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Improve Math Models &
Improve Math Models &
Increase Productivity
Increase Productivity
By Phil Brubaker
By Phil Brubaker
for
for
Scientists,
Scientists,
Engineers & Management
Engineers & Management
Optimal Designs Enterprise
Optimal Designs Enterprise
goal-driven.net
goal-driven.net
Improve Models &
Improve Models &
Increase Productivity
Increase Productivity
Agenda
Agenda
Design Objectives
Design Objectives
Language Background
Language Background
Example Code
Example Code
Optimization of Optimizations
Optimization of Optimizations
Code Definition
Code Definition
Optimization
Optimization
=
=
Simulation
Simulation
+
+
Objective (Function)
Objective (Function)
Why Optimization
Why Optimization
over Simulation?
over Simulation?
Maximize Parameter Tolerance for Mfg.
Maximize Parameter Tolerance for Mfg.
Solutions are at min./max. locations where
Solutions are at min./max. locations where
their partial derivatives = 0 & thus allow largest
their partial derivatives = 0 & thus allow largest
delta errors.
delta errors.
Minimize # of Executions for a solution
Minimize # of Executions for a solution
Introduction
Introduction
to
to
FortranCalculus
FortranCalculus
a Calculus-level Computer Language
a Calculus-level Computer Language
1974 First Commercial Release (PROSE)
1974 First Commercial Release (PROSE)
I taught PROSE in 1975-79.
I taught PROSE in 1975-79.
Solved problems within 4 hours (each)
Solved problems within 4 hours (each)
Present version is FortranCalculus
Present version is FortranCalculus
Main obstacle: no design objective
Main obstacle: no design objective
Objective-Driven Engineering
Objective-Driven Engineering
What are your
What are your
Goals/Objectives
Goals/Objectives
for
for
a given Project?
a given Project?
What are you building?
What are you building?
Mr. Arithmetic Mr. Algebra Mr. Calculus
Mr. Arithmetic Mr. Algebra Mr. Calculus
------------------- --------------- ----------------
------------------- --------------- ----------------
Slide Rule Simulate Optimize
Slide Rule Simulate Optimize
Objective-Driven Cuts
Objective-Driven Cuts
Sawmill's operation
Objective for cutting log?
Objective for cutting log?
Objective-Driven Cuts
Objective-Driven Cuts
Sawmill's operation
Objective for cutting log?
Objective for cutting log?
Maximize company's profits
Maximize company's profits
Minimize pollution
Minimize pollution
Minimize 'waste'
Minimize 'waste'
Maximize 2" x 4"s
Maximize 2" x 4"s
Sawmill's operation
Parameters to consider:
Parameters to consider:
Log-to-lumber Processing:
Log-to-lumber Processing:
Size of log
Size of log: diameter, length, taper, knots, etc.
: diameter, length, taper, knots, etc.
Time required
Time required to: cut log, sharpen blades, lubricate
to: cut log, sharpen blades, lubricate
machinery, etc.
machinery, etc.
Strength or flexibility
Strength or flexibility desired of various sizes.
desired of various sizes.
Inventory
Inventory
Market trends
Market trends
Objective-Driven Cuts
Objective-Driven Cuts
Chaotic Design Process
Chaotic Design Process
Control Box
Control Box
Bingo, a design!
Bingo, a design!
Objective-Driven Design
Objective-Driven Design
Surveillance Design Objective:
Surveillance Design Objective:
Maximize surveillance coverage
Maximize surveillance coverage
Minimize number of satellites
Minimize number of satellites
Solution? (Not realistic; Objective needs work.)
Solution? (Not realistic; Objective needs work.)
Objective-Driven Design
Objective-Driven Design
Thin-Film-Head (TFH) for Magnetic Recording
TFH with coil
TFH with coil
Disc platter cross-section
Disc platter cross-section
Typical Readback Pulse
Typical Readback Pulse Optimum pulse shape
Optimum pulse shape
versus
versus
Objective-Driven Design
Objective-Driven Design
Thin-Film-Head (TFH) for Magnetic Recording
Thin-Film-Head (TFH) for Magnetic Recording
Design Objective?
Design Objective?
Maximize Profit
Maximize Profit
Maximize Pulse Symmetry
Maximize Pulse Symmetry
Minimize Pulse Width
Minimize Pulse Width
Minimize Pollution
Minimize Pollution
Maximize User Satisfaction
Maximize User Satisfaction
???
