AMDL

Coupled Adjoint-based Rotor Design using a Fluid StructureInteraction in Time Spectral Form

Seongim Choi1, Hyunsoon Kim2, and Rachit Prasad2

Department of Mechanical Engineering
Gwangju Institute of Science and Technology (GIST)
Virginia Polytechnic Institute and State University
1

Contents
I.

Introduction

II. Fluid-Structure Interaction in Time Spectral Form

III. Adjoint Sensitivity for Time Spectral Form
IV. Sensitivity Analysis Results
V. Conclusions

2

Introduction: Motivation▣ Motivation➢ The system involving multi-physics is difficult to analyze with high fidelity analysis.
➢ For example, difficulties associated with simulation of helicopter flight• Aerodynamic challenges: Complexity of unsteady and vortical air flows• Structural dynamic challenges: Long and flexible rotor blades• Coupled physics of aerodynamics and structures▣ Objective➢ Develop an accurate but efficient analysis and design method for rotor blade.

Source: https://qph.ec.quoracdn.net/main-qimg-85267c6f4bb30dfa7bd5
e86ba427fa8f-c?convert_to_webp=true

3

Source: https://en.wikipedia.org/wiki/File:UH-60_2nd_Squadron,_2nd_Cavalry_Regiment_(cropped).jpg

Introduction: Literature Review▣ Literature Review for Rotor Design➢ A comprehensive rotor design code
• CAMRAD (NASA Ames Research Center / U.S. Army, 1980)– Lifting-line theory, Lifting-surface theory• CAMRADII (Johnson Aeronautics,1994)• UMARC (University of Maryland, 1990)– FEM formulation + Quasi-steady 2-D strip theory➢ CFD/CA coupling• CAMRAD II, RCAS and UMARC started to include main rotor 3D-CFD coupling. (in 2000s)• Hybrid solver: OVERFLOW + CHARM (by CDI, 2016)

Smith, Marilyn J., et al. "An assessment of CFD/CSD prediction state-of-the-art using the HART II international workshop data." 68th Annual Forum of
the American Helicopter Society, Ft. Worth, TX. 2012.

4

Introduction: Contributions▣ Literature Review➢ Adjoint-based shape optimization for static aeroelastic problem• Kenway, Kennedy and Martins (U.Mich, 2014)➢ Time Accurate approach with unsteady adjoint-based shape optimization of rotor (Dynamic FSI problem)• Mishra, Mani and Mavriplis (Wyoming University, 2014)• Contributions of th present study- Time-Spectral form of fluid and structural equations of motion(steady form of governing equations)- Steady adjoint formulation for unsteady, dynamic problems in time-spectral form.
- Fluid-structure interface (FSI) and coupled adjoint solution
1. Kenway, Gaetan KW, Graeme J. Kennedy, and Joaquim RRA Martins. "Scalable parallel approach for high-fidelity steady-state aeroelastic
analysis and adjoint derivative computations." AIAA journal 52.5 (2014): 935-951.
2. Mishra, Asitav, et al. "Time-dependent adjoint-based aerodynamic shape optimization applied to helicopter rotors." Rn 3 (2014): 2.

5

Contents
I.

Introduction

II. Fluid-Structure Interaction in Time Spectral Form

III. Adjoint Sensitivity for Time Spectral Form
IV. Sensitivity Analysis Results
I. Conclusions

6

Traditional FSI Approach▣ Partitioned and Staggered Approach¶u
V + R(u) = 0
Fluid:¶tComputational Fluid
Dynamics(CFD, R(W,x)=0)

Pressure Field

Displacement Field

Structure:
Finite Element
Method(FEM, S(U,x)=0)

𝑀𝒙ሷ 𝜓 + 𝐶 𝜓 𝒙ሶ 𝜓 + 𝐾 𝜓 𝒙 𝜓= 𝐹(𝑥, 𝑥,ሶ 𝜓)➢ For the unsteady problems, physics of the problem limits the time step

serial staggered
partitioned approach

𝑊𝑛 : fluid state vector
𝑈𝑛 : structural state vector
𝑥𝑛 : position of grid point
𝑃𝑛 : fluid pressure
𝑢𝑛 : structural displacement

Source: C. Farhat, M. Lesoinne, “Two efficient staggered
7 algorithms for the serial and parallel solution of three-dimensional nonlinear transient aeroelastic
problems”, Computational Methods in Applied Mechanics and Engineering, Vol. 182, 2000, Pg. 499-515

Time Spectral Formulation (Fluids)• For collocation method, the equation should be satisfied ateach time instance (Fourier collocation point)
N /2
N
T
¶u
ikt
¶u j
u jN = å u~k e( j = 0,… , N - 1), t j =jN
V + R(u) = 0
V
+
R(u
)
=
0
(
j
=
0,…
,N
-1)
N
j
t=
t
k =- N / 2¶t¶t
1 N -1
-ikt
j

j

u~k =

• Fourier collocation derivative in (physical) time space

åu e
N
j =0

N
j

j

( k = - N / 2,… , N / 2 - 1)

N /2-1¶ulN= å ikuk eiktl (l = 0,..., N -1)¶tk=- N /2=

N /2-1å ek=- N /2

iktl

ik N -1 N -ikt jå u j e (l = 0,..., N -1)
N j=0

æ 1 N /2-1
ö N
ik (t j -tl )= åçike÷ u j (l = 0,..., N -1)
å
N
è
ø
j=0
k=- N /2
N -1

