5/18/2005 ICCS-05 1
On the Fundamental Tautology of ValidatingData-Driven Models and Simulations
John Michopoulos & Sam Lambrakos Naval Research Laboratory
Materials Science and Technology DivisionWashington DC, 20375, USA
International Conference on Computational Sciences 2005 Atlanta, GA USA,, 22-25 May 2005
This material is based upon work supported by the National Science Foundation under Grant No. ITR-0205663
5/18/2005 ICCS-05 2
OUTLINE
• Motivational Aspects• Qualification Verification Validation In general• “Embedded Validation”• Epistemological Aspects• Example• Conclusions
5/18/2005 ICCS-05 3
Motivational Aspects
Answer Some Questions for systems where BOTH input and output are measurable:
Are data-driven modeling and simulation practices equivalent to the non data-driven (or model driven) practices the same from the QV&V perspective?
Do data-driven models require validation in the model-driven modeling sense?
5/18/2005 ICCS-05 4
Term Definitions
Modeling: Establishing, a conceptual (analytical, mathematical) and computational (discretization, algorithmic, programmatic, visualization) representation of the system such that its behavior is the same with that of the actual physical system.Simulation: Generating predictive behavior of the system through exercising the a previously established model of the system.Model-Driven: Modeling uses some a priori concept of how the system works from an inside-out perspective (Bottom-Up)Data-Driven: Modeling uses only behavioral data and ignores internal composition (Top-Down)
5/18/2005 ICCS-05 5
System-Model-Behavior
System External Appearance Behavior AppearanceInternal Appearance
Phys
ical
Wor
ldCo
ncep
tual
Wor
ld
Hybrid 10 Load versus StrokeOpen Hole, Net Tension
-20
0
20
40
60
80
100
120
140
160
180
200
0 1 2 3 4 5 6
Stroke (mm)
Load
(kN)
Modeled System
?
P
P
δ
Physical System
Hybrid 10 Load versus StrokeOpen Hole, Net Tension
-20
0
20
40
60
80
100
120
140
160
180
200
0 1 2 3 4 5 6
Stroke (mm)
Load
(kN)
)(PδP δ )(δP
0),( =δPRPδ
5/18/2005 ICCS-05 6
MODELING A MULTIPHYSICS SYSTEM
Homogeneous SystemHomogeneous Fieldsi.e. Deformable Media
Multiple SystemHomogeneous Fieldsi.e. Deformable Media
In contact
Homogeneous System
Multiple Fieldsi.e. Electro-Thermo-
elastic Media
Multiple SystemMultiple Fields
i.e. Thermoelasticmedia
In Fluids
FieldCardinality
Multiple DomainsOne Domain
One
Fie
ldM
ultip
le F
ield
s
DomainCardinality
aV∂
aV
bV∂
bV
2aΓ
3aΓ
1bΓ
1aΓ
2bΓ
kp
lp
kq
lq
bkp
nI mOPhysical system
Analytical model
mFnJ mP
Obs.-Math. Op. Obs.-Math. Op.
nJ mPComputational
model
mCF
Math.+Coding. Op. Math.+ Coding. Op.
nJ mPVisualization
model
mDF
Discr.+Coding Op. Discr.+Coding Op.
MO
DELIN
G
SIMU
LATIO
N
5/18/2005 ICCS-05 7
MODELING A MULTIPHYSICS SYSTEM
I O
mmnnmn
OIOI ℜ⊆ℜ⊆→ ,,:~ F Behavior Functional relating the output to the inputs of the system.
p r
0)~,~,~( =pq ξR Relational restriction over dependent, independent, parameter variables.
)~,~ ~ ~ pq p ξ(Ξ∇= Vector Function Storage Mechanism:Potential or Energy Density function.
0,...)~ ,~( =
∂∂
∇ℵ qt
q mm
n
i Composition Behavior through Conservation Law Relations: Dependence, of dependent variables on position in the structure, and time.
Structuralstate
engine
)~,~~~ pq ξ(C= Bulk Constitutive Relations:Functional dependence, of dependentvariables on independent variablesand parameters.