???
Solution: Determine TFH geometry
Solution: Determine TFH geometry
parameters A, B, C, etc. to achieve objective.
parameters A, B, C, etc. to achieve objective.
Objective-Driven Design
Objective-Driven Design
Matched Filter for Magnetic Recording
Matched Filter for Magnetic Recording
Electrical Filter
Electrical Filter
Yin(t) ==>
Yin(t) ==>
Transfer
Transfer
Function
Function
-------------
-------------
H(s)
H(s)
==> Yout(t)
==> Yout(t)
versus
versus
Typical Input Pulse, Yin(t)
Typical Input Pulse, Yin(t) Desired Output Pulse, Yout(t)
Desired Output Pulse, Yout(t)
Objective-Driven Design
Objective-Driven Design
Matched Filter for Magnetic Recording
Matched Filter for Magnetic Recording
Results:
Results:
Textbook problem solved in 4 hours
Textbook problem solved in 4 hours
Design required 2 years to acquire a true practical
Design required 2 years to acquire a true practical
objective function
objective function
Development time dropped
Development time dropped from 12 to 1 week
from 12 to 1 week
Design was mathematically optimal
Design was mathematically optimal
"One Step from First Principles to Solutions"
"One Step from First Principles to Solutions"
Enhancing
Enhancing
Scientific & Engineering
Scientific & Engineering
Productivity
Productivity
FortranCalculus Background
FortranCalculus Background
Optimal Designs Enterprise
Optimal Designs Enterprise
goal-driven.net
goal-driven.net
Industry Issue
Industry Issue
A Proven Approach
A Proven Approach
Summ
Summary
ary
Fortran Calculus
Fortran Calculus
Language
Language
Agenda
Agenda
Scientific & Engineering
Scientific & Engineering
Productivity
Productivity
Industry Issue
Industry Issue
Costly Problem/Solution Cycle ...
Costly Problem/Solution Cycle ...
Model Married to Algorithm
Model Married to Algorithm
Validation Delayed
Validation Delayed
Long Problem/Solution Cycle
Long Problem/Solution Cycle
Problem "Understanding" Delayed
Problem "Understanding" Delayed
Basic, Fortran, MACSYMA, etc. Languages
Basic, Fortran, MACSYMA, etc. Languages
Formulate
Formulate
Problem
Problem
Approximations,
Approximations,
Methods, etc.
Methods, etc.
Programming of
Programming of
Reduced Problem
Reduced Problem
Debug Problem,
Debug Problem,
Math, & Program
Math, & Program
Engineer
Engineer
Mathematician
Mathematician
Programmer
Programmer
All
All
Rapid Prototyping for Adaptive
Rapid Prototyping for Adaptive
Engineering
Engineering
Rapid Prototyping
Rapid Prototyping
for Adaptive Engineering
for Adaptive Engineering
Basic, Fortran, MACSYMA, etc. Languages (Cont.)
Basic, Fortran, MACSYMA, etc. Languages (Cont.)
Engineering:
Engineering: Quickly Frozen
Quickly Frozen
Commitment:
Commitment: Large
Large
Cost:
Cost: High
High
Delay:
Delay: Long
Long
Algebra Level Summary
Algebra Level Summary
FC Technology History
FC Technology History
Late 1960's - Pioneered - TRW / NASA
Late 1960's - Pioneered - TRW / NASA
Mid 1970's - Validated - PROSE, Inc.
Mid 1970's - Validated - PROSE, Inc.
Late 1980's - Migrated - Du Pont
Late 1980's - Migrated - Du Pont
Today -
Today - FortranCalculus
FortranCalculus
Pioneered
Pioneered
Validated
Validated
Migrated
Migrated
Operational
Operational
Scientific & Engineering
Scientific & Engineering
Productivity
Productivity
FortranCalculus
FortranCalculus
A Proven Approach
A Proven Approach ...
...
Allows Rapid Prototyping
Allows Rapid Prototyping
Decouples Models from Algorithms
Decouples Models from Algorithms
Allows Interchangeable Algorithms
Allows Interchangeable Algorithms
Accelerates Problem "Understanding"
Accelerates Problem "Understanding"
Enabled by "Automatic Differentiation"
Enabled by "Automatic Differentiation"
Rapid Prototyping for Adaptive
Rapid Prototyping for Adaptive
Engineering
Engineering
Calculus-level Languages: Prose & fortranCalculus
Calculus-level Languages: Prose & fortranCalculus
Formulate
Formulate
Problem
Problem
Debug Problem
Debug Problem
Engineer
Engineer
Engineer
Engineer
Rapid Prototyping
Rapid Prototyping
for Adaptive Engineering
for Adaptive Engineering
FortranCalculus Language (Cont.)
