Nicholas Zabaras (PI), Swagato Acharjee, Veera Sundararaghavan NSF Grant Number: DMI- 0113295

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.

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

PREFORM DESIGN TO FILL DIE CAVITY

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 with no flash for a fixed stroke – Initial void fraction 5%

Material:Fe-2%Si at 1273 K

Iterations

Nor

mal

ized

obj

ectiv

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

Initial 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

Material:Al 1100-O at 673 K

Iterations

Nor

mal

ized

obj

ectiv

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

Optimal die

DIE DESIGN FOR UNIFORM MATERIAL STATE AT EXIT

Additional support from AFOSR and ARO. Computing facilities provided by Cornell Theory Center

[6] S. Acharjee and N. Zabaras, "Uncertainty propagation in finite deformations -- A spectral stochastic Lagrangian approach", Computer Methods in Applied Mechanics and Engineering, submitted for publication.

[1] 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, Vol. 41/7, pp. 2011-2037, 2004

[2] Swagato 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

[3] V. Sundararaghavan and N. Zabaras, "A dynamic material library for the representation of single phase polyhedral microstructures", Acta Materialia, Vol. 52/14, pp. 4111-4119, 2004

[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] 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, Vol. 193 (45-47), pp. 5017-5034, 2004

[7] V. Sundararaghavan and N. Zabaras, "Classification of three-dimensional microstructures using support vector machines", Computational Materials Science, Vol. 32, pp. 223-239, 2005 .

[8] Velamur Asokan Badri Narayanan and N. Zabaras, "Stochastic inverse heat conduction using a spectral approach", International Journal for Numerical Methods in Engineering, Vol. 60/9, pp. 1569-1593, 2004

[9] S. Ganapathysubramanian and N. Zabaras, "Modeling the thermoelastic-viscoplastic response of polycrystals using a continuum representation over the orientation space", International Journal of Plasticity, Vol. 21/1 pp. 119-144, 2005

[10] V. Sundararaghavan and N. Zabaras, "On the synergy between classification of textures and deformation process sequence selection", Acta Materialia, in press

Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory

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

sub-problemsub-problem

Contact sensitivity sensitivity

sub-problemsub-problem

Remeshingsensitivity sensitivity

sub-problemsub-problem

Kinematic Kinematic sensitivity sensitivity

sub-problemsub-problem

Sensitivity Sensitivity Problem Problem (Linear)(Linear)

Design Design SimulatorSimulator

OptimizationOptimization

Current capabilities

- Thermomechanical deformation process design in the presence of ductile damage

-Thermomechanical deformation process design in the presence of dynamic recrystallization

-Multi-stage deformation process design

-Implementation of 3D continuum sensitivity analysis algorithm. Mathematically rigorous computation of gradients - good convergence observed within few optimization iterations

Continuum sensitivity method - broad outline

• Discretize infinite dimensional design space into a finite dimensional space

• Differentiate the continuum governing equations with respect to the design variables

• Discretize the equations using finite elements

• Solve and compute the gradients

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

Press forcePress force

Processing temperatureProcessing temperature

Press speedPress speed

Product qualityProduct quality

Geometry restrictionsGeometry restrictions

CostCost

CONSTRAINTSCONSTRAINTSOBJECTIVESOBJECTIVES

Material usageMaterial usage

Plastic workPlastic work

Uniform deformationUniform deformation

MicrostructureMicrostructure

Desired shapeDesired shape

Residual stressesResidual stresses Thermal parametersThermal parameters

Identification of stagesIdentification of stages

Number of stagesNumber of stages

Preform shapePreform shape

Die 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)

Micro problem driven by the velocity gradient F

Macro problem driven by the macro-design variable β

Bn+1

Ω = Ω (r, t; F)~Polycrystal

plasticity

x = x(X, t; β) F = F (X, t; β)

ODF: 1234567

F = deformation gradient

Fn+1

B0 X

Material: 99.987% pure polycrystalline f.c.c Aluminum

Process: Upset forging

Forging rate = 0.01 /s

Total deformation = 15%

YX

Z

6 74.63085 70.12274 65.61463 61.10652 56.59841 52.0903

Eq. stress(MPa)

XY

Z

6 302.2155 301.854 301.4853 301.1192 300.7541 300.388

Temperature (oC)

Stroke (mm)

Fo

rce

(N)

0 0.05 0.1 0.150

10

20

30

40

50

60

70

80

90

100

MULTI-LENGTH SCALE FORGING

Ongoing efforts

Extension to complex multistage forging and extrusion processes

-Incorporate remeshing using CUBIT (Sandia) and interface with suitable data transfer schemes

-Computational issues – Parallel implementation using PETSC (ANL)

-Extension to constitutive modeling and process design of Titanium alloys.

-Development of a multiscale version of the design simulator employing a polycrystal plasticity based constitutive model involving a novel two length scale sensitivity analysis for process and materials design

Synergistic research activities

-Design and analysis of deformation processes in the presence of uncertainty

-Statistical learning techniques for process sequence selection

-Microstructure classification and reconstruction

-Model reduction techniques in multiscale modeling

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Displacement (mm)

SD

Loa

d (N

)

Homogeneous materialHeterogeneous material

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

2

4

6

8

10

12

14

Displacement (mm)

Load

(N)

Mean

Spectral stochastic simulation: A tension test modeled using a GPCE-based approach. The internal state variable is assumed to be uncertain and derived from an assumed covariance kernel: (a) The initial and mean deformed configuration of the tension specimen (b) The mean load versus displacement curve and a set of embedded sample realizations (c) The standard deviation of the response

Recent publications

[110] pole figure

FeatureDATABASE OF ODFsUniaxial (z-axis)

Compression Texture

z-axis <110> fiber (BB’)

0 10 20 30 40 50 60 70 80 90143.6

143.8

144

144.2

144.4

144.6

144.8

145

145.2

145.4

Angle from the rolling direction

Youn

gs M

odul

us (G

Pa)

Desired property distributionInitialOptimal (reduced)

Multi-stage reduced order control of Young’s Modulus

Stage: 2 Tension

( = 0.17339)

Stage: 1 Shear

( = -0.03579)

Classification

Adaptive reduced basis selection

Process – 2 Plane strain compression a = 0.3515

Process – 1 Tension a = 0.9539

Initial Conditions: Stage 1

DATABASE

Higher dimensional feature Space x

Design parameter (