a gradient optimization method for efficient design of three-dimensional deformation processes

A gradient optimization method for A gradient optimization method for efficient design of three-dimensional efficient design of three-dimensional

deformation processesdeformation processes

Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory

CCOORRNNEELLLL U N I V E R S I T Y


Swagato Acharjee

and

Nicholas Zabaras

Materials Process Design and Control LaboratorySibley School of Mechanical and Aerospace Engineering

188 Frank H. T. Rhodes HallCornell University

Ithaca, NY 14853-3801

Email: [email protected]: http://www.mae.cornell.edu/zabaras/




RESEARCH SPONSORS

U.S. AIR FORCE PARTNERS

Materials Process Design Branch, AFRL

Computational Mathematics Program, AFOSR

CORNELL THEORY CENTER

ARMY RESEARCH OFFICE

Mechanical Behavior of Materials Program

NATIONAL SCIENCE FOUNDATION (NSF)

Design and Integration Engineering Program




COMPUTATIONAL DESIGN OF DEFORMATION PROCESSESCOMPUTATIONAL DESIGN OF DEFORMATION PROCESSES

Press forcePress force

Processing temperatureProcessing temperaturePress speedPress speed

Product qualityProduct qualityGeometry restrictionsGeometry restrictions

CostCost

CONSTRAINTSCONSTRAINTSOBJECTIVESOBJECTIVESMaterial usageMaterial usage

Plastic workPlastic work

Uniform deformationUniform deformation

MicrostructureMicrostructure

Desired shapeDesired shape

Residual stressesResidual stresses Thermal parametersThermal parameters

Identification of stagesIdentification of stagesNumber of stagesNumber of stagesPreform shapePreform shapeDie shape Die shape Mechanical parametersMechanical parameters

VARIABLESVARIABLES

COMPUTATIONAL PROCESS DESIGN

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)

DESIGN OPTIMIZATION FRAMEWORKDESIGN OPTIMIZATION FRAMEWORK

Gradient methods

Finite differences (Kobayashi et al.) Multiple direct (modeling) steps Expensive, insensitive to small perturbations

Direct differentiation technique (Chenot et al., Grandhi et al.)

Discretization sensitive Sensitivity of boundary condition Coupling of different phenomena

Automatic differentiation technique

Continuum sensitivity method(Zabaras et al.)

Design differentiate continuum equations Complex physical system Linear systems

Heuristic methods Genetic algorithms(Ghosh et al.)

Multiple direct (modeling) steps

Response surface methods(Grandhi et al., Shoemaker et al.)

Complex response Numerous direct steps

Continuum equations

Design differentiate

Discretize





COMPONENTS OF A DEFORMATION PROCESS DESIGN ENVIRONMENT

Kinematic Kinematic sub-problemsub-problem

Direct Direct problemproblem

(Non Linear)(Non Linear)

Constitutive sub-problemsub-problem

Contact sub-problemsub-problem

Thermal Thermal sub-problemsub-problem

Remeshing sub-problemsub-problem

Constitutive sensitivitysensitivity

sub-problemsub-problem

Thermal Thermal sensitivity sensitivity


Contact sensitivity sensitivity


Remeshingsensitivity sensitivity


Kinematic Kinematic sensitivity sensitivity


Sensitivity Sensitivity Problem Problem (Linear)(Linear)

Design Design SimulatorSimulator

OptimizationOptimization







KINEMATIC AND CONSTITUTIVE FRAMEWORKKINEMATIC AND CONSTITUTIVE FRAMEWORK

BBn BB

FF e

FF p

FF

FF

Initial configurationInitial configuration Temperature: n

void fraction: fn

Deformed configurationDeformed configuration Temperature: void fraction: f

Intermediate thermalIntermediate thermalconfigurationconfiguration Temperature:

void fraction: fo

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

(1) Multiplicative decomposition framework(1) Multiplicative decomposition framework

(3) Radial return-based implicit integration algorithms(3) Radial return-based implicit integration algorithms(2) State variable rate-dependent models(2) State variable rate-dependent models

(4) Damage and thermal effects(4) Damage and thermal effects

Governing equation – Deformation problemGoverning equation – Deformation problem

Governing equation – Coupled thermal problemGoverning equation – Coupled thermal problem

Thermal expansion:Thermal expansion:

.

