2. A gradient optimization method for efficient design of three-dimensional deformation processes 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: zabaras@cornell.edu URL: http://www.mae.cornell.edu/zabaras /

3. 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

4. Materials Process Design and Control Laboratory COMPUTATIONAL DESIGN OF DEFORMATION PROCESSES 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 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)

5. DESIGN OPTIMIZATION FRAMEWORK Gradient methods  Finite differences (Kobayashi et al.)  Multiple direct (modeling) steps  Expensive, insensitive to small perturbations  Direct differentiation technique (Chenot et al., Grandhi et al.)  Discretization sensitive  Sensitivity of boundary condition  Coupling of different phenomena  Automatic differentiation technique  Continuum sensitivity method (Zabaras et al.)  Design differentiate continuum equations  Complex physical system  Linear systems Heuristic methods  Genetic algorithms (Ghosh et al.)  Multiple direct (modeling) steps  Response surface methods (Grandhi et al., Shoemaker et al.)  Complex response  Numerous direct steps Continuum equations Design differentiate Discretize Materials Process Design and Control Laboratory

6. COMPONENTS OF A DEFORMATION PROCESS DESIGN ENVIRONMENT 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 Materials Process Design and Control Laboratory

7. KINEMATIC AND CONSTITUTIVE FRAMEWORK 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:. Hyperelastic-viscoplastic constitutive laws

8. Materials Process Design and Control Laboratory 3D CONTACT PROBLEM Impenetrability Constraints Coulomb Friction Law Coulomb Friction Law  Continuum implementation of die-workpiece contact.  Augmented Lagrangian regularization to enforce impenetrability and frictional stick conditions  Workpiece-die interface assumed to be a continuous surface. Die surface parametrized using polynomial curves Inadmissible region n τ1τ1 Reference configuration Current configuration Admissible region Contact/friction model τ2τ2

9. Materials Process Design and Control Laboratory DEFINITION OF SENSITIVITIES o F n + F n X xnxn FnFn BoBo x+x o o F r + F r x B x n + x n = x (X, t n ;  p +   p ) o ~ Q n + Q n = Q (X, t n ;  p +   p ) o ~ x = x (x n, t ;  p ) ^ B’ n x n = x (X, t n ;  p ) ~ Ω n = Ω (X, t n ;  p ) ~ I+L n FrFr x + x = x (x+x n, t ;  p +   p ) ^ o o o x n +x n B n B’  Shape sensitivity design parameters – Preform shape  Parameter sensitivity design parameters – Die shape, ram speed, material parameters, initial state

10. Materials Process Design and Control Laboratory SENSITIVITY KINEMATIC PROBLEM 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

11. Materials Process Design and Control Laboratory 3D CONTINUUM SENSITIVITY CONTACT PROBLEM  Continuum approach for computing traction sensitivities  Accurate computation of traction derivatives using augmented Lagrangian approach. Key issue Contact tractions are inherently non-differentiable due to abrupt slip/stick transitions Regularization assumptions A particle that lies in the admissible (or inadmissible) region for the direct problem also lies in the admissible (or inadmissible) region for the sensitivity problem. A point that is in a state of slip (or stick) in the direct problem is also in the same state in the sensitivity problem. y = y + y υ τ1τ1 υ + υ o τ 1 + τ 1 o x + x o X Die o o y + [y] x = x ( X, t, β p ) ~ x = x ( X, t, β p + Δ β p ) ~ B0B0 B΄B΄ B x τ 2 + τ 2 o τ2τ2

12. Materials Process Design and Control Laboratory SENSITIVITY ANALYSIS OF CONTACT/FRICTION Sensitivity of contact tractions Sensitivity of inelastic slip Normal traction: Stick: Slip: Remarks 1.Sensitivity deformation is a linear problem 2.Iterations are preferably avoided within a single time increment 3.Additional augmentations are avoided by using large penalties in the sensitivity contact problem Sensitivity of gap

13. Materials Process Design and Control Laboratory CONTINUUM SENSITIVITY METHOD - BROAD OUTLINE 1.Discretize infinite dimensional design space into a finite dimensional space 2.Differentiate the continuum governing equations with respect to the design variables 3.Discretize the equations using finite elements 4.Solve and compute the gradients 5.Combine with a gradient optimization framework to minimize the objective function defined

14. 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 VALIDATION OF CSM

15. Materials Process Design and Control Laboratory Equivalent Stress Sensitivity Thermo-mechanical shape sensitivity analysis- Perturbation to the preform shape CSM FDM Equivalent Stress Sensitivity Temperature Sensitivity Reference problem – Open die forging of a cylindrical billet VALIDATION OF CSM

16. Materials Process Design and Control Laboratory Equivalent Stress Sensitivity Thermo-mechanical parameter sensitivity analysis- Forging velocity perturbed CSM FDM Equivalent Stress Sensitivity Temperature Sensitivity Reference problem – Open die forging of a cylindrical billet VALIDATION OF CSM

17. Materials Process Design and Control Laboratory PREFORM DESIGN TO MINIMIZE BARRELING IN FINAL PRODUCT 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

18. Materials Process Design and Control Laboratory PREFORM DESIGN TO MINIMIZE BARRELING IN FINAL PRODUCT 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

19. Materials Process Design and Control Laboratory PREFORM DESIGN TO FILL DIE CAVITY Optimal preform shape Final optimal forged productFinal forged product Initial preform shape Objective: Design the initial preform such that the die cavity is fully filled with no flash for a fixed stroke – Initial void fraction 5% Material: Fe-2%Si at 1273 K Iterations Normalized objective

20. Materials Process Design and Control Laboratory DIE DESIGN TO MINIMIZE DEVIATION OF STATE VARIABLE AT EXIT Optimal dieInitial die Objective: Design the extrusion die for a fixed reduction such that the deviation in the state variable at the exit cross section is minimized Material: Al 1100-O at 673 K Iterations Normalized objective

21. Materials Process Design and Control Laboratory IN CONCLUSION - 3D continuum shape and parameter sensitivity analysis - Implementation of 3D continuum sensitivity contact with appropriate regularization - Mathematically rigorous computation of gradients - good convergence observed within few optimization iterations - Extension to polycrystal plasticity based multi-scale process modeling Issues to be addressed: -Incorporate remeshing and suitable data transfer schemes – essential for simulating complicated forging and extrusion processes -Computational issues – Parallel implementation using Petsc Reference Swagato Acharjee and N. Zabaras, "The continuum sensitivity method for the computational design of three-dimensional deformation processes", Computer Methods in Applied Mechanics and Engineering, accepted for publication.