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NSF Grant Number: DMI- 0113295PI: Prof. Nicholas Zabaras Institution: Cornell University
Title: Development of a robust computational design simulator for industrial deformation processes
Research Objectives:To develop a mathematically and computationally rigorous gradient-based optimization methodology for virtual materials process design that is based on quantified product quality and accounts for process targets and constraints including economic aspects.
Current capabilities:•Development of a general purpose continuum sensitivity method for the design of multi-stage industrial deformation processes•Deformation process design for porous materials•Design of 3D realistic preforms and dies•Extension to polycrystal plasticity based constitutive models with evolution of crystallographic texture
Materials Process Design and Control Laboratory, Cornell University
(Minimal barreling)Initial guess Optimal preform
ODF: 1234567
Macro - continuum Micro-scale
Polycrystal plasticity
Future research•Multiscale metal forming design with reduced order modeling of microstructure •Design of formed products with desired directional microstructure dependent properties•Probabilistic design using spectral methods with specification of robustness limits in the design variables
Iteration 3
Iteration 6
0.993
0.994
0.995
0.996
0.997
0.998
0.999
1
1.001
1.002
0 20 40 60 80
Angle from rolling direction
Initial
Intermediate
OptimalDesired
Design for desired yield stress at a material point
Broader Impact: A virtual laboratory for realistic materials process design is developed that will lead to reduction in lead time for process development, trimming the cost of an extensive experimental trial-and-error process development effort, developing processes for tailored material properties and increasing volume/time yield. The design simulator under development provides a robust and handy industrial tool to carry out real-time metal forming design.
Nor
mal
ized
yie
ld s
tres
s
Deformation Process Design for Tailored Material Properties
Difficult Insertion of new materials and
processes into production
Numerical SimulationTrial-and-error and with no design information
Conventional Design Tools
Material ModelingIncremental improvements
in specific areas
Development of designer knowledge base
Time consuming and costly
Computational Material Process Design Simulator
Sensitivity Information
points to most influential parameters so as to
optimally design the process
Virtual Material Process Laboratory
ReliabilityBased Design
for material/tool variability & uncertainties
in mathematical & physical models
Data Mining of Designer Knowledge
for rapid solution to complex problems and to
further drive use of knowledge
Materials Process Design
control of microstructure using various length and time scale computational
tools
Accelerated Insertion of new materials and
processes
Innovative Processes
for traditional materials
Reliability based design
Sensitivity information
Designer knowledge
Materials process design
Virtual Materials Virtual Materials Process LaboratoryProcess Laboratory
Selection of a virtual direct process model
Selection of the sequence of processes
(stages) and initial process parameter
designs
Selection of the design variables like
die and preform parameterization
Continuum multistage process sensitivity analysis consistent with the direct process model
Optimization algorithms
Interactive optimization environment
Virtual Deformation Process Design SimulatorVirtual Deformation Process Design Simulator
Description of parameter
sensitivities: Take FR
= I with the design velocity gradient L0 = 0.
