1. Materials Process Design and Control Laboratory STOCHASTIC AND DETERMINISTIC TECHNIQUES FOR COMPUTATIONAL DESIGN OF DEFORMATION PROCESSES Swagato Acharjee B-Exam Date: April 13, 2006 Sibley School of Mechanical and Aerospace Engineering Cornell University

2. Materials Process Design and Control Laboratory 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) ACKNOWLEDGEMENTS

3. Materials Process Design and Control Laboratory OUTLINE  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

4. Materials Process Design and Control Laboratory Part I - Deterministic design of deformation processes

5. Materials Process Design and Control Laboratory METAL FORMING PROCESSES Extrusion Forging Rolling Boeing 747 18,600 forgings

6. Materials Process Design and Control Laboratory Press force Processing temperature Press speed Product quality Geometry restrictions Cost CONSTRAINTS OBJECTIVES Material usage Plastic work Uniform deformation Microstructure Desired shape Residual stresses Thermal parameters Identification of stages Number of stages Preform shape Die shape Mechanical parameters VARIABLES 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) COMPUTATIONAL DESIGN OF DEFORMATION PROCESSES

7. Materials Process Design and Control Laboratory 1.Discretize infinite dimensional design space into a finite dimensional space 2.Differentiate the continuum governing equations with respect to the design variables to obtain the sensitivity problem 3.Discretize the direct and sensitivity equations using finite elements 4.Solve and compute the gradients 5.Combine with a gradient optimization framework to minimize the objective function defined DEFORMATION PROCESS DESIGN - BROAD OUTLINE

8. Materials Process Design and Control Laboratory B n B F e F p F  F Initial configuration Temperature:  n void fraction: f n Deformed configuration Temperature:  void fraction: f Intermediate thermal configuration Temperature:  void fraction: f o Stress free (relaxed) configuration Temperature:  void fraction: f (1) Multiplicative decomposition framework (3) Radial return-based implicit integration algorithms (2) State variable rate-dependent models (4) Damage and thermal effects Governing equation – Deformation problem Governing equation – Coupled thermal problem Thermal expansion: F  =   I. F  –1. Hyperelastic-viscoplastic constitutive laws CONSTITUTIVE FRAMEWORK

9. Materials Process Design and Control Laboratory Impenetrability Constraints Coulomb Friction Law Coulomb Friction Law Inadmissible region n τ1τ1 Reference configuration Current configuration Admissible region Contact/friction model τ2τ2  Continuum implementation of die-workpiece contact.  Augmented Lagrangian regularization to enforce impenetrability and frictional stick conditions  Contact surface smoothing using Gregory Patches 3D CONTACT PROBLEM

10. Materials Process Design and Control Laboratory Continuum problem Differentiate Discretize Design sensitivity of equilibrium equation Calculate such that x = x (x r, t, β, ∆β ) o o Variational form - F r and x o o o λ and x o P r and F, o  o o Constitutive problem Regularized contact problem Kinematic problem SENSITIVITY DEFORMATION PROBLEM

11. Materials Process Design and Control Laboratory Curved surface parametrization – Cross section can at most be an ellipse Model semi-major and semi-minor axes as 6 degree bezier curves Design vector a b (x,y) =(acosθ, bsinθ) H PREFORM DESIGN TO MINIMIZE BARRELING

12. Materials Process Design and Control Laboratory Optimal preform shape Final optimal forged product Final forged product Initial preform shape 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 Iterations Normalized objective PREFORM DESIGN TO MINIMIZE BARRELING IN FINAL PRODUCT

13. Materials Process Design and Control Laboratory 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. EXTENSION TO COMPLEX SIMULATIONS

14. Materials Process Design and Control Laboratory Reference problem – large flashDie/Workpiece Setup Objective: Design the initial preform such that the die cavity is fully filled with minimum flash for a fixed stroke Objective Function: PREFORM DESIGN FOR A STEERING LINK

15. Materials Process Design and Control Laboratory Preform design for a steering link First iteration – underfillIntermediate iteration – underfill PREFORM DESIGN FOR CLOSED DIE FORGING

16. Materials Process Design and Control Laboratory Preform design for a steering link Final iteration flash minimized and complete fill Objective function PROCESS DESIGN

17. Materials Process Design and Control Laboratory PREFORM DESIGN FOR AN AUTOMOTIVE CROSS SHAFT Initial Setup Material Ti-6 Al 4-V Power law model

18. Materials Process Design and Control Laboratory PREFORM DESIGN FOR AN AUTOMOTIVE CROSS SHAFT Flash Underfill Initial iteration

