A Short Course in Applied Mathematics 2 February 2004 – 7 February 2004 N ∞M∞T Series Two Course Canisius College, Buffa

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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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## A Short Course in Applied Mathematics 2 February 2004 – 7 February 2004 N ∞M∞T Series Two Course Canisius College, Buffa

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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

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

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.

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

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

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

h, k

APPROXIMATE

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

(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

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)

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:

h

q

q

h

h

NOT CONVERGENT

Convergence Issues

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

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

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

(DE)

(SE)

(DE)

(SE)

Special Structure of SE’s

FIRST: SOLVE (DE)

SECOND: SOLVE (SE)

• 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.

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

(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

“SOLVE”

h

h

h

h

h

h

METHOD OF MAPPINGS

S

=T(x,q)

(q)

x=M(,q)

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.

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

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

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 #2

? WHAT HAPPENS ?

Linear Finite Elements

q = 1.5

q = 1.5

T

w(x ,q )

z( ,q )

MODEL PROBLEM #2

H1 - ERROR FOR w(x ,q )

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 )

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 #2

s(x ,q )

M by (2)

r( ,q )

THE HYBRID CONTINUOUS SENSITIVITY METHOD

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

the cost function

q

q

q

q

q

q

q

q

q

OR ...

q

q

q

1D Interface Problem

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() ?

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()

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?

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

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

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

2D Sensitivity Computations

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

!! WORKS IN 2D !!

2D Sensitivity Computations

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

!! WORKS IN 2D !!

FOR COMPLEX GEOMETRY

NUMERICAL METHODS

(IVP)

x0

t0

Numerical Methods

FORWARD DIFFERENCE

x0

t0

Implicit Euler Method

BACKWARD DIFFERENCE

Numerical Methods Matter

DIFFERENTIATE THEEQUATIONWITH RESPECT TOq

Numerical Methods Matter

INTERCHANGE THE ORDER OF DIFFERENTIATION

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

END OF SHORT COURSE

BUT…

NOW A WORD FROM