PI: Prof. Nicholas Zabaras Participating student: Swagato Acharjee

Materials Process Design and Control Laboratory, Cornell University

Robust design and analysis of deformation processes

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.

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Iterations

Final iteration – Flash reduced , no underfill

First iteration Underfill

Reference problem Large Flash

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

I - Deterministic Design of Deformation Processes

Equilibrium equation

Design derivative of equilibrium

equation

Material constitutive laws

Time & space discretized weak form

Sensitivity weak form

Contact & frictionconstraints

Incremental sensitivity contact

sub-problem

Conservation of energy

Schematic of the continuum sensitivity method (CSM)

Continuum problemDesign

differentiateDiscretize

Incrementalthermal sensitivity

sub-problemIncremental

sensitivity constitutive sub-problem

II – Uncertainty modeling in inelastic deformation processes

III– Ongoing work - Robust design with explicit consideration of uncertainty

MOTIVATION - All physical systems have an inherent associated randomness

•Uncertainties in process conditions

•Input data

•Model formulation

•Material heterogeneity

•Errors in simulation software

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.

• 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.

SOURCES OF UNCERTAINTIES

Engineering component

Random Meso-Scale

featuresDesign variables

Fail Safe

http://mpdc.mae.cornell.edu

Objective Function

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Displacement (mm)

Lo

ad

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Mean

Preform Optimization of a Steering LInk

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

UNCERTAINTY DUE TO MATERIAL HETEROGENEITY

NON INTRUSIVE STOCHASTIC GALERKIN (NISG) MODELING OF PROCESS UNCERTAINTY IN UPSETTING

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Displacement (mm)

SD

Lo

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Similar homogeneos material

Heterogeneous material

Effect of heterogeneities at linear-nonlinear transition

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.

Financial support from NSF, AFOSR and ARO. Computing facilities provided by Cornell Theory Center

Input support space

SD

E

PDF of design Objective

Adaptive discretization of the PDF of the design objective based on Smooth (S) Extreme (E) and Discontinuous (D) regions

Void fraction: 0.002 0.004 0.007 0.009 0.011 0.013 0.016

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an

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rce

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Stroke (mm)0 0.2 0.4 0.6 0.8 1

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Stroke (mm)

Std

. D

ev.

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Uncertainty in die/workpiece friction and initial shape

Random realizations