19. Materials Process Design and Control Laboratory PREFORM DESIGN FOR AN AUTOMOTIVE CROSS SHAFT Intermediate iteration

20. Materials Process Design and Control Laboratory PREFORM DESIGN FOR AN AUTOMOTIVE CROSS SHAFT Final iteration Reduced Flash Minimum Underfill

21. Materials Process Design and Control Laboratory PREFORM DESIGN FOR AN AUTOMOTIVE CROSS SHAFT Objective Function:

22. Materials Process Design and Control Laboratory Kinematicsub-problem Direct problem (Non Linear) Constitutivesub-problem Contactsub-problem Thermalsub-problem Remeshingsub-problem Constitutivesensitivitysub-problem Thermalsensitivitysub-problem Contactsensitivitysub-problem Remeshingsensitivitysub-problem Kinematicsensitivitysub-problem Sensitivity Problem (Linear) Design Simulator Optimization DEFORMATION PROCESS DESIGN ENVIRONMENT

23. Materials Process Design and Control Laboratory Part II - Stochastic modeling of inelastic deformations

24. Materials Process Design and Control Laboratory SOURCES OF UNCERTAINTIES Uncertainties in process conditions Input data Model formulation – approximations, assumptions. Errors in simulation softwares 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 All physical systems have an inherent associated randomness MOTIVATION

25. Materials Process Design and Control Laboratory Two way flow of statistical information 11e21e41e61e9 Engineering Length Scales ( ) Physics Chemistry Materials 0 A Information flow Statistical filter Electronic Nanoscale Microscale Mesoscale Continuum Material information – inherently statistical in nature. Atomic scale – Kinetic theory, Maxwell’s distribution etc. Microstructural features – correlation functions, descriptors etc. Information flow across scales Material heterogeneity MOTIVATION

26. Materials Process Design and Control Laboratory 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 Material Model 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 Dependance Nature and degree of correlation Process MOTIVATION:UNCERTAINTY IN METAL FORMING PROCESSES

27. Materials Process Design and Control Laboratory 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. 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 UNCERTAINTY IN METAL FORMING PROCESSES

28. Materials Process Design and Control Laboratory The statistical average of a function   Ω X dyyfygXgE)()()]([ For a stochastic process W ( x,t,  ) Covariance 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. )],','(),,([)',',,(  txWtxWtxtx   C RANDOM VARIABLES

29. Materials Process Design and Control Laboratory    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 polynomialSupport spaceRandom 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) GENERALIZED POLYNOMIAL CHAOS EXPANSION - OVERVIEW

30. Materials Process Design and Control Laboratory Key features Total Lagrangian formulation – (assumed deterministic initial configuration) Spectral decomposition of the current configuration leading to a stochastic deformation gradient B n+1 (  ) x n+1 (  )=x(X,t n+1, , ) B0B0 X x n+1 (  ) F()F() FINITE DEFORMATION UNCERTAINTY ANALYSIS USING SSFEM

31. Materials Process Design and Control Laboratory 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(  ) = A -1 (  ) Galerkin projection Formulate and solve linear system for B j (PC expansion) TOOLBOX FOR ELEMENTARY OPERATIONS ON RANDOM VARIABLES

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

33. Materials Process Design and Control Laboratory 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. UNCERTAINTY DUE TO MATERIAL HETEROGENEITY

34. 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

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

36. Materials Process Design and Control Laboratory B n+1 (  ) B0B0 X()X() x n+1 (  ) F()F() x n+1 (  )=x(X R,t n+1, , ) XRXR F*(  ) BRBR FR()FR() 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. MODELING INITIAL CONFIGURATION UNCERTAINTY

37. Materials Process Design and Control Laboratory 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. INITIAL CONFIGURATION UNCERTAINTY

38. Materials Process Design and Control Laboratory Stochastic simulation Results plotted in mean deformed configuration INITIAL CONFIGURATION UNCERTAINTY

39. Materials Process Design and Control Laboratory Point at top Point at centerline INITIAL CONFIGURATION UNCERTAINTY

40. Materials Process Design and Control Laboratory 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 ? MERITS AND PITFALLS OF GPCE

41. Materials Process Design and Control Laboratory Non Intrusive Stochastic Galerkin Method (NISG)

42. Materials Process Design and Control Laboratory 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 NISG - FORMULATION

43. Materials Process Design and Control Laboratory 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. True PDF Interpolant FE Grid Convergence rate identical to usual finite elements, depends on order of interpolation, mesh size (h, p versions). NISG - DETAILS

44. Materials Process Design and Control Laboratory Mean InitialFinal Using 6x6 uniform support space grid Uniform 0.02 Material heterogeneity induced by random distribution of micro-voids modeled using KLE and an exponential kernel. Gurson type model for damage evolution EFFECT OF RANDOM VOIDS ON MATERIAL BEHAVIOR

