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.