FortranCalculus Language (Cont.)
Engineering:
Engineering: Adaptive
Adaptive
Commitment:
Commitment: Small
Small
Cost:
Cost: Low
Low
Delay:
Delay: Short
Short
Calculus Level Summary
Calculus Level Summary
Enabling Technology
Enabling Technology
FortranCalculus
FortranCalculus
Symbolic Differentiation Evaluated at a Point
Symbolic Differentiation Evaluated at a Point
Generates
Generates
Gradient Vectors
Gradient Vectors
Jacobian Matrices
Jacobian Matrices
Hessian Matrices
Hessian Matrices
... of Any Programmed Model
... of Any Programmed Model
Automatic Differentiation
Automatic Differentiation
Enabling Technology
Enabling Technology
FortranCalculus
FortranCalculus
Enables
Enables
Inverse Problem Solving
Inverse Problem Solving
Nonlinear Optimization
Nonlinear Optimization
Optimization of Differential Equations
Optimization of Differential Equations
Structured Nesting of Optimization Algorithm
Structured Nesting of Optimization Algorithms
s
Automatic Differentiation (Cont.)
Automatic Differentiation (Cont.)
FortranCalculus
FortranCalculus
,
,
a calculus level language
a calculus level language
Increases Science/Engineering Productivity
Increases Science/Engineering Productivity
Allows Rapid Model Prototyping
Allows Rapid Model Prototyping
Reduces Costly Problem/Solution Cycle
Reduces Costly Problem/Solution Cycle
Accelerates Problem "Understanding"
Accelerates Problem "Understanding"
Proven Concept Since 1968
Proven Concept Since 1968
Provides a Competitive Technical Edge
Provides a Competitive Technical Edge
Summary
Summary
Explicit & Implicit Equations
Explicit & Implicit Equations
Inverse & Optimization Problems
Inverse & Optimization Problems
Differential Equations
Differential Equations
IVP & BVP Problems
IVP & BVP Problems
Limits & Constraints
Limits & Constraints
FortranCalculus
FortranCalculus
Language
Language
Example Code
Example Code
Agenda
Agenda
Complete Example Code
Complete Example Code
global all
global all
problem rocket ! three stage rocket design optimization
problem rocket ! three stage rocket design optimization
dimension spi(3),spivac(3),tburn(3),thrust(3),xip(3),wprop(3),
dimension spi(3),spivac(3),tburn(3),thrust(3),xip(3),wprop(3),
& ratio(3),wstage(3),strfac(3),delv(3),g(2)
& ratio(3),wstage(3),strfac(3),delv(3),g(2)
thrust(1)=350 : thrust(2)=1500 : thrust(3)=4100
thrust(1)=350 : thrust(2)=1500 : thrust(3)=4100
tburn(1)=110 : tburn(2)=100 : tburn(3)=180
tburn(1)=110 : tburn(2)=100 : tburn(3)=180
xip(1)=5d-3 : xip(2)=0 : xip(3)=0
xip(1)=5d-3 : xip(2)=0 : xip(3)=0
spivac(1)=315 : spivac(2)=315 : spivac(3)=315
spivac(1)=315 : spivac(2)=315 : spivac(3)=315
FIND
FIND thrust(1),thrust(2),tburn(2),tburn(3); in stages;
thrust(1),thrust(2),tburn(2),tburn(3); in stages;
* by Hera; reporting dlvtot,tbtot; to minimize weight
* by Hera; reporting dlvtot,tbtot; to minimize weight
end
end
model stages
model stages
Find
Find thrust(3),tburn(1); in eqns; by Ajax; to match g
thrust(3),tburn(1); in eqns; by Ajax; to match g
end
end
model eqns
model eqns
