation : ∂tu = u∂xu + ν∂2xu

1SRIRAM K. RALLABHANDI, Sonic Boom Adjoint Methodology and its Applications, Journal ofAircraft 48, 4 (2011)

The Sonic-Boom issue S. Vincent

Augmented Burgers Equation

∂σP = P∂τP + 1

Γ∂2τP +∑

ν

Cν∂2τ

1 + θν∂τP − ∂σG

2GP + ∂σ(ρ0c0)

2ρ0c0P

▸ P∂τP : Non-linearity

▸1Γ∂2τP : Viscosity

▸ ∑νCν∂

2τ

1+θν∂τP : Molecular Relaxation

▸ − ∂σG2G

P : Ray Tube Spreading

▸∂σ(ρ0c0)

2ρ0c0P : Atmosphere Stratification

Parameters▸ P : dimensionless pressure▸ σ : dimensionless distance▸ τ : dimensionless time

Burgers-like Equation : ∂tu = u∂xu + ν∂2xu

1SRIRAM K. RALLABHANDI, Sonic Boom Adjoint Methodology and its Applications, Journal ofAircraft 48, 4 (2011)

The Sonic-Boom issue S. Vincent

Augmented Burgers Equation

∂σP = P∂τP + 1

Γ∂2τP +∑

ν

Cν∂2τ

1 + θν∂τP − ∂σG

2GP + ∂σ(ρ0c0)

2ρ0c0P

▸ P∂τP : Non-linearity

▸1Γ∂2τP : Viscosity

▸ ∑νCν∂

2τ

1+θν∂τP : Molecular Relaxation

▸ − ∂σG2G

P : Ray Tube Spreading

▸∂σ(ρ0c0)

2ρ0c0P : Atmosphere Stratification

Parameters▸ P : dimensionless pressure▸ σ : dimensionless distance▸ τ : dimensionless time

Burgers-like Equation : ∂tu = u∂xu + ν∂2xu

1SRIRAM K. RALLABHANDI, Sonic Boom Adjoint Methodology and its Applications, Journal ofAircraft 48, 4 (2011)

The Sonic-Boom issue S. Vincent

Augmented Burgers Equation

∂σP = P∂τP + 1

Γ∂2τP +∑

ν

Cν∂2τ

1 + θν∂τP − ∂σG

2GP + ∂σ(ρ0c0)

2ρ0c0P

▸ P∂τP : Non-linearity

▸1Γ∂2τP : Viscosity

▸ ∑νCν∂

2τ

1+θν∂τP : Molecular Relaxation

▸ − ∂σG2G

P : Ray Tube Spreading

▸∂σ(ρ0c0)

2ρ0c0P : Atmosphere Stratification

Parameters▸ P : dimensionless pressure▸ σ : dimensionless distance▸ τ : dimensionless time

Burgers-like Equation : ∂tu = u∂xu + ν∂2xu

1SRIRAM K. RALLABHANDI, Sonic Boom Adjoint Methodology and its Applications, Journal ofAircraft 48, 4 (2011)

The Sonic-Boom issue S. Vincent

Augmented Burgers Equation

∂σP = P∂τP + 1

Γ∂2τP +∑

ν

Cν∂2τ

1 + θν∂τP − ∂σG

2GP + ∂σ(ρ0c0)

2ρ0c0P

▸ P∂τP : Non-linearity

▸1Γ∂2τP : Viscosity

▸ ∑νCν∂

2τ

1+θν∂τP : Molecular Relaxation

▸ − ∂σG2G

P : Ray Tube Spreading

▸∂σ(ρ0c0)

2ρ0c0P : Atmosphere Stratification

Parameters▸ P : dimensionless pressure▸ σ : dimensionless distance▸ τ : dimensionless time

Burgers-like Equation : ∂tu = u∂xu + ν∂2xu

1SRIRAM K. RALLABHANDI, Sonic Boom Adjoint Methodology and its Applications, Journal ofAircraft 48, 4 (2011)

