A gradient optimization method for efficient design of three-dimensional deformation processes

1. A gradient optimization method for efficient design of three-dimensional deformation processes Swagato Acharjee and Nicholas Zabaras Materials Process Design and Control Laboratory Sibley School of Mechanical and Aerospace Engineering188 Frank H. T. Rhodes Hall Cornell University Ithaca, NY 14853-3801 Email: zabaras@cornell.edu URL: http://www.mae.cornell.edu/zabaras/ Materials Process Design and Control Laboratory

2. RESEARCH SPONSORS • U.S. AIR FORCE PARTNERS • Materials Process Design Branch, AFRL • Computational Mathematics Program, AFOSR ARMY RESEARCH OFFICE Mechanical Behavior of Materials Program NATIONAL SCIENCE FOUNDATION (NSF) Design and Integration Engineering Program • CORNELL THEORY CENTER Materials Process Design and Control Laboratory

3. COMPUTATIONAL DESIGN OF DEFORMATION PROCESSES • 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 Materials Process Design and Control Laboratory

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

5. Design Simulator Optimization Direct problem (Non Linear) Sensitivity Problem (Linear) COMPONENTS OF A DEFORMATION PROCESS DESIGN ENVIRONMENT Kinematic sub-problem Kinematic sensitivity sub-problem Constitutive sub-problem Constitutive sensitivity sub-problem Contact sub-problem Contact sensitivity sub-problem Materials Process Design and Control Laboratory Thermal sub-problem Thermal sensitivity sub-problem Remeshing sub-problem Remeshing sensitivity sub-problem Materials Process Design and Control Laboratory

6. F B B F F F n  e p KINEMATIC AND 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 Materials Process Design and Control Laboratory

7. 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 • Workpiece-die interface assumed to be a continuous surface. Die surface parametrized using polynomial curves Materials Process Design and Control Laboratory

8. B’ B n n o ^ o x + x = x (x+xn, t ; p+  p) DEFINITION OF SENSITIVITIES ~ xn= x (X, tn; p ) ~ Ωn= Ω (X, tn; p ) B x Fr xn Fn ^ x = x (xn, t ; p) X I+Ln Bo o Fn + Fn o xn+xn x+x o o Fr + Fr ~ B’ o xn + xn= x (X , tn; p+  p) ~ o Qn + Qn= Q (X, tn; p+  p) • Shape sensitivity design parameters – Preform shape • Parameter sensitivity design parameters – Die shape, ram speed, material parameters, initial state Materials Process Design and Control Laboratory

9. 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 KINEMATIC PROBLEM Design sensitivity of equilibrium equation Variational form - Calculate such that o o o Pr and F,  Constitutive problem Regularized contact problem Materials Process Design and Control Laboratory

10. o x + x • Continuum approach for computing traction sensitivities • Accurate computation of traction derivatives using augmented Lagrangian approach. o y + [y] Die υ B τ1 x τ2 ~ x = x ( X, t, β p+ Δβ p ) x = x ( X, t, β p ) ~ o τ1 + τ1 υ + υ B0 o X B΄ y = y + y o τ2 + τ2 o 3D CONTINUUM SENSITIVITY CONTACT PROBLEM • 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. Materials Process Design and Control Laboratory

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

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

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

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

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

16. 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 Final forged product Final optimal forged product Materials Process Design and Control Laboratory

17. 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 Materials Process Design and Control Laboratory

18. PREFORM DESIGN TO FILL DIE CAVITY 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 Initial preform shape Optimal preform shape Normalized objective Final forged product Final optimal forged product Iterations Materials Process Design and Control Laboratory

19. DIE DESIGN TO MINIMIZE DEVIATION OF STATE VARIABLE AT EXIT 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 Initial die Optimal die Normalized objective Iterations Materials Process Design and Control Laboratory

20. 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. Materials Process Design and Control Laboratory