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Sensitivity Analysis

- Multi-disciplinary Design Optimization
- Centre for Aerospace Systems Design Engineering
- Department of Aerospace Engineering
- Indian Institute of Technology
- Mumbai 400 076

Sensitivity Analysis

- Engineering Analysis for MDO is usually

computationally intensive - Evolutionary algorithms for optimization require

too many function calls (analysis) - Efficient, accurate evaluation of gradients is

required by gradient based optimizer - Sensitivity Analysis (generating gradients) is an

important component in MDO

Sensitivity Analysis

- Sensitivity Analysis
- Derivative of output w.r.t. input
- Gradient of output w.r.t. inputs
- Jacobian of outputs w.r.t. inputs

Sensitivity Analysis

- Methods for sensitivity estimation
- Finite Difference Method
- Analytical
- Direct
- Adjoint Continuous, Discrete
- Complex Variables Method
- Automatic Differentiation
- ADIFOR
- ADIC
- Global Sensitivity Equation (GSE)

Finite Difference Method

Finite Difference Methods

- Forward, backward or central differences may be

used - Derivative of yj w.r.t. xi by forward difference

is given by - Note all other elements of vector x are held

fixed at some value - N derivatives will require N1 function

evaluations

Finite Difference Methods

- Advantages
- Almost nil effort to implement
- Most optimizers support finding derivatives by

finite difference from within (requiring no

effort from user) - Well behaved with smooth functions that are

evaluated using non-iterative methods

Finite Difference Methods

- Disadvantages
- Noisy functions yield poor results. Most

iterative analysis with termination criteria are

noisy. Most engineering analysis are non-linear

and iterative in nature. - Accuracy sensitive to step size. 1 pp3 pp8
- N derivatives require N1 analysis.

Computationally intensive. 2 pp 87

Noisy Function, Step Size

f

y

Step size? Large Averaging Small round off in

df

An Example

- f(y, z) y0.1 z 0.2 , ?f/?y 0.1 y -0.9

z 0.2 - y 0.2, z 0.3
- f 0.669156, Analytical ?f/?y 0.334578
- Error in FD derivative
- dy ?f/?y by FD err
- 0.04000000 0.307801 -8.003
- 0.00250000 0.332713 -0.557
- 0.00015625 0.334167 -0.123
- 0.00000977 0.329590 -1.491
- 0.00000061 0.292969 -12.436

Check Derivative Routine

- f(xd) f(x) d f od2
- g(d) f(xd) f(x) d f od2
- g(d) / g(d/2) od2 / o(d/2)2
- Use function and derivative routines to compute

g(d) / g(d/2). If this value is of the order of

4 then the 2 routines are consistent

Check Derivative Routine

- x1 0.2 x2 0.3
- Function f 0.669156, Analytical ?f/?x1

0.334578 - Verification of FD Derivatives
- FD Step ?f/?x1 g(d)/g(d/2) d0.03
- 0.010000 0.327283E00 7.659
- 0.002500 0.332713E00 4.213
- 0.000625 0.334072E00 3.920
- 0.000156 0.334167E00 3.903
- 0.000039 0.334167E00 3.903
- 0.000010 0.329590E00 5.406
- 0.000002 0.317383E00

-1.077 - 0.000001 0.292969E00 1.360
- 0.334578E00 3.989 analytical
- derivative

Analytical Derivatives

Derivative of Response of an Analysis

- Evaluate f(x),
- Subject to h(x) 0
- x, f, h are all vectors
- x ? ?nx f ? ?nf h ? ?nz
- nx gt nz
- for any arbitrary x, h(x) ? 0
- Only (nx nz) values of x can be
- prescribed
- nz values of x get determined
- by h(x) 0

h(x) ? 0 violates some Conservation. It is not

OK to look at f(x) when x violates Physics.

