STOCHASTIC AND DETERMINISTIC TECHNIQUES FOR COMPUTATIONAL DESIGN OF DEFORMATION PROCESSES Swagato Acharjee B-Exam Date: April 13, 2006 Sibley School of Mechanical and Aerospace EngineeringCornell University
ACKNOWLEDGEMENTS • 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)
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
METAL FORMING PROCESSES Forging Boeing 747 18,600 forgings Extrusion Rolling
COMPUTATIONAL DESIGN OF DEFORMATION PROCESSES • 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) OBJECTIVES VARIABLES CONSTRAINTS Material usage Identification of stages Press force Plastic work Number of stages Press speed Preform shape Uniform deformation Processing temperature Die shape Microstructure Geometry restrictions Mechanical parameters Desired shape Product quality Thermal parameters Residual stresses Cost
DEFORMATION PROCESS DESIGN - BROAD OUTLINE • Discretize infinite dimensional design space into a finite dimensional space • Differentiate the continuum governing equations with respect to the design variables to obtain the sensitivity problem • Discretize the direct and sensitivity equations using finite elements • Solve and compute the gradients • Combine with a gradient optimization framework to minimize the objective function defined
B B F F F F F F n . e p –1 = I CONSTITUTIVE FRAMEWORK (1) Multiplicative decomposition framework (2) State variable rate-dependent models (3) Radial return-based implicit integration algorithms (4) Damage and thermal effects Initial configuration Temperature: n void fraction: fn Deformed configuration Temperature: void fraction: f Governing equation – Deformation problem Governing equation – Coupled thermal problem Thermal expansion: . Intermediate thermal configuration Temperature: void fraction: fo Stress free (relaxed) configuration Temperature: void fraction: f Hyperelastic-viscoplastic constitutive laws
Contact/friction model Current configuration Reference configuration Admissible region n τ1 τ2 Inadmissible region 3D CONTACT PROBLEM ImpenetrabilityConstraints Coulomb Friction Law • Continuum implementation of die-workpiece contact. • Augmented Lagrangian regularization to enforce impenetrability and frictional stick conditions • Contact surface smoothing using Gregory Patches
Continuum problem Differentiate Discretize o o o o Fr and x λ and x o o x = x (xr, t, β, ∆β ) Kinematic problem SENSITIVITY DEFORMATION PROBLEM Design sensitivity of equilibrium equation Variational form - Calculate such that o o o Pr and F, Constitutive problem Regularized contact problem
(x,y) =(acosθ, bsinθ) b Design vector a PREFORM DESIGN TO MINIMIZE BARRELING Curved surface parametrization – Cross section can at most be an ellipse Model semi-major and semi-minor axes as 6 degree bezier curves H
PREFORM DESIGN TO MINIMIZE BARRELING IN FINAL PRODUCT 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 Initial preform shape Optimal preform shape Normalized objective Iterations Final forged product Final optimal forged product
EXTENSION TO COMPLEX SIMULATIONS • 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.
PREFORM DESIGN FOR A STEERING LINK Objective: Design the initial preform such that the die cavity is fully filled with minimum flash for a fixed stroke Objective Function: Die/Workpiece Setup Reference problem – large flash
PREFORM DESIGN FOR CLOSED DIE FORGING Preform design for a steering link First iteration – underfill Intermediate iteration – underfill
PROCESS DESIGN Preform design for a steering link Final iteration flash minimized and complete fill Objective function
PREFORM DESIGN FOR AN AUTOMOTIVE CROSS SHAFT Material Ti-6 Al 4-V Power law model Initial Setup
PREFORM DESIGN FOR AN AUTOMOTIVE CROSS SHAFT Initial iteration Underfill Flash
PREFORM DESIGN FOR AN AUTOMOTIVE CROSS SHAFT Intermediate iteration
PREFORM DESIGN FOR AN AUTOMOTIVE CROSS SHAFT Final iteration Reduced Flash Minimum Underfill
PREFORM DESIGN FOR AN AUTOMOTIVE CROSS SHAFT Objective Function:
Kinematic sub-problem Kinematic sensitivity sub-problem Constitutive sub-problem Constitutive sensitivity sub-problem Sensitivity Problem (Linear) Contact sub-problem Contact sensitivity sub-problem Design Simulator Optimization Direct problem (Non Linear) Thermal sub-problem Thermal sensitivity sub-problem Remeshing sub-problem Remeshing sensitivity sub-problem DEFORMATION PROCESS DESIGN ENVIRONMENT
MOTIVATION All physical systems have an inherent associated randomness Engineering component • SOURCES OF UNCERTAINTIES • Uncertainties in process conditions • Input data • Model formulation – approximations, assumptions. • Errors in simulation softwares Heterogeneous random Microstructural features Why uncertainty modeling ? Assess product and process reliability. Estimate confidence level in model predictions. Identify relative sources of randomness. Provide robust design solutions. Component reliability Safe Fail
Engineering Information flow Materials Continuum Chemistry Mesoscale Statistical Two way flow of statistical information filter Physics 0 Microscale Length Scales ( ) A 1 1e2 1e4 1e6 1e9 Nanoscale Electronic MOTIVATION Information flow across scales Material heterogeneity • Material information – inherently statistical in nature. • Atomic scale – Kinetic theory, Maxwell’s distribution etc. • Microstructural features – correlation functions, descriptors etc.
