Transcript S. Acharjee - Professor Nicholas Zabaras

STOCHASTIC AND DETERMINISTIC
TECHNIQUES FOR COMPUTATIONAL
DESIGN OF DEFORMATION PROCESSES
Swagato Acharjee
B-Exam
Date: April 13, 2006
Sibley School of Mechanical and Aerospace
Engineering
Cornell University
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ACKNOWLEDGEMENTS
SPECIAL COMMITTEE:
 Prof. Nicholas Zabaras
 Prof. Subrata Mukherjee
 Prof. Leigh Phoenix
FUNDING SOURCES:
 Air Force Office of Scientific Research (AFOSR), National Science
Foundation (NSF), Army Research Office (ARO)
 Cornell Theory Center (CTC)
 Sibley school of Mechanical & Aerospace Engineering
Materials Process Design and Control Laboratory (MPDC)
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OUTLINE
Deterministic design of deformation processes
•Overview of direct and sensitivity deformation problems
•Applications
Stochastic modeling of inelastic deformations
•Probability and stochastic processes
•Generalized Polynomial Chaos Expansions (GPCE)
•Non Intrusive Stochastic Galerkin Approximation
Stochastic optimization
•Robust design of deformation processes
•Applications
Suggestion for future work
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Part I - Deterministic design of deformation
processes
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METAL FORMING PROCESSES
Forging
Boeing 747
18,600 forgings
Extrusion
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Rolling
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COMPUTATIONAL DESIGN OF DEFORMATION PROCESSES
BROAD DESIGN OBJECTIVES
Given raw material, obtain final product with desired microstructure
and shape with minimal material utilization and costs
COMPUTATIONAL PROCESS DESIGN
Design the forming and thermal process sequence
Selection of stages (broad classification)
Selection of dies and preforms in each stage
Selection of mechanical and thermal process parameters in each stage
Selection of the initial material state (microstructure)
OBJECTIVES
Material usage
Plastic work
Uniform deformation
Microstructure
Desired shape
Residual stresses
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VARIABLES
CONSTRAINTS
Press force
Press speed
Processing temperature
Geometry restrictions
Product quality
Cost
Identification of stages
Number of stages
Preform shape
Die shape
Mechanical parameters
Thermal parameters
Materials Process Design and Control Laboratory
DEFORMATION PROCESS DESIGN - BROAD OUTLINE
1. Discretize infinite dimensional design space into a finite dimensional space
2. Differentiate the continuum governing equations with respect to the design
variables to obtain the sensitivity problem
3. Discretize the direct and sensitivity equations using finite elements
4. Solve and compute the gradients
5. Combine with a gradient optimization framework to minimize the objective
function defined
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CONSTITUTIVE FRAMEWORK
(1) Multiplicative decomposition framework
(3) Radial return-based implicit integration algorithms
Initial configuration
Temperature: n
void fraction: fn
Deformed configuration
Temperature: 
void fraction: f
F
Bn
F
B
(2) State variable rate-dependent models
(4) Damage and thermal effects
Governing equation – Deformation problem
Governing equation – Coupled thermal problem
Fe
Thermal expansion:
.
Fp
F F
Intermediate thermal
configuration
Temperature: 
void fraction: fo
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Stress free (relaxed)
configuration
Temperature: 
void fraction: f
 –1
.
=I
Hyperelastic-viscoplastic constitutive laws
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3D CONTACT PROBLEM
Contact/friction model
Impenetrability Constraints
Current
configuration
Reference
configuration
Admissible region
n
Coulomb Friction Law
τ1
τ2
Inadmissible region
 Continuum implementation of die-workpiece contact.
 Augmented Lagrangian regularization to enforce impenetrability and frictional stick conditions
Contact surface smoothing using Gregory Patches
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SENSITIVITY DEFORMATION PROBLEM
Continuum problem
Differentiate
Discretize