N -1

= å ( DN ) lj u (l = 0,..., N -1)
j=0, whereNjì 1
ï (-1)l+ j cosec( (l - j)p )(DN )lj = í 2
N
ï
0
î

l¹0
l= j

:
:

æ 0 d1 … d N -1 -d N -1 … -d1 öæ u1 öç÷ç 2 ÷
22…
…
ç-d 0 d1 d2-d2 ÷ç u ÷
Dt u n = ç .1 .
.
.
..
.. ... ÷ç ... ÷.. ... ...
.
.
ç÷ç N ÷-d -dç÷è u ø
…
…
0
d
d
2
è 1
ø
2
1

• Final form of the time-spectral equation in a time-domain and steady state !
VDNU + R(U) = 0, where U = (u0N , u1N ,… , u NN -1 )¶ u jNV+ VDN u jN + R(u jN ) = 0 ( j = 0,… N - 1)
¶t

8

8

Time Spectral Formulation (Structures)▣ Dynamic Structural Analysis➢ Governing equation𝑀𝒙ሷ 𝜓 + 𝐶 𝜓 𝒙ሶ 𝜓 + 𝐾 𝜓 𝒙 𝜓 = 𝐹(𝑥, 𝑥,ሶ 𝜓)
𝒙: nodal displacements➢ First order ODE form

𝝍: azimuth angle (0  ~ 360)𝑦ሶ 𝜓 = 𝐴𝑦 𝜓 + 𝐵𝑓(𝜓)0where, 𝑦 𝜓 = 𝑥(𝜓)
,
𝑓
𝜓
=
,𝐴=
𝐹
ሶ
𝑥(𝜓)
0
𝑀−1
9

0
−𝑀−1 𝐾𝐼, and 𝐵 =−1
−𝑀 𝐶

Time Spectral Formulation (Structures)
Spectral Method▣ Dynamic Structural Analysis using Spectral Formulation➢ State-vector form equation𝑦ሶ 𝜓 = 𝐴𝑦 𝜓 + 𝐵𝑓(𝜓)➢ Assuming the solution with a Fourier series❖ solution approximation ❖ error minimization- Trigonometric (Fourier)- Galerkin- Chebychev- Collocation- Legendre
- Tau

𝑁𝐻

𝑦 𝑡 =𝑦ෞ0 + ෍ 𝑦ෞ
𝑐𝑛 cos 𝜔𝑛𝑡 + 𝑦ෞ
𝑠𝑛 sin 𝜔𝑛𝑡
𝑛=1➢ The time derivative can be converted into a Matrix Vector product. It can be solved by applying pseudo-time
stepping
𝜕𝑦𝑇𝑆+ 𝐷𝑁 𝑦𝑇𝑆 − 𝐴𝑦𝑇𝑆 − B𝑓𝑇𝑆 = 0
𝜕𝜏where,𝑦𝑇𝑆 =𝑦(𝜓0 + ∆𝜓)𝑦(𝜓0 + 2∆𝜓)⋮𝑦(𝜓0 + 2𝜋)

𝑓𝑇𝑆 =
10

𝑓(𝜓0 + ∆𝜓)𝑓(𝜓0 + 2∆𝜓)⋮𝑓(𝜓0 + 2𝜋)

𝜏: pseudo timeProposed FSI in Time-Spectral Formulation▣ Time Spectral form of Governing Equations
Pressure Field
Displacement Field

Fluid Equation

𝑉

Structural Equation

𝜕𝒘𝒕𝒔+ 𝑉𝝎𝑫𝒘𝒕𝒔 + 𝑹 𝒘𝒕𝒔 = 0
𝜕𝜏
𝑤𝑗𝑘

𝜕𝒚𝒕𝒔
+ 𝜔𝑫𝒚𝒕𝒔 − 𝑨𝒚𝒕𝒔 − 𝑩𝒇𝒕𝒔 = 0
𝜕𝜏
𝑤𝑗𝑘+𝑟

𝑃𝑗𝑘+𝑟

𝑤𝑗𝑘+2𝑟

𝑃𝑗𝑘+2𝑟11

𝑤𝑗𝑘+3𝑟𝑃𝑗𝑘+3𝑟𝑤𝑗𝑘 : fluid state vector
𝑢𝑗𝑙 : structural state vector
𝑥𝑗𝑘 : position of grid point
𝑃𝑗𝑘 : fluid pressure𝛿𝑗𝑘 : structural displacement

TS FSI Validation: Sectional Airloads Comparison (CFD/CSD)
86.5% R

0

-50

90

180

270

360

C9017 (dynamic stall)
86.5% R

0

180

270

360

Pitching moment

50
0

90

86.5% R

Spectra
-50
l

-50

-100
10

-100
360
0

90
180 270
Azimuth, degs.

Time-Acc90
180 270
Azimuth, degs.

360

180

270

360

-50

0

90

180

270

360

50

UR
UR
UR

0

0

-50

-50
0

90

180

270

360

12
0

90

180

270

360

lb/ft

lb/ft

chord force
50

Spectral

100

400

200

96.5% R

200

Time-Acc

0

90

180

Flig
ht

270

0
-200

360

-400
0

90

180

270

360

90
180 270
Azimuth, degs.

360

600

400

Fligh
t
90

400

600
0

-50
0

600

-400
0

50
0

600

-200

50

96.5% R

Normal force

0

lb/ft

ft-lb/ft¼ -c

Time-A
-50
cc
Fligh
t
-100

50

100

96.5% R

50

0

-100
0

lb/ft

UH-60A
C8534
with 7 harmonics

ft-lb/ft

50

400

200

200

Spectr
al

0
-200

-400
0

90
180 270
Azimuth, degs.