data gc,wpayld,delvip,tbip/32.174,50,2.8e4,400/
data gc,wpayld,delvip,tbip/32.174,50,2.8e4,400/
dlvtot=0 : tbtot=0
dlvtot=0 : tbtot=0
weight=wpayld
weight=wpayld
do 10 i=1,3
do 10 i=1,3
spi(i)=spivac(i)*(1-xip(i))
spi(i)=spivac(i)*(1-xip(i))
wprop(i)=thrust(i)*tburn(i)/spi(i)
wprop(i)=thrust(i)*tburn(i)/spi(i)
wstage(i)=0.0234*thrust(i)+wprop(i)+1.255*wprop(i)**0.704+4
wstage(i)=0.0234*thrust(i)+wprop(i)+1.255*wprop(i)**0.704+4
strfac(i)=wprop(i)/wstage(i)
strfac(i)=wprop(i)/wstage(i)
weight=weight+wstage(i)
weight=weight+wstage(i)
ratio(i)=weight/(weight-wprop(i))
ratio(i)=weight/(weight-wprop(i))
delv(i)=gc*spi(i)*log(ratio(i))
delv(i)=gc*spi(i)*log(ratio(i))
dlvtot=dlvtot+delv(i)
dlvtot=dlvtot+delv(i)
tbtot=tbtot+tburn(i)
tbtot=tbtot+tburn(i)
10 continue
10 continue
g(1)=dlvtot-delvip ! total delta v constraint
g(1)=dlvtot-delvip ! total delta v constraint
g(2)=tbtot-tbip
g(2)=tbtot-tbip ! total burn time constraint
! total burn time constraint
end
end
Math
Model
Example Convergence Report
Example Convergence Report
o o o
o o o
LOOP NUMBER .... [INITIAL] 5 6
LOOP NUMBER .... [INITIAL] 5 6
UNKNOWNS
UNKNOWNS
A ( 1) 1.000000E+00 4.432149E-01 3.737358E-01
A ( 1) 1.000000E+00 4.432149E-01 3.737358E-01
B ( 1) 1.000000E+00 4.040783E+00 4.284183E+00
B ( 1) 1.000000E+00 4.040783E+00 4.284183E+00
C ( 1) 1.230000E+02 4.305000E+02 4.920000E+02
C ( 1) 1.230000E+02 4.305000E+02 4.920000E+02
OBJECTIVE
OBJECTIVE
ERRSUM 8.189812E+00
ERRSUM 8.189812E+00 4.870502E-02 3.879211E-02
4.870502E-02 3.879211E-02
o o o
o o o
Explicit Equation
Explicit Equation
o o o
o o o
Find
Find A,B,C
A,B,C; In
; In Engine
Engine; to ...
; to ...
o o o
o o o
Model
Model Engine
Engine
Y = Function( X;
Y = Function( X; A,B,C
A,B,C)
)
End Model
End Model
Math
Model
Implicit Equation
Implicit Equation
o o o
o o o
Find
Find A,B,C
A,B,C; In
; In Engine
Engine; to
; to Match
Match
G
G
o o o
o o o
Model
Model Engine
Engine
G
G = Y - Function( X, Y;
= Y - Function( X, Y; A,B,C
A,B,C)
)
End Model
End Model
Inverse Problem
Inverse Problem
o o o
o o o
Ydesired = 123.456 ! target value
Ydesired = 123.456 ! target value
Find
Find A,B,C
A,B,C; In
; In Engine
Engine; to
; to Match
Match
G
G
o o o
o o o
Model
Model Engine
Engine
Y = Function( X;
Y = Function( X;
A,B,C
A,B,C)
)
G
G = Ydesired - Y
= Ydesired - Y
End Model
End Model
Optimization Problem
Optimization Problem
o o o
o o o
Find
Find A,B,C
A,B,C; In
; In Engine
Engine; to
; to Minimize
Minimize
G
G
o o o
o o o
Model
Model Engine
Engine
G
G = Y - Function( X;
= Y - Function( X; A,B,C
A,B,C)
)
End Model
End Model
Explicit Differental Equations
Explicit Differental Equations
o o o
o o o
Initiate
Initiate
ISIS
ISIS; For
; For Engine
Engine;
;
Equations Y2Dot/YDot, YDot/Y; ...
Equations Y2Dot/YDot, YDot/Y; ...
o o o
o o o
Integrate
Integrate
Engine
Engine; By
; By ISIS
ISIS
o o o
o o o
Model
Model Engine
Engine
Y2Dot
Y2Dot = Function( YDot, Y
= Function( YDot, Y)
)
End Model
End Model
Implicit Differental Equations
Implicit Differental Equations
Initiate ...