Design sensitivity of equilibrium equation
Variational form - Calculate
o
o
Fr and x
Kinematic problem
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o
o
o
x = x (xr, t, β, ∆β )
o o
Pr and F, 
Constitutive problem
o
such that
o
λ and x
Regularized
contact problem
Materials Process Design and Control Laboratory
PREFORM DESIGN TO MINIMIZE BARRELING
Curved surface parametrization – Cross section
can at most be an ellipse
H
Model semi-major and semi-minor axes as 6
degree bezier curves
x(  )  a cos 
y (  )  b sin 
6
6
a (  )   ii
b(  )   i  6i
i 1
i 1
1  (1.0   )5 (1.0  5.0 ) 2  15.0(1.0   ) 4  2
3  20.0(1.0   )3 3
4  15.0(1.0   ) 2  4
5  6.0(1.0   ) 2  5
6   6
  2z / H
(x,y) =(acosθ, bsinθ)
b
a
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Design vector
β  {1 ,  2 , 3 ,  4 ,  5 ,  6 ,  7 , 8 ,  9 , 10 , 11 , 12 }T
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PREFORM DESIGN TO MINIMIZE BARRELING IN FINAL PRODUCT
Objective:
Design the initial preform for a
fixed reduction so that the
barreling in the final product in
minimized
Material:
Al 1100-O at 673 K
Normalized objective
Initial preform shape
Optimal preform shape
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
2
Iterations
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4
6
8
Final forged product
Final optimal forged product
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EXTENSION TO COMPLEX SIMULATIONS
Remeshing
•Advanced THEX algorithm for unstructured hexahedral remeshing using CUBIT
(Sandia).
•Interface CUBIT with C++ code using NETCDF arrays and FAN utilities
Speed
•Fast solution using Block Jacobi\ ILU preconditioned GMRES solver (PetSc).
•Fully parallel assembly.
•Fully parallel remeshing and data transfer.
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PREFORM DESIGN FOR A STEERING LINK
Objective:
Design the initial preform such that the die cavity is fully filled with minimum flash
for a fixed stroke
Objective Function:
Die/Workpiece Setup
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Reference problem – large flash
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PREFORM DESIGN FOR CLOSED DIE FORGING
Preform design for a steering link
First iteration – underfill
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Intermediate iteration – underfill
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PROCESS DESIGN
Preform design for a steering link
Final iteration flash minimized and
complete fill
Objective function
Normalized Objective Function
1.2
1
0.8
0.6
0.4
0.2
0
0
5
10
15
20
Iterations
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PREFORM DESIGN FOR AN AUTOMOTIVE CROSS SHAFT
Material Ti-6 Al 4-V
Power law model
Initial Setup
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PREFORM DESIGN FOR AN AUTOMOTIVE CROSS SHAFT
Initial iteration
Underfill
Flash
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PREFORM DESIGN FOR AN AUTOMOTIVE CROSS SHAFT
Intermediate iteration
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PREFORM DESIGN FOR AN AUTOMOTIVE CROSS SHAFT
Final iteration
Reduced
Flash
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Minimum
Underfill
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PREFORM DESIGN FOR AN AUTOMOTIVE CROSS SHAFT
Objective Function:
1.2
Normalized Objective Function
1
0.8
0.6
0.4
0.2
0
0
2
4
6
8
10
12
14
Iterations
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DEFORMATION PROCESS DESIGN ENVIRONMENT
Design
Simulator
Constitutive
sub-problem
Contact
sub-problem
Thermal
sub-problem
Kinematic
sensitivity
sub-problem
Kinematic
sub-problem
Direct problem
(Non Linear)
Sensitivity
Problem (Linear)
Remeshing
sensitivity
sub-problem
Remeshing
sub-problem
Constitutive
sensitivity
sub-problem
Contact
sensitivity
sub-problem
Thermal
sensitivity
sub-problem
Optimization
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Part II - Stochastic modeling of inelastic
deformations
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MOTIVATION
All physical systems have an inherent associated randomness
SOURCES OF UNCERTAINTIES
Engineering
component
•Uncertainties in process conditions
