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Computational Methods for Design Lecture 5 - Design and Optimization Problems John A. Burns C enter for O ptimal D esign A nd C ontrol I nterdisciplinary C enter for A pplied M athematics Virginia Polytechnic Institute and State University Blacksburg, Virginia 24061-0531

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Computational Methods for DesignLecture 5 - Design and Optimization ProblemsJohn A. BurnsCenterforOptimalDesignAndControlInterdisciplinaryCenterforAppliedMathematicsVirginia Polytechnic Institute and State UniversityBlacksburg, Virginia 24061-0531

A Short Course in Applied Mathematics

2 February 2004 – 7 February 2004

N∞M∞T Series Two Course

Canisius College, Buffalo, NY


1d model problem l.jpg

OPTIMAL DESIGN PROBLEM: Find the parameter 1 < q0, to

minimize the cost function

q

q

q

q

(S)

q

q,

Given data , 0 < x < 1the goal is to match by

solving the following

1D Model Problem

LET 1 < q <  and consider the boundary value problem


Model problem 1 l.jpg

q

q

q

q

q

q

(S)

q,

q

q

q

Model Problem #1

SENSITIVITY

The sensitivity equation fors(x, q) = qw(x , q)in the

“physical” domain(q) = (0,q) is given by

Can be made “rigorous” by the method of mappings.

MORE ABOUT THIS NEAR THE END


Typical cost function l.jpg

q

q

q

q

CONTINUOUS

SENSITIVITY

q

q

q

q

q

DISCRETE

SENSITIVITY

h

h

h

h

q

q

q

q

q

Typical Cost Function

WHERE w( x , q )USUALLY SATISFIES A DIFFERENTIAL EQUATION AND q IS A PARAMETER (OR VECTOR OF PARAMETERS)

THE CHAIN RULE PRODUCES

OR (Reality) USING NUMERICAL SOLUTIONS


Computing gradients l.jpg

h

q

TYPICAL APPROACHES TO COMPUTE

q=q0

(I) BY FINITE DIFFERENCES

h

h

q0

q

q0

h

q0

q

(II) BYDISCRETESENSITIVITIES

h

h

h

h

q0

q0

q0

q0

q0

Computing Gradients


Computing gradients6 l.jpg

DISCRETE SENSITIVITIES

FINITE DIFFERENCES

  • REQUIRES THE EXISTENCE OF THE

  • DISCRETE SENSITIVITY

  • REQUIRES 2 NON-LINEAR

  • SOLVES

  • IF SHAPE IS A DESIGN

  • VARIABLE, FD REQUIRES 2

  • MESH GENERATIONS

  • IF SHAPE IS A DESIGN VARIABLE,

  • THE DISCRETE SENSITIVITY LEADS TO

  • MESH DERIVATIVES COMPUTATIONS

WHAT IS THE “CONTINUOUS / HYBRID”

SENSITIVITY EQUATION METHOD? --- SEM

h

h

q0

q0

q0

Computing Gradients

h, k

APPROXIMATE


A sensitivity equation method l.jpg

w(x)

w h(x) = Finite Element Approximation

x

x=0

x=q

x=1

NUMERICAL APPROXIMATION

h

(S)

h

h

h

h

q,

q

A Sensitivity Equation Method

FORq> 1 ANDh=q/(N+1) CONSIDER (FORMAL)

DISCRETE STATE EQUATION


A sensitivity equation method8 l.jpg

(S)

h

h

q,

q

h

q

q

h

(S)

h

h

h

h

q,

q

A Sensitivity Equation Method

  • IMPORTANT OBSERVATIONS

    • The sensitivity equations are linear

    • The sensitivity equation “solver” can be constructed independently of the forward solver -- SENSE™

    • When done correctly “mesh gradients” are not required


A sensitivity equation method9 l.jpg

h

q,

q

(S)

h

h

q

q

s(x)= qw(x,q)

(S)

h

s h,k(x) = Finite Element Approximation of

x

x=0

x=q

x=1

2nd NUMERICAL APPROXIMATION

h,k

h,k

q

q

A Sensitivity Equation Method

FOR q> 1 ANDk= q/(M+1) CONSIDER (FORMAL)


Convergence issues l.jpg

h

h,k

h

q

q

q

q

h

k

h

q

q

h

k

h

k

a trust region method should (might?) converge.