Bulkstate
engine q
~ )~,~~~ εεξσ ⋅= (C
5/18/2005 ICCS-05 8
MODELING A MULTIPHYSICS SYSTEM
Methodologies for Developing Potential Functions
Potential Function
Lagrangean/Hamiltonian Thermodynamic
Equilibrium Non-Equilibrium
Compensating Fields Internal Variables
5/18/2005 ICCS-05 9
MODELING A MULTIPHYSICS SYSTEM
Model driven coupled field methodology
Is # of equations >from # of variable pairs
Consider theoretical physicalsystem characteristics
& define state parameters
Define dependent &independent variable pairs
Employ conservation laws
Postulate constitutiveequations
Employ additional axioms
Rewrite previous restrictions to field eqtns.
Define BV problem forgiven structure
Solve field equations
Evaluate & Display resultsSimulate system response
Determine material properties constants
yes
no
Keep and Stop
Validated?
yes
no
5/18/2005 ICCS-05 10
MODELING A MULTIPHYSICS SYSTEM
AXIOMS OF CONSTITUTIVE BEHAVIOR DERIVATION PROCESSAxiom (I) Causality: The motion, temperature, electric field, and magnetic induction of the material points of a body are self-evident and observable in any thermoelectromechanical behavior of a body. The remaining quantities (other than those derivable from the motion, temperature, electric field, and magnetic induction) excluding the body force, energy supply, and free charge density that enter the balance laws and the entropy inequality, are the dependent variables.
Axiom (II) Determinism: The value of any depen-dent variable, at material point X of the body B at time t, is determined by the history of all material points of B.
Axiom (III) Equipresence: At the outset, all constitutive response functionals are to be considered to depend on the same list of constitutive variables, until the contrary is deduced.
Axiom (IV) Objectivity: The constitutive response functionals are form-invariant under arbitrary rigid motions of the spatial frame of reference and a constant shift of the origin of time.
Axiom (V) Time Reversal: The entropy product-ion must be nonnegative under time reversal.
Axiom (VI) Material Invariance: The constitutive response functionals must be form-invariant with respect to a group of transformations of the material frame of reference {X ---> X } and "microscopic time reversal" as {t ---> -t } representing the material symmetry conditions. These transformations must leave the density and charge at { X, t } unchanged.
Axiom (VII) Neighborhood: The values of the response functionals at X are not affected appreciably by the values of the independent constitutive variables at distant points from X.
Axiom (VIII) Memory: The values of the constitutive variables, at a distant past from the present, do not affect appreciably the values of the constitutive response functionals at present.
Axiom (IX) Admissibility: Constitutive equations must be consistent with the balance laws and the entropy inequality.
5/18/2005 ICCS-05 11
MODELING A MULTIPHYSICS SYSTEM
MAIN SCIENTIFIC CHALLENGES OF MODEL DRIVEN APPROACH• Impossible Uncoupled Experiments for Coupling coefficients
Determination?EXAMPLE: Isotropic Nonlinear Electromagnetic Solids. Two of the 20 apriori
derived constitutive relations for the spatial vectors for heat and current define some of the known “effects”:
• Potentially NON TERMINATING
Thermal conductivity
Ohm’s effect
Peltier’s effect
Seebeck’s effect
Anisotropic Peltier’s effect
Righi-Leduc’s effect Ettinghaussen’seffect
Anisotropic Seebeck’s effect
Hall’s effect Nerst’s effect
]~)~()~(~[]~)~~()~~(~[
~)~(~)~~(~~~
~~~~~~~~
1112
1111
1092
82
7
651
41
321
BTeBTeBEeBEe
BTBBEBTeEe
BTBETeEeTEJ
×∇−×∇+×−×+
∇⋅+⋅+∇++
×∇+×+∇++∇+=
−−−−
−−
−−
λλ
λλλλ
λλλλλλ
]~)~~()~~(~[]~)~()~(~[
~)~~(~)~(~~~
~~~~~~~~
1112
1111
1092
82
7
651
41
321
BEeBEeBTeBTe
BEBBTBEeTe
BEBTEeTeETq
×−×+×∇−×∇+
⋅+∇⋅++∇+
×+×∇++∇++∇=
−−−−
−−
−−
κκ
κκκκ
κκκκκκ
5/18/2005 ICCS-05 12
MODELING A MULTIPHYSICS SYSTEM
Data driven Modeling methodology
Consider state parameters
Define dependent &independent variable sets
Add detail by employing conservation laws
Generate constitutive lawsand field equations
Define BV problem forgiven structure
Solve field equations
Evaluate & Display resultsSimulate system response
Keep and Stop
Experimentally Collectinput/output pairs
Define relation between i/o pairs trough potential
function(s) with free coefficients
Construct & solve over-determined system of equations
5/18/2005 ICCS-05 13
QV&V Definitions
Intersecting definitions according to AIAA, DMSO, ASME, DOE/DP-ASCI
• Qualification: determination that a conceptual model implementation represents correctly a real physical system.