•Input data
Heterogeneous
random
Microstructural
features
•Model formulation – approximations,
assumptions.
•Errors in simulation softwares
Why uncertainty modeling ?
Assess product and process reliability.
Component
reliability
Fail
Safe
Estimate confidence level in model predictions.
Identify relative sources of randomness.
Provide robust design solutions.
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MOTIVATION
Material
heterogeneity
Engineering
Information flow across scales
•Atomic scale – Kinetic theory,
Maxwell’s distribution etc.
•Microstructural features – correlation
functions, descriptors etc.
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Microscale
Nanoscale
Physics
Material information – inherently
statistical in nature.
Mesoscale
Statistical
filter
Chemistry
Materials
Continuum
0
Length Scales ( A )
Electronic
1
1e2
1e4
1e6
1e9
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MOTIVATION:UNCERTAINTY IN METAL FORMING PROCESSES
Material
Process
Model
Forging rate
Stereology/Grain texture
Yield surface changes
Die/Billet shape
Dynamic recrystallization
Friction
Phase transformation
Isotropic/Kinematic
hardening
Cooling rate
Phase separation
Stroke length
Internal fracture
Billet temperature
Other heterogeneities
Softening laws
Rate sensitivity
Internal state variables
Dependance
Nature and degree
of correlation
Forging velocity
Small change in preform shape
could lead to underfill
Die shape
Die/workpiece friction
Initial preform
shape
Material
properties/models
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Texture, grain sizes
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UNCERTAINTY IN METAL FORMING PROCESSES
Earlier works
1. Kleiber et. al. – IJNME 2004
Response surface method for analysis of sheet forming processes
2. Sluzalec et. al. – IJMS 2000
Perturbation type methods
3. Doltsinis et. al. – CMAME 2003,2005
Perturbation type methods – avoided all strong nonlinearities
Issues with stochastic analysis
Extremely complex phenomena – nonlinearities at all stages - large deformation plasticity,
microstructure evolution, contact and friction conditions, thermomechanical coupling and
damage accumulation – standard RBDO methods do not work well.
Lack of robust and efficient uncertainty analysis tools specific to metal forming.
High levels of uncertainty in the system
Possibility of reusing already developed legacy codes.
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RANDOM VARIABLES
Definition – Probability space
The sample space Ω, the collection of all possible events in a sample space F and
the probability law P that assigns some probability to all such combinations
constitute a probability space (Ω, F, P )
Stochastic process – function of space, time and random dimension.
W ( x, t ,  )
x  X , t T ,  
The statistical average of a function
E[ g ( X )]   g ( y ) f X ( y )dy
For a stochastic process W (x,t, )
Covariance
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Ω
C ( x, t , x' , t ' )  E[W ( x, t ,  )W ( x' , t ' , )]
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GENERALIZED POLYNOMIAL CHAOS EXPANSION - OVERVIEW
n
~
W ( x, t ,)   Wi ( x, t ) i ( )
(Wiener,Karniadakis,Ghanem)
i 0
Chaos
polynomials
Stochastic
process
(random variables)
Reduced order representation of a stochastic processes.
Subspace spanned by orthogonal basis functions from the Askey series.
Chaos polynomial
Support space
Random variable
Legendre
[,]
Uniform
Jacobi
[,]
Beta
Hermite
[-∞,∞]
Normal, LogNormal
Laguerre
[0, ∞]
Gamma
Number of chaos polynomials used to represent output uncertainty depends on
- Type of uncertainty in input
- Number of terms in KLE of input
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- Distribution of input uncertainty
- Degree of uncertainty propagation desired
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FINITE DEFORMATION UNCERTAINTY ANALYSIS USING SSFEM
F()
X
B0
B n 1 ( )  Bin 1 i ( )
x n 1 ( )  xin 1 i ( )
xn+1()=x(X,tn+1, ,)
x ( X, tn 1 ,  )
X
F ( )  P ( xn 1 ,  )Q 1 ( X )
F ( )   0 x ( X, tn 1 ,  ) 
F  Fi  i = P Q 1  ( Pi  i )Q 1
xn+1()
Bn+1()
xn1 ( )
P ( xn 1 ,  ) 
 
