On the analysis of finite deformations On the analysis of finite deformations and continuum damage in materials with and continuum damage in materials with random properties random properties Materials Process Design and Control Laborator Materials Process Design and Control Laborator C C O O R R N N E E L L L L U N I V E R S I T Y Swagato Acharjee and Nicholas Zabaras Materials Process Design and Control Laboratory Sibley School of Mechanical and Aerospace Engineering 188 Frank H. T. Rhodes Hall Cornell University Ithaca, NY 14853-3801 Email: [email protected][email protected]URL: http://mpdc.mae.cornell.edu/
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On the analysis of finite deformations and continuum damage in materials with random properties Materials Process Design and Control Laboratory Swagato.
Materials Process Design and Control Laboratory OUTLINE Motivation Overview of GPCE GPCE Solution methodology GPCE based Applications Merits and pitfalls of GPCE Overview of Support space/Stochastic Galerkin method Solution scheme using Support space method Extension to Continuum Damage Applications Conclusions/Future work
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On the analysis of finite deformations On the analysis of finite deformations and continuum damage in materials with and continuum damage in materials with
random propertiesrandom properties
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
OUTLINE
•Motivation
•Overview of GPCE
•GPCE Solution methodology
•GPCE based Applications
•Merits and pitfalls of GPCE
•Overview of Support space/Stochastic Galerkin method
•Solution scheme using Support space method
•Extension to Continuum Damage
•Applications
•Conclusions/Future work
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
Two w
ay flow
of sta
tistica
l infor
mation
1 1e2 1e4 1e6 1e9
Eng
inee
ring
Length Scales ( )
Phy
sics
Che
mis
tryM
ater
ials
0 A
Informati
on flo
w
Statistical filter
Electronic
Nanoscale
Microscale
Mesoscale
Continuum
MOTIVATION: THE BIG PICTURE
Material information – inherently statistical in nature.
•Atomic scale – Kinetic theory, Maxwell’s distribution etc.
•Microstructural features – correlation functions, descriptors etc.
Information flow across scales
Need to develop efficient tools for incorporating statistical information for a complete characterization of material behavior.
Materials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
MOTIVATION:UNCERTAINTY IN FINITE DEFORMATION PROBLEMS
Metal formingForging velocity
Lubrication – friction at die-workpiece interface
Intermediate material state variation over a multistage sequence –residual-stresses, temperature, change in microstructure, expansion/contraction of the workpiece
Die shape – is it constant over repeated forgings ?
Damage evolution through processing stages
Preform shapes (tolerances)
Composites – fiber orientation, fiber spacing, constitutive model
Biomechanics – material properties, constitutive model, fibers in tissues
Material heterogeneity
Materials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
OVERVIEW OF FINITE DEFORMATION DETERMINISTIC PROBLEM
Number of chaos polynomials used to represent output uncertainty depends on Number of chaos polynomials used to represent output uncertainty depends on
- Type of uncertainty in input- Type of uncertainty in input - Distribution of input uncertainty- Distribution of input uncertainty- Number of terms in KLE of input - Degree of uncertainty propagation desired- Number of terms in KLE of input - Degree of uncertainty propagation desired
Materials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
UNCERTAINTY ANALYSIS USING SSFEM
Key features
Total Lagrangian formulation – (assumed deterministic initial configuration)
Spectral decomposition of the current configuration leading to a stochastic deformation gradient
Bn+1(θ)
xn+1(θ)=x(X,tn+1, θ,)
B0
Xxn+1(θ)
F(θ)
1 1( ) ( )n ni i B B
10 1
( , )( ) ( , )
nn
tt
x X,F x X,
X1
1( ) ( , ) ( ) nx QF P X
1 1( )i i i iQ Q F F = P P
11
( )( , )
n
nP xx
1 1( ) ( )n ni i x x
11( )
i
i i
nnP
xx
( )
Q XX
Materials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
UNCERTAINTY ANALYSIS USING SSFEM
Linearized PVW
On integration (space) and further simplification
( ) ( ) ( )i j i jP
f d Galerkin projection Inner product
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
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.
Orthotropic hyperelastic material model with uncertain angle of orthotropy modeled using KL expansion with exponential
covariance and uniform random variables
Two independent random variables with order 4 PCE (Legendre Chaos)
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CCOORRNNEELLLL U N I V E R S I T Y
EFFECT OF UNCERTAIN FIBER ORIENTATION – MC COMPARISON
Nozzle tip displacement
MC results from 1000 samples generated using Latin Hypercube Sampling
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
Bn+1(θ)B0
X(θ) xn+1(θ)F(θ)
xn+1(θ)=x(XR,tn+1, θ,)
XR
F*(θ)
MODELING INITIAL CONFIGURATION UNCERTAINTY
BR
FR(θ)
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.