Derivative of Response of an Analysis

Analysis

- Analysis, h(y,z) 0
- Input y design variable(s)
- Output z state variable(s)
- Analysis for non-linear system requires
- Value of y and initial guess for z
- evaluator for h in terms of y z
- iterator to update z, so that h(y,z) is driven to

0. - Convergence criteria to arrive at z(y)

Analysis

Response of Analysis

- Evaluate f(y, z)
- Subject to, h(y, z) 0
- Note
- y alone is design variable
- z is output of analysis, zy
- df/dy is required ? total derivative

Analytical Derivatives Across Analysis

- Sensitivity required, df/dy ? (Let y, z, f, h

be all scalars) - df/dy ?f/?y ?f/?z dz/dy
- dh ?h/?y dy ?h/?z dz 0
- dz/dy - (?h/?z)-1 ?h/?y
- df/dy ?f/?y - ?f/?z (?h/?z)-1 ?h/?y
- Requires all partial derivatives at (y, z)
- Analytical expressions for partial derivatives?

Analytical Derivatives Across Analysis

- Sensitivity Required
- df/dy ?f/?y - ?f/?z (?h/?z)-1 ?h/?y
- Direct Method
- df/dy ?f/?y - ?f/?z (?h/?z)-1 ?h/?y
- OR
- Adjoint Method
- df/dy ?f/?y - ?f/?z (?h/?z)-1 ?h/?y
- Difference!
- In the effort to compute bracketed quantities
- (solution of linear algebraic equations)
- when f and y are vectors

An Example

- f(y, z) y0.1 z 0.2
- ?f/?y 0.1 y -0.9 z 0.2 , ?f/?z

0.2 y 0.1 z -0.8 - h(y, z) y0.3 z 0.4 - 1.0
- ?h/?y 0.3 y -0.7 z 0.4, ?h/?z 0.4 y

0.3 z -0.6 - Y 0.2, Analysis No of iters 62, z

3.343695 - ?f/?y 0.541899 ?f/?z 0.064826
- ?h/?y 1.499999 ?h/?z 0.119628
- df/dy ?f/?y - ?f/?z (?h/?z)-1 ?h/?y -

0.27094

An Example

- Comparison With FD Across Analysis
- y 0.2, ? z 3.343695
- f 1.08380 Analytical deriv, df/dy -

0.27094 - dy df/dy by FD
- 0.00400000 -.26813149
- 0.00025000 -.27084351
- 0.00001563 -.26702881
- 0.00000391 -.21362303
- 0.00000024 -.48828122
- 0.00000006 1.95312488

An Example - Timings

- Let time for evaluation T
- ie. evaluation of h, f and their analytical

derivatives - FD across Analysis
- 2 (Niter T T) 2 T (Niter 1)
- (Analysis evaluation of f) to be done twice
- Analytical Derivative
- Niter T 4 T T T (Niter 5)
- 1 analysis 4 derivatives f
- Note Niter 62 in this case

When f, h, y, z Are All Vectors

- Direct Method

When f, h, y, z Are All Vectors

- Adjoint Method

Analytical Derivatives

- Direct and Adjoint Methods require partial

derivatives of f and h - Direct method requires solution of nznz system

for ny RHS vectors - Adjoint method requires solution of nznz syetm

for nf RHS vector - The nznz matrix is often the same as used in

analysis

Direct / Adjoint Methods

- Based on Control Theory Concept

Introduction

- Proposed by Antony Jameson (Stanford)
- Aerodynamic Design via Control Theory Journal

of Scientific Computing, vol. 3, 233-260, 1988. - Optimization is treated as a control problem with

constraints as the control variables

Continuous Adjoint Method

Discrete Adjoint Equations-1

Continuous Adjoint Equations

Continuous Field Equations h(y,z)

Discrete Field Equations

Discrete Adjoint Equations-2

Discrete Adjoint Method

Basic Formulation

(Adjoint Equation)

If ? is such that

then dI G. dy where

- Unlike FDM, solution of h(y,z)0 is not required

for each design variable

Quasi 1D Nozzle Flow

Objective function

Quasi 1D Nozzle Flow(contd.)

Equating terms with dU, Adjoint equation is

Adjoint boundary conditions

- Observations
- Similar procedure for higher dimensional flows
- Form of adjoint equation is strongly linked to

the field equation - Adjoint equations may be solved using similar

discretization and numerical solution strategies

of the field equations

Algorithm

Calculation of ?f/?y

Solution of Field Eq. h(y,z)0

Flow variables (z)

Solution of Adjoint Eq.

