NSF DMII Conference 2005

1
Robust design and analysis of deformation
processes
PI Prof. Nicholas Zabaras Participating
student Swagato Acharjee Materials Process
Design and Control Laboratory, Cornell University
http//mpdc.mae.cornell.edu
Research Objectives To develop a
mathematically and computationally rigorous
methodology for virtual materials process design
that is based on quantified product quality and
accounts for process targets and constraints with
explicit consideration of uncertainty in the
process.
II Uncertainty modeling in inelastic
deformation processes
I - Deterministic Design of Deformation Processes
III Ongoing work - Robust design with explicit
consideration of uncertainty
MOTIVATION - All physical systems have an
inherent associated randomness
Object oriented, parallel MPI based software for
Lagrangian finite element analysis and design of
3D hyperelastic-viscoplastic metal forming
processes. Implementation of 3D continuum
sensitivity analysis algorithm. Mathematically
rigorous computation of gradients - good
convergence observed within few optimization
iterations Advanced unstructured hexahedral
remeshing using the meshing software CUBIT
(Sandia). Thermomechanical deformation process
design in the presence of ductile damage and
dynamic recrystallization Multi-stage
deformation process design
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.
SOURCES OF UNCERTAINTIES

• Uncertainties in process conditions
• Input data
• Model formulation
• Material heterogeneity
• Errors in simulation software

UNCERTAINTY DUE TO MATERIAL HETEROGENEITY
Uncertainty modeling in a tension test using
Generalized Polynomial Chaos Expansions (GPCE).
The input uncertainty is assumed in the state
variable (deformation resistance) a random
heterogeneous parameter
Schematic of the continuum sensitivity method
(CSM)
Discretize
Design differentiate
Continuum problem
Contact friction constraints
Equilibrium equation
Sensitivity weak form
Design derivative of equilibrium equation
Adaptive discretization of the PDF of the design
objective based on Smooth (S) Extreme (E) and
Discontinuous (D) regions
Incremental sensitivity contact sub-problem
Material constitutive laws

• 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.
• Design problem is a multi-objective and
  multi-constraint optimization problem.

Incremental thermal sensitivity sub-problem
Time space discretized weak form
Incremental sensitivity constitutive
sub-problem
Effect of heterogeneities at linear-nonlinear
transition
Random realizations
Conservation of energy
Preform Optimization of a Steering LInk
NON INTRUSIVE STOCHASTIC GALERKIN (NISG) MODELING
OF PROCESS UNCERTAINTY IN UPSETTING
Reference problem Large Flash
First iteration Underfill
SELECTED PUBLICATIONS

• 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 support-based
  stochastic Galerkin approach for modeling
  uncertainty propagation in deformation
  processes", Computers and Structures, submitted.
• S. Acharjee and N. Zabaras, "A gradient
  optimization method for efficient design of
  three-dimensional deformation processes",
  NUMIFORM, Columbus, Ohio, 2004.
• N. Zabaras and S. Acharjee, "An efficient
  sensitivity analysis for optimal 3D deformation
  process design", 2005 NSF Design, Service and
  Manufacturing Grantees Conference, Scottsdale,
  Arizona, 2005.
• S. Acharjee and N. Zabaras "Modeling uncertainty
  propagation in large deformations", 8th US
  National Congress in Computational Mechanics,
  Austin, TX, 2005.
• S. Acharjee and N. Zabaras, "On the analysis of
  finite deformations and continuum damage in
  materials with random properties", 3nd M.I.T.
  Conference on Computational Fluid and Solid
  Mechanics, Cambridge, MA, 2005.

Final iteration Flash reduced , no underfill
Objective Function
Uncertainty in die/workpiece friction and initial
shape
Financial support from NSF, AFOSR and ARO.
Computing facilities provided by Cornell Theory
Center