Main features: Gateaux differential referred to the fixed configuration Y Rigorous definition of sensitivity Driving force for the sensitivity problem is LR=FR FR
-1 o
Shape and Parameter Continuum Sensitivity Analysis
Equilibrium equation
Design derivative of equilibrium
equation
Material constitutive
laws
Design derivative of the material
constitutive laws
Design derivative ofassumed kinematics
Assumed kinematics
Incremental sensitivityconstitutive sub-problem
Time & space discretizedmodified weak form
Time & space discretized weak form
Sensitivity weak form
Contact & frictionconstraints
Regularized designderivative of contact &frictional constraints
Incremental sensitivity contact
sub-problem
Conservation of energy
Design derivative of energy equation
Incrementalthermal sensitivity
sub-problem
Schematic of the continuum sensitivity method (CSM)
Continuum problemDesign
differentiate Discretize
3D Continuum sensitivity contact sub-problem
Continuum approach for computing traction sensitivities – In line with the continuum sensitivity approach Accurate computation of traction derivatives using augmented Lagrangian regularization. Traction derivatives computed without augmentation using oversize penalties
Regularization assumptions•No slip/stick transition between direct/perturbed problem
•No admissible/inadmissible region transition between direct/perturbed problem
y = y + y
υτ1
υ + υo τ1 + τ1 o
x + x o
X
DieDie
o
oy + [y]
x = x ( X, t, β p )~
x = x ( X, t, β p+ Δ β p )~
B0 B΄
Bx
ParameterParametersensitivitysensitivityanalysisanalysis
υ
r
υ
r
x = x ( X, t, β s )B0
B’0
BR
X + X
X
o
x = x ( X + X , t, β s+ Δ β s )~
oX = X (Y ; β s+ Δ β s )
~
Y
X = X (Y ; β s )
~
~
x + xB΄
o
B
Die
x
ShapeShapesensitivitysensitivityanalysisanalysis
τ2 + τ2 o
τ2 1 2( , )y y
1 2( , )y y
1 2
1 2
, ,
o o
y y y
First reported 3D regularized
contact sensitivity algorithm
Equivalent stress sensitivity
Perturbation of the preform shape parametersCSM FDM
Equivalent stress sensitivity
Temperature sensitivity Temperature sensitivity
Open die forging of a cylindrical billetValidation of 3D thermo-mechanical shape sensitivity analysis
Senst Temp (FDM)0.00012.5E-05
-5E-05-0.000125-0.0002
Senst Temp (CSM)0.00012.5E-05
-5E-05-0.000125-0.0002
Senst Stress (FDM)0.00010
-0.0001-0.0002-0.0003
Senst Stress (CSM)0.00010
-0.0001-0.0002-0.0003
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0 10 20 30Iteration Numeber
Obj
etiv
e Fu
nctio
n
Minimize the flash and the deviation between the die and the workpiece through a preforming shape design
Unfilledcavity
FlashThe samematerial in a
conventional design
The samematerial withan optimum
design
Noflash
Fully filledcavity
Process design for the manufacture of an engine disk – two possible approaches
Minimize the gap between the finishing die and the workpiece in a two stage forging, with given finishing die;unknown die but prescribed stroke in the preforming stage.
Initial design Unfilledcavity
Optimal design
0.0
2.0
4.0
6.0
8.0
0 1 2 3 4 5 6
Iteration Number
Ob
ject
ive
Fu
nct
ion
(x1
.0E
-05)
Al 1100-O Initially at
673K;Preform and
die parameterizati-
-on
Objective: Minimize the flash and the
deviation between the die and the workpiece for a preforming shape and volume designMaterial:- 2024-T351Al, 300K,
5% initial void fraction, varying elastic properties (using Budiansky method),
co-efficient of frictionbetween die & workpiece = 0.1
Product using guess preform
Product using optimal preform
Distribution of the void fraction in
product
r - axis
z-ax
is
0 0.5 1
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1 Initialprefrom
Optimalpreform
Variation of preform shape with
optimization iterations
Iteration number
Non
-dim
ensi
onal
obj
ectiv
ePreform design for porous material
4
5
6
7
8 0.05117 0.04436 0.03755 0.03074 0.02393 0.01702 0.01021 0.0034
Void fraction
1
1
2
2 3 5
5
68
8 0.04507 0.03886 0.03265 0.02644 0.02013 0.01392 0.00771 0.0015
Void fraction
0 2 4 6 8 10 120.003
0.004
0.005
0.006
0.007
0.008
3D Preform design to fill die cavity for forging a circular disk
Optimal preform shape
Final optimal forged productFinal forged product
Initial preform shape
Objective: Design the initial preform such that the die cavity is fully filled for a fixed strokeMaterial:Al 1100-O at 673 K
0
0.10.2
0.3
0.40.5
0.6
0.7
0.80.9
1
0 2 4 6 8
Iterations
Nor
mal
ized
obj
ecti
ve
(1) Continuum framework
(3) Desired effectiveness in terms of state variables
(2) State variable evolution laws
Initial configuration
Bo B
F e
Fp
F
FDeformed configuration
Intermediate thermalconfiguration
Stress free (relaxed) configuration
Phenomenology Polycrystal plasticityInitial configuration
Bo BF *F p
FDeformed configuration
Stress free (relaxed) configuration
(1) Single crystal plasticity
(3) Ability to tune microstructure for desired properties
(2) State evolves for each crystal
The effectiveness of design for desired product properties is limited by the ability
of phenomenological state-variables to capture the dynamics of the underlying
microstructural mechanisms
Polycrystal plasticity provides us with the ability to capture material properties in
terms of the crystal properties. This approach is essential for realistic design
leading to desired microstructure-sensitive properties
From phenomenology to polycrystal plasticityFrom phenomenology to polycrystal plasticity
Need for polycrystalline analysis Challenges in polycrystalline analysis
Infinite microstructural degrees of freedom limits the scope of design
Optimal design
Initial design
2
3
4
5
6
7
3
3
3
4
78
?