xn1i
Pi ( xn 1i ) 

X
Q ( X ) 

Key features
Total Lagrangian formulation – (assumed deterministic initial configuration)
Spectral decomposition of the current configuration leading to a stochastic deformation
gradient
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TOOLBOX FOR ELEMENTARY OPERATIONS ON RANDOM VARIABLES
Scalar operations
Use
precomputed
expectations
of basis
functions and
direct
manipulation
of basis
coefficients
Non-polynomial
function evaluations
1. Addition/Subtraction
1. Square root
2. Multiplication
2. Exponential
3. Inverse
3. Higher powers
Matrix\Vector operations
Matrix Inverse
Compute B() = A-1()
1. Addition/Subtraction
2. Multiplication
Use direct
integration
over support
space
(PC expansion)
Galerkin projection
3. Inverse
4. Trace
5. Transpose
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Formulate and solve linear
system for Bj
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UNCERTAINTY ANALYSIS USING SSFEM
Linearized PVW
On integration (space) and further simplification
Galerkin projection
Inner product
 i  j   i ( ) j ( ) f ( )d
P
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UNCERTAINTY DUE TO MATERIAL HETEROGENEITY
State variable based power law model.
State variable – Measure of deformation resistance- mesoscale property
Material heterogeneity in the state variable assumed to be a second order
random process with an exponential covariance kernel.
Eigen decomposition of the kernel using KLE.
Initial and mean
deformed config.
Eigenvectors
 
f  0  
s
n
 r 

b 

2
s( p)  s0 (1   i i vi ( p))
R ( p1 , 0, p2 , 0)   2 exp 
i 1
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V1
0.409396
0.395813
0.38223
0.368646
0.355063
0.341479
0.327896
0.314313
0.300729
0.287146
V2
0.339819
0.239033
0.138247
0.0374605
-0.0633257
-0.164112
-0.264898
-0.365684
-0.466471
-0.567257
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UNCERTAINTY DUE TO MATERIAL HETEROGENEITY
Dominant effect of
material heterogeneity on
response statistics
Load vs Displacement
SD Load vs Displacement
14
1.6
12
1.4
1.2
8
SD Load (N)
10
Load (N)
Homogeneous material
Heterogeneous material
Mean
6
1
0.8
0.6
4
0.4
2
0.2
0
0
0.1
0.2
0.3
0.4
0.5
Displacement (mm)
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0.6
0.7
0.8
0
0
0.1
0.2
0.3
0.4
0.5
Displacement (mm)
0.6
0.7
0.8
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UNCERTAINTY DUE TO MATERIAL HETEROGENEITY-MC RESULTS
MC results from 1000 samples generated using Latin Hypercube
Sampling (LHS). Order 4 PCE used for SSFEM
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MODELING INITIAL CONFIGURATION UNCERTAINTY
xn+1()=x(XR,tn+1, ,)
F()
X()
xn+1()
FR()
Bn+1()
B0
XR
F*()
BR
Introduce a deterministic reference configuration BR which maps
onto a stochastic initial configuration by a stochastic reference
deformation gradient FR(θ). The deformation problem is then solved
in this reference configuration.
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INITIAL CONFIGURATION UNCERTAINTY
Deterministic simulationUniform bar under tension
with effective plastic strain
of 0.7 . Power law
constitutive model.
Initial configuration assumed to vary
uniformly between two extremes with
strain maxima in different regions in the
stochastic simulation.
Eq. strain
1.26656
1.19446
1.12235
1.05024
0.978136
0.906029
0.833922
0.761816
0.689709
0.617602
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Eq. strain
0.749556
0.708158
0.666759
0.625361
0.583962
0.542564
0.501165
0.459767
0.418368
0.37697
Materials Process Design and Control Laboratory
INITIAL CONFIGURATION UNCERTAINTY
Stochastic simulation
Eq. strain
0.756608
0.749791
0.742975
0.736158
0.729342
0.722525
0.715709
0.708892
0.702076
0.69526
SD Eq. strain
0.268086
0.242507
0.216928
0.191349
0.16577
0.140191
0.114612
0.0890327
0.0634537
0.0378747
SD uy
0.117879
0.106091
0.0943028
0.082515
0.0707271
0.0589393
0.0471514
0.0353636
0.0235757
0.0117879
SD ux
0.0504606
0.0454145
0.0403685
0.0353224
0.0302764
0.0252303
0.0201842
0.0151382
0.0100921
0.00504606
Results plotted in mean deformed configuration
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INITIAL CONFIGURATION UNCERTAINTY
Point at centerline
Point at top
Outer boundary plot
4
3.5
y (mm)
3
2.5
Mean-MC
Mean - SSFEM - o4
Mean -Deterministic
2
1.5
1
0.5
0
0.264
0.266
0.268
0.27
0.272
0.274
0.276
x (m m )
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MERITS AND PITFALLS OF GPCE
•
Reduced order representation of uncertainty
•
Faster than mc by at least an order of magnitude
•
Exponential convergence rates for many problems
•
Provides complete response statistics and convergence in
distribution
But….
•
Behavior near critical points.
•
Requires continuous polynomial type smooth response.
•
Performance for arbitrary PDF’s.
•
How do we represent inequalities, eigenvalues spectrally ?
•
Can we afford to rewrite complex metal forming codes ?
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Non Intrusive Stochastic Galerkin
Method (NISG)
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NISG - FORMULATION
Parameters of interest in stochastic analysis are the moment information (mean,
standard deviation, kurtosis etc.) and the PDF.
For a stochastic process
g ( x, t ,  )  g ( x , t ,  )
x  X , t T ,   
Definition of moments
M p   ( g ( x, t ,  )) p f ( )d

