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STOCHASTIC AND DETERMINISTIC TECHNIQUES FOR COMPUTATIONAL DESIGN OF DEFORMATION PROCESSES

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STOCHASTIC AND DETERMINISTIC TECHNIQUES FOR COMPUTATIONAL DESIGN OF DEFORMATION PROCESSES

Swagato Acharjee

B-Exam

Date: April 13, 2006

Sibley School of Mechanical and Aerospace EngineeringCornell University

- SPECIAL COMMITTEE:
- Prof. Nicholas Zabaras
- Prof. Subrata Mukherjee
- Prof. Leigh Phoenix

- FUNDING SOURCES:
- Air Force Office of Scientific Research (AFOSR), National Science

Foundation (NSF), Army Research Office (ARO)

- Cornell Theory Center (CTC)
- Sibley school of Mechanical & Aerospace Engineering

Materials Process Design and Control Laboratory (MPDC)

- Deterministic design of deformation processes
- Overview of direct and sensitivity deformation problems
- Applications
- Stochastic modeling of inelastic deformations
- Probability and stochastic processes
- Generalized Polynomial Chaos Expansions (GPCE)
- Non Intrusive Stochastic Galerkin Approximation
- Stochastic optimization
- Robust design of deformation processes
- Applications
- Suggestion for future work

COMPUTATIONAL DESIGN OF DEFORMATION PROCESSES

- BROAD DESIGN OBJECTIVES
- Given raw material, obtain final product with desired microstructure
- and shape with minimal material utilization and costs

- COMPUTATIONAL PROCESS DESIGN
- Design the forming and thermal process sequence
- Selection of stages (broad classification)
- Selection of dies and preforms in each stage
- Selection of mechanical and thermal process parameters in each stage
- Selection of the initial material state (microstructure)

OBJECTIVES

VARIABLES

CONSTRAINTS

Material usage

Identification of stages

Press force

Plastic work

Number of stages

Press speed

Preform shape

Uniform deformation

Processing temperature

Die shape

Microstructure

Geometry restrictions

Mechanical parameters

Desired shape

Product quality

Thermal parameters

Residual stresses

Cost

DEFORMATION PROCESS DESIGN - BROAD OUTLINE

- Discretize infinite dimensional design space into a finite dimensional space
- Differentiate the continuum governing equations with respect to the design variables to obtain the sensitivity problem
- Discretize the direct and sensitivity equations using finite elements
- Solve and compute the gradients
- Combine with a gradient optimization framework to minimize the objective function defined

B

F

F

F

F

F

F

n

.

e

p

–1

= I

CONSTITUTIVE FRAMEWORK

(1) Multiplicative decomposition framework

(2) State variable rate-dependent models

(3) Radial return-based implicit integration algorithms

(4) Damage and thermal effects

Initial configuration

Temperature: n

void fraction: fn

Deformed configuration

Temperature:

void fraction: f

Governing equation – Deformation problem

Governing equation – Coupled thermal problem

Thermal expansion:

.

Intermediate thermal

configuration

Temperature:

void fraction: fo

Stress free (relaxed)

configuration

Temperature:

void fraction: f

Hyperelastic-viscoplastic constitutive laws

Current

configuration

Reference

configuration

Admissible region

n

τ1

τ2

Inadmissible region

3D CONTACT PROBLEM

ImpenetrabilityConstraints

Coulomb Friction Law

- Continuum implementation of die-workpiece contact.
- Augmented Lagrangian regularization to enforce impenetrability and frictional stick conditions
- Contact surface smoothing using Gregory Patches

Differentiate

Discretize

o

o

o

o

Fr and x

λ and x

o

o

x = x (xr, t, β, ∆β )

Kinematic problem

SENSITIVITY DEFORMATION PROBLEM

Design sensitivity of equilibrium equation

Variational form -

Calculate such that

o

o

o

Pr and F,

Constitutive problem

Regularized

contact problem

b

Design vector

a

PREFORM DESIGN TO MINIMIZE BARRELING

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

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

H

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

Material:

Al 1100-O at 673 K

Initial preform shape

Optimal preform shape

Normalized objective

Iterations

Final forged product

Final optimal forged product

EXTENSION TO COMPLEX SIMULATIONS

- Remeshing
- Advanced THEX algorithm for unstructured hexahedral remeshing using CUBIT (Sandia).
- Interface CUBIT with C++ code using NETCDF arrays and FAN utilities
- Speed
- Fast solution using Block Jacobi\ ILU preconditioned GMRES solver (PetSc).
- Fully parallel assembly.
- Fully parallel remeshing and data transfer.