45. Materials Process Design and Control Laboratory Load displacement curves EFFECT OF RANDOM VOIDS ON MATERIAL BEHAVIOR

46. Materials Process Design and Control Laboratory 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 VALIDATION

47. Materials Process Design and Control Laboratory 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 Random ? Shape Random ? friction PROCESS UNCERTAINTY

48. Materials Process Design and Control Laboratory Force SD Force PROCESS STATISTICS

49. Materials Process Design and Control Laboratory ParameterMonte Carlo (20000 LHS samples) Support space 10x10 Mean2.2859e32.2863e6 SD297.912299.59 m3-8.156e6 - -9.545e6 m41.850e101.979e10 Final force statisticsConvergence study PROCESS STATISTICS Relative Error

50. Materials Process Design and Control Laboratory FORM Approximation SORM Approximation Actual limit state surface Full order reliability method g Design point Safe state Z(g)>0 Unsafe state Z(g)<0 β Objective: Design the forging press for the process on the basis of the maximum force required based on a probability of failure of 0.0002.- β = 3.54 Minimum required force capacity vs Stroke for a press failure probability of 0.0002 Minimum design force = 2843 N Limit state function Probability of failure RELIABILITY BASED DESIGN

51. Materials Process Design and Control Laboratory Axisymmetric flashless closed die forging Same process with initial void fraction 0.03 Deterministic Simulation Decrease in void fraction in the billet during the process leads to unfilled die cavity Initial preform volume same as volume of die cavity STOCHASTIC ESTIMATION OF DIE UNDERFILL

52. Materials Process Design and Control Laboratory Stochastic Simulation Assumed void fraction using KLE PDF of die underfill Using 10x10 uniform support space grid STOCHASTIC ESTIMATION OF DIE UNDERFILL

53. Materials Process Design and Control Laboratory 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. NISG is the way to go REVIEW OF NISG AND GPCE

54. Materials Process Design and Control Laboratory Part III - Robust Design of Deformation Processes

55. Materials Process Design and Control Laboratory 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. KEY ISSUES 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. ROBUST DESIGN ENVIRONMENT

56. Materials Process Design and Control Laboratory Design Objective Probability Constraint Norm Constraint SPDE Constraint Augmented Objective ROBUST DESIGN PROBLEM FORMULATION

57. Materials Process Design and Control Laboratory CSSM problem decomposed into a set of CSM problems Compute sensitivities of parameters with respect to stochastic design variables by defining perturbations to the PDF of the design variables. Decomposition based on the fact that perturbations to the PDF are local in nature A CONTINUUM STOCHASTIC SENSITIVITY SCHEME (CSSM)

58. Materials Process Design and Control Laboratory Design Objective – unconstrained case Set of Nel E *n objective functions NISG APPROXIMATION FOR OBJECTIVE FUNCTION

59. Materials Process Design and Control Laboratory BENCHMARK APPLICATION 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 Flat die upsetting of a cylinder

60. Materials Process Design and Control Laboratory Deterministic problem - optimal solution Deterministic problem 1D problem 2D problem OBJECTIVE FUNCTION

61. Materials Process Design and Control Laboratory DESIGN PARAMETERS Deterministic problem 2D problem 1D problem Initial guess parameters Mean SD Mean SD

62. Materials Process Design and Control Laboratory OBJECTIVE FUNCTION

63. Materials Process Design and Control Laboratory FINAL FREE SURFACE SHAPE CHARACTERISTICS Mean SD

64. Materials Process Design and Control Laboratory Suggestions for future work

65. Materials Process Design and Control Laboratory Fine scale heterogeneities Coarse scale heterogeneities Nature of randomness differs significantly between scales, though not fully uncorrelated. Need a multiscale evaluation of the Correlation Kernels Present method Assume correlation between macro points Decompose using KLE grain size, texture, dislocations macro-cracks, phase distributions MULTISCALE NATURE OF MATERIAL HETEROGENEITIES

66. Materials Process Design and Control Laboratory As the number of random variables increases, problem size rises exponentially. (assume 10 evaluations per random dimension) CURSE OF DIMENSIONALITY

67. Materials Process Design and Control Laboratory 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 PROPOSED SOLUTIONS

68. Materials Process Design and Control Laboratory 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. Support-space of input Importance spaced grid PROPOSED SOLUTIONS

69. Materials Process Design and Control Laboratory DIMENSION ADAPTIVE QUADRATURE (Gerstner et. al. 2003) Full grid SchemeSparse grid SchemeDimension adaptive Scheme Very popular in computational finance applications. Has been used in as high as 256 dimensions. PROPOSED SOLUTIONS

70. Materials Process Design and Control Laboratory 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. JOURNAL PUBLICATIONS

71. Materials Process Design and Control Laboratory Thank You