• Verification: determination that a computational model implementation represents correctly a conceptual model of the physical system.
• Validation: determination of the degree to which a computer model is an accurate representation of the real physical system from the perspective of the intended uses of the model.
5/18/2005 ICCS-05 14
Model-Driven Approach for Q&VV
Expe
rim
enta
tion
Physical Behavior
(data)
ACTUAL PHYSICAL SYSTEM
ANALYTICAL SYSTEM MODEL
COMPUTATIONAL SYSTEM MODEL
Programming
PrototypingAnalysis
Simulated Behavior
(data)Simulation
verification test
no: -> readjust
qualification test (=) no: -
>adapt
validation test (=)no: -
>adapt
Simulated Behavior
(data)Simulation
5/18/2005 ICCS-05 15
General Optimization Approach
EXPERIMENTATION
SIMULATION
System Model
Physical System
OPTIMIZATION
PhysicalBehavior
SimulatedBehavior
closeenough?
Modify designvariables
Final designvariables
yes
no
5/18/2005 ICCS-05 16
Data-Driven Approach for Q&VV
Expe
rim
enta
tion
Physical Behavior
(data)
ACTUAL PHYSICAL SYSTEM
ANALYTICAL SYSTEM MODEL
COMPUTATIONAL SYSTEM MODEL
Programming
PrototypingAnalysis
Simulated Behavior
(data)Simulation
verification test
no: -> readjust
qualification test (=) no: -
>adapt
Simulated Behavior
(data)Simulation
5/18/2005 ICCS-05 17
EPISTEMOLOGICAL BACKGROUND
Identify Observables
Collect Control andBehavior Facts
Formulate Theory T torepresent these facts
inductively
Use T to predictunobserved behavior with
certainty
Industrialized-InductiveScientific Method
PhysicalWorld
ConceptualWorldfrom from
in
Make some observations
Make Hypothesis H
Formulate Theory T
Use logic to deducepredicted observation O
Do experiment to observeO or ~O
~O isobserved
H or T is falsified(Popper)
O isobserved
true
H under T is confirmed(Hempel)
false
OR
true
Use T with noconfidence
Use T with lowconfidence
Strong Approach Weak Approach
Hypothetico-deductive Scientific Method
false
Hypothetico-deductive(rationalist)
inductive(empiricist)
Hypothetico-deductive
Industrializedinductive
LogicalPositivism
1800s 1920-30 Today
Scientific Method Evolution
5/18/2005 ICCS-05 18
EXAMPLE: IONIC POLYMER COMPOSITE ARTIFICIAL MUSCLES
5/18/2005 ICCS-05 19
Ionic Polymer Metal Composite (IPMC) Large Deflection PlatesNon-linear electro-elasto-dynamic field PDEs:
For Electric and Mechanical activation onlywithout mass transport:
2 2 2,22 ,11 ,12 ,12 ,11 ,22
2 2 2 2,12 ,11 ,22
2 20
(1 ) ( 2 ),
[( ) ]1 2{ }
k k
k k
i k k c
h qw p E F w F w F wN h
F Ep E E w w w
V p F p EEh
ν
νε ρ
∇ ∇ − + ∇ = + − +
∇ ∇ − ∇ = −
−∇ + ∇ − =
or equivalently:2 2 2
1 ,22 ,11 ,12 ,12 ,11 ,22
2 2 2 22 ,12 ,11 ,22
2 2 23 4 5
( 2 ),
[( ) ]