When the error

is small, then

h

q

q

h

k

R. G. Carter, “On the Global Convergence of Trust-Region Algorithms Using Inexact Gradient Information”, SIAM J. Num. Anal., Vol 28 (1991), 251-265.

J. T. Borggaard, “The Sensitivity Equation Method for Optimal Design”, Ph.D. Thesis, Virginia Tech, Blacksburg, VA, 1995.

J. T. Borggaard and J. A. Burns, “A PDE Sensitivity Equation Method for Optimal Aerodynamic Design”, Journal of Computational Physics, Vol.136 (1997), 366-384.

Convergence Issues

THEOREM. The finite element scheme is asymptotically consistent.

IDEA:


Convergence issues11 l.jpg
Convergence Issues

N=16, M=32


Convergence issues12 l.jpg

h

q

q

h

h

NOT CONVERGENT

Convergence Issues

THE CASE k = h is often used, but may not be “good enough”


Timing issues l.jpg

THE CASE k = 2h offers flexibility and

h

2h

q

q

h

convergence.

Timing Issues

But, what about timings?

Approximately 96 .6% of cpu time spent in function evaluations

Approximately 02 .4% of cpu time spent in gradient evaluations


Mathematics impacts practically l.jpg

480 CPU HRS ~3 WEEKS

Mathematics Impacts “Practically”

UNDERSTANDING THE PROPER MATHEMATICAL FRAMEWORK CAN BE EXPLOITED TO PRODUCE BETTER SCIENTIFIC COMPUTING TOOLS

  • A REAL JET ENGINE WITH 20 DESIGN VARIABLES

    • PREVIOUS ENGINEERING DESIGN METHODOLOGY REQUIRED 8400 CPU HRS ~ 1YEAR

    • USING A HYBRID SEM DEVELOPED AT VA TECH AS IMPLEMENTED BY AEROSOFT IN SENSE™REDUCED THE DESIGN CYCLE TIME FROM ...

8400 CPU HRS ~ 1YEAR TO

NEW MATHEMATICSWAS THEENABLING TECHNOLOGY


Special structure of se s l.jpg

(DE)

(SE)

(DE)

(SE)

Special Structure of SE’s

FIRST: SOLVE (DE)

SECOND: SOLVE (SE)


General comments l.jpg
General Comments

  • THERE ARE MANY VARIATIONS THAT CAN IMPROVE THE BASIC IDEA

    • COMBINING AUTOMATIC DIFFERENTIATION AND SEM

    • SMOOTHING AND GRADIENT PROJECTIONS

    • ADAPTIVE GRID GENERATION

  • THE ORDER OF THINGS MATTER

    • DIFFERENTIATE-THEN-APPROXIMATE

    • DERIVE SENSITIVITY EQUATION BEFORE MAPPING TO A “COMPUTATIONAL DOMAIN”

      • DOES NOT REQUIRE MESH DERIVATIVES

      • REQUIRES A MORE SOPHISTICATED MATHEMATICAL FRAMEWORK

      • NEEDS A “DIFFERENT THEORY”

J. A. Burns and L. G. Stanley, “A Note on the Use of Transformations in Sensitivity

Computations for Elliptic Systems”, Journal of Mathematical & Computer Modeling,

Vol. 33, pp. 101-114, 2001.