360

0
-200

-400
0

FlightTime-Accurate (TA)Spectral (TS)

Speed: 155 kts
Thrust: 17,500 lbs

Contents
I.

Introduction

II. Fluid-Structure Interaction in Time Spectral Form

III. Adjoint Sensitivity for Time Spectral Form
IV. Sensitivity Analysis Results
I. Conclusions

13

Sensitivities and Design▣ Approach: Sensitivity Analysis for unsteady problems.
➢ Shape Optimization requires gradient with respect to a large number of design variables.
With N design variables

Finite Difference Method(FDM)

N+1 Aeroelastic Simulations

𝑓 𝑥 + ∆𝑥 − 𝑓(𝑥)∆𝑥
➢ Adjoint Method is extremely handy in such cases
𝑓′ 𝑥 ≈

1 Aeroelastic+
Solution

Adjoint Method

1 Adjoint
Solution𝜕𝑅𝜕𝐼( )𝑇 𝜆 = −𝜕𝑤
𝜕𝑤➢ Disadvantage: For unsteady problems, physics residuals and adjoint variables need to be stored at each physical time step→ Memory and time intensive.
14

Coupled Adjoint for Coupled Sensitivity Analysis▣ Coupled Adjoint Analysis➢ Gradient computation using adjoint method for a coupled FSI problem➢ Objective function: 𝐼 = 𝐼(𝑤, 𝑦, 𝑥, 𝑏)

𝒘: fluid state variables
𝒚: structural state variables
𝒙: mesh state variables
𝒃: design variables➢ Total derivative of objective function w.r.t design variables
𝑑𝐼 𝜕𝐼 𝜕𝐼 𝜕𝑤 𝜕𝐼 𝜕𝑦 𝜕𝐼 𝜕𝑥=+++
𝑑𝑏 𝜕𝑏 𝜕𝑤 𝜕𝑏 𝜕𝑦 𝜕𝑏 𝜕𝑥 𝜕𝑏➢ State equations for fluid and structural system and their derivatives w.r.t design variables
Fluid: 𝑅 𝑤, 𝑦, 𝑥, 𝑏 = 0 →

𝑑𝑅
𝜕𝑅
𝜕𝑅 𝜕𝑤
𝜕𝑅 𝜕𝑦
𝜕𝑅 𝜕𝑥
= +
+
+
=0
𝑑𝑏
𝜕𝑏
𝜕𝑤 𝜕𝑏
𝜕𝑦 𝜕𝑏
𝜕𝑥 𝜕𝑏

Structure: 𝑆 𝑤, 𝑦, 𝑥, 𝑏 = 0 →

𝑑𝑆
𝜕𝑆
𝜕𝑆 𝜕𝑤
𝜕𝑆 𝜕𝑦
𝜕𝑆 𝜕𝑥
= +
+
+
=0
𝑑𝑏
𝜕𝑏
𝜕𝑤 𝜕𝑏
𝜕𝑦 𝜕𝑏
𝜕𝑥 𝜕𝑏

➢ As the residual derivatives are zero, they can be multiplied with an adjoint vector and added to the objectivefunction derivative.
𝑑𝐼
𝑑𝑏𝜕𝐼 𝜕𝐼 𝜕𝑤 𝜕𝐼 𝜕𝑦 𝜕𝐼 𝜕𝑥
𝜕𝑅 𝜕𝑅 𝜕𝑤 𝜕𝑅 𝜕𝑦 𝜕𝑅 𝜕𝑥
𝜕𝑆 𝜕𝑆 𝜕𝑤 𝜕𝑆 𝜕𝑦 𝜕𝑆 𝜕𝑥
=
+
+
+
+ 𝜆𝑇
+
+
+
+ 𝜙𝑇 ( +++)𝜕𝑏 𝜕𝑤 𝜕𝑏 𝜕𝑦 𝜕𝑏 𝜕𝑥 𝜕𝑏
𝜕𝑏 𝜕𝑤 𝜕𝑏 𝜕𝑦 𝜕𝑏 𝜕𝑥 𝜕𝑏
𝜕𝑏
𝜕𝑤
𝜕𝑏
𝜕𝑦
𝜕𝑏
𝜕𝑥
𝜕𝑏
=
Zero
= Zero𝝀: adjoint vector for flow state
𝝓: adjoint vector for structural state15

Coupled Adjoint for Coupled Sensitivity Analysis▣ Coupled Adjoint Analysis➢ Rearranging the terms
𝑑𝐼 𝜕𝐼
𝜕𝑅
𝜕𝑆
=
+ 𝜆𝑇
+ 𝜙𝑇
𝑑𝑏 𝜕𝑏
𝜕𝑏
𝜕𝑏
+

𝜕𝐼
𝜕𝑅
𝜕𝑆 𝜕𝑤
𝜕𝐼
𝜕𝑅
𝜕𝑆 𝜕𝑦
+ 𝜆𝑇
+ 𝜙𝑇
+ ( + 𝜆𝑇
+ 𝜙𝑇 )
𝜕𝑤
𝜕𝑤
𝜕𝑤 𝜕𝑏
𝜕𝑦
𝜕𝑦
𝜕𝑦 𝜕𝑏
= ZeroR : fluid residuals
S : Structural residuals
𝒘: fluid state variables
𝒚: structural state variables
𝒙: mesh state variables
𝒃: design variables= Zero➢ The arbitrary adjoint vector can be chosen to be independentof the derivative of state variables w.r.t.