PREFORM DESIGN FOR A STEERING LINK

Objective:

Design the initial preform such that the die cavity is fully filled with minimum flash for a fixed stroke

Objective Function:

Die/Workpiece Setup

Reference problem – large flash

PREFORM DESIGN FOR CLOSED DIE FORGING

Preform design for a steering link

First iteration – underfill

Intermediate iteration – underfill

Preform design for a steering link

Final iteration flash minimized and complete fill

Objective function

PREFORM DESIGN FOR AN AUTOMOTIVE CROSS SHAFT

Intermediate iteration

PREFORM DESIGN FOR AN AUTOMOTIVE CROSS SHAFT

Objective Function:

sub-problem

Kinematic

sensitivity

sub-problem

Constitutive

sub-problem

Constitutive

sensitivity

sub-problem

Sensitivity Problem (Linear)

Contact

sub-problem

Contact

sensitivity

sub-problem

Design Simulator

Optimization

Direct problem

(Non Linear)

Thermal

sub-problem

Thermal

sensitivity

sub-problem

Remeshing

sub-problem

Remeshing

sensitivity

sub-problem

DEFORMATION PROCESS DESIGN ENVIRONMENT

All physical systems have an inherent associated randomness

Engineering component

- SOURCES OF UNCERTAINTIES
- Uncertainties in process conditions
- Input data
- Model formulation – approximations, assumptions.
- Errors in simulation softwares

Heterogeneous random Microstructural features

Why uncertainty modeling ?

Assess product and process reliability.

Estimate confidence level in model predictions.

Identify relative sources of randomness.

Provide robust design solutions.

Component reliability

Safe

Fail

Information flow

Materials

Continuum

Chemistry

Mesoscale

Statistical

Two way flow of statistical information

filter

Physics

0

Microscale

Length Scales ( )

A

1

1e2

1e4

1e6

1e9

Nanoscale

Electronic

MOTIVATION

Information flow across scales

Material heterogeneity

- Material information – inherently statistical in nature.
- Atomic scale – Kinetic theory, Maxwell’s distribution etc.
- Microstructural features – correlation functions, descriptors etc.

Model

Process

Forging rate

Die/Billet shape

Friction

Cooling rate

Stroke length

Billet temperature

Stereology/Grain texture

Dynamic recrystallization

Phase transformation

Phase separation

Internal fracture

Other heterogeneities

Yield surface changes

Isotropic/Kinematic hardening

Softening laws

Rate sensitivity

Internal state variables

Forging velocity

Die shape

Die/workpiece friction

Initial preform shape

Dependance Nature and degree of correlation

Texture, grain sizes

Material properties/models

MOTIVATION:UNCERTAINTY IN METAL FORMING PROCESSES

Small change in preform shape could lead to underfill

UNCERTAINTY IN METAL FORMING PROCESSES

Earlier works

1. Kleiber et. al. – IJNME 2004

Response surface method for analysis of sheet forming processes

2. Sluzalec et. al. – IJMS 2000

Perturbation type methods

3. Doltsinis et. al. – CMAME 2003,2005

Perturbation type methods – avoided all strong nonlinearities

Issues with stochastic analysis

Extremely complex phenomena – nonlinearities at all stages - large deformation plasticity, microstructure evolution, contact and friction conditions, thermomechanical coupling and damage accumulation – standard RBDO methods do not work well.

Lack of robust and efficient uncertainty analysis tools specific to metal forming.

High levels of uncertainty in the system

Possibility of reusing already developed legacy codes.

The statistical average of a function

ò

=

E

[

g

(

X

)]

g

(

y

)

f

(

y

)

dy

X

Ω

=

C

(

x

,

t

,

x

'

,

t

'

)

E

[

W

(

x

,

t

,

)

W

(

x

'

,

t

'

,

)]

RANDOM VARIABLES

Definition – Probability space

The sample space Ω, the collection of all possible events in a sample space F and the probability law P that assigns some probability to all such combinations constitute a probability space (Ω, F, P )

Stochastic process – function of space, time and random dimension.

For a stochastic process W (x,t, )

Covariance

~

å

=

W

(

x

,

t

,

)

W

(

x

,

t

)

(

)

i

i

=

0

i

Chaos polynomials

(random variables)

Stochastic process

GENERALIZED POLYNOMIAL CHAOS EXPANSION - OVERVIEW

(Wiener,Karniadakis,Ghanem)

Reduced order representation of a stochastic processes.

Subspace spanned by orthogonal basis functions from the Askey series.

