1. Modeling uncertainty propagation in large deformations Materials Process Design and Control Laboratory 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: swagato@cornell.edu zabaras@cornell.edu URL: http://mpdc.mae.cornell.edu/

2. people Materials 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

3. 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

4. Materials Process Design and Control Laboratory MOTIVATION All physical systems have an inherent associated randomness SOURCES OF UNCERTAINTIES Multiscale material information – inherently statistical in nature. Uncertainties in process conditions Input data Model formulation – approximations, assumptions. Why uncertainty modeling ? Assess product and process reliability. Estimate confidence level in model predictions. Identify relative sources of randomness. Provide robust design solutions. Engineering component Heterogeneous random Microstructural features Fail Safe Component reliability

5. Materials Process Design and Control Laboratory MOTIVATION:UNCERTAINTY IN FINITE DEFORMATION PROBLEMS Metal forming Composites – fiber orientation, fiber spacing, constitutive model Biomechanics – material properties, constitutive model, fibers in tissues Initial preform shape Material properties/models Forging velocity Texture, grain sizes Die/workpiece friction Die shape Small change in preform shape could lead to underfill

6. Materials Process Design and Control Laboratory OVERVIEW OF FINITE DEFORMATION DETERMINISTIC PROBLEM Linearized principle of virtual work equationB 0 B F e F p F Initial configuration Deformed configuration Governing equation (1) Multiplicative decomposition framework (2) State variable based rate-dependent constitutive models (3) Total Lagrangian formulation (4) Semi-implicit stress update scheme (Ortiz,1990) Strain measure – Green strain Conjugate stress measure – PKII stress

7. Materials Process Design and Control Laboratory GENERALIZED POLYNOMIAL CHAOS EXPANSION - OVERVIEW    n i ii txWtxW 0 )(),( ~ ),,(   Stochastic process Chaos polynomials (random variables) Reduced order representation of a stochastic processes. Subspace spanned by orthogonal basis functions from the Askey series. Chaos polynomial Support space Random variable Legendre [  ] Uniform Jacobi Beta Hermite [-∞,∞] Normal, LogNormal Laguerre [0, ∞] Gamma 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 (Wiener,Karniadakis,Ghanem)

8. Materials Process Design and Control Laboratory 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 B n+1 (θ) x n+1 (θ)=x(X,t n+1, θ, ) B0B0 X x n+1 (θ) F(θ)F(θ)

9. Materials Process Design and Control Laboratory TOOLBOX FOR ELEMENTARY OPERATIONS ON RANDOM VARIABLES Scalar operations Matrix\Vector operations 1.Addition/Subtraction 2.Multiplication 3.Inverse 1.Addition/Subtraction 2.Multiplication 3.Inverse 4.Trace 5. Transpose Non-polynomial function evaluations 1.Square root 2.Exponential 3.Higher powers Use precomputed expectations of basis functions and direct manipulation of basis coefficients Use direct integration over support space Matrix Inverse Compute B(θ) = A -1 (θ) Galerkin projection Formulate and solve linear system for B j (PC expansion)

10. Materials Process Design and Control Laboratory UNCERTAINTY ANALYSIS USING SSFEM Linearized PVW On integration (space) and further simplification Galerkin projection Inner product

11. Materials Process Design and Control Laboratory 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 Initial and mean deformed config.

12. Materials Process Design and Control Laboratory Load vs Displacement SD Load vs Displacement Dominant effect of material heterogeneity on response statistics UNCERTAINTY DUE TO MATERIAL HETEROGENEITY

13. Materials Process Design and Control Laboratory UNCERTAINTY DUE TO MATERIAL HETEROGENEITY-MC RESULTS MC results from 1000 samples generated using Latin Hypercube Sampling (LHS). Order 4 PCE used for SSFEM

14. Materials Process Design and Control Laboratory EFFECT OF UNCERTAIN FIBER ORIENTATION Aircraft nozzle flap – composite material, subjected to pressure on the free end 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)