the design variables.
➢ The adjoint equation to be solved

𝜕𝑅 𝜕𝑅
𝜕𝑤 𝜕𝑦
𝜕𝑆 𝜕𝑆
𝜕𝑤 𝜕𝑦

𝑇

𝜕𝐼
𝜆
𝜕𝑤
= − 𝜕𝐼𝜙𝜕𝑦➢ Total derivative of objective function w.r.t. design variables is
𝑑𝐼 𝜕𝐼 𝜕𝐼 𝜕𝑤 𝜕𝐼 𝜕𝑦 𝜕𝐼
𝜕𝑅
𝜕𝑆
𝑇
𝑇
=
+
+
=
+𝜆
+𝜙
𝑑𝑏 𝜕𝑏 𝜕𝑤 𝜕𝑏 𝜕𝑦 𝜕𝑏 𝜕𝑏
𝜕𝑏
𝜕𝑏
16

Design Framework

Source: S. Choi, K. Lee, J.J. Alonso “Helicopter Rotor Design using a Time Spectral and Adjoint Based Method”, Journalof Aircraft, Vol. 51, No. 2, March-April, 2014,17

Aero-Only Design at Forward Flight (Flight 8534 of UH-60A)
Design condition:• At each iteration, new control angles and aeroelastic deformation sh
ould be provided from the Comprehensive Analysis• Simplifications for “aerodynamic design”- constant aeroelastic deformation, and constant shaft angle.
- only two constraints.
• Objective function : reduce Torque ( CQ).
• Constraint 1 : same Thrust (CT ).
• Constraint 2 : same or less Drag Force ( CD).
• A total of 4 harmonics (9 time instances) are used
• Using NPSOL (Nonlinear Programming SOLver).
18

Design variables••••

Chord length and position of leading edge / sweep at 84.7% and 94.2%, 100% = 6
Twist angle at the 10 span locations = 10
Airfoil camber/thickness changes at 10 locations around airfoil along 9 sections on
the span (using Hicks-Henne bump functions) = 90
A total of 106 design variables
chord length, sweep

Twist at 10 radial locations

Example of bump19
19

Sensitivity Analysis Results
𝜕𝐼
𝐷𝑟𝑎𝑔
𝜕𝑏

▣ Sensitivity Analysis Results (Aero only)1➢ Object Function : Drag

9
2

𝜕𝑅 𝜕𝑅
𝜕𝑤 𝜕𝑦
𝜕𝑆 𝜕𝑆
𝜕𝑤 𝜕𝑦

𝑇

𝜓 = 0°

𝜕𝐼𝜆𝜕𝑤= − 𝜕𝐼3𝜙𝜕𝑦𝜓 = 40°8

𝜕𝑅: (368,640 x 368,640) M
𝜕𝑤

atrix

most sensitive locat 4

𝜓 = 80°𝜓 = 280°

AdvancingSide

RetreatingSide7

𝜓 = 240°ion

𝜓 = 120°6 78
54

𝜓 = 320°

𝜓 = 200°

𝜓 = 160°

9
3

2

1
10

6

5
Forward20

Sensitivity Analysis Results
𝜕𝐼
𝐿𝑖𝑓𝑡
𝜕𝑏

▣ Sensitivity Analysis Results (Aero only)➢ Object Function : Lift

1
9

2
𝜕𝑅 𝜕𝑅
𝜕𝑤 𝜕𝑦
𝜕𝑆 𝜕𝑆
𝜕𝑤 𝜕𝑦

𝑇

𝜓=0

𝜕𝐼
𝜆
𝜕𝑤
= − 𝜕𝐼3𝜙𝜕𝑦𝜓 = 40°𝜓 = 320°8
𝜓 = 80°𝜓 = 280°

AdvancingSide

RetreatingSide7

𝜓 = 240°
𝜓 = 120°6 78
54

𝜓 = 160°

4
9
3

2

1
10

𝜓 = 200°

5

6
Forward21

Optimized Resultsbaseline

optimized

2222

CFD/CSD Coupled Validation• After 5 CFD/CSD coupling iterations.
CFD : Time-accurate, Navier-Stokes computation
CSD : UMARC (3 degree trim - thrust, rolling, pitching M constrained, shaft angle fixed)

normal force (M2Cn)
Aero-optimization validation
7.4% reduction in torque.
almost constant in thrust.
but 30% increase in rolling M.
15% increase in pitching M.

flow
direction

chord force (M2Cc )CFD/CSD coupled
5% decrease in torque.
Constant in thrust.
3% increase in rolling M.
2% increase in pitching M.

23

23

Contents
I.

Introduction

II. Fluid-Structure Interaction in Time Spectral Form

III. Adjoint Sensitivity for Time Spectral Form
IV. Sensitivity Analysis Results
I. Conclusions

24Computational Modelling: Comprehensive Analysis (CA)▣ UMARC : University of Maryland Advanced Rotorcraft Code [Ref.6]

UMARC (CA)Aerodynamic Model(lifting line theory + wake model)

Structural Model(E-B beam model)

Flight Dynamics

Coupled Trim Analysis

Stability Analysis
25

Computational Modelling: Structures▣ Coupled Trim Analysis Results (UMARC)
Iteration No.

𝑐𝑡 /𝜎: disc loading
𝜎: solidity

Iteration No.
(deg.)

Iteration No.

