DEVELOPMENT OF A ROBUST COMPUTATIONAL DESIGN SIMULATOR FOR INDUSTRIAL

DEFORMATION PROCESSES

Nicholas Zabaras (PI) and Shankar Ganapathysubramanian

URL: http://www.mae.cornell.edu/zabaras/

Email: zabaras@cornell.edu

Optimization based design of deformation processes Mathematically consistent & accurate continuum sensitivity finite element analysis Unified approach towards shape and parameter sensitivity analysis Oriented towards the design of multi-stage processes

Optimumdeformation process

Billet Product

DEFORMATION PROCESS DESIGN FOR TAILORED DEFORMATION PROCESS DESIGN FOR TAILORED MATERIAL PROPERTIESMATERIAL PROPERTIES

Minimal overall cost:force, energy, etc.

MaterialsMaterialsProcessProcessDesignDesign

SimulatorSimulator

Tailored material properties in the

final product Desired microstructural

features

Desired spatialdistributions of state variables

Controlled texture,recrystallization,

fracture & porosity

Desired shape withminimal material

utilization

Accelerated processsequence design

APPROACH

Interactive Optimization Environment

Given process constraints &parameters

Desired productproperties

VIRTUAL DEFORMATION PROCESS DESIGN SIMULATORVIRTUAL DEFORMATION PROCESS DESIGN SIMULATOR

MaterialMaterialProcessProcessDesignDesign

SimulatorSimulator

Selection of the sequence of processes (stages) and initial process parameter designs

• knowledge based expert systems• microstructure evolution paths• ideal forming techniques

Selection of the design variables (e.g. die and

preform parametrization)

Optimization algorithms

Continuum multistage process sensitivity analysis consistent with the direct process model

Assessment of automatic process optimization

Reliability of the design to uncertainties in the physical and computational models

Mathematical representation of the design objective(s) &

constraints Selection of a virtual direct process model

Interactive Interactive Optimization Optimization EnvironmentEnvironment

Node: Intermediate

preform

Arc: Processing

Stage

FinalProduct

Initial Product

Optimal Path (pth)Feasible Paths (jth)

1st Stage

FinishingStage(nth)

ith Stage

DESIGN OF MULTI STAGE DEFORMATION PROCESSES

CostFunction = + +Cost

of DiesEnergy

ConsumptionMaterialUsage

i=1 n

minm

Evaluate number of stages n and select a process sequence p from all feasible paths (j=1 … m), such that:

Ideal forming & microstructure evolution paths based initial designs

Advanced knowledge-based algorithms for process sequence selection

Process sequenceselection

Given raw material, obtain product of desired microstructure and shape with minimal material utilization and cost

Design the forming and thermal process sequenceSelection of stages (broad classification)Selection of dies and preforms in each stageSelection of mechanical and thermal process parameters in each stageSelection of the initial material state (microstructure)

Press force

Processing temperaturePress speed

Product qualityGeometry restrictions

Cost

CONSTRAINTSOBJECTIVES

Material usage

Plastic work

Uniform deformation

MicrostructureDesired shapeResidual stresses Thermal parameters

Identification of stagesNumber of stagesPreform shapeDie shape Mechanical parameters

VARIABLES

COMPUTATIONAL DESIGN OF METAL FORMING PROCESSES

BROAD DESIGN OBJECTIVES

COMPUTATIONAL PROCESS DESIGN

GOVERNING PHYSICS

UPDATED LAGRANGIAN FRAMEWORK OF ANALYSIS

BBo BB

FF e

FF p

FF

FF

Initial configuration Temperature: o

void fraction: fo

Deformed configuration Temperature: void fraction: f

Intermediate thermalconfiguration Temperature:

void fraction: fo

Stress free (relaxed) configuration Temperature: void fraction: f

Referenceconfiguration

r

n

Inadmissible region

Currentconfiguration

Admissible region

CONSTITUTIVE MODEL CONTACT/FRICTION MODEL

Multiplicative decomposition framework State variable rate-dependent models Hyperelastic constitutive law Thermal and damage effects