Hyperelastic-viscoplastic constitutive laws




3D CONTACT PROBLEM3D CONTACT PROBLEM

ImpenetrabilityImpenetrability ConstraintsConstraints

Coulomb Friction LawCoulomb Friction Law

Continuum implementation of die-workpiece contact. Augmented Lagrangian regularization to enforce impenetrability and frictional stick conditions Workpiece-die interface assumed to be a continuous surface. Die surface parametrized using polynomial curves

Inadmissible region

n

τ1

Referenceconfiguration

Currentconfiguration

Admissible region

Contact/friction model

τ2




DEFINITION OF SENSITIVITIES

o

Fn + Fn

X

xn

Fn

Bo

x+xo

oFr + Fr

xB

xn + xn = x (X , tn ; p + p)o ~

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

x = x (xn, t ; p)^

B’n

xn = x (X, tn ; p )~

Ω n = Ω (X, tn ; p )~

I+Ln

Fr

x + x = x (x+xn , t ; p + p)^o o

oxn+xn

Bn

B’

Shape sensitivity design parameters – Preform shape Parameter sensitivity design parameters – Die shape, ram speed, material

parameters, initial state




SENSITIVITY KINEMATIC PROBLEMSENSITIVITY KINEMATIC PROBLEM

Continuum problem Differentiate Discretize

Design sensitivity of equilibrium equation

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

Variational form -

FFrr and and xxoo o

λ and x o

Pr and F,o

o o

Constitutive problem

Regularized contact problem

Kinematic problem


3D CONTINUUM SENSITIVITY CONTACT PROBLEM3D CONTINUUM SENSITIVITY CONTACT PROBLEM

Continuum approach for computing traction sensitivities Accurate computation of traction derivatives using augmented Lagrangian approach.

Key issue

Contact tractions are inherently non-differentiable due to abrupt slip/stick transitions

Regularization assumptions

•A particle that lies in the admissible (or inadmissible) region for the direct problem also lies in the admissible (or inadmissible) region for the sensitivity problem.

•A point that is in a state of slip (or stick) in the direct problem is also in the same state in the sensitivity 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

τ2 + τ2 o

τ2 1 2( , )y y






SENSITIVITY ANALYSIS OF CONTACT/FRICTIONSENSITIVITY ANALYSIS OF CONTACT/FRICTION

Sensitivity of contact tractionsSensitivity of contact tractions

Sensitivity of inelastic slipSensitivity of inelastic slip

Normal traction:Normal traction:

Stick:Stick:

Slip:Slip:

RemarksRemarks

1.1. Sensitivity Sensitivity deformation is a deformation is a linear problemlinear problem

2.2. Iterations are Iterations are preferably avoided preferably avoided within a single time within a single time incrementincrement

3.3. Additional Additional augmentations are augmentations are avoided by using avoided by using large penalties in the large penalties in the sensitivity contact sensitivity contact problemproblem

1 1 2 2

______ ___________

( ) ( ) ( ) ( ) ( ) ( )

o ooo o o o

T T T TN N 1 1 2 2λ ν y + ν y τ y τ y τ y τ y

1( )n

o o o

N N nN g x

1o o

TT

. .