Lagrange Multipliers (?)

Calculation of ?h/ ?y (?h/ ?zi)(?zi/?aj)(?aj/?y

)

Calculation of Gradients (G)

?ai/?y

Grid Generator

a Grid variables (Cell areas/lengths)

Road Ahead

- Groups actively involved in further research
- Stanford University (Jameson et al)1,5,6,7
- Oxford University (Mike Giles)8
- Virginia Tech. (Dr. Eugene Cliff)3
- Old Dominian University (Baysal et al)
- Applications
- Extension to unstructured grids 2
- Optimization of a supersonic aircraft

configuration9 - Further areas of research
- A new reduced formulation for adjoint methods

proposed by Jameson7 to eliminate grid

sensitivity of gradients.

Complex Step Method

Complex Variable Method

- For f(x) ? R, it can be shown using

Cauchy-Riemann equality that - As no subtraction is involved, step size can be

as small as the machine precision allows - It has been found that usually a step size lt 10-8

gives near analytic solutions2 - First proposed by Lyness Moler1 in 1967

CVM An example

CVM Implementation

Declare all the variables as complex type

- All the algorithms can be broken into basic

operations - Operations that require redefinition (usually)
- Relational Operators
- Arithmetic functions operators
- FORTRAN-90 intrinsically supports most of the

complex operations - Two options for CVM implementation
- Source modification
- Operator Overloading

Redefine all the functions with complex

definitions

Modification of the I/O statements

Add complex step (h) to the desired variable

Gradient Calculation

CVM General Remarks

- Advantages
- Experimentation with step size not required
- Easy implementation
- Disadvantages
- Separate simulations for each gradient required
- Implementation of CVM as an automatic

differentiation tool3 - Groups working in this area
- Squire Trapp4
- Anderson Newman 5

An Example

- f(y, z) y0.1 z 0.2 , ?f/?y 0.1 y -0.9

z 0.2 - y 0.2, z 0.3
- Comparison with CVM
- f complex f anal df/dx1

dx1 comp df/dx1 - 0.669156 0.670339 0.3345779 0.4000E-01

0.3308477 - 0.669156 0.669168 0.3345779 0.4000E-02

0.3345398 - 0.669156 0.669156 0.3345779 0.4000E-04

0.3345779 - 0.669156 0.669156 0.3345779 0.4000E-05

0.3345779 - 0.669156 0.669156 0.3345779 0.4000E-06

0.3345779 - 0.669156 0.669156 0.3345779 0.4000E-08

0.3345779 - 0.669156 0.669156 0.3345779 0.4000E-10

0.3345779

An Example Timing Issues

- Complex arithmetic takes less than 4 times real

arithmetic - Small enough value of h can yield both f and

?f/?y in single execution

Automatic Differentiation

User Supplied Gradients

User Supplied Gradients

User Supplied Gradients

Gradients by ADIFOR /ADIC

How ADI uses chain rule?

- function1 ( x, y, z )
- function2 ( x, y, w, r )
- function3 ( w, r )
- function4 ( w, r, z )
- We are interested in
- ?z/?x and ?z/?y

- For function3 we can write
- ?r/?w
- If we want to use ?r/?w in
- calculation of ?r/?x, we need
- to use chain rule.
- ?r/?x ?r/?w ?w/?x
- In general we can write
- r ?r/?w w

How to use ADI generated code?

- function1 ( x, y, z )
- function2 ( x, y, w, r )
- function3 ( w, r )
- function4 ( w, r, z )

- f1 ( x, y, z, x, y, z )
- f2 ( x, y, w, r, x, y, w, r )
- f3 ( w, r, w, r )
- f4 ( w, r, z, w, r, z )

New function f1 evaluates the values and

directional gradient specified by direction (x,

y) z ?z/?x ?z/?y x yT To evaluate the

gradient vector we do the above evaluation in

directions (1, 0) and (0, 1)

ADI Sample Code

- To evaluate following function
- function abc ( y, z, w )
- w -y / (z z z)
- end
- ADI software usually breaks
- down the complex expressions
- into simple operations. This
- requires introduction of
- temporary variables to store
- intermediate results