Material Model Process 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 Forging velocity Die shape Die/workpiece friction Initial preform shape Dependance Nature and degree of correlation Texture, grain sizes Material properties/models MOTIVATION:UNCERTAINTY IN METAL FORMING PROCESSES Small change in preform shape could lead to underfill
UNCERTAINTY IN METAL FORMING PROCESSES 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 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.
The statistical average of a function ò = E [ g ( X )] g ( y ) f ( y ) dy X Ω = C ( x , t , x ' , t ' ) E [ W ( x , t , ) W ( x ' , t ' , )] RANDOM VARIABLES 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. For a stochastic process W (x,t, ) Covariance
n ~ å = W ( x , t , ) W ( x , t ) ( ) i i = 0 i Chaos polynomials (random variables) Stochastic process GENERALIZED POLYNOMIAL CHAOS EXPANSION - OVERVIEW (Wiener,Karniadakis,Ghanem) Reduced order representation of a stochastic processes. Subspace spanned by orthogonal basis functions from the Askey series. 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
FINITE DEFORMATION UNCERTAINTY ANALYSIS USING SSFEM F() xn+1() X Bn+1() B0 xn+1()=x(X,tn+1, ,) Key features Total Lagrangian formulation – (assumed deterministic initial configuration) Spectral decomposition of the current configuration leading to a stochastic deformation gradient
TOOLBOX FOR ELEMENTARY OPERATIONS ON RANDOM VARIABLES Non-polynomial function evaluations Scalar operations Use direct integration over support space • Square root • Exponential • Higher powers • Addition/Subtraction • Multiplication • Inverse Use precomputed expectations of basis functions and direct manipulation of basis coefficients Matrix Inverse Matrix\Vector operations ComputeB() = A-1() • Addition/Subtraction • Multiplication • Inverse • Trace • 5. Transpose (PC expansion) Galerkin projection Formulate and solve linear system for Bj
UNCERTAINTY ANALYSIS USING SSFEM Linearized PVW On integration (space) and further simplification Inner product Galerkin projection
Initial and mean deformed config. 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
UNCERTAINTY DUE TO MATERIAL HETEROGENEITY Dominant effect of material heterogeneity on response statistics Load vs Displacement SD Load vs Displacement
UNCERTAINTY DUE TO MATERIAL HETEROGENEITY-MC RESULTS MC results from 1000 samples generated using Latin Hypercube Sampling (LHS). Order 4 PCE used for SSFEM
MODELING INITIAL CONFIGURATION UNCERTAINTY xn+1()=x(XR,tn+1, ,) F() X() xn+1() FR() Bn+1() B0 XR F*() BR 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.
INITIALCONFIGURATIONUNCERTAINTY 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.
INITIALCONFIGURATIONUNCERTAINTY Stochastic simulation Results plotted in mean deformed configuration
INITIALCONFIGURATIONUNCERTAINTY Point at centerline Point at top
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 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 ?
NISG - FORMULATION 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
True PDF Interpolant FE Grid NISG - DETAILS 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. Convergence rate identical to usual finite elements, depends on order of interpolation, mesh size (h , p versions).
Initial Final EFFECT OF RANDOM VOIDS ON MATERIAL BEHAVIOR Material heterogeneity induced by random distribution of micro-voids modeled using KLE and an exponential kernel. Gurson type model for damage evolution Mean Uniform 0.02 Using 6x6 uniform support space grid
Load displacement curves EFFECT OF RANDOM VOIDS ON MATERIAL BEHAVIOR
Comparison of statistical parameters Final load values VALIDATION
PROCESS UNCERTAINTY Random ? friction Random ? Shape 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
PROCESS STATISTICS SD Force Force
Final force statistics Convergence study PROCESS STATISTICS Relative Error
RELIABILITY BASED DESIGN Objective: Design the forging pressfor the process on the basis of the maximum force required based on a probability of failure of 0.0002.- β = 3.54 Actual limit state surface Full order reliability method Unsafe state Z(g)<0 Design point SORM Approximation g β FORM Approximation Safe state Z(g)>0 Limit state function Probability of failure Minimum required force capacity vs Stroke for a press failure probability of 0.0002 Minimum design force = 2843 N