Number of chaos polynomials used to represent output uncertainty depends on

- Type of uncertainty in input - Distribution of input uncertainty- Number of terms in KLE of input - Degree of uncertainty propagation desired

FINITE DEFORMATION UNCERTAINTY ANALYSIS USING SSFEM

F()

xn+1()

X

Bn+1()

B0

xn+1()=x(X,tn+1, ,)

Key features

Total Lagrangian formulation – (assumed deterministic initial configuration)

Spectral decomposition of the current configuration leading to a stochastic deformation gradient

TOOLBOX FOR ELEMENTARY OPERATIONS ON RANDOM VARIABLES

Non-polynomial function evaluations

Scalar operations

Use direct integration over support space

- Square root
- Exponential
- Higher powers

- Addition/Subtraction
- Multiplication
- Inverse

Use precomputed expectations of basis functions and direct manipulation of basis coefficients

Matrix Inverse

Matrix\Vector operations

ComputeB() = A-1()

- Addition/Subtraction
- Multiplication
- Inverse
- Trace
- 5. Transpose

(PC expansion)

Galerkin projection

Formulate and solve linear system for Bj

UNCERTAINTY ANALYSIS USING SSFEM

Linearized PVW

On integration (space) and further simplification

Inner product

Galerkin projection

Initial and mean deformed config.

UNCERTAINTY DUE TO MATERIAL HETEROGENEITY

State variable based power law model.

State variable – Measure of deformation resistance- mesoscale property

Material heterogeneity in the state variable assumed to be a second order random process with an exponential covariance kernel.

Eigen decomposition of the kernel using KLE.

Eigenvectors

UNCERTAINTY DUE TO MATERIAL HETEROGENEITY

Dominant effect of material heterogeneity on response statistics

Load vs Displacement

SD Load vs Displacement

UNCERTAINTY DUE TO MATERIAL HETEROGENEITY-MC RESULTS

MC results from 1000 samples generated using Latin Hypercube Sampling (LHS). Order 4 PCE used for SSFEM

MODELING INITIAL CONFIGURATION UNCERTAINTY

xn+1()=x(XR,tn+1, ,)

F()

X()

xn+1()

FR()

Bn+1()

B0

XR

F*()

BR

Introduce a deterministic reference configuration BR which maps onto a stochastic initial configuration by a stochastic reference deformation gradient FR(θ). The deformation problem is then solved in this reference configuration.

INITIALCONFIGURATIONUNCERTAINTY

Deterministic simulation- Uniform bar under tension with effective plastic strain of 0.7 . Power law constitutive model.

Initial configuration assumed to vary uniformly between two extremes with strain maxima in different regions in the stochastic simulation.

- Reduced order representation of uncertainty
- Faster than mc by at least an order of magnitude
- Exponential convergence rates for many problems
- Provides complete response statistics and convergence in distribution
- But….
- Behavior near critical points.
- Requires continuous polynomial type smooth response.
- Performance for arbitrary PDF’s.
- How do we represent inequalities, eigenvalues spectrally ?
- Can we afford to rewrite complex metal forming codes ?

Parameters of interest in stochastic analysis are the moment information (mean, standard deviation, kurtosis etc.) and the PDF.

For a stochastic process

Definition of moments

NISG - Random space discretized using finite elements to

Output PDF computed using local least squares interpolation from function evaluations at integration points.

Deterministic evaluations at fixed points

Interpolant

FE Grid

NISG - DETAILS

Finite element representation of the support space.

Inherits properties of FEM – piece wise representations, allows discontinuous functions, quadrature based integration rules, local support.

Provides complete response statistics.

Decoupled function evaluations at element integration points.

Convergence rate identical to usual finite elements, depends on order of interpolation, mesh size (h , p versions).

Final

EFFECT OF RANDOM VOIDS ON MATERIAL BEHAVIOR

Material heterogeneity induced by random distribution of micro-voids modeled using KLE and an exponential kernel. Gurson type model for damage evolution

Mean

Uniform 0.02

Using 6x6 uniform support space grid

EFFECT OF RANDOM VOIDS ON MATERIAL BEHAVIOR

Random ?

friction

Random ?

Shape

Axisymmetric cylinder upsetting – 60% height reduction (Initial height 1.5 mm)

Random initial radius – 10% variation about mean (1 mm)– uniformly distributed

Random die workpiece friction U[0.1,0.5]

Power law constitutive model

Using 10x10 support space grid

Objective:

Design the forging pressfor

the process on the basis of the maximum force required based on a probability

of failure of 0.0002.- β = 3.54

Actual limit state surface Full order reliability method

Unsafe state Z(g)<0

Design point

SORM Approximation

g

β

FORM Approximation

Safe state Z(g)>0

Limit state function

Probability of failure

Minimum required force capacity vs Stroke for a press failure probability of 0.0002

Minimum design force = 2843 N

STOCHASTIC ESTIMATION OF DIE UNDERFILL

Deterministic Simulation

Axisymmetric flashless closed die forging

Same process with initial void fraction 0.03

Initial preform volume same as volume of die cavity

Decrease in void fraction in the billet during the process leads to unfilled die cavity

STOCHASTIC ESTIMATION OF DIE UNDERFILL

Stochastic Simulation

Assumed void fraction using KLE

PDF of die underfill

Using 10x10 uniform support space grid

Both provide complete response statistics and convergence in distribution.