Initiate ...
o o o
o o o
Integrate
Integrate
Engine
Engine; By
; By ISIS
ISIS
o o o
o o o
Model
Model Engine
Engine
Find
Find
Y2Dot
Y2Dot; In IDE; To Match
; In IDE; To Match G
G
End Model
End Model
Model IDE
Model IDE
G
G =
= Y2Dot
Y2Dot
-
-
Fun(
Fun(Y2Dot
Y2Dot,
, YDot, Y
YDot, Y)
)
End Model
End Model
Initial Value Problems
Initial Value Problems
Explicit Equations
Explicit Equations
Initiate
Initiate
YDot = 123.456 : Y2Dot = 234.567 : Y3Dot = ...
YDot = 123.456 : Y2Dot = 234.567 : Y3Dot = ... ! Initial Values
! Initial Values
o o o
o o o
Integrate
Integrate ...
...
o o o
o o o
Model ...
Model ...
YnDot
YnDot = Function( Yn
= Function( Yn1
1Dot, ... , YDot, Y
Dot, ... , YDot, Y)
)
End Model
End Model
Initial Value Problems
Initial Value Problems
Implicit Equations
Implicit Equations
o o o
o o o
Integrate ...
Integrate ...
o o o
o o o
Find
Find YnDot; ...; To Match
YnDot; ...; To Match G
G
o o o
o o o
Model ...
Model ...
G
G = YnDot - Function( YnDot, ... , YDot, Y
= YnDot - Function( YnDot, ... , YDot, Y)
)
End Model
End Model
Boundary Value Problems
Boundary Value Problems
Explicit Equations
Explicit Equations
o o o
o o o
Find
Find YDot0, ... To Match
YDot0, ... To Match H
H
o o o
o o o
Integrate
Integrate ...
...
o o o
o o o
H
H = (Y0 - Y0_desired)**2 +
= (Y0 - Y0_desired)**2 +
(Ylast - Ylast_desired)**2 ! boundary values
(Ylast - Ylast_desired)**2 ! boundary values
Boundary Value Problems
Boundary Value Problems
Implicit Equations
Implicit Equations
o o o
o o o
Find
Find YDot0, ...; To Match
YDot0, ...; To Match H
H
o o o
o o o
Integrate
Integrate ...
...
o o o
o o o
Find
Find YnDot; ...; To Match
YnDot; ...; To Match G
G
o o o
o o o
Model ...
Model ...
G
G = YnDot - Function( YnDot, ... , YDot, Y
= YnDot - Function( YnDot, ... , YDot, Y)
)
End Model
End Model
Limits & Inequality Constraints
Limits & Inequality Constraints
o o o
o o o
Find
Find ... With Lowers ... And Uppers ... Holding ...
... With Lowers ... And Uppers ... Holding ...
o o o
o o o
Tweak
Tweak
Tweak
Tweak
Tweak
Tweak
o o o
o o o
Find
Find E,F,G
E,F,G ...
... ! Add to any problem
! Add to any problem
o o o
o o o ! in order to tweak
! in order to tweak E,F,G
E,F,G ...
...
Nested Calculus Processes
Nested Calculus Processes
o o o
o o o
Find
Find ...
...
o o o
o o o
Integrate
Integrate ...
...
o o o
o o o
Integrate
Integrate ...
...
o o o
o o o
Find
Find ...
...
How are problems solved?
How are problems solved?
o o o
o o o
Find
Find ...
... by
by 'Solver'
'Solver'
o o o
o o o
'Solver' is a numerical method using
'Solver' is a numerical method using
Automatic Differention to calculate
Automatic Differention to calculate
necessary derivatives. The
necessary derivatives. The
available solvers are in a MC
available solvers are in a MC
library; e.g. Ajax, Mars, Neptune,
library; e.g. Ajax, Mars, Neptune,
etc.
etc.
Objective-Driven Engineering
Objective-Driven Engineering
Optimizations
Optimizations
within
within
Optimization
Optimization
A nesting example
Optimal Designs Enterprise
Optimal Designs Enterprise
goal-driven.net
goal-driven.net
Automotive Mfg. Company
Automotive Mfg. Company
Optimization Levels
Optimization Levels
Optimal Designs Enterprise
Optimal Designs Enterprise
Level 1
Company
Company
Design
Dept.
Mfg.
Dept.
Engine
Mfg.