Iteration No.
control angle: αs

ϕs

θ0

θ1c

θ1s

θtr
26

Computational Modelling: Loosely Coupled FSI Analysis▣ FSI Analysis Procedure

UMARC (CA)

Lifting Line Theory

Initial Aerodynamic Force
Deformation(𝜹) : update CFD Grid
Structural + Trim
Analysis

CFD Analysis

Pressure(𝒑) : update air loads

27

Computational Modelling: Loosely Coupled FSI Analysis▣ FSI Analysis Results

Azimuth Angle (𝝍)Azimuth Angle (𝝍)Azimuth Angle (𝝍)28

Iteration No.

Sensitivity Analysis Results▣ Coupled Sensitivity Analysis: Aero + Structure(368,640) (3,348)
Aerodynamic Jacobian(368,640)(3,348)
Coupled Cross Jacobians
✓

𝜕𝑅 𝜕𝑅
𝜕𝑤 𝜕𝑦
𝜕𝑆 𝜕𝑆
𝜕𝑤 𝜕𝑦

𝑇

𝜕𝐼
𝜆
𝜕𝑤
= − 𝜕𝐼𝜙𝜕𝑦Structural Jacobian

𝜕𝑅 𝜕𝑆
𝜕𝑆
𝜕𝑅
, and
are sparse matrices,
is densely populated matrix
𝜕𝑤 𝜕𝑦
𝜕𝑤
𝜕𝑦

✓ Due to the large size of the adjoint matrix, GMRES, a Krylov subspace solver has been used to solve the system.
✓ This has been implemented using PETSC, a suite of scalable and parallel routines for the solution of large scale PDEs.
✓ The above system takes around ~1500 iterations with 600 restart iterations to converge
29

Sensitivity Analysis Results
72

73

71

▣ Coupled Sensitivity Analysis: Aero + Structure

70

76

69
68

I (objective function) = Sum of all displacement

4

75

67

69th design variabl
edesign variable No.

▣ Future Design
I (objective function) = torqueC (constraints) = moments and thrust
30

74Contents
I.

Introduction

II. Fluid-Structure Interaction in Time Spectral Form

III. Adjoint Sensitivity for Time Spectral Form
IV. Sensitivity Analysis Results
I. Conclusions

31

Conclusion▣ Time spectral and adjoint-based method is effective for the design
involving multiphysics problems.
▣ An accurate but efficient coupled sensitivity analysis method for rotor
design is developed.
▣ Aerodynamics only and Coupled sensitivity analysis is performed andvalidated by comparing with FDM results.
▣ Fluid-Structure coupled adjoint-based sensitivity analysis will be used
to optimize the shape of rotor blade.

32

AMDL

Thank you for your attention !

33

33

Adjoint based Sensitivity Analysis▣ Coupled Adjoint Analysis➢ Gradient computation using adjoint method for a coupled FSI problem➢ Objective function: 𝐼 = 𝐼(𝑤, 𝑦, 𝑥, 𝑏)
𝒘: fluid state variables
𝒚: structural state variables
𝒙: mesh state variables
𝒃: design variables➢ Total derivative of objective function w.r.t design variables
𝑑𝐼 𝜕𝐼 𝜕𝐼 𝜕𝑤 𝜕𝐼 𝜕𝑦 𝜕𝐼 𝜕𝑥=+++
𝑑𝑏 𝜕𝑏 𝜕𝑤 𝜕𝑏 𝜕𝑦 𝜕𝑏 𝜕𝑥 𝜕𝑏➢ State equations for fluid and structural system and their derivatives w.r.t design variables
𝑑𝑅

𝜕𝑅

𝜕𝑅 𝜕𝑤

𝜕𝑅 𝜕𝑦

𝜕𝑅 𝜕𝑥

𝑑𝑆

𝜕𝑆

𝜕𝑆 𝜕𝑤

𝜕𝑆 𝜕𝑦

𝜕𝑆 𝜕𝑥

Fluid: 𝑅 𝑤, 𝑦, 𝑥, 𝑏 = 0 → 𝑑𝑏 = 𝜕𝑏 + 𝜕𝑤 𝜕𝑏 + 𝜕𝑦 𝜕𝑏 + 𝜕𝑥 𝜕𝑏 = 0
Structure: 𝑆 𝑤, 𝑦, 𝑥, 𝑏 = 0 → 𝑑𝑏 = 𝜕𝑏 + 𝜕𝑤 𝜕𝑏 + 𝜕𝑦 𝜕𝑏 + 𝜕𝑥 𝜕𝑏 = 0➢ As the residual derivatives are zero, they can be multiplied with an adjoint vector and added to the objectivefunction derivative.
𝑑𝐼 𝜕𝐼 𝜕𝐼 𝜕𝑤 𝜕𝐼 𝜕𝑦 𝜕𝐼 𝜕𝑥
𝜕𝑅 𝜕𝑅 𝜕𝑤 𝜕𝑅 𝜕𝑦 𝜕𝑅 𝜕𝑥
𝜕𝑆 𝜕𝑆 𝜕𝑤 𝜕𝑆 𝜕𝑦 𝜕𝑆 𝜕𝑥
=
+
+
+
+ 𝜆𝑇
+
+
+
+ 𝜙𝑇 ( +++)𝑑𝑏 𝜕𝑏 𝜕𝑤 𝜕𝑏 𝜕𝑦 𝜕𝑏 𝜕𝑥 𝜕𝑏
𝜕𝑏 𝜕𝑤 𝜕𝑏 𝜕𝑦 𝜕𝑏 𝜕𝑥 𝜕𝑏
𝜕𝑏 𝜕𝑤 𝜕𝑏 𝜕𝑦 𝜕𝑏 𝜕𝑥 𝜕𝑏
𝝀: adjoint vector for flow state
𝝓: adjoint vector for structural state34= Zero= ZeroAdjoint based Sensitivity Analysis▣ Coupled Adjoint Analysis➢ Rearranging the terms
𝑑𝐼 𝜕𝐼
𝜕𝑅
𝜕𝑆
=
+ 𝜆𝑇
+ 𝜙𝑇
𝑑𝑏 𝜕𝑏
𝜕𝑏
𝜕𝑏
+

