DEVELOPMENT OF A ROBUST COMPUTATIONAL DESIGN SIMULATOR FOR INDUSTRIAL DEFORMATION PROCESSES

1. DEVELOPMENT OF A ROBUST COMPUTATIONAL DESIGN SIMULATOR FOR INDUSTRIAL DEFORMATION PROCESSES Nicholas Zabaras (PI) and Shankar Ganapathysubramanian URL: http://www.mae.cornell.edu/zabaras/ Email: zabaras@cornell.edu

2. DEFORMATION PROCESS DESIGN FOR TAILORED MATERIAL PROPERTIES Tailored material properties in the final product Desired microstructural features Interactive Optimization Environment Given process constraints & parameters Desired product properties Desired spatial distributions of state variables Optimum deformation process Materials Process Design Simulator Controlled texture, recrystallization, fracture & porosity Billet Product Desired shape with minimal material utilization APPROACH • Optimization based design of deformation processes • Mathematically consistent & accurate continuum sensitivity finite element analysis • Unified approach towards shape and parameter sensitivity analysis • Oriented towards the design of multi-stage processes Minimal overall cost: force, energy, etc. Accelerated process sequence design

3. Interactive Optimization Environment VIRTUAL DEFORMATION PROCESS DESIGN SIMULATOR Mathematical representation of the design objective(s) & constraints • knowledge based expert systems • microstructure evolution paths • ideal forming techniques Selection of a virtual direct process model Selection of the design variables (e.g. die and preform parametrization) Selection of the sequence of processes (stages) and initial process parameter designs Material Process Design Simulator Optimization algorithms Assessment of automatic process optimization Continuum multistage process sensitivity analysis consistent with the direct process model Reliability of the design to uncertainties in the physical and computational models

4. Cost of Dies Energy Consumption Material Usage • Ideal forming & microstructure evolution paths based initial designs • Advanced knowledge-based algorithms for process sequence selection DESIGN OF MULTI STAGE DEFORMATION PROCESSES Initial Product Node: Intermediate preform 1st Stage Evaluate number of stages n and select a process sequence p from all feasible paths (j=1 … m), such that: Arc:Processing Stage n Cost Function + +  min = m i=1 ith Stage Process sequence selection Finishing Stage(nth) Final Product Optimal Path (pth) Feasible Paths (jth)

5. COMPUTATIONAL DESIGN OF METAL FORMING PROCESSES 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 BROAD DESIGN OBJECTIVES • Given raw material, obtain product of desired microstructure • and shape with minimal material utilization and cost 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)

6. Admissible region Current configuration F B F F F B o  e p n r Inadmissible region UPDATED LAGRANGIAN FRAMEWORK OF ANALYSIS CONSTITUTIVE MODEL CONTACT/FRICTION MODEL Initial configuration Temperature: o void fraction: fo Deformed configuration Temperature:  void fraction: f Reference configuration Intermediate thermal configuration Temperature:  void fraction: fo Stress free (relaxed) configuration Temperature:  void fraction: f GOVERNING PHYSICS • Mechanical dissipation • Augmented Lagrangian approach • Coulomb friction • Multiplicative decomposition framework • State variable rate-dependent models • Hyperelastic constitutive law • Thermal and damage effects

7. B B B’ n n n o o Fn + Fn FR + FR o o o Fr + Fr Fr + Fr Fn + Fn DEFINITIONS OF SENSITIVITY FIELDS IN AN UPDATED LAGRANGIAN FRAMEWORK ~ xn= x (X, tn; p ) Parameter sensitivity analysis ~ Qn = Q (X, tn ; p ) B Fr xn x ^ Fn x = x (xn, t ; p) Design parameters I+Ln X • Ram speed • Shape of die surface • Material parameters • Initial state Bo o xn+xn x+x o o ~ B’ xn + xn= x (Y , tn; p+  p ) ~ o ~ Qn + Qn= Q (Y, tn; p+  p ) xn= x (X, tn; s ) ~ __ Qn= Q (X, tn; s ) Fr Shape sensitivity analysis X = X (Y; s ) xn x B Fn X Bo ^ FR x = x (xn, t ; s) BR I+Lo Y I+Ln Main features o X+X • Gateaux differential referred to the fixed configuration Y • Rigorous definition of sensitivity • Key element: LR=FRFR-1 __ x+x xn+xn o o o X + X= X (Y; s +  s) o ~ xn + xn= x (Y , tn; s+  s) o ~ o Qn + Qn= Q (Y, tn; s+  s)

8. SCHEMATIC OF THE CONTINUUM SENSITIVITY METHOD (CSM) Equilibrium equation Contact & friction constraints Design derivative of equilibrium equation Sensitivity weak form Regularized design derivative of contact & frictional constraints Material constitutive laws Incremental sensitivity contact sub-problem Design derivative of the material constitutive laws Time & space discretized weak form Incremental thermal sensitivity sub-problem Incremental Sensitivity constitutive sub-problem Assumed kinematics Design derivative of energy equation Time & space discretized modified weak form Design derivative of assumed kinematics Conservation of energy

9. Continuum problem Differentiate Discretize THE CONTINUUM SENSITIVITY METHOD SUB-PROBLEMS o o o λ and x Pr and F, o o x = x (xr, t, β, ∆β ) Design sensitivity of equilibrium equation o  Calculate and such that o o o o o o   Fr and x Constitutive problem Regularized contact problem Thermal problem Kinematic problem