0
h qw c V F w F w F wN h
F c V E w w w
V c F c V c
∇ ∇ − ∇ = + − +
∇ ∇ − ∇ = −
∇ + ∇ ∇ + + =
5/18/2005 ICCS-05 20
IPMC Large Deflection Plates: Parameter Identification
Multi-Objective Function Optimization2 2 2
1 1 1min ( ) min{ [ ( ) ] [ ( ) ] [ ( ) ] }
subject to constrains:
where: , , are the experimentally observed state variables a
n n no s e s e s e
j i j i i j i i j ii i i
u lj j j
e e ei i i
f c w c w F c F V c V
t c t
w F V
= = =
= − + − + −
≥ ≥
∑ ∑ ∑
t each point
( ), ( ), ( ) are the simulated state variables at each point
are the uknown constants to be identified
, are the upper and lowe
s s si j i j i j
j
u lj j
i
w c F c V c i
c
t t r limits constraining each uknown j
To fully characterize ci we need a family of experiments that sweeps across boundary values of w, F, V and measuresthem on a grid superimposed on the domain of the plate
5/18/2005 ICCS-05 21
IPMC Large Deflection Plates: Experimental Results (M. Shahinpoor UNM)
Voltage, Displacement vs. Time of IPMC plate
-0.1
0
0.1
0 2 4 6
Time (s)
Volta
ge (1
00v)
, D
ispl
acem
ent
(mm
)
Voltage Displacement
Voltage vs. Displacement of an IPMC plate (40mm x 40mm x 0.21mm)
012345678
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14
Displacement (mm)
Vol
tage
(v)
5/18/2005 ICCS-05 22
IPMC Large Deflection Plates: Biharmonic Single-Field Bases
4 41
6 2 2 2 2 21 1
16 (2 ) (2 )( , ) sin sin( ) 2 2
s
m n
q a b m x a m y bw x yD mn b m a n a b
π ππ
∞ ∞
= =
+ +=
+∑∑4 ( 1) / 2
25 5
1,3,5,...
tanh 24 ( 1)( , ) cos (1 cosh2cosh
1 sinh )2cosh
2
ms m m
m m
m
m
qa m x m yw x yD m a a
m y m ya a
m ba
α απ ππ α
π πα
πα
−∞
=
+−= − +
+
=
∑
(1,2)where: ( , ) single-field plate deflection at point ( , ) ( , ) mechanical load distribution on the plate Flexural rigility of the plate , length and wi
sw x y x yq q x yDa b
==
dth of the plate along x and y axes
5/18/2005 ICCS-05 23
IPMC Large Deflection Plates: Comparison (1)
Deflection Time Histories in 3D carpet plots
EXPERIMENT COMPUTED
5/18/2005 ICCS-05 24
IPMC Large Deflection Plates: Comparison (2)
Deflection Time Histories in Contour plots
EXPERIMENT COMPUTED
5/18/2005 ICCS-05 25
Conclusions
• Data driven models and simulations contain validation
• Model-driven models have epistemologicorigins
• When data exist “data-driven” is preferable
5/18/2005 ICCS-05 26
THANK YOU FOR YOUR ATTENTIONQUESTIONS?
5/18/2005 ICCS-05 27
Data-Driven Approach: ExampleIdentify material from small specimens to predict behavior of large system
Data Driven Composite Materials & Structures WorkbenchData Driven Composite Materials & Structures Workbench
1.0”
0.5”
0.5”
0.5”
0.6”
0.04”
0.1”
grip area
grip area
?