NISG - Random space  discretized using finite elements to  h
M
h
p

nel
n
h
p h
(
g
(
x
,
t
,

))
f
(

)
d


w
(
g
(
x
,
t
,

))
f i ( ie )

i
e
ie

h
h
nel nint
p
h
M h p   wi ( g eih ( x, t )) p f eih
e 1 i 1
e 1 i 1
Deterministic evaluations
at fixed points  ie
Output PDF computed using local least squares interpolation from function
evaluations at integration points.
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NISG - DETAILS
Finite element representation of the support
space.
True PDF
Interpolant
Inherits properties of FEM – piece wise
representations, allows discontinuous
functions, quadrature based integration rules,
local support.
Provides complete response statistics.
Decoupled function evaluations at element
integration points.
FE Grid
Convergence rate identical to usual finite elements, depends on order of
interpolation, mesh size (h , p versions).
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EFFECT OF RANDOM VOIDS ON MATERIAL BEHAVIOR
Material heterogeneity induced by random distribution of micro-voids modeled
using KLE and an exponential kernel. Gurson type model for damage evolution
2
fˆ ( p)  fˆ0 (1   i n vi ( p))
Initial
i 1
Final
Mean
Uniform 0.02
SD Eq. strain
SD-Void fraction
0.268086
0.0098
0.242507
0.0096
0.216928
0.0094
0.191349
0.0092
0.16577
0.0091
0.140191
0.0089
0.114612
0.0087
0.0890327
0.0634537
0.0378747
Void fraction
0.0419
0.0388
0.0357
0.0325
0.0294
0.0263
0.0231
SD-Void fraction
0.0186
0.0172
0.0158
0.0143
0.0129
0.0115
0.0101
Using 6x6 uniform support space grid
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EFFECT OF RANDOM VOIDS ON MATERIAL BEHAVIOR
Load displacement curves
Mean
0.7
6
0.6
SD Load (N)
Load (N)
5
4
3
2
0.5
0.4
0.3
0.2
Mean +/- SD
0.1
1
0.1
0.2
0.3
Displacement (mm)
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0.4
0
0.1
0.2
0.3
0.4
Displacement (mm)
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VALIDATION
Comparison of statistical parameters
Final load values
Parameter
Monte Carlo Support
(1000 LHS
space 6x6
samples)
uniform grid
Support
space 7x7
uniform grid
Mean
6.1175
6.1176
6.1175
SD
0.799125
0.798706
0.799071
m3
0.0831688
0.0811457
0.0831609
m4
0.936212
0.924277
0.936017
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Materials Process Design and Control Laboratory
PROCESS UNCERTAINTY
Random ?
friction
Random ?
Shape
Axisymmetric cylinder upsetting
– 60% height reduction (Initial
height 1.5 mm)
Random initial radius – 10%
variation about mean (1 mm)–
uniformly distributed
Random die workpiece friction
U[0.1,0.5]
Power law constitutive model
Using 10x10 support space grid
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Void fraction: 0.002 0.004 0.007 0.009 0.011 0.013 0.016
Materials Process Design and Control Laboratory
PROCESS STATISTICS
Force
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SD Force
Materials Process Design and Control Laboratory
PROCESS STATISTICS
Final force statistics
Parameter
Monte Carlo Support
(20000 LHS space 10x10
samples)
Mean
2.2859e3
2.2863e6
SD
297.912
299.59
m3
-8.156e6
-9.545e6
m4
1.850e10
1.979e10
3.50E-04
3.00E-04
Convergence study
Relative Error
Relative error
2.50E-04
2.00E-04
1.50E-04
1.00E-04
5.00E-05
0.00E+00
0
2
4
6
8
10
12
14
Grid resolution (Number of elements per dimension)
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RELIABILITY BASED DESIGN
Objective:
Unsafe state
Z(g)<0
Design the forging press for
the process on the basis of the maximum
force required based on a probability
of failure of 0.0002.- β = 3.54
Design point
g
Actual limit state surface
Full order reliability
method
SORM Approximation
β
FORM Approximation
Safe state
Z(g)>0
Limit state function
Probability of failure
Minimum required force capacity vs Stroke
for a press failure probability of 0.0002
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Minimum design force = 2843 N
Materials Process Design and Control Laboratory
STOCHASTIC ESTIMATION OF DIE UNDERFILL
Deterministic Simulation
Axisymmetric flashless closed die forging
Same process with initial void fraction
0.03
Void frac: 3.3E-03 6.6E-03 9.9E-03 1.3E-02 1.6E-02
Initial preform volume
same as volume of die
cavity
Decrease in void fraction in the billet during the process leads to unfilled die cavity
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STOCHASTIC ESTIMATION OF DIE UNDERFILL
Stochastic Simulation
Assumed void fraction using KLE
 r 