Solution: Develop microstructure model reduction
n0
s0
s0
n0
ns
0.992
1.012
1.032
0 20 40 60 80
Angle from rolling direction
InitialIntermediateOptimalDesired
R v
alue
Nor
mal
ized
obj
ecti
ve f
unct
ion
Desired value: α = {1.2,0,0,0,0}T,
Initial guess: α = {0.5,0,0,0,0}T
Converged reduced order solution:
α = {1.19,0.05,0.001,0,0}T
Design problem: Ξ = {F,G,H,N}T
(from Hill’s anisotropic yield criterion)
Design for microstructure sensitive property – R value
0
0.10.2
0.3
0.4
0.5
0.60.7
0.8
0.9
1
0 10 20 30 40 50 60 70
Iterations
Desired value: α = {1,0,0,0,0}T
Initial guess: α = {0.5,0,0,0,0}T
Converged solution: α = {0.987,0.011,0,0,0}T0.985
0.99
0.995
1
1.005
1.01
1.015
1.02
1.025
1.03
1.035
0 10 20 30 40 50 60 70 80 90Angle from rolling direction
InitialIntermediateOptimalDesired
h
Nor
mal
ized
hys
tere
sis
loss
Nor
mal
ized
obj
ecti
ve f
unct
ion
Design Problem
Hysteresis loss
Crystal <100> direction.
Easy direction of
magnetization – zero power
loss
External magnetization direction
Materials by designMaterials by designDesign of microstructure and deformation for minimal hysteresis lossDesign of microstructure and deformation for minimal hysteresis loss
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 5 10 15Iterations
Process design for tailored material properties
Time
Equiv
alen
tstr
ess
(MP
a)
10 20 30 40 50
5
10
15
20
25
30
35
40
iteration 1
iteration 2
iteration 3iteration 4
iteration 5
Converged solution
Desired response
12 0.210611 0.199610 0.18859 0.17748 0.16647 0.15536 0.14435 0.13324 0.12223 0.11112 0.10001 0.0890
Grain size (mm)
Initial grain size = 0.091mm
12 0.150011 0.143610 0.13739 0.13098 0.12457 0.11826 0.11185 0.10554 0.09913 0.09272 0.08641 0.0800
Initial grain size = 0.091mm
Grain size (mm)
The guess die shape resulted in large grains along
the exit cross-section of the
extruded product. Using the optimal
die shape, presence of such
large grains is eliminated.