𝜕𝐼
𝜕𝑅
𝜕𝑆 𝜕𝑤
𝜕𝐼
𝜕𝑅
𝜕𝑆 𝜕𝑦
+ 𝜆𝑇
+ 𝜙𝑇
+ ( + 𝜆𝑇
+ 𝜙𝑇 )
𝜕𝑤
𝜕𝑤
𝜕𝑤 𝜕𝑏
𝜕𝑦
𝜕𝑦
𝜕𝑦 𝜕𝑏

R : fluid residuals
S : Structural residuals
𝒘: fluid state variables
𝒚: structural state variables
𝒙: mesh state variables
𝒃: design variables= Zero= Zero➢ The arbitrary adjoint vector can be chosen to be independent of the derivative of state variables w.r.t. thedesign variables.
𝑇➢ The adjoint equation to be solved
𝜕𝑅 𝜕𝑅
𝜕𝐼
𝜆
𝜕𝑤 𝜕𝑦
𝜕𝑤
=
−
𝜕𝑆 𝜕𝑆
𝜕𝐼
𝜙
𝜕𝑤 𝜕𝑦
𝜕𝑦
➢ Total derivative of objective function w.r.t. design variables is
𝑑𝐼 𝜕𝐼 𝜕𝐼 𝜕𝑤 𝜕𝐼 𝜕𝑦 𝜕𝐼
𝜕𝑅
𝜕𝑆
=
+
+
=
+ 𝜆𝑇
+ 𝜙𝑇
𝑑𝑏 𝜕𝑏 𝜕𝑤 𝜕𝑏 𝜕𝑦 𝜕𝑏 𝜕𝑏
𝜕𝑏
𝜕𝑏
35

Contents
I. Introduction (Motivation, Approach & Contribution)
II. Unsteady Dynamic Analysis using Time Spectral Form
III. Adjoint Based Sensitivity Analysis➢ Adjoint Equations➢ Derivation of Jacobian Terms

IV. Computational Modelling➢➢➢

Aerodynamics
Structures
Fluid-Structure Interaction

V. Sensitivity Analysis Results
VI. Conclusion and Future Work36

Derivation of Jacobian sub Matrices▣

𝜕𝑅computed by differentiating the internal flux and boundary condition functions.
𝜕𝑤

𝜕𝑅 𝜕𝑅
𝜕𝑤 𝜕𝑦
𝜕𝑆 𝜕𝑆
𝜕𝑤 𝜕𝑦

▣

𝑇

𝜕𝑅

𝜕𝑤

differentiate central flux and dissipationdifferentiate the contribution of boundary condition

𝜕𝑆
calculated by differentiating structural equations of motion (S) with respect to structural state (y).
𝜕𝑦
𝑇

𝜕𝑅 𝜕𝑅
𝜕𝑤 𝜕𝑦
𝜕𝑆 𝜕𝑆
𝜕𝑤 𝜕𝑦

▣

𝜕𝐼
𝜆
𝜕𝑤
= − 𝜕𝐼
𝜙
𝜕𝑦

𝜕𝐼
𝜆
𝜕𝑤
= − 𝜕𝐼𝜙𝜕𝑦

𝑆 = 𝜔𝐷𝑦 − 𝐴𝑦 − B𝑓

𝜕𝑆
= 𝜔𝐷 − 𝐴𝜕𝑦𝜕𝑆
calculated by differentiating structural equations of motion (S) with respect to fluid state (w) by
𝜕𝑤

using chain rule.
𝜕𝑅 𝜕𝑅
𝜕𝑤 𝜕𝑦
𝜕𝑆 𝜕𝑆
𝜕𝑤 𝜕𝑦

𝑇

𝜕𝐼
𝜆
𝜕𝑤
= − 𝜕𝐼𝜙𝜕𝑦𝜕𝑆 𝜕𝑆 𝜕𝑓 𝜕𝑝=
𝜕𝑤 𝜕𝑓 𝜕𝑝 𝜕𝑤
37

𝜕𝑆

𝑆 = 𝜔𝐷𝑦 − 𝐴𝑦 − B𝑓, 𝜕𝑓 = −𝐵Derivation of Jacobian sub Matrices▣

𝜕𝑅computed by differentiating the internal flux and boundary condition functions.
𝜕𝑤

𝜕𝑅 𝜕𝑅
𝜕𝑤 𝜕𝑦
𝜕𝑆 𝜕𝑆
𝜕𝑤 𝜕𝑦

▣

𝑇

𝜕𝑅

𝜕𝑤

𝜕𝑆
calculated by differentiating structural equations of motion (S) with respect to structural state (y).
𝜕𝑦
𝑇