MaterialCharacterization
MaterialCharacterization
Material/StructuralComposition
Material/StructuralComposition
Material/StructuralBehavior Simulation
Material/StructuralBehavior Simulation
SpecimenTesting
SpecimenTesting
5/18/2005 ICCS-05 28
MECHATRONICALLY AUTOMATED APPROACH: Overview
General Case:3 displacements + 3 rotations + 3 forces + 3 moments + Np x 3 strains + Np x Nf = 12+ (3+Nf)xNp Datastreams
u0
u1
u2
u0
u1
u2
p1
p2
p3
p4
p5
p6
p7
p8
p9
p10
p11
p12
p13
p14
p15
u0
u1
u2
p1
p2
p3
p4
p5
p6
p7
p8
p9
p10
p11
p12
p13
p14
p15
p1
p2
p3
p4
p5
p6
p7
p8
p9
p10
p11
p12
p13
p14
p15
u0
u1
u2
p1
p2
p3
p4
p5
p6
p7
p8
p9
p10
p11
p12
p13
p14
p15p11
Θ2
Θ1
NRL’s Automated 6-D Loader
Specimen Special Case:2 displacements + 1 rotation=3DOFs=6Datastreams
5/18/2005 ICCS-05 29
Approach
Data driven methodology for PMCs: Data Collection
Hexcel Rd5129, (+/-15), path 2Pure Opening
HexcelRd5129pc 815,- 15<, Load case ;2
-0.002-0.001 00.001d1
-0.03
-0.02
-0.01
0
d2
00.00050.001
0.00150.002d3
-0.002-0.001 00.001d1
u0u1
u2
HexcelRd5129pc 815,- 15<, Load case ;2
-1001020
f1
-1500
-1000
-500
0
f2
0
200
400
f3
-1001020
f1
-1500
-1000
-500
0
f2
f0f1
f2
0.005 0.01 0.015 0.02 0.025 0.03 0.035disp
250
500
750
1000
1250
1500
f HexcelRd5129pc 815,- 15<, Load case ;2
u
f
5/18/2005 ICCS-05 30
Approach
-2
0
2
0 0.005 0.01 0.015 0.02-500050010001500
-500050010001500
1350-1350-1800 1800
Θ2
Θ1
r
900
900
-900
-900 450
450
-450
-450
00
2 5 8 11 1458 811
Data driven methodology for PMCs: Data Reduction
r
f
Θ1
5/18/2005 ICCS-05 31
Approach
Data driven methodology for PMCs: Data Reduction
-0.01
0
0.01 x
-0.01
0
0.01
y0
1000
2000
3000
4000
f@x, yD0
1000
2000
3000
4000
f@x, yD
u0
u1
f-0.02
-0.01
0
0.01
0.02
-0.02-0.01
00.01
0.02
-500050010001500
-0.02
-0.01
0
0.01
0.02u1
u0
Load
-0.02
-0.01
0
0.01
0.02
-0.02-0.01
00.01
0.02
0
50000
100000
-0.02
-0.01
0
0.01
0.02u1
u0
Stiffness
-0.02
-0.01
0
0.01
0.02
-0.02-0.01
00.01
0.02
0510
15
-0.02
-0.01
0
0.01
0.02u1
u0
Dissipated energy
5/18/2005 ICCS-05 32