Mechanical dissipation Augmented Lagrangian approach Coulomb friction

DEFINITIONS OF SENSITIVITY FIELDS IN AN UPDATED LAGRANGIAN FRAMEWORK

x = x (xn, t ; p)^

xn = x (X, tn ; p )

Qn = Q (X, tn ; p )

oFn + Fn

X

xn

Fn

Bo

x+xoo

Fr + Fr

xB

xn + xn = x (Y , tn ; p + p )

Qn + Qn = Q (Y, tn ; p + p )o ~

B’n

I+Ln

Fr

oxn+xn

B n

B’

X = X (Y; s )

oFR + FR

Y

X

X+Xo

xn+xno

xn

oFn + Fn

FR

Fn

BR

Bo

I+Lo

x+xoo

Fr + Fr

x B

xn + xn = x (Y , tn ; s + s)

Qn + Qn = Q (Y, tn ; s + s)

x = x (xn, t ; s)

B n

xn = x (X, tn ; s )Qn = Q (X, tn ; s )

I+Ln

X + X= X (Y; s + s)

Fr

Parameter sensitivity analysis

Design parameters Ram speed Shape of die surface Material parameters Initial state

Shape sensitivity analysis

Main features Gateaux differential referred to the fixed configuration Y Rigorous definition of sensitivity Key element: LR=FR FR

-1 oo

~o

~

~

__

o

~~

__

~o

o

~

^

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 problem Differentiate Discretize

Design sensitivity of equilibrium equation

Calculate and such that x = x (xr, t, β, ∆β )oo

o

FFrr and and xxoo

Kinematic problem

oλ and x o

Regularized contact problem

Pr and F,o

o

Constitutive problem

THE CONTINUUM SENSITIVITY METHOD SUB-PROBLEMS

Thermal problem

o

o

o

Consider the non-differentiability of contact and frictionconditions

Sensitivity deformation is a linear problem

Iterations are avoided within a single time increment

Additional augmentations are avoided by using large penalties in the sensitivity contact problem

THE CONTINUUM SENSITIVITY CONTACT SUB-PROBLEM

y = y + y

υ

r

υ + υo

r + rox + x o

X

y = y ( ξ )

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

y,ξ ξy

o

+

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

By = y ( ξ )Die

y = y ( ξ )

x

ShapeShapeSensitivitySensitivityAnalysisAnalysis

REGULARIZATION

Contact and friction

sensitivity assumptions

REMARKS

Stress Sensitivity100.00

85.0070.0055.0040.0025.0010.00-5.00

-20.00-35.00-50.00

Convection/ Radiation

Conduction

Rigid DieForging rate

Unfilled die cavity

Flash

Damage/microstructure

A ONE-STAGE HOT FORMING PREFORM DESIGN PROBLEM

MATERIAL SYSTEM

1100-Al workpieceInitial temperature 673 KAxisymmetric problem Standard ambient conditions

Design objectivesFind preform shape of minimum volume such that the die is filled completely and the flash is minimized

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0 10 20 30Iteration Numeber

Ob

jeti

ve F

un

ctio

n

Unfilled cavityand flash!

Optimal design

Fully filledcavity

Initial design

Iteration number

Ob

ject

ive

(mm

2)

THE CONTINUUM SENSITIVITY METHOD FOR MULTI-STAGE DEFORMATION PROCESSES

Sequential transfer of sensitivities from one stage to the next

Design Objective

Knowledge-based methods

Shapesensitivity analysis

Die and process parameter sensitivity analysis

Selection of stages

Design of preforms

Design of dies

Generic Forming Stage

1100-Al workpieceInitial temperature 673 KAxisymmetric problem Standard ambient conditions 2 pre-defined stages - preforming + finishing