Optimal solution
Guess solution
Design the extrusion die for a fixed
reduction so that the variation in grain size (at the exit) is
minimized
Material:0.2%C steel ,
friction coefficient of 0.01
Phenomenological approachPhenomenological approach Polycrystal approachPolycrystal approachDesign for the
strain rate such that a desired
material response is achieved
Material:99.98% pure
f.c.c Al Iteration index
Ob
ject
ive
fun
ctio
n(M
Pa2
)
1 2 3 4 5 6 70
100
200
300
400
500
600
700
800
Hi – peformance computing
USER INTERFACE
Robust product
specifications
Control and reduced order
modeling
Stochastic optimization,
Spectral/Bayesian framework
Design database,
simulations and experiments
User update
Output design
Input
Modifications in objectives
• Starting with robust product specifications, you compute not only the full statistics
of the design variables but also the acceptable variability in the system parameters
• Directly incorporate uncertainties in the system into the design analysis
• Experimentation and testing driven by product design specifications
• Improve overall design performance
Robust design simulatorRobust design simulator
Suppose we had a collection of data (from experiments or simulations) for the ODF:
such that it is optimal for the ODF represented as
Is it possible to identify a basis
POD technique – Proper orthogonal decomposition
Solve the optimization problem
Method of snapshots
where
Applications of microstructure model reduction
GE 90Boeing 747Modern aircraft engine design and materials selection is an extremely challenging area. Desired directional properties include: strength at high temperatures, R-values elastic, fatigue, fracture properties thermal expansion, corrosion resistance, machinability properties Developing advanced materials for gas turbine engines is expensive – Is it possible to control material properties and product performance through deformation processes?
Further developments for multi-stage designs – 3D geometries [Ref. 1-5] Simultaneous thermal & mechanical design Sensitivity analysis for multi-body deformations
Design across length scales [Ref. 3-6]
Coupled length scale analysis with control of grain size, phase distribution and orientation
Microstructure development through stochastic processes Generate universal snap shots for reduced order modeling Develop algorithms for real-time microstructural reduced order model mining - Ref. [6]
Robust design algorithms [Ref. 7-8] Can we design a process with desired robustness limits in the objective? Work includes a spectral method for the design of thermal systems – Ref. [7]
Develop a spectral stochastic FE approach towards robust deformation process design Introduce Bayesian material parameter estimation – Ref. [8]
Couple materials process design with required materials testing selection
Develop an integrated approach to materials process design and materials testing selection: Materials testing driven by design objectives!
With given robustness limits on the desired product attributes, a virtual design simulator
can point to the required materials testing that can obtain material properties with the needed level of accuracy.
Forthcoming research efforts
ACKNOWLEDGEMENTSThe work presented here was funded
by NSF grant DMI-0113295 with additional support from AFOSR and
AFRL.
[6] S. Ganapathysubramanian and N. Zabaras, "Modeling the thermoelastic-viscoplastic response of polycrystals using a continuum representation over the orientation space", International Journal of Plasticity, submitted for publication.
References[1] S. Ganapathysubramanian and N. Zabaras, "Computational design of deformation processes for materials with ductile damage", Computer Methods in Applied Mechanics and Engineering, Vol. 192, pp. 147--183, 2003.
[2] S. Ganapathysubramanian and N. Zabaras, "Deformation process design for control of microstructure in the presence of dynamic recrystallization and grain growth mechanisms", International Journal for Solids and Structures, in press.
[3] S. Ganapathysubramanian and N. Zabaras, "Design across length scales: A reduced-order model of polycrystal plasticity for the control of microstructure-sensitive material properties", Computer Methods in Applied Mechanics and Engineering, submitted for publication..[4] 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. 51/18, pp. 5627-5646, 2003.
[5] V. Sundararaghavan and N. Zabaras, "A dynamic material library for the representation of single phase polyhedral microstructures", Acta Materialia, submitted for publication.
[7] Velamur Asokan Badri Narayanan and N. Zabaras, "Stochastic inverse heat conduction using a spectral approach", International Journal for Numerical Methods in Engineering, in press.
[8] Jingbo Wang and N. Zabaras, "A Bayesian inference approach to the stochastic inverse heat conduction problem", International Journal of Heat and Mass Transfer, accepted for publication.