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COMPUTATIONAL DESIGN OF MULTISTAGE DEFORMATION PROCESSES

COMPUTATIONAL DESIGN OF MULTISTAGE DEFORMATION PROCESSES. Prof. Nicholas Zabaras & Shankar Ganapathysubramanian. Materials Process Design and Control Laboratory Sibley School of Mechanical and Aerospace Engineering 188 Frank H. T. Rhodes Hall Cornell University

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COMPUTATIONAL DESIGN OF MULTISTAGE DEFORMATION PROCESSES

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  1. COMPUTATIONAL DESIGN OF MULTISTAGE DEFORMATION PROCESSES Prof. Nicholas Zabaras & Shankar Ganapathysubramanian Materials Process Design and Control Laboratory Sibley School of Mechanical and Aerospace Engineering188 Frank H. T. Rhodes Hall Cornell University Ithaca, NY 14853-3801Email: zabaras@cornell.edu URL: http://www.mae.cornell.edu/zabaras/ NATIONAL SCIENCE FOUNDATION Design and Integration Engineering Program Computational Mathematics Program Materials Process Design and Control Laboratory

  2. U.S. Air Force Partners • Materials Process Design Branch, AFRL • Computational Mathematics Program, AFOSR • NATIONAL SCIENCE FOUNDATION (NSF) • Design and Integration Engineering Program FEDERAL & INDUSTRIAL SPONSORS/COLLABORATORS Materials Process Design & Control Laboratory • Industrial Sponsors • ALCOA, ATC-Materials Process Design Program Materials Process Design and Control Laboratory

  3. FROM MATERIALS PROCESS MODELING TO COMPUTATIONAL DESIGN • Accelerated Insertionto new materials and processes • Difficult Insertion • of new materials and processes into production • Innovative Processesfor traditional materials Data Mining of Designer Knowledge for rapid solutions to complex problems and to further drive use of knowledge • Materials Process Design • control of microstructure using various length and time scale computational models • Development of Designer Knowledge Base • time consuming and costly endeavor Virtual Material Process Laboratory • Materials Modeling • incremental improvements in specific areas Sensitivity Information points to most influential parameters so as to optimally design the process Reliability Based Design for material/tool variability & uncertainties in mathematical and physical models • Numerical Simulation • Trial-and-error and with no design information • Conventional Design Tools • computational material process design simulator Materials Process Design and Control Laboratory

  4. 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 Interactive Optimization Environment • Materials Process Design • control of microstructure using various length and time scale computational models Designer Knowledge for rapid mining of solutions to complex problems and to further update the digital library Virtual Materials Process Laboratory Optimization algorithms Continuum multistage process sensitivity analysis consistent with the direct process model Sensitivity Information points to most influential parameters so as to optimally design the process Reliability Based Design for material/tool variability & uncertainties in mathematical and physical models A Virtual Laboratory for Materials Process Design people Materials Process Design and Control Laboratory

  5. Mathematical representation of the design objective(s) & constraints • knowledge based expert systems • microstructure evolution paths • ideal forming techniques Interactive Optimization Environment Continuum multistage process sensitivity analysis consistent with the direct process model Reliability of the design to uncertainties in the physical and computational models A VIRTUAL MATERIALS PROCESS DESIGN SIMULATOR 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 Materials Process Design and Control Laboratory

  6. CONSTRAINTS VARIABLES Press force Identification of stages Press speed Number of stages Processing temperature Preform shape Geometry restrictions Die shape Product quality Mechanical parameters Thermal parameters ? ? DESIGN OF MULTI STAGE DEFORMATION PROCESSES Given raw material and desired hardware component performance, compute optimal manufacturing process sequence(s) Material Process Design Simulator Ideal forming & microstructure evolution paths based initial designs Advanced knowledge-based algorithms for process sequence selection Shape and parameter sensitivity analysis Desired Final Shape Selection of Stages Design of Preforms Design of Dies Stage 1 ? ? Stage 2 ? Stage 3 Materials Process Design and Control Laboratory

