Computational Methods for Design Lecture 5 Design and Optimization Problems John A' Burns Center for

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Computational Methods for Design Lecture 5 -
Design and Optimization Problems John A.
BurnsCenter for Optimal Design And
ControlInterdisciplinary Center for Applied
MathematicsVirginia Polytechnic Institute and
State UniversityBlacksburg, Virginia 24061-0531

• A Short Course in Applied Mathematics
• 2 February 2004 7 February 2004
• N8M8T Series Two Course
• Canisius College, Buffalo, NY

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1D Model Problem
LET 1 lt q lt ? and consider the boundary value
problem
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Model Problem 1
SENSITIVITY
The sensitivity equation for s(x, q ) ?q w(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
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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
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Computing Gradients
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Computing Gradients
h, k
APPROXIMATE
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A Sensitivity Equation Method
FOR q gt 1 AND hq/(N1) CONSIDER (FORMAL)
DISCRETE STATE EQUATION
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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

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A Sensitivity Equation Method
FOR q gt 1 AND k q/(M1) CONSIDER (FORMAL)
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Convergence Issues
THEOREM. The finite element scheme is
asymptotically consistent.
IDEA
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Convergence Issues
N16, M32
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Convergence Issues
THE CASE k h is often used, but may not be
good enough
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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
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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 1 YEAR
• USING A HYBRID SEM DEVELOPED AT VA TECH AS
  IMPLEMENTED BY AEROSOFT IN SENSE REDUCED THE
  DESIGN CYCLE TIME FROM ...

8400 CPU HRS 1 YEAR TO
NEW MATHEMATICS WAS THE ENABLING TECHNOLOGY
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Special Structure of SEs
FIRST SOLVE (DE)
SECOND SOLVE (SE)
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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.
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MODEL PROBLEM 2
LET 1 lt q lt ? and consider the boundary value
problem
DERIVE THE SENSITIVITY EQUATION
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MODEL PROBLEM 2
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MODEL PROBLEM 2
The sensitivity equation for s(x, q ) ?q w(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
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METHOD OF MAPPINGS
S
?T(x,q)
?
?(q)
xM(?,q)
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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.
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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
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MODEL PROBLEM 2
FOR q gt 1 AND hq/(N1) CONSIDER (FORMAL)

h
h
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MODEL PROBLEM 2
FOR q gt 1 AND k q/(M1) CONSIDER (FORMAL)
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MODEL PROBLEM 2
? WHAT HAPPENS ? Linear Finite Elements
q 1.5
q 1.5
T
w(x ,q )
z(? ,q )
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MODEL PROBLEM 2
H1 - ERROR FOR w(x ,q )
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MODEL PROBLEM 2
?q z(? ,q) p(? ,q )
M by (1)
s(x ,q )
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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)
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MODEL PROBLEM 2
s(x ,q )
M by (2)
r(? ,q )
THE HYBRID CONTINUOUS SENSITIVITY METHOD
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1D Interface Problem
ELLIPTIC PROCESS MODEL - 2 MATERIALS
CONTINUITY
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1D Interface Problem
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1D Interface Problem
THE SOLUTION AND SENSITIVITY IS GIVEN BY
HOW SMOOTH IS s(x, q ) ?q w(x , q) ?
s( , q ) ? H1(?) ?
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1D Interface Problem
s( , q ) ? H1(?)
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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?

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Formal Derivation
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Formal Derivation
LIKEWISE ...
WEAKEST FORM OF THE ELLIPTIC PROBLEM
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Sensitivity Computations
Petrov-Galerkin FE Method (Burns, Lin STanley
- 03)
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2D Sensitivity Computations
Petrov-Galerkin FE Method (Burns, Lin STanley
- 03)
!! WORKS IN 2D !!
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2D Sensitivity Computations
Petrov-Galerkin FE Method (Burns, Lin STanley
- 03)
!! WORKS IN 2D !!
FOR COMPLEX GEOMETRY
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WHAT ABOUT NUMERICAL METHODS
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Numerical Methods
FORWARD DIFFERENCE
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Explicit Euler
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Implicit Euler Method
BACKWARD DIFFERENCE
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Implicit Euler
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Numerical Methods Matter
DIFFERENTIATE THE EQUATION WITH RESPECT TO q
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Numerical Methods Matter
INTERCHANGE THE ORDER OF DIFFERENTIATION
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Numerical Methods Matter
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SOME RUNS
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Numerical Methods Matter
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Numerical Methods Matter
FORWARD EULER
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Numerical Methods Matter
BACKWARD EULER
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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

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