Model problem 2 l.jpg

OPTIMAL DESIGN PROBLEM: Find the parameter 0 < q0< 1, to minimize

the cost function

q

(S)

q

q

q

q,

q

where

MODEL PROBLEM #2

LET 1 < q <  and consider the boundary value problem

DERIVE THE SENSITIVITY EQUATION



Model problem 219 l.jpg

(S)

q ,

q

(S)

q,

q

q

1

x

0

q

0

1

 = (0,1)

(q) = (0,q)

MODEL PROBLEM #2

The sensitivity equation for s(x, q) = qw(x , q) in the

“physical” domain (q) = (0,q) is given by

APPROXIMATIONS and CHANGE OF VARIABLES

(METHOD OF MAPPINGS)

 = T(x,q) = x/q


Method of mappings l.jpg

“SOLVE”

h

h

h

h

h

h

METHOD OF MAPPINGS

S

=T(x,q)

(q)

x=M(,q)


Model problem 221 l.jpg

M(S)

q

q

q

-q2

q

MODEL PROBLEM #2

Map (0,q) to (0,1) by  = T(x,q) = x/q and note that the

inverse mapping M( ,q) = q maps (0,1) to (0, q).

Define

z( ,q) = w(M( ,q), q) = w(q , q) - transformed state

p( , q) = q z( ,q) - sensitivity of the transformed state

and

r ( , q) = s(M( ,q), q) = s(q, q) - transformed sensitivity.


Model problem 222 l.jpg

M(S)

M(S)

q

q

q

MODEL PROBLEM #2

To compute s(x, q) one has two choices

Solve M( S) for r( , q) and transform back to get

(1) s(x, q) = r( , q) = r(T(x,q), q) = r(x/q , q)

Solve M(S) for p( , q) and transform back to get

(2) s(x, q) = p(x/q , q) - z(x/q , q)[ M (x/q , q)]-1[qM (x/q , q)]

MESH DERIVATIVE


Model problem 223 l.jpg

w(x)

w h(x) = Finite Element Approximation

x

x=0

x=q

x=1

NUMERICAL APPROXIMATION

h

(S)

h

h

h

q,

q

q,

(S)

q

q

MODEL PROBLEM #2

FOR q> 1 AND h=q/(N+1) CONSIDER (FORMAL)

h

h


Model problem 224 l.jpg

h

h

s(x)= qw(x,q)

(S)

h

s h,k(x) = Finite Element Approximation of

x

x=0

x=q

x=1

q,

h,k

2nd NUMERICAL APPROXIMATION

(S)

h,k

q

q

q

q

MODEL PROBLEM #2

FOR q> 1 ANDk= q/(M+1) CONSIDER (FORMAL)


Model problem 225 l.jpg
MODEL PROBLEM #2

? WHAT HAPPENS ?

Linear Finite Elements

q = 1.5

q = 1.5

T

w(x ,q )

z( ,q )


Model problem 226 l.jpg
MODEL PROBLEM #2

H1 - ERROR FOR w(x ,q )


Model problem 227 l.jpg

0

N = 03

-0.05

N = 05

N = 09

-0.1

-0.15

-0.2

-0.25

-0.3

-0.35

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

MODEL PROBLEM #2

q [z( ,q)] = p( ,q )

M by (1)

s(x ,q )


Model problem 228 l.jpg

0.7

0.6

[z(x/q , q)]

0.5

[zh(x/q , q)]

Finite Element Approximation

of the Spatial Derivative

0.4

0.3

0.2

0.1

0

-0.1

-0.2

0

0.5

1

1.5

MODEL PROBLEM #2

(2) s(x, q) = p(x/q , q) - z(x/q , q)[ M (x/q , q)]-1[q M (x/q , q)]


Model problem 229 l.jpg
MODEL PROBLEM #2

s(x ,q )

M by (2)

r( ,q )

THE HYBRID CONTINUOUS SENSITIVITY METHOD


1d interface problem l.jpg

x

1

 = 2

q

q

 = 1

q

0

q

()

q

q

q

q

q

q

q

q

1D Interface Problem

ELLIPTIC PROCESS MODEL - 2 MATERIALS

CONTINUITY


1d interface problem31 l.jpg

OPTIMAL DESIGN PROBLEM: Find the parameter 0 < q0< 1, to minimize

the cost function

q

q

q

q

q

q

q

q

q

OR ...

q

q

q

1D Interface Problem


1d interface problem32 l.jpg

q

q

q

q

q

q

q

q

q

q

1D Interface Problem

THE SOLUTION AND SENSITIVITY IS GIVEN BY

HOW SMOOTH ISs(x, q ) = q w(x , q)?

s( · , q ) H1() ?