- GPCE fails for systems with sharp discontinuities. (inequalities).
- Seamless integration of NISG into existing codes. Ideal for complex simulations with strong nonlinearities. (Finite deformations – eigen strains, inequalities, complex constitutive models).
- GPCE needs explicit spectral expansion and repeated Galerkin projections.
- NISG can handle completely empirical probability density functions due to local support with no change in the convergence properties (convergence is based on number of elements used to discretize the support-space and the order of interpolation).
- Curse of dimensionality – both methods are susceptible.

REVIEW OF NISG AND GPCE

NISG is the way to go

PROBLEM STATEMENT

Compute the predefined random process design parameters which lead to a desired objectives with acceptable (or specified) levels of uncertainty in the final product and satisfying all constraints.

KEY ISSUES

- Robustness limits on the desired properties in the product –
- acceptable range of uncertainty.
- Design in the presence of uncertainty/ not to reduce uncertainty.
- Design variables are stochastic processes or random variables.
- Consider all ‘important’ process and material data to be random
- processes – by itself a design decision.
- Design problem is a multi-objective and multi-constraint
- optimization problem.

ROBUST DESIGN PROBLEM FORMULATION

Design Objective

Probability Constraint

Norm Constraint

SPDE Constraint

Augmented Objective

A CONTINUUM STOCHASTIC SENSITIVITY SCHEME (CSSM)

Compute sensitivities of parameters with respect to stochastic design variables by defining perturbations to the PDF of the design variables.

CSSM problem decomposed into a set of CSM problems

Decomposition based on the fact that perturbations to the PDF are local in nature

NISG APPROXIMATION FOR OBJECTIVE FUNCTION

Design Objective – unconstrained case

Set of NelE*n objective functions

Flat die upsetting of a cylinder

Case 1 – Deterministic problem

Case 2 – 1 random variable (uniformly distributed) – friction – 66% variation about mean (0.3) (10x1 grid) – 1D problem

Case 3 – 2 random variables (uniformly distributed) – friction(66%) and desired shape (10% about mean) (10x10 grid) - 2D problem

Nature of randomness differs significantly between scales, though not fully uncorrelated.

- Need a multiscale evaluation of the Correlation Kernels

MULTISCALE NATURE OF MATERIAL HETEROGENEITIES

Present method

Assume correlation between macro points

Decompose using KLE

Fine scale heterogeneities

grain size, texture, dislocations

Coarse scale heterogeneities

macro-cracks, phase distributions

As the number of random variables increases, problem size rises exponentially.

(assume 10 evaluations per random dimension)

A PRIORI ADAPTIVITY

- Initial sensitivity analysis with respect to random parameters.
- Sensitivities used to a priori refine/coarsen grid discretization along each random dimension.
- Easily implemented using version of earlier CSM analysis

ADAPTIVE DISCRETIZATION BASED ON OUTPUT STOCHASTIC FIELD

- Refine/Coarsen input support space grid based on output defined control parameter (Gradients, standard deviations etc.)
- Applicable using standard h,p adaptive schemes.

Importance spaced grid

Support-space of input

PROPOSED SOLUTIONS

1

1

0.8

0.8

0.8

0.6

0.6

0.6

0.4

0.4

0.4

0.2

0.2

0.2

0

0

0

-0.2

-0.2

-0.2

-0.4

-0.4

-0.4

-0.6

-0.6

-0.6

-0.8

-0.8

-0.8

-1

-1

-1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0.8

PROPOSED SOLUTIONS

DIMENSION ADAPTIVE QUADRATURE (Gerstner et. al. 2003)

Full grid Scheme

Sparse grid Scheme

Dimension adaptive Scheme

Very popular in computational finance applications.

Has been used in as high as 256 dimensions.

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, pp. 5627-5646, 2003.

S. 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.

S. Acharjee and N. Zabaras, "Uncertainty propagation in finite deformation plasticity -- A spectral stochastic Lagrangian approach", Computer Methods in Applied Mechanics and Engineering, in press.

S. Acharjee and N. Zabaras, "A concurrent model reduction approach on spatial and random domains for stochastic PDEs", International Journal for Numerical Methods in Engineering, in press.

S. Acharjee and N. Zabaras, "A non-intrusive stochastic Galerkin approach for modeling uncertainty propagation in deformation processes", Computers and Structures, in preparation.

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