MECHATRONICALLY AUTOMATED APPROACH: Inverse Model
u0
u1
u2
t0
t1
t2n discrete elements
)~()())~(~( ~~)~,~ εχεχεφ ii mcxcc =⋅=(
Energy Balance: jjV
iv
tsv
v
u
u dxxuutdqqtr ))(( 21
0 ∫∫ =−∂
εφ
General Problem: Build a function out of knowing some of its values trough a collocation method
kkk VD φ=Total DE in
element k of n
∑ ∫=
=n
kjj
Vikk dxxV
1
))(( ∂
εφφ
Total DE inwhole structure
εr
εθ
εφ
Strain Space125,,2,1,for
,0 ,1
)~( …=
⎭⎬⎫
⎩⎨⎧
≠=
= ji
jiji
ji εχ
εθ εφ
χ
εθ εφ
χ
εθ εφ
χ
εθ εφ
χ
εθ εφ
χ)(0)(0)~
)~()(1)()~()()~ 11
mcmc
mcmcmc
iii
immiii
=++++=
++++=
……
……
εφ
εχεχεφ(
(
5/18/2005 ICCS-05 33
MECHATRONICALLY AUTOMATED APPROACH:Optimization
jjV
iv
tsv
v
u
u dxxuutdqqtr ))(( 21
0 ∫∫ =−∂
εφ
[ ] 0~ ≥cM
DED monotonicity: 100 additional constrains
pp
e
ep
ep
ii DeVmc =+∑=110
1
)~()( εχFor a loading point p:
Final Problem: Minimize subject to
e~ [ ] 0~ ≥cMUsing 17 points per loading pathgenerates 255 loading points leading to 255 equations for 125 unknowns
[ ] decX ~~~ =+For all selected loading points:
Solution Method: Least Squares with Linear Constrains
5/18/2005 ICCS-05 34
MECHATRONICALLY AUTOMATED APPROACH:Simulation Synthesis
Pr. JGM 13
UTILIZATION OF BASIS LOADING CASES RESPONSE THROUGH LINEAR COMBINATION OF RESULTING STRAIN FIELDS
a a axx
yy
xy sliding
xx
yy
xy open close
xx
yy
xy bending
0 1 2
~~~
~~~
~~~
/
εεε
εεε
εεε
⎧
⎨⎪
⎩⎪
⎫
⎬⎪
⎭⎪
+
⎧
⎨⎪
⎩⎪
⎫
⎬⎪
⎭⎪
+
⎧
⎨⎪
⎩⎪
⎫
⎬⎪
⎭⎪
u0
u1
u2
a1
a2
a0
~ ~ ~ ~u a u a u a up = + +0 0 1 1 2 2
~up
Compute Dissipated EnergyDensity Distribution
bending basis case
opening/sliding basis case
sliding basis case
u0
~~~
εεε
xx
yy
xy sliding
⎧
⎨⎪
⎩⎪
⎫
⎬⎪
⎭⎪
u1~~~
/
εεε
xx
yy
xy open close
⎧
⎨⎪
⎩⎪
⎫
⎬⎪
⎭⎪
~~~
εεε
xx
yy
xy bending
⎧
⎨⎪
⎩⎪
⎫
⎬⎪
⎭⎪
u2
5/18/2005 ICCS-05 35
MECHATRONICALLY AUTOMATED APPROACH:Simulation Synthesis
u0
u1
u2
(opening)
(bending)
(shearing)
5/18/2005 ICCS-05 36
Approach
Data driven methodology for PMCs: Material Characterization
SpecimenManufacturing
SpecimenManufacturing
Material, Geometry,Loading Spec.
Material, Geometry,Loading Spec.