- function abc ( y, z, w )
- t1 -y
- t2 z z
- t3 t2 z
- w t1 / t3
- end
- ADI will use the above code
- to use symbolic differentiation
- to calculate gradients
- Forward Mode
- Reverse Mode

ADI Forward Mode

- Uses successive chain rule to derive the

gradients - abc ( y, z, w )
- t1 -y
- t2 z z
- t3 t2 z
- w t1 / t3

- abc ( y, y, z, z, w, w )
- t1 -y
- t1 -y
- t2 z z
- t2 z z z z
- t3 t2 z
- t3 t2 z z t2
- w t1 / t3
- w ( t1 t3 w ) / t3

ADI Reverse Mode

- Set of operations can be looked as series of

operators working on input variables and

generating new variables

- abc ( y, z, w )
- t1 -y
- t2 z z
- t3 t2 z
- w t1 / t3

w W( T3( T2( T1( y, z ) ) ) w ?W/?T3

?T3/?T2 ?T2/?T1 (?T1/?y y ?T1/?z z)

If gradient is calculated this way, it is

actually in the reverse direction of the basic

computation. Hence the term reverse mode.

ADI Reverse Mode

- Basic Rules
- y f(s)
- sbar sbar ?y/?s ybar
- y f(s, t)
- sbar sbar ?y/?s ybar
- tbar tbar ?y/?t ybar
- abc ( y, z, w )
- t1 -y
- t2 z z
- t3 t2 z
- w t1 / t3

- abc ( y, y, z, z, w, w )
- t1 - y
- t2 z z
- t3 t2 z
- w t1 / t3
- t1bar 1 / t3
- t3bar -t1 / (t3 t3)
- t2bar t3bar z
- zbar t3bar t2
- zbar zbar t2bar z
- zbar zbar t2bar z
- ybar - t1bar
- w ybar y zbar z

ADI generated code

- FORWARD MODE
- t1 - y
- t1 - y
- t2 z z
- t2 z z z z
- t3 t2 z
- t3 t2 z t2 z
- w t1 / t3
- w (t1 - t3 w) / t3

- REVERSE MODE
- t1 - y
- t2 z z
- t3 t2 z
- w t1 / t3
- t1bar 1 / t3
- t3bar -t1 / (t3 t3)
- t2bar t3bar z
- zbar t3bar t2
- zbar zbar t2bar z
- zbar zbar t2bar z
- ybar - t1bar
- w ybar y zbar z

ADI Comments

- The forward mode can be used to produce the

partial derivatives of all dependent variables

w.r.t. a single independent variable time

proportional to the evaluation of F - The reverse mode can be used to produce the

partial derivatives of a single dependent

variable w.r.t. all independent variables in time

proportional to the evaluation of F - ? y1, , yn ? F (? x1, , xm ?)
- For values of n ?? m, the forward mode is more

efficient - For values of m ?? n, the reverse mode is more

efficient

Global Sensitivity Equation

Global Sensitivity (GS)

- Interest is in Multi-Disciplinary Analysis (MDA)
- MDA are coupled analysis
- GS -gt
- GSE requires running
- Analysis to convergence
- twice

Local Sensitivity (LS)

- LS -gt ?? -gt

GS LS

Global Sensitivity Equation

- Requires sensitivity of disciplinary analysis

only - Can be computed after MDA converges

Time Savings

- Suppose
- Analysis-1 and Analysis-2 require n1 and n2

iterations respectively, - T is the time per evaluation for both the

analyses, and - N iterations are required for MDA convergence,

then - Time for FD across MDA
- 2N(n1 n2)T
- Time using GSE
- (N1)(n1 n2)T

Global Sensitivity Equations

- First introduced by Sobieski1 in 1990
- Successful implementations
- HSCT project by Anthony Giunta2
- Supersonic Transport Aircraft by Barthelemy

Dovi3 - Forward swept wing by Kapania Eldred4

Sensitivity Landscape

General References

- James Newman, et. al. Overview of Sensitivity

Analysis and Shape Optimization for Complex

Aerodynamic Configurations, J A/C, Vol 36, No.