Power
Train
Engine
Design
Power
Train
Level 2
Dept.s
Level 3
Groups
Design Department's
Design Department's
Engine Design Code
Engine Design Code
(Get Iron, Rubber, etc values from Co. database)
(Get Iron, Rubber, etc values from Co. database)
Find
Find EngineSize
EngineSize, etc; In
, etc; In Engine
Engine; to Minimize
; to Minimize
Pollution
Pollution; and Maximize
; and Maximize GasEfficiency
GasEfficiency...
...
o o o
o o o
Model
Model Engine
Engine
HorsePower = ... Iron ...
HorsePower = ... Iron ... EngineSize
EngineSize ...
...
GasEfficiency
GasEfficiency = ... TerrainType ... HorsePower
= ... TerrainType ... HorsePower
Pollution
Pollution = ... CarWeight ... HorsePower ...
= ... CarWeight ... HorsePower ...
GasInEfficiency ... Rubber
GasInEfficiency ... Rubber
End Model
End Model
Level 3
Design Department's Code
Design Department's Code
(Get Iron, Rubber, etc values from Co. database)
(Get Iron, Rubber, etc values from Co. database)
Find
Find CarWeight
CarWeight, etc; In
, etc; In CarDesign
CarDesign; to Minimize
; to Minimize
CarPollution
CarPollution; and Maximize
; and Maximize CarSafety
CarSafety ...
...
o o o
o o o
Model
Model CarDesign
CarDesign
Call EngineDesign
Call EngineDesign !
! Another Optimization
Another Optimization
Call PowerTrain
Call PowerTrain !
! Another Optimization
Another Optimization
CarPollution
CarPollution = ...
= ... CarWeight
CarWeight ... HorsePower ...
... HorsePower ...
GasInEfficiency ... Rubber ... Coal
GasInEfficiency ... Rubber ... Coal
CarSafety
CarSafety = ... Iron ... CarWeight ...
= ... Iron ... CarWeight ...
End Model
End Model
Level 2
Company's Code
Company's Code
(Get present Iron, Rubber, etc values from Co. database)
(Get present Iron, Rubber, etc values from Co. database)
Find
Find Iron
Iron,
, Rubber
Rubber, etc; In
, etc; In Company
Company; to Minimize
; to Minimize
Time2Market
Time2Market; and Maximize
; and Maximize Profit
Profit ...
...
o o o
o o o
Model
Model Company
Company
Call Design
Call Design !
! Another Optimization
Another Optimization
Call Manufacturing
Call Manufacturing !
! Another Optimization
Another Optimization
Call Sales
Call Sales
!
! Another Optimization
Another Optimization
Time2Market
Time2Market = ...
= ... Iron
Iron ...
...
Profit
Profit = ...
= ... Iron
Iron ...
... Rubber
Rubber ... Coal ...
... Coal ...
End Model
End Model
Level 1
Code for Optimizations
Code for Optimizations
Level 3 Code ... Groups
Level 3 Code ... Groups
Most Important ... fundamental equations
Most Important ... fundamental equations
Can Run Independently
Can Run Independently
Contains Math Models to Simulate Design/Mfg.
Contains Math Models to Simulate Design/Mfg.
Level 2 Code ... Dept.s
Level 2 Code ... Dept.s
Runs Latest Level 3 code too
Runs Latest Level 3 code too
Level 1 Code ... Company
Level 1 Code ... Company
Runs Latest of All Lev
Runs Latest of All Levels
els
All L 3s
All L 2s
L 1
Some L 3s
l L 2
1 L 3
Easy to Update
Easy to Update
Objective-Driven Employees
Objective-Driven Employees
for
for
Entire Company
Entire Company
Optimization
Optimization
of
of
Optimizations
Optimizations
What are you building?
What are you building?
Mr. Arithmetic Mr. Algebra Mr. Calculus
Mr. Arithmetic Mr. Algebra Mr. Calculus
------------------- --------------- ----------------
------------------- --------------- ----------------
Slide Rule Simulate Optimize
Slide Rule Simulate Optimize
Conclusion
Conclusion
Use
Use
FortranCalculus
FortranCalculus
to improve
to improve
Math Models
Math Models
Design Productivity
Design Productivity
Manufacturing Tolerances
Manufacturing Tolerances
Optimal Designs Enterprise
Optimal Designs Enterprise
goal-driven.net
goal-driven.net