  7. Materials Process Design Simulator ESSENTIAL FEATURES OF A DESIGN SIMULATOR OF INDUSTRIAL PROCESSES Application Aspects Theoretical Aspects • Efficiency – avoid extensive direct forming simulations (as in surface response methods) • Provide consistent coupling of direct & sensitivity analyses with • knowledge based expert systems • microstructure evolution paths • ideal forming techniques • Oriented towards the design of multi-stage processes • Allow for realistic polycrystalline material constitutive models • Allow for hot forming design and intermediate thermal stages • Interface with commercial solid modelers and optimization tools • Allow consistent application of remeshing, data transfer & adaptivity techniques to sensitivity analysis • Contact/frictional conditions drive most forming design processes and need careful consideration • Extend assumed strain methods to sensitivity analyses (preserve volume) • Mathematically consistent and accurate computation of sensitivity fields • Provide a unified approach to parameter and shape sensitivity / optimization Materials Process Design and Control Laboratory

  8. COMPUTATIONAL DESIGN OF FORMING 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 Uniform deformation Preform shape Processing temperature Die shape Microstructure Geometry restrictions Mechanical parameters Desired shape Product quality Thermal parameters Residual stresses Cost Materials Process Design and Control Laboratory

  9. Cost of Dies Energy Consumption Material Usage Cost Function + + = n  i=1 m min J=1 DESIGN OF MULTI-STAGE PROCESSES Initial Product Based on the `designer knowledge’, evaluate practicable stage number (n) and select a process sequence p from all feasible paths (j=1 … m), such that: Node: Intermediate preform 1st Stage Arc:Processing Stage • such that: • Equipment constraint (press force, ram speed, maximum stroke, etc) • Process temperature constraint • Other process constraints ith Stage • Number of stages - n • Force constraints for each stage • Stroke allocation for each stage • Stage temperature, etc. Finishing Stage(nth) Final Product Optimal Path (pth) Feasible Paths (jth) Materials Process Design and Control Laboratory

  10. ? ? COMPUTATIONAL MULTI-STAGE FORMING DESIGN Knowledge-based methods Design of Sequences Design Objective Shape and parameter sensitivity analysis Design of Preforms Die and process parameter sensitivity analysis Design of Dies Desired Final Shape Selection of Stages Design of Preforms Design of Dies Stage 1 ? ? Stage 2 Stage 3 ? Materials Process Design and Control Laboratory

  11. B F F F F B F F F F D o .   e p p  –1 p –1 =   I . p . = dT  = sym(Lp) = MACROSCOPIC 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: o void fraction: fo Thermal expansion: Deformed configuration Temperature:  void fraction: f . • Inelastic response: • Flow rule: •  Is the viscoplastic potential • Internal variable evolution • Damage evolution Hyper-elastic constitutive law Intermediate thermal configuration Temperature:  void fraction: fo Stress free (relaxed) configuration Temperature:  void fraction: f Mechanical dissipation Materials Process Design and Control Laboratory

  12. THE DIRECT CONTACT PROBLEM Admissible region ImpenetrabilityConstraints Current configuration Reference configuration Coulomb Friction Law n r Inadmissible region • Coulomb friction law assumed at the die-work piece interface • Augmented Lagrangian approach to enforce impenetrability and frictional stick conditions Materials Process Design and Control Laboratory

  13. ~ xn= x (X, tn; p ) ~ Qn= Q (X, tn; p ) ^ x = x (xn, t ; p) B’ B n n o o ^ x + x = x (x+xn, t ; p+  p) o Fn + Fn o Fr + Fr ~ o xn + xn= x (Y , tn; p+  p) o ~ Qn + Qn= Q (Y, tn; p+  p) DEFINITION OF PARAMETER SENSITIVITY B Fr xn x Design Parameters • Ram speed • Shape of die surfaces • Material parameters • Initial state Fn I+Ln X Bo o xn+xn x+x o B’ Two stage state variable sensitivity contour w.r.t. parameter change Materials Process Design and Control Laboratory

  14. ~ xn= x (X, tn; s ) ~ Qn= Q (X, tn; s ) Main Features • Mathematically rigorous definition of sensitivity fields • Gateaux differentials (directional derivatives) referred to fixed Y in the configuration BR • Key element: LR=FRFR-1(velocity design gradient) ^ x = x (xn, t ; s) B n o o ^ x + x = x (x+xn, t ; s+  s) o FR + FR o o o Fn + Fn Fr + Fr o ~ X + X= X (Y; s +  s) ~ o xn + xn= x (Y , tn; s+  s) o ~ Qn + Qn= Q (Y, tn; s+  s) DEFINITION OF SHAPE SENSITIVITY Fr ~ X = X (Y; s ) xn x B Fn X Bo FR BR I+Lo Y I+Ln o X+X x+x xn+xn o o Stress sensitivity contour w.r.t preform shape change Materials Process Design and Control Laboratory