1, Jan-Feb 1999

References FDM

- Hicks Henne, Wing design by numerical

optimization, Journal of Aircraft vol. 15, no.

7, July 1978.

References Complex Variable

- Lyness, J N Moler C B, Numerical

differentiation of analytic functions,SIAM J.

Numerical Analysis, Vol. 4, 1967, pp. 202-210. - Joaquim Martins, Sensitivity Analysis, Internal

Report, Aero. Astro. Dept. Stanford University. - Martins, Sturdza, Alonso, The Connection between

the Complex-Step Derivative Approximation and

Algorithmic Differentiation, Proceedings of the

39th Aerospace Sciences Meeting and Exhibit,

Reno, NV, Jan. 2001. - Squire, W. and Trapp, G., Using Complex

Variables to Estimate Derivatives of Real

Functions, SIAM Review, Vol. 10, No. 1, 1998 pp

110-112. - Anderson, Newman et. Al., Sensitivity Analysis

for the Navier-Stokes Equations on Unstructured

Meshes Using Complex Variables, AIAA Journal,

Vol. 39, No. 1, 2000 p. 56-62.

References Adjoint Method

- Antony Jameson Aerodynamic Design via control

theory, J. of scientific Computing Vol. 3

233-260, 1988. - Anderson, Venkatakrishnan, Aerodynamic Design

Optimization on Unstructured Grids with a

Continuous Adjoint Formulation, AIAA 97-0643. - Lei Xei, Gradient-based Optimum Aerodynamic

Design using adjoint method, PhD Thesis,

Virginia Polytechnic Institute and state

university, Aerospace Engineering, 2002. - Dadone, Grossman, Fast convergence of inviscid

fluid dynamic design problems, Computers and

fluids Vol. 32, 607-627, 2003. - Antony Jameson, Introduction to Adjoint

Methods, Lectures in Von Karmen Institute of

Fluid Mechanics, Feb. 2003. - Siva Nadarajah Antony Jameson, A Comparison of

the Continuous Discrete Adjoint Approach to

Automatic Aerodynamic Optimization,

AIAA-2000-0667 - Antony Jameson Sangho Kim, Reduction of the

Adjoint Gradient Formula for Aerodynamic Shape

Optimization Problems, AIAA Journal, Vol. 41,

No. 11, Nov. 2003. - M.B. Giles and N.A. Pierce. Improved lift and

drag estimates using adjoint Euler equations'.

AIAA Paper 99-3293, 1999. - Alonso et al,Aerodynamic Shape Optimization fo

Supersonic Aircraft configurations via an Adjoint

Formulation on Parallel Computers, Computers and

Fluids, Vol. 28. No. 4, 1999, pp. 675-700.

References Automatic Differentiation

- Griewank A., On Automatic Differentiation, in

Mathematical Programming Recent developments and

Applications, M. Iri and K. Tanabe, ets., Kluwer

Academic Publishers, 1989, pp. 83-108. - Bischof C., Corliss G., Green L., Griewank A.,

Haigler K. and Newman P., Automatic

Differentation of Advanced CFD Codes for

Multidisciplinary Design, Journal of Computing

systems in Engineering, Vol. 3., 1992, pp.

625-638. - Jueses David, A Taxonomy of Automatic

Differentiation Tools, Proceedings of the

Workshop on Automatic Differentiation of

Algorithms Theory, Implementation and

Application, SIAM, 1991, pp. 315-330. - Computational Differentiation Technical Reports

http//www-fp.mcs.anl.gov/autodiff/tech_reports.h

tml

References GSE

- Sobieszczanski-Sobieski, J., Sensitivity of

Complex, Internally Coupled Systems, AIAA J.,

vol. 28, no. 1, Jan. 1990, pp. 153-160. - Anthony Giunta, Sensitivity Analysis for Coupled

Aero-structural Systems, NASA/TM-1999-209367. - Barthelemy et. al., Supersonic Transport Wing

Minimum Weight Design Integrating Aerodynamics

and Structures, J. Aircraft, 31(2), 330-338

(1994). - Kapania, R. K. and Eldred, L. B., Sensitivity

Analysis of Wing Aeroelastic Response, J.

Aircraft, 30(4), 496-504 (1993).

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