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
Deterministic simulation- Uniform bar under tension with effective plastic strain of 0.7 . Power law constitutive model.
STRAIN LOCALIZATION DUE TO INITIAL CONFIGURATION UNCERTAINTY
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
Point at top
Plastic strain 0.7
Outer boundary plot
00.5
11.5
2
2.53
3.54
0.264 0.266 0.268 0.27 0.272 0.274 0.276
x (mm)
y (m
m)
Mean-MC
Mean - SSFEM - o4
Mean -Deterministic
STRAIN LOCALIZATION DUE TO INITIAL CONFIGURATION UNCERTAINTY
Point at centerline
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
MERITS AND PITFALLS OF GPCE
• 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
But….
• Behavior near critical points.
• Requires continuous polynomial type smooth response.
• Performance for arbitrary PDF’s.
• How do we represent inequalities spectrally ?
• How do we compute eigenvalues spectrally ?
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
SUPPORT SPACE METHOD - INTRODUCTION
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.Convergence rate identical to usual finite elements, depends on order of interpolation, mesh size (h , p versions).
Easily extend to updated Lagrangian formulations.
Constitutive problem fully deterministic – deterministic evaluation at quadrature points – trivially extend to damage problems.
True PDF
Interpolant
FE Grid
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CCOORRNNEELLLL U N I V E R S I T Y
SUPPORT SPACE METHOD – SOLUTION SCHEME
Linearized PVW
Galerkin projection0 0
. .j mi i j k k m
B B
dP dV P dV
u uX X
. .B B
dP dFdV P dFdV
( ) ( )
( ) ( )( ). ( ).( ) ( )
n nn nB B
dP dV P dV
u ux x
( ) ( )
( ) ( )( ). ( ) ( ). ( )( ) ( )
n nn nP B P B
dP dVd P dVd
u ux x
Galerkin projection
GPCE
Support space
1 ( )
1 ( )
( )( ). ( )( )
( ) ( ). ( )( )
s
e n
s
e n
nel
e nP B
nel
e nP B
dP dVd
P dVd
ux
ux
2i j i j l l lu K B
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
UNCERTAINTY IN INITIAL CONFIGURATION- NONPOROUS MATERIAL
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CCOORRNNEELLLL U N I V E R S I T Y
PROBLEM 2: EFFECT OF RANDOM VOIDS ON MATERIAL BEHAVIOR
Load displacement curves
Displacement (mm)
Load
(N)
0.1 0.2 0.3 0.4
1
2
3
4
5
6Mean
Mean +/- SD
Displacement (mm)
SD
Load
(N)
0.1 0.2 0.3 0.40
0.1
0.2
0.3
0.4
0.5
0.6
0.7
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CCOORRNNEELLLL U N I V E R S I T Y
FURTHER VALIDATION
Comparison of statistical parameters
Parameter Monte Carlo (1000 LHS samples)
Support space 6x6 uniform grid
Support space 7x7 uniform grid
Mean 6.1175 6.1176 6.1175
SD 0.799125 0.798706 0.799071
Skewness 0.0831688 0.0811457 0.0831609
Kurtosis 0.936212 0.924277 0.936017
Final load values
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
Demonstration of two non-statistical methods for modeling uncertainty in finite deformation problems.
• Both provide complete response statistics and convergence in distribution.
• The support-space approach incurs a larger computation cost in comparison to the GPCE approach for a given stochastic simulation of systems with smooth inputs.
• GPCE fails for systems with sharp discontinuities. (inequalities).
• Easier to integrate the support space method into existing codes. Only change global assembly routine. Ideal for complex simulations with strong nonlinearities. (Finite deformations – eigen strains, inequalities, complex constitutive models).
• GPCE needs explicit spectral expansion and repeated Galerkin projections.
IN CONCLUSION
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CCOORRNNEELLLL U N I V E R S I T Y
• The support-space approach 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).
• GPCE on the other hand loses its convergence properties if the Askey chaos chosen does not correspond to the input distribution.
• Curse of dimensionality – both methods are susceptible. More research needed on intelligent approximations.
IN CONCLUSION
Relevant Publication
S. Acharjee and N. Zabaras, "Uncertainty propagation in finite deformations -- A spectral stochastic Lagrangian approach", Computer Methods in Applied Mechanics and Engineering, in press
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CCOORRNNEELLLL U N I V E R S I T Y
FUTURE WORK
Linkage?
Information Theory
• Field of mathematics founded by Shannon in 1948
•Try to transfer as much information as possible about parameters of interest (displacements, stresses, strains etc)
• Extend to metal forming simulations. (Strong nonlinearities – contact)
• Examine effect of process parameters/ material randomness on design objectives.