𝜕𝑅 𝜕𝑅
𝜕𝑤 𝜕𝑦
𝜕𝑆 𝜕𝑆
𝜕𝑤 𝜕𝑦

▣

𝜕𝐼
𝜆
𝜕𝑤
= − 𝜕𝐼
𝜙
𝜕𝑦

𝜕𝐼
𝜆
𝜕𝑤
= − 𝜕𝐼𝜙𝜕𝑦

𝑆 = 𝜔𝐷𝑦 − 𝐴𝑦 − B𝑓

𝜕𝑆
= 𝜔𝐷 − 𝐴𝜕𝑦𝜕𝑆
calculated by differentiating structural equations of motion (S) with respect to fluid state (w) by
𝜕𝑤

using chain rule.
𝜕𝑅 𝜕𝑅
𝜕𝑤 𝜕𝑦
𝜕𝑆 𝜕𝑆
𝜕𝑤 𝜕𝑦

𝑇

𝜕𝐼
𝜆
𝜕𝑤
= − 𝜕𝐼𝜙𝜕𝑦𝜕𝑆 𝜕𝑆 𝜕𝑓 𝜕𝑝=
𝜕𝑤 𝜕𝑓 𝜕𝑝 𝜕𝑤
38

𝜕𝑆

𝑆 = 𝜔𝐷𝑦 − 𝐴𝑦 − B𝑓, 𝜕𝑓 = −𝐵Derivation of Jacobian sub Matrices▣

𝜕𝑅computed by differentiating the internal flux and boundary condition functions.
𝜕𝑤

𝜕𝑅 𝜕𝑅
𝜕𝑤 𝜕𝑦
𝜕𝑆 𝜕𝑆
𝜕𝑤 𝜕𝑦

▣

𝑇

𝜕𝑅

𝜕𝑤

𝜕𝑆
calculated by differentiating structural equations of motion (S) with respect to structural state (y).
𝜕𝑦
𝑇

𝜕𝑅 𝜕𝑅
𝜕𝑤 𝜕𝑦
𝜕𝑆 𝜕𝑆
𝜕𝑤 𝜕𝑦

▣

𝜕𝐼
𝜆
𝜕𝑤
= − 𝜕𝐼
𝜙
𝜕𝑦

𝜕𝐼
𝜆
𝜕𝑤
= − 𝜕𝐼𝜙𝜕𝑦

𝑆 = 𝜔𝐷𝑦 − 𝐴𝑦 − B𝑓

𝜕𝑆
= 𝜔𝐷 − 𝐴𝜕𝑦𝜕𝑆
calculated by differentiating structural equations of motion (S) with respect to fluid state (w) by
𝜕𝑤

using chain rule.
𝜕𝑅 𝜕𝑅
𝜕𝑤 𝜕𝑦
𝜕𝑆 𝜕𝑆
𝜕𝑤 𝜕𝑦

𝑇

𝜕𝐼
𝜆
𝜕𝑤
= − 𝜕𝐼𝜙𝜕𝑦𝜕𝑆 𝜕𝑆 𝜕𝑓 𝜕𝑝=
𝜕𝑤 𝜕𝑓 𝜕𝑝 𝜕𝑤
39

𝜕𝑆

𝑆 = 𝜔𝐷𝑦 − 𝐴𝑦 − B𝑓, 𝜕𝑓 = −𝐵Derivation of Jacobian sub Matrices▣

𝜕𝑅computed by differentiating the internal flux and boundary condition functions.
𝜕𝑤

𝜕𝑅 𝜕𝑅
𝜕𝑤 𝜕𝑦
𝜕𝑆 𝜕𝑆
𝜕𝑤 𝜕𝑦

▣

𝑇

𝜕𝑅

𝜕𝑤

𝜕𝑆
calculated by differentiating structural equations of motion (S) with respect to structural state (y).
𝜕𝑦
𝑇

𝜕𝑅 𝜕𝑅
𝜕𝑤 𝜕𝑦
𝜕𝑆 𝜕𝑆
𝜕𝑤 𝜕𝑦

▣

𝜕𝐼
𝜆
𝜕𝑤
= − 𝜕𝐼
𝜙
𝜕𝑦

𝜕𝐼
𝜆
𝜕𝑤
= − 𝜕𝐼𝜙𝜕𝑦

𝑆 = 𝜔𝐷𝑦 − 𝐴𝑦 − B𝑓

𝜕𝑆
= 𝜔𝐷 − 𝐴𝜕𝑦𝜕𝑆
calculated by differentiating structural equations of motion (S) with respect to fluid state (w) by
𝜕𝑤
𝜌
𝑤
1

using chain rule.
𝜕𝑅 𝜕𝑅
𝜕𝑤 𝜕𝑦
𝜕𝑆 𝜕𝑆
𝜕𝑤 𝜕𝑦

𝑇

𝜕𝐼
𝜆
𝜕𝑤
= − 𝜕𝐼𝜙𝜕𝑦𝜕𝑆 𝜕𝑆 𝜕𝑓 𝜕𝑝=,
𝜕𝑤 𝜕𝑓 𝜕𝑝 𝜕𝑤
δp = 𝛾 − 1

p= 𝛾−1

δ𝑤5 +

1
𝑤5 −
2𝑤1

𝑤2 2 + 𝑤3 2 + 𝑤4 2

𝑤2
, 𝑤 = 𝑤3
𝑤4
𝑤5

𝜌𝑢
= 𝜌𝑣
𝜌𝑤
𝜌𝐸

1
1
2 + 𝑤 2 + 𝑤 2 δ𝑤 −𝑤
𝑤2 𝛿𝑤2 + 𝑤3 𝛿𝑤3 + 𝑤4 𝛿𝑤4
2
3
4
1
2
40
𝑤
2𝑤1
1

Derivation of Jacobian sub Matrices▣ The fluid residual, R, is not explicitly dependent on the structural state, y, but through the wallboundary condition.
𝜕𝑅 𝜕𝑅
𝜕𝑤 𝜕𝑦
𝜕𝑆 𝜕𝑆
𝜕𝑤 𝜕𝑦