Strain FieldDetermination
Strain FieldDetermination
SpecimenSpecimen
Strain FieldsStrain Fields
Multi-dTesting
Multi-dTesting
Analytic DEDDetermination
Analytic DEDDetermination
Minimize Difference|Meas.-Anal.|DE
Minimize Difference|Meas.-Anal.|DE
Derivation ofMeasured DED
Derivation ofMeasured DEDMeasured Boundary
Displ. & Loads
Measured BoundaryDispl. & Loads
Measured DEMeasured DE
Set of Analytic DEsSet of Analytic DEs
Actual DEDActual DED
5/18/2005 ICCS-05 37
How - ApproachSTRUCTURAL SYSTEM IDENTIFICATION-CHARACTERIZATION
)()()()( .. δψδδδδ +=+== mPPPP irecrec
)~(~~)~( ucu χψ ⋅=
∫∫∂
⋅+=V
kjjkiii uducmduuPi ~)~(~)(21)(
0χδδ
δ
iiij bcA =
)()()( .. δδ
δδ
δδ irecrectotal W
ddW
ddW
dd
+=
1-D Load Space Partitioned Model of Behavior
∫∫∂
+=V
kjjkiii udumduuPi ~)~()(21)(
0ψδδ
δ
n-D Load Space Partitioned Model of Behavior
duumduuP ∫∫ +=δδψδδ
00)()(
21)(
)()()( .. δδδ irecrectotal WWW +=
5/18/2005 ICCS-05 38
How - Approach:GENERAL CONTINUOUM SYSTEMS MODELING
Data driven methodology for PMCs
AXIOMS OF ENRICHMENT
• All state variables are varying in a locally flat fashion both in space and time (continuity)
• The behavior of the whole structure is equivalent to the composition of the behaviors of structural discretization units (composition behavior)
• The observed behavior is repeatable when observed under identical conditions in various times (first order of reality)
ASSUMPTIONS
• Loading is either static or slowly varying• The material behavior will be non-viscous, and independent of rate and load history• The constitutive relation is continuous both in input and output variables• Deformations are sufficiently small so that the infinitesimal stress and strain tensors may be
employed
5/18/2005 ICCS-05 39
MODELING A MULTIPHYSICS SYSTEM
I O
mmnnmn
OIOI ℜ⊆ℜ⊆→ ,,:~ F Behavior Functional relating the output to the inputs of the system.
p r
0)~,~,~( =pq ξR Relational restriction over dependent, independent, parameter variables.
0,...)~ ,~( =
∂∂
∇ℵ qt
q mm
n
i Composition Behavior through Conservation Law Relations: Dependence, of dependent variables on position in the structure, and time.
Structuralstate
engine
)~,~~~ pq ξ(C= Bulk Constitutive Relations:Functional dependence, of dependentvariables on independent variablesand parameters.
Bulkstate
engine q
~ )~,~~~ εεξσ ⋅= (C
)~,~ ~ ~ pq p ξ(Ξ∇= Vector Function Storage Mechanism:Potential or Energy Density function.
5/18/2005 ICCS-05 40
MODELING A MULTIPHYSICS SYSTEM
Data driven Modeling methodology
)~,~ ~ ~ pq p ξ(Ξ∇= Vector Function Storage Mechanism:Potential or Energy Density function.
)~,~ )~,~)~,~ ppp ξϕξξ ((( +Φ=Ξ Additivity of recoverable and non-recoverable components.
ppqp ~)~(~2
1)~,~ ⋅=Φ ξ(
Recoverable energy definition.
))~(~,~( ~~)~,~ xpcp ξχξϕ ⋅=(
Non-recoverable energy definition.
5/18/2005 ICCS-05 41
Expe
rim
enta
tion
Traditional vs. Data-Driven Approach of Q&VV
ACTUAL PHYSICAL SYSTEM
ANALYTICAL SYSTEM MODEL
COMPUTATIONAL SYSTEM MODEL
Programming
PrototypingAnalysis
Physical Behavior
(data)
Simulated Behavior
(data)
Simulated Behavior
(data)
validation test (=)
verification test
qualification test (=)
Simulation
no: -> readjustno: -
>adapt
no: ->
adapt
Simulation
DATA DRIVEN INVERSEMODELING ENCAPSULATION
DATA DRIVEN CODEGENERATION
5/18/2005 ICCS-05 42
Data-Driven System Identification and Simulation
Simulator
Physical system
System Model
Optimizer
ExperimentalFRAME
System Model
Sensor or DesignFRAME
5/18/2005 ICCS-05 43
ACTUAL PHYSICAL SYSTEM
CONCEPTUAL SYSTEM MODEL
COMPUTATIONAL SYSTEM MODEL
Programming
SimulationAnalysis
Physical Behavior
(data)
Simulated Behavior
(data)
Simulated Behavior
(data)
validation test (=)
Experimentation
Pred
ictio
n
Prediction
no: ->
adapt
DATA DRIVEN CONCEPTUAL SYSTEM MODEL