10. Die υ Parameter Sensitivity Analysis B Consider the non-differentiability of contact and friction conditions r x REGULARIZATION Contact and friction x = x ( X, t, β p ) ~ y = y ( ξ ) sensitivity assumptions o υ + υ B0 o X y = y + y THE CONTINUUM SENSITIVITY CONTACT SUB-PROBLEM B΄ o y = y ( ξ ) r + r o x + x o y + [y] o y,ξ ξ y + ~ υ Die x = x ( X, t, β p+ Δβ p ) Shape Sensitivity Analysis o ~ x = x ( X + X , t, β s+ Δβ s ) y = y ( ξ ) r B X x ~ B0 ~ ~ x = x ( X, t, β s) X = X (Y ;β s+ Δβ s ) X = X (Y ;β s ) Y υ BR B΄ r o o X + X x + x B’0 Sensitivity deformation is a linear problem Iterations are avoided within a single time increment Additional augmentations are avoided by using large penalties in the sensitivity contact problem REMARKS

11. A ONE-STAGE HOT FORMING PREFORM DESIGN PROBLEM Rigid Die Forging rate Convection/ Radiation Flash Conduction Unfilled die cavity Damage/microstructure Unfilled cavity and flash! Initial design MATERIAL SYSTEM 1100-Al workpiece Initial temperature 673 K Axisymmetric problem Standard ambient conditions Design objectives Find preform shape of minimum volume such that the die is filled completely and the flash is minimized Fully filled cavity Optimal design 0.12 0.10 0.08 Objective (mm2) 0.06 Objetive Function 0.04 0.02 0.00 0 10 20 30 Iteration number Iteration Numeber

12. Design Objective Selection of stages Design of preforms Knowledge-based methods Design of dies Shape sensitivity analysis Die and process parameter sensitivity analysis Sequential transfer of sensitivities from one stage to the next THE CONTINUUM SENSITIVITY METHOD FOR MULTI-STAGE DEFORMATION PROCESSES Generic Forming Stage

13. Finishing stage Rigid Die PREFORMING DIE DESIGN PROBLEM FOR SHAPE CONTROL Preforming stage Preforming Stage Finishing Stage Unfilled cavity MATERIAL SYSTEM Initial design 1100-Al workpiece Initial temperature 673 K Axisymmetric problem Standard ambient conditions 2 pre-defined stages - preforming + finishing Design objective Design the preforming die for a fixed volume of the workpiece such that the finishing die is filled Fully filled cavity Optimal design Flash 8.0 6.0 Objective Function (x1.0E-05) Objective (mm2) 4.0 2.0 0.0 0 1 2 3 4 5 6 Iteration number Iteration Number

14. PREFORMING DIE DESIGN FOR CONTROL OF MICROSTRUCTURE 1 . 5 5 7 1 . 5 6 5 1 . 4 5 ) m 1 . 4 m ( 4 h , t h 3 g i 1 . 3 5 e H 2 1 1 . 3 1 . 2 5 1 . 2 0 0 . 5 1 R a d i u s , r ( m m ) Preforming stage Finishing stage Preforming Stage Finishing Stage MATERIAL SYSTEM 1100-Al workpiece Initial temperature 673 K Axisymmetric problem Standard ambient conditions 2 pre-defined stages - preforming + finishing Design objective Design the preforming die for a fixed volume of the workpiece such that the variation in state in the product is minimum Scalar state variable (MPa) Initial design Height (mm) Scalar state variable (MPa) Optimal design Radius (mm) Design In MPa Initial Optimal Average state Objective Objective Function 50.2 52.3 Deviation 3.73 1.88 Iteration number

15. Need to couple grain growth/ orientation and recrystallization simulation models with CSM based computational design for explicit control of micro-structural features in deformation processes FUTURE EXTENSIONS TO MULTI-SCALE PROCESS DESIGN: PCG CONTROL DURING EXTRUSION Peripheral coarse grain (PCG) Extrusion

16. USING COMPUTATIONAL DESIGN TO DEVELOP A DIGITAL MATERIALS PROCESS LIBRARY Alloy flow stress Profile output data Billet input data Material point data

17. FORTHCOMING RESEARCH EFFORTS • Testing and further developments for single-stage designs - complex 2D geometries • Regularized contact/ friction sensitivity modeling • Simultaneous thermal & mechanical design • Sensitivity analysis for multi-body deformations Multi length scale design • Control of grain growth, texture and recrystallization Multi-stage forming design • Coupling with ideal forming & microstructure evolution paths based initial designs • Framework for web-based forming design Development of a 3D forming design simulator • Industrial design applications Robust design algorithms REFERENCES • Srikanth, A., et.al. “Continuum Lagrangian sensitivity analysis for metal forming processes with applications to die design”, Int. J. Numer. Methods Engr., (2000) 679-720. • Srikanth, A. and N. Zabaras. “Shape optimization and preform design in metal forming processes”, Comput. Methods Appl. Mech. Engr., (2000) 1859-1901. • Ganapathysubramanian, S. and N. Zabaras. “Continuum sensitivity method for finite thermo-inelastic deformations with applications to the design of hot forming processes”, Int. J. Numer. Methods Engr., (submitted) • Zabaras, N., et.al. “Continuum sensitivity method for the design of multi-stage metal forming processes”, Int. J. Mech. Sciences (submitted) ACKNOWLEDGEMENTS The work presented here was funded by NSF grant DMI-0113295 with additional support from AFOSR, AFRL and ALCOA.