15. Materials Process Design and Control Laboratory EFFECT OF UNCERTAIN FIBER ORIENTATION – MC COMPARISON Nozzle tip displacement MC results from 1000 samples generated using Latin Hypercube Sampling

16. Materials Process Design and Control Laboratory B n+1 (θ) B0B0 X(θ)X(θ) x n+1 (θ) F(θ)F(θ) x n+1 (θ)=x(X R,t n+1, θ, ) XRXR F*(θ) MODELING INITIAL CONFIGURATION UNCERTAINTY BRBR FR(θ)FR(θ) Introduce a deterministic reference configuration B R which maps onto a stochastic initial configuration by a stochastic reference deformation gradient F R (θ). The deformation problem is then solved in this reference configuration.

17. Materials Process Design and Control Laboratory Deterministic simulation- Uniform bar under tension with effective plastic strain of 0.7. Power law constitutive model. Plastic strain 0.7 Initial configuration assumed to vary uniformly between two extremes with strain maxima in different regions in the stochastic simulation. STRAIN LOCALIZATION DUE TO INITIAL CONFIGURATION UNCERTAINTY

18. Materials Process Design and Control Laboratory Stochastic simulation Plastic strain 0.7 Results plotted in mean deformed configuration STRAIN LOCALIZATION DUE TO INITIAL CONFIGURATION UNCERTAINTY

19. Materials Process Design and Control Laboratory Point at top Plastic strain 0.7 STRAIN LOCALIZATION DUE TO INITIAL CONFIGURATION UNCERTAINTY Point at centerline

20. Materials Process Design and Control Laboratory 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 ?

21. Materials Process Design and Control Laboratory 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

22. Materials Process Design and Control Laboratory SUPPORT SPACE METHOD – SOLUTION SCHEME Linearized PVW Galerkin projection GPCE Support space

23. Materials Process Design and Control Laboratory EXTENSION TO CONTINUUM DAMAGE Stochastic finite deformation damage evolution based on Gurson- Tvergaard-Needleman model. Updated Lagrangian formulation (Anand, Zabaras et. al.). Material heterogeneity induced by random distribution of micro-voids modeled using KLE and an exponential kernel. Constitutive model

24. Materials Process Design and Control Laboratory PROBLEM 2: EFFECT OF RANDOM VOIDS ON MATERIAL BEHAVIOR Mean InitialFinal Using 6x6 uniform support space grid Uniform 0.02

25. Materials Process Design and Control Laboratory PROBLEM 2: EFFECT OF RANDOM VOIDS ON MATERIAL BEHAVIOR Load displacement curves

26. Materials Process Design and Control Laboratory FURTHER VALIDATION Comparison of statistical parameters ParameterMonte Carlo (1000 LHS samples) Support space 6x6 uniform grid Support space 7x7 uniform grid Mean6.11756.11766.1175 SD0.7991250.7987060.799071 m30.0831688 0.08114570.0831609 m40.9362120.9242770.936017 Final load values

27. Materials Process Design and Control Laboratory PROCESS UNCERTAINTY Axisymmetric cylinder upsetting – 60% height reduction Random initial radius – 10% variation about mean – uniformly distributed Random die workpiece friction U[0.1,0.5] Power law constitutive model Using 10x10 support space grid Random ? Shape Random ? friction

28. Materials Process Design and Control Laboratory PROCESS STATISTICS Force SD Force ParameterMonte Carlo (7000 LHS samples) Support space 10x10 Mean2.2859e32.2863e6 SD297.912299.59 m3-8.156e6 - -9.545e6 m41.850e101.979e10 Final force statistics

29. Materials Process Design and Control Laboratory 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. 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

30. Materials Process Design and Control Laboratory 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

31. Materials Process Design and Control Laboratory 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 reliability of processes. Examine effect of process parameters/ material randomness on design objectives. Robust design applications Incorporate microscale statistical information. Information theoretic correlation kernels