Burgers-like Equation S. Vincent

Outline

The Sonic-Boom issue

Burgers-like EquationFinite ElementsFinite Differences

Optimization with IPOptObjective Function and ConstraintsNumerical Experiments

Augmented Burgers EquationConjugate Gradient DescentGradient ComputationNumerical Experiments

Credit: NASA, Boeing

Burgers-like Equation S. Vincent

Burgers-like Equation Solver: Finite Elements Method

Equation: ∂tu = u∂xu

Implementation on Fenics:

▸ Finite elements▸ Continuous Galerkin

Burgers-like Equation S. Vincent

Outline

The Sonic-Boom issue

Burgers-like EquationFinite ElementsFinite Differences

Optimization with IPOptObjective Function and ConstraintsNumerical Experiments

Augmented Burgers EquationConjugate Gradient DescentGradient ComputationNumerical Experiments

Credit: NASA, Boeing

Burgers-like Equation S. Vincent

Burgers-like Equation Solver: Finite Differences

▸ Equation:∂tu = ∂x(f(u)) + ν∂xxu, f(u) = u2/2

▸ Engquist-Osher Numerical Flux:

f̂(unj , unj+1) =unj (unj − ∣unj ∣) + unj+1(unj+1 + ∣unj+1∣)

4

▸ Discretized Equation:

un+1j = unj +

∆t

∆x(f̂(unj , unj+1) − f̂(unj−1, u

nj )) + ν

∆t

∆x2(unj+1 − 2unj + unj−1)

▸ CFL Condition:

∆t ≤ ∆x2

∥(unj )j,n∥∞∆x + 2ν

Burgers-like Equation S. Vincent

Numerical Experiment

Optimization with IPOpt S. Vincent

Outline

The Sonic-Boom issue

Burgers-like EquationFinite ElementsFinite Differences

Optimization with IPOptObjective Function and ConstraintsNumerical Experiments

Augmented Burgers EquationConjugate Gradient DescentGradient ComputationNumerical Experiments

Credit: NASA, Boeing

Optimization with IPOpt S. Vincent

Problem Formulation

The objective function to minimize is:

J((unj )j=0..N,n=0..T ) =N

∑j=0

∣uTj − utargetj ∣2

The constraints are:▸ The discretized equation (T ×N constraints)▸ The boundary conditions (∀n un0 = 0 = unN )

Implementation: AMPL + IPOpt

Optimization with IPOpt S. Vincent

Outline

The Sonic-Boom issue

Burgers-like EquationFinite ElementsFinite Differences

Optimization with IPOptObjective Function and ConstraintsNumerical Experiments

Augmented Burgers EquationConjugate Gradient DescentGradient ComputationNumerical Experiments

Credit: NASA, Boeing

Optimization with IPOpt S. Vincent

First numerical implementation

∂tu + u∂xu = ν∂xxu

▸ The target is a smoothN-Wave

▸ There are no constraintson the initial data

Optimization with IPOpt S. Vincent

Initial Data: the F-Function

The F-Function▸ Near-field signature represented

using Whitham’s theory (1952)▸ First applied to Aircrafts in 1972▸ Generalized by Plotkin, Li &

Rallabhandi in 2009

F (x) =

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

2H xyf

if 0 ≤ x ≤ yf2

C( 2xyf

− 1) −H( 2xyf

− 2) if yf2≤ x ≤ yf

B(x − yf) +C if yf ≤ x ≤ λB(x − yf) −D if λ ≤ x ≤ L

Optimization with IPOpt S. Vincent

Recovering a Generalized F-Function with IPOpt

Equation : ∂tu = u∂xu + ν∂xxu

0.0 0.5 1.0 1.5 2.0 2.5 3.00.2

0.0

0.2

0.4

0.6

0.8

1.0 Two F-functions

u0 used to create the target

u0 found by IPOpt

Optimization with IPOpt S. Vincent

Major Computational Obstacles

▸ AMPL is not a very flexible software

▸ Refining the mesh Ô⇒ more memory required

▸ Size of the problem (number of cells × time steps)