1d interface problem33 l.jpg

0.3

PLOT OF w(x, q) AT q = .5

PLOT OF SENSITIVITY s(x,q) AT q = .5

1

0.2

0.9

0.8

0.1

0.7

0.6

0

0.5

-0.1

0.4

0.3

-0.2

0.2

-0.3

0.1

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-0.4

-0.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1D Interface Problem

s( · , q ) H1()


1d interface problem34 l.jpg

q

q

q

q

()

q

q

q

q

(C)

(J)

1D Interface Problem

  • HOWEVER, THE SENSITIVITY EQUATION IS GIVEN BY THE BOUNDARY VALUE PROBLEM

  • HOW DID WE DERIVE THIS SYSTEM?

  • WHAT DO WE MEAN BY A SOLUTION?

  • CAN THIS BE MADE RIGOROUS?


Formal derivation l.jpg

q

q

LET

q

q

q

q

q

q

q

q

TAKE THE TOTAL DERIVATIVE OF

q

q

q

q

q

q

q

q

q

q

q

q

q- q

q- q

q+ q

q+ q

q q

q q

q q

q q

Formal Derivation


Formal derivation36 l.jpg

CONTINUITY

JUMP

q

q

q

q

q

q

q

q

q

q

in [W (q)]’

?

()

Formal Derivation

LIKEWISE ...

WEAKEST FORM OF THE ELLIPTIC PROBLEM


Sensitivity computations l.jpg
Sensitivity Computations

Petrov-Galerkin FE Method (Burns, Lin & STanley - 03)


2d sensitivity computations l.jpg
2D Sensitivity Computations

Petrov-Galerkin FE Method (Burns, Lin & STanley - 03)

!! WORKS IN 2D !!


2d sensitivity computations39 l.jpg
2D Sensitivity Computations

Petrov-Galerkin FE Method (Burns, Lin & STanley - 03)

!! WORKS IN 2D !!

FOR COMPLEX GEOMETRY


Slide40 l.jpg

WHAT ABOUT

NUMERICAL METHODS


Numerical methods l.jpg

(IVP)

x0

t0

Numerical Methods

FORWARD DIFFERENCE


Explicit euler l.jpg

x0

t0

Explicit Euler


Implicit euler method l.jpg

x0

t0

Implicit Euler Method

BACKWARD DIFFERENCE


Implicit euler l.jpg

x0

t0

Implicit Euler


Numerical methods matter l.jpg
Numerical Methods Matter

DIFFERENTIATE THEEQUATIONWITH RESPECT TOq


Numerical methods matter46 l.jpg
Numerical Methods Matter

INTERCHANGE THE ORDER OF DIFFERENTIATION




Numerical methods matter49 l.jpg

SOLUTION

Numerical Methods Matter



Numerical methods matter51 l.jpg

FOR ALL AND ALL

Numerical Methods Matter

BACKWARD EULER


Why sensitivities l.jpg
Why Sensitivities?

  • USEFUL IN OPTIMIZATION BASED DESIGN

  • SENSITIVITIES HAVE MANY OTHER USES

    • PRIORITIZE DESIGN & CONTROL VARIABLES

    • EVALUATE DESIGNS & CONTROL LAWS

    • NON- OPTIMIZATION BASED DESIGN

    • FAST SOLVERS

    • ANALYZE UNCERTAINTIES

    • PREDICT “FAILURE” (FLOW SEPARATION, ETC.)

MAY REQUIRE COMPLEX

MATHEMATICAL THEORIES

-------

DIFFERENTIATION OF

SET-VALUED FUNCTIONS

  • SOME OBSERVATIONS

    • DO NUMERICS CAREFULLY

    • “ORDER’’ MATTERS


Slide53 l.jpg

END OF SHORT COURSE

BUT…

NOW A WORD FROM

MY SPONSORS

VIRGINIA TECH


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