_____________

|| ||

o

trialo

TT N

Tλ

.o o

C a x b

( ).( )o o o

g y y x

,( ). .C

,

y x y

( ). .i

o o

i ib ,y x y yi ia

Sensitivity of gapSensitivity of gap




CONTINUUM SENSITIVITY METHOD - BROAD OUTLINECONTINUUM SENSITIVITY METHOD - 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

3. Discretize the equations using finite elements

4. Solve and compute the gradients

5. Combine with a gradient optimization framework to minimize the objective function defined




Curved surface parametrization – Cross section can at most be an ellipse

Model semi-major and semi-minor axes as 6 degree bezier curves

6

1

51

3 33

2 55

( ) cos

( )

(1.0 ) (1.0 5.0 )

20.0(1.0 )

6.0(1.0 )

i ii

x a

a

6

61

4 22

2 44

66

( ) sin

( )

15.0(1.0 )

15.0(1.0 )

i ii

y b

b

2 /z H

Design vector

1 2 3 4 5 6 7 8 9 10 11 12{ , , , , , , , , , , , }T βa

b

(x,y) =(acosθ, bsinθ)

H

VALIDATION OF CSMVALIDATION OF CSM




Equivalent Stress Sensitivity

Thermo-mechanical shape sensitivity analysis- Perturbation to the preform shapeCSM FDM


Temperature Sensitivity Temperature Sensitivity

Reference problem – Open die forging of a cylindrical billet

VALIDATION OF CSMVALIDATION OF CSM

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





Thermo-mechanical parameter sensitivity analysis- Forging velocity perturbedCSM FDM


Temperature Sensitivity Temperature Sensitivity

Reference problem – Open die forging of a cylindrical billet

VALIDATION OF CSM

Senst Temp (CSM)0.060.0450.030.0150

Senst Temp ( FDM)0.060.0450.030.0150

Senst Stress (FDM)0.030.02250.0150.00750

Senst Stress (CSM)0.030.02250.0150.00750




PREFORM DESIGN TO MINIMIZE BARRELING IN FINAL PRODUCT

Optimal preform shape

Final optimal forged productFinal forged product

Initial preform shape

0

0.1

0.2

0.3

0.40.5

0.6

0.7

0.8

0.9

1

0 1 2 3 4 5 6

Iterations

Nor

mal

ized

obj

ecti

ve f

unct

ion

a

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




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


0

0.1

0.20.3

0.4

0.5

0.6

0.70.8

0.9

1

0 2 4 6 8

Iterations

No

rma

lize

d o

bje

ctiv

e




PREFORM DESIGN TO FILL DIE CAVITY




Objective: Design the initial preform such that the die cavity is fully filled with no flash for a fixed stroke – Initial void fraction 5%

Material:Fe-2%Si at 1273 K

Iterations

No

rma

lize

d o

bje

ctiv

e

0

0.10.2

0.3

0.40.5

0.6

0.7

0.80.9

1

0 1 2 3 4 5 6




DIE DESIGN TO MINIMIZE DEVIATION OF STATE VARIABLE AT EXIT

Optimal dieInitial die

Objective: Design the extrusion die for a fixed reduction such that the deviation in the state variable at the exit cross section is minimized


Iterations

No

rma

lize

d o

bje

ctiv

e

State Var (MPa)37.273736.756936.240235.723435.206634.689934.173133.656333.139532.622832.106

0

0.1

0.20.3

0.4

0.5

0.6

0.70.8

0.9

1

0 2 4 6 8 10

State Var (MPa)37.620737.081236.541836.002335.462834.923334.383933.844433.304932.765532.226

State Var (MPa)37.337.244437.188937.133337.077837.022236.966736.911136.855636.8

State Var (MPa)37.337.244437.188937.133337.077837.022236.966736.911136.855636.8




IN CONCLUSIONIN CONCLUSION

- 3D continuum shape and parameter sensitivity analysis

- Implementation of 3D continuum sensitivity contact with appropriate regularization

- Mathematically rigorous computation of gradients - good convergence observed within few optimization iterations

- Extension to polycrystal plasticity based multi-scale process modeling

Issues to be addressed: -Incorporate remeshing and suitable data transfer schemes – essential for simulating complicated forging and extrusion processes

-Computational issues – Parallel implementation using Petsc

ReferenceSwagato Acharjee and N. Zabaras, "The continuum sensitivity method for the computational design of three-dimensional deformation processes", Computer Methods in Applied Mechanics and Engineering, accepted for publication.

a gradient optimization method for efficient design of three-dimensional deformation processes

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