 b 
R ( p1 , 0, p1 , 0)   2 exp 
2
fˆ ( p)  fˆ0 (1   i n vi ( p))
i 1
PDF of die underfill
Using 10x10 uniform support space grid
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REVIEW OF NISG AND GPCE
•
Both provide complete response statistics and convergence in distribution.
•
GPCE fails for systems with sharp discontinuities. (inequalities).
•
Seamless integration of NISG into existing codes. Ideal for complex
simulations with strong nonlinearities. (Finite deformations – eigen strains,
inequalities, complex constitutive models).
•
GPCE needs explicit spectral expansion and repeated Galerkin projections.
•
NISG can handle completely empirical probability density functions due to
local support with no change in the convergence properties (convergence is
based on number of elements used to discretize the support-space and the
order of interpolation).
•
Curse of dimensionality – both methods are susceptible.
NISG is the way to go
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Part III - Robust Design of Deformation Processes
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ROBUST DESIGN ENVIRONMENT
PROBLEM STATEMENT
Compute the predefined random process design parameters which lead to a
desired objectives with acceptable (or specified) levels of uncertainty in the
final product and satisfying all constraints.
KEY ISSUES
•
Robustness limits on the desired properties in the product –
acceptable range of uncertainty.
•
Design in the presence of uncertainty/ not to reduce uncertainty.
•
Design variables are stochastic processes or random variables.
•
Consider all ‘important’ process and material data to be random
processes – by itself a design decision.
•
Design problem is a multi-objective and multi-constraint
optimization problem.
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ROBUST DESIGN PROBLEM FORMULATION
Design Objective
Probability Constraint
Norm Constraint
SPDE Constraint
Augmented Objective
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A CONTINUUM STOCHASTIC SENSITIVITY SCHEME (CSSM)
Compute sensitivities of parameters with respect to stochastic design variables
by defining perturbations to the PDF of the design variables.
CSSM problem decomposed into a set of CSM problems
Decomposition based on the fact that perturbations to the PDF are local in
nature
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NISG APPROXIMATION FOR OBJECTIVE FUNCTION
Design Objective – unconstrained case
Set of NelE*n objective
functions
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BENCHMARK APPLICATION
Flat die upsetting of a cylinder
Case 1 – Deterministic problem
Case 2 – 1 random variable (uniformly distributed) – friction – 66% variation
about mean (0.3) (10x1 grid) – 1D problem
Case 3 – 2 random variables (uniformly distributed) – friction(66%) and
desired shape (10% about mean) (10x10 grid) - 2D problem
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OBJECTIVE FUNCTION
Deterministic problem - optimal solution
Deterministic problem
1D problem
2D problem
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DESIGN PARAMETERS
Initial guess parameters
Deterministic problem
2D problem
Mean
SD
1D problem
Mean
SD
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Materials Process Design and Control Laboratory
OBJECTIVE FUNCTION
2.50E-02
Objective Function
2.00E-02
1.50E-02
1D
Deterministic
1.00E-02
2D
5.00E-03
0.00E+00
1
2
3
4
5
6
7
8
9
10
Iterations
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U N I V E R S I T Y
Materials Process Design and Control Laboratory
FINAL FREE SURFACE SHAPE CHARACTERISTICS
1.6
1.4
1.2
1
y (mm)
Mean
0.8
0.6
1D
Deterministic
0.4
2D
0.2
0
6.50E-01
7.50E-01
8.50E-01
9.50E-01
1.05E+00
1.15E+00
1.25E+00
x (mm)
1.6
1.4
1.2
SD
y (mm)
1
0.8
0.6
1D
0.4
2D
0.2
0
4.00E-02 5.00E-02 6.00E-02 7.00E-02
8.00E-02 9.00E-02 1.00E-01 1.10E-01
x (mm)
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Materials Process Design and Control Laboratory
Suggestions for future work
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U N I V E R S I T Y
Materials Process Design and Control Laboratory
MULTISCALE NATURE OF MATERIAL HETEROGENEITIES
Present method
Assume correlation between macro points
n
Decompose using KLE
 r 