Problem Solved on IPOpt Real Problem300 × 2500 1500 × 744000

Augmented Burgers Equation S. Vincent

Outline

The Sonic-Boom issue

Burgers-like EquationFinite ElementsFinite Differences

Optimization with IPOptObjective Function and ConstraintsNumerical Experiments

Augmented Burgers EquationConjugate Gradient DescentGradient ComputationNumerical Experiments

Credit: NASA, Boeing

Augmented Burgers Equation S. Vincent

Algorithm: Conjugate Gradient Descent Method

1. Solve the forward problem

2. Solve the adjoint equations

3. Compute the gradient

4. Compute the descent direction and updatethe initial data

5. Check that the objective function decreased"correctly"

▸ If not: Reduce the step size and go back to 4.▸ Otherwise go back to 1.

Stopping criteria:

Maximum IterationsMinimal Step Size

Augmented Burgers Equation S. Vincent

Augmented Burgers Equation Solver: Splitting Method

∂σP = P∂τP + 1

Γ∂2τP +∑

ν

Cν∂2τ

1 + θν∂τP − ∂σG

2GP + ∂σ(ρ0c0)

2ρ0c0P

▸ Diffusion: sn = wn−1 + ∆σ

2Γ∂τ(sn +wn−1)

▸ Inviscid Burgers Equation: un = sn +∆σsn∂τsn

▸ Ray Tube Spreading & Atmosphere Stratification: v̂nj = unj knj

▸ First Molecular Relaxation: (1 + θO∂τ)(vn − v̂n) = ∆σCO2∂2τ(vn + v̂n)

▸ Second Molecular Relaxation: (1+θN∂τ)(wn −vn) = ∆σCN2∂2τ(wn +vn)

Augmented Burgers Equation S. Vincent

Initial Data and Target

Augmented Burgers Equation S. Vincent

Outline

The Sonic-Boom issue

Burgers-like EquationFinite ElementsFinite Differences

Optimization with IPOptObjective Function and ConstraintsNumerical Experiments

Augmented Burgers EquationConjugate Gradient DescentGradient ComputationNumerical Experiments

Credit: NASA, Boeing

Augmented Burgers Equation S. Vincent

Gradient Computation : Adjoint Method

▸ Lagrangian: takes into account the splitting

L =∑j

(wNj −w∗

j )2/2 +∑

j

µj(w0j − FH,B,C,D,yf ,λ(xj))

+k=N−1

∑k=0,j

multipliers × constraints

▸ The lagrangian derivatives provide the adjoint equationsÔ⇒ calculation of µj

▸ The derivatives with respect to the design variables provides the gradient

Augmented Burgers Equation S. Vincent

Outline

The Sonic-Boom issue

Burgers-like EquationFinite ElementsFinite Differences

Optimization with IPOptObjective Function and ConstraintsNumerical Experiments

Augmented Burgers EquationConjugate Gradient DescentGradient ComputationNumerical Experiments

Credit: NASA, Boeing

Augmented Burgers Equation S. Vincent

Numerical Experiments: Results

▸ Polack-Ribiere’s Conjugate Gradient Descent Method▸ Gradient Estimated with Finite Differences

S. Vincent

Conclusion

Summary▸ Forward Solver with Fenics

▸ Forward Solver with Engquist-Osher flux

▸ Optimization with IPOpt+Ampl

▸ Conjugate Gradient Descent Method

Areas of Improvement▸ Fianlize the adjoint code

▸ Test other gradient descent methods

▸ Perform an optimization for the full propagation

0 200 400 600 80060

40

20

0

20

40

60

S. Vincent

Thanks for your attention!

Credit: NASA