Design objectiveDesign the preforming die for a fixed volume of theworkpiece such that the finishing die is filled

PREFORMING DIE DESIGN PROBLEM FOR SHAPE CONTROL

MATERIAL SYSTEM

Preforming stage

Finishing stage

Rigid Die

FlashFlash

Preforming Stage Finishing StageUnfilledcavity

Fullyfilledcavity

0.0

2.0

4.0

6.0

8.0

0 1 2 3 4 5 6Iteration NumberO

bje

ctiv

e F

un

ctio

n

(

x1.0

E-0

5)

Iteration number

Optimal design

Initial design

Ob

ject

ive

(mm

2)

Preforming Stage Finishing Stage

State variable ( MPa )55.21053.48751.76450.04048.31746.594

State variable ( MPa )54.43151.72949.02846.32643.62540.923

I t e r a t i o n i n d e x

O b j e c t i

v e f u n c t i

o n

0 1 2 3 4 5 6 7 8

0 . 0 5

0 . 1

0 . 1 5

0 . 2

0 . 2 5

Ob

ject

ive

Fu

nct

ion

1100-Al workpieceInitial temperature 673 KAxisymmetric problem Standard ambient conditions 2 pre-defined stages - preforming + finishing

Design objectiveDesign the preforming die for a fixed volume of the workpiece such that the variation in state in the product is minimum

PREFORMING DIE DESIGN FOR CONTROL OF MICROSTRUCTURE

MATERIAL SYSTEM

Preforming stage

Radius, r (mm)

He

igh

t,h

( mm

)

0 0.5 11.2

1.25

1.3

1.35

1 .4

1.45

1.5

1.55

1

2

3

4

5

6

7

Average state

Initial Optimal

Deviation

50.2 52.3

3.73 1.88Iteration number

Ob

ject

ive

Scalar statevariable (MPa)

Scalar statevariable (MPa)

Radius (mm)

Hei

gh

t (m

m)

Optimal design

Initial design

DesignIn MPa

Finishing stage

Extrusion

Peripheral coarse grain (PCG)

FUTURE EXTENSIONS TO MULTI-SCALE PROCESS DESIGN:PCG CONTROL DURING EXTRUSION

Need to couple grain growth/ orientation and recrystallization simulation models with CSM based computational design for explicit control of micro-structural features in deformation processes

Alloy flow stress

Material point data

Profile output data

Billet input data

USING COMPUTATIONAL DESIGN TO DEVELOP A DIGITAL MATERIALS PROCESS LIBRARY

Srikanth, A., et.al. “Continuum Lagrangian sensitivity analysis for metal forming processes with applications to die design”, Int. J. Numer. Methods Engr., (2000) 679-720.

Srikanth, A. and N. Zabaras. “Shape optimization and preform design in metal forming processes”, Comput. Methods Appl. Mech. Engr., (2000) 1859-1901.

Ganapathysubramanian, S. and N. Zabaras. “Continuum sensitivity method for finite thermo-inelastic deformations with applications to the design of hot forming processes”, Int. J. Numer. Methods Engr., (submitted)

Testing and further developments for single-stage designs - complex 2D geometries Regularized contact/ friction sensitivity modeling Simultaneous thermal & mechanical design Sensitivity analysis for multi-body deformations

Multi length scale design Control of grain growth, texture and recrystallization

Multi-stage forming design Coupling with ideal forming & microstructure evolution paths based initial designs Framework for web-based forming design

Development of a 3D forming design simulator Industrial design applications

Robust design algorithms

ACKNOWLEDGEMENTS

The work presented here was funded by NSF grant DMI-0113295 with additional support from AFOSR, AFRL and ALCOA.

FORTHCOMING RESEARCH EFFORTS

Zabaras, N., et.al. “Continuum sensitivity method for the design of multi-stage metal forming processes”, Int. J. Mech. Sciences (submitted)

REFERENCES