 b 
R ( p1 , 0, p2 , 0)   2 exp 
s( p)  s0 (1   i i vi ( p))
i 1
Fine scale heterogeneities
grain size, texture, dislocations
Coarse scale heterogeneities
macro-cracks, phase distributions
•Nature of randomness differs significantly between scales, though not
fully uncorrelated.
•Need a multiscale evaluation of the Correlation Kernels
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CURSE OF DIMENSIONALITY
As the number of random variables increases, problem size rises exponentially.
1E+20
Function evaluations
1E+16
1E+12
1E+08
10000
1
0
5
10
15
20
25
No. of variables
(assume 10 evaluations per random dimension)
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PROPOSED SOLUTIONS
A PRIORI ADAPTIVITY
•
Initial sensitivity analysis with respect to random parameters.
•
Sensitivities used to a priori refine/coarsen grid discretization
along each random dimension.
•
Easily implemented using version of earlier CSM analysis
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PROPOSED SOLUTIONS
ADAPTIVE DISCRETIZATION BASED ON OUTPUT
STOCHASTIC FIELD
•
Refine/Coarsen input support space grid based on output
defined control parameter (Gradients, standard deviations etc.)
•
Applicable using standard h,p adaptive schemes.
Support-space of input
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U N I V E R S I T Y
Importance spaced grid
Materials Process Design and Control Laboratory
PROPOSED SOLUTIONS
DIMENSION ADAPTIVE QUADRATURE (Gerstner et. al. 2003)
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
-1 -0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1
Full grid Scheme
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
-1 -0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1
Sparse grid Scheme
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
-1 -0.8-0.6 -0.4-0.2 0 0.2 0.4 0.6 0.8 1
Dimension adaptive
Scheme
Very popular in computational finance applications.
Has been used in as high as 256 dimensions.
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JOURNAL PUBLICATIONS
S. Acharjee and N. Zabaras "A proper orthogonal decomposition approach to
microstructure model reduction in Rodrigues space with applications to the control
of material properties", Acta Materialia, Vol. 51, pp. 5627-5646, 2003.
S. Acharjee and N. Zabaras "The continuum sensitivity method for the
computational design of three-dimensional deformation processes", Computer
Methods in Applied Mechanics and Engineering, in press.
S. Acharjee and N. Zabaras, "Uncertainty propagation in finite deformation
plasticity -- A spectral stochastic Lagrangian approach", Computer Methods in
Applied Mechanics and Engineering, in press.
S. Acharjee and N. Zabaras, "A concurrent model reduction approach on spatial
and random domains for stochastic PDEs", International Journal for Numerical
Methods in Engineering, in press.
S. Acharjee and N. Zabaras, "A non-intrusive stochastic Galerkin approach for
modeling uncertainty propagation in deformation processes ", Computers and
Structures, in preparation.
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U N I V E R S I T Y
Materials Process Design and Control Laboratory
Thank You
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U N I V E R S I T Y
Materials Process Design and Control Laboratory