𝑇

𝜕𝐼
𝜆
𝜕𝑤
= − 𝜕𝐼
𝜙
𝜕𝑦

y

𝑥𝑠

R R sx xv xs=
y sx xv xs y
xs

xv

𝑥𝑣

sx

R

𝑦 structural state (deflection)
𝑥𝑠 surface mesh𝑥𝑣 volume mesh𝑠𝑥 mesh metrics
R aero residual41

Sensitivity Analysis Results▣ Sensitivity Analysis Results (Aero only)➢ Comparison with FDM (Finite Difference Method) analysis

𝜕𝐼
𝜕𝑏

𝑑𝐼
𝑑𝑏𝐴𝐷𝐽

𝜕𝐼
𝜕𝑏

𝑑𝐼
𝑑𝑏𝐹𝐷𝑀

𝑑𝐼
𝑑𝑏𝐹𝐷𝑀

Azimuth Angle (𝝍)𝑑𝐼 𝜕𝐼 𝜕𝐼 𝜕𝑤=+
𝑑𝑏 𝜕𝑏 𝜕𝑤 𝜕𝑏

𝜕𝐼
𝜕𝑅
=
+𝜆
𝜕𝑏
𝜕𝑏
42

𝑑𝐼
𝑑𝑏𝐴𝐷𝐽

Azimuth Angle (𝝍)Sensitivity Analysis Results▣ Sensitivity Analysis Results (Aero only)➢ Comparison with FDM (Finite Difference Method) analysis (updated after dissertation proposal)

𝜕𝐼
𝜕𝑏

𝑑𝐼
𝑑𝑏𝐴𝐷𝐽

𝜕𝐼
𝜕𝑏
𝑑𝐼
𝑑𝑏𝐹𝐷𝑀

𝑑𝐼
𝑑𝑏𝐴𝐷𝐽(𝑂𝐿𝐷)
𝑑𝐼
𝑑𝐼
𝑑𝑏𝐹𝐷𝑀
𝑑𝑏𝐴𝐷𝐽(𝑁𝐸𝑊)Azimuth Angle (𝝍)Azimuth Angle (𝝍)43Sensitivity Analysis Results▣ Sensitivity Analysis Results (Aero only)➢ Comparison with FDM (Finite Difference Method) analysis (updated after dissertation proposal)

𝜕𝐼
𝜕𝑏

𝜕𝐼
𝜕𝑏

Azimuth Angle (𝝍)𝑑𝐼
𝑑𝐼
𝑑𝑏𝐹𝐷𝑀 𝑑𝐼
𝑑𝑏𝐴𝐷𝐽(𝑂𝐿𝐷)
𝑑𝑏𝐴𝐷𝐽(𝑁𝐸𝑊)Azimuth Angle (𝝍)

𝑑𝐼
𝑑𝑏𝐹𝐷𝑀

𝑑𝐼
𝑑𝑏𝐴𝐷𝐽

44

Sensitivity Analysis Results▣ Finite Difference Method

𝑓′ 𝑥➢ Step Size Study (3 time instances)DRAG

𝑓 𝑥 + ∆𝑥 − 𝑓(𝑥)
≈
∆𝑥

LIFT

MOMENT

1st Time Instance

2nd Time Instance

3rd Time Instance
delP=

0.1

1.0

10.0

45

0.1

1.0

10.0

Sensitivity Analysis Results▣ Finite Difference Method➢ Step Size Study (3 time instances)46

Publications▣ Kim, Hyunsoon, et al. "Towards the optimal operation of a thermal-recharging float in the ocean." Ocean Engineering156 (2018): 381-395. (published)▣ Im, Dong Kyun, Hyunsoon Kim, and Seongim Choi. "Mapped Chebyshev Pseudo-Spectral Method for Dynamic Aero-Elastic P
roblem of Limit Cycle Oscillation." International Journal of Aeronautical and Space Sciences 19, no. 2 (2018): 316-329. (publis
hed)▣ Prasad, Rachit, Hyunsoon Kim, Seongim Choi, and Seulgi Yi. "High fidelity prediction of flutter/LCO using time spectral method." In 2018 AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, p. 0459. 2018. (published)▣ Prasad, Rachit, Hyunsoon Kim, and Seongim Choi. "Flutter Related Design Optimization using the Time Spectral and Coupled
Adjoint Method." In 2018 AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, p. 0101. 2018. (published)▣ Prasad, Rachit, Hyunsoon Kim, and Seongim Choi. "Adjoint based Finite Element Model Updating and Validation using Time D
omain based FSI Analysis." In 58th AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, p. 1125
. 2017. (published)▣ Prasad, Rachit, Hyunsoon Kim, Dongkyun Im, Seongim Choi, and Seulgi Yi. "Analysis and Sensitivity Calculation using High Fi
delity Spectral Formulation-Based FSI and Coupled Adjoint Method." In 17th AIAA/ISSMO Multidisciplinary Analysis and Optim
ization Conference, p. 3993. 2016. (published)▣ “Coupled Adjoint-based Rotor Design using a Fluid Structure Interaction in Time Spectral Form” (under preparation)

47

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