S. Vincent June 2014 · The Sonic-Boom issue S. Vincent Augmented Burgers Equation @ ˙P=P@ ˝P+ 1...
Transcript of S. Vincent June 2014 · The Sonic-Boom issue S. Vincent Augmented Burgers Equation @ ˙P=P@ ˝P+ 1...
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