  15. FRAMEWORK OF CONTINUUM SENSITIVITY ALGORITHM 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 and 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 Materials Process Design and Control Laboratory

  16. o x o o x = x (xr, t, β, ∆β ) KINEMATIC SENSITIVITY ANALYSIS Continuum Lagrangian configuration Continuum equilibrium equation (Updated Lagrangian) Bn Design sensitivity of equilibrium equation Direct differentiation Calculate such that   Bn • Parameter sensitivity LR = 0 Driving Force o • Shape sensitivity LR = FR FR-1 Finite element discretization Discrete linear sensitivity equilibrium equation Bn o K x = f Materials Process Design and Control Laboratory

  17. o o o o o o Relation between T and Ee , Ee and Fe and finally Fe and F o o o Evolution of Fp , s and F  SENSITIVITY CONSTITUTIVE FRAMEWORK What do we need from the sensitivity constitutive sub-problem to solve the sensitivity kinematic problem ? o o o Calculate Linear relationship between T and F ,  o o c  Evolution of the state sensitivity as a linear function of F , where V = T, s, Fe Materials Process Design and Control Laboratory

  18. y = y ( ξ ) o x + x o y + [y] o y,ξ ξ y + ~ x = x ( X, t, β p+ Δβ p ) o ~ x = x ( X + X , t, β s+ Δβ s ) ~ X = X (Y ;β s+ Δβ s ) SENSITIVITY ANALYSIS OF CONTACT/FRICTION Die υ Parameter Sensitivity Analysis B r x • Regularization introduced • Contact sensitivity assumption • Friction sensitivity assumption x = x ( X, t, β p ) ~ y = y ( ξ ) o υ + υ B0 o X y = y + y B΄ o r + r υ Die Shape Sensitivity Analysis y = y ( ξ ) r B X x ~ B0 ~ x = x ( X, t, β s) X = X (Y ;β s ) Y υ BR B΄ r o o X + X x + x B’0 Materials Process Design and Control Laboratory

  19. 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 gap and inelastic slip Materials Process Design and Control Laboratory

  20. F-bar method B-bar method Without stabilization (mesh A) o ave F J 1 o o o o h h 3 F F F F h h h h 1 - ε -1 F tr J + ε With stabilization (mesh A) h NINT 1 - ε Fh ∑ -1 -1 ξ J ξ ξ F F N tr - J 3 a h h h F a a a ha a = 1 h With stabilization (mesh B) + = PERFORMANCE OF ASSUMED STRAIN ANALYSIS Fh Bn Fvol Fhvol Fndev Fh=FhFh vol dev Modified sensitivity weak form (stabilized F-bar method) Sensitivity of the assumed deformation gradient Materials Process Design and Control Laboratory

  21. 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 MULTISTAGE CONTINUUM SENSITIVITY ANALYSIS Generic Forming Stage Materials Process Design and Control Laboratory

  22. X x ~ x = x (X, t ; X, Y ) FX X = X (Y, to ;  ) X o X + X= X (Y, to ; Y +  Y) B Bo Q = Q (Y, to ;  ) o Y Q + Q= Q (Y, to ; Y +  Y) FY I+Lo Y Bi o FY + FY Y +  Y o o X+X B’ FX + FX B’ o x+x o Y o ~ o x + x = x (X + X, t ; X, Y+  Y) MULTISTAGE SENSITIVITY ANALYSIS • Multi-stage sensitivity features • Sequential transfer of sensitivities • Shape sensitivities • Parameter sensitivities Validation of multistage sensitivities State Stress DDM DDM FDM FDM Materials Process Design and Control Laboratory

  23. Distorted elements in the old mesh INTRODUCTION TO REMESHING & DATA TRANSFER Mesh Quality Criteria Criteria for Remeshing • Criterion for inner angles • Aspect ratio • Diagonal ratio • Interference with die • Mesh distortion criterion • Error criterion • Conditioning of stiffness matrix Generation of New Mesh • Better-conditioned mesh quality • Accurately describe evolving boundary • Reasonable size of elements Data Transfer Requirements • Consistency with constitutive equations • Satisfy equilibrium • Compatibility of history variables • Compatibility with boundary conditions • Minimization of numerical diffusion New mesh Data Transfer Methods Developed • Shape function based method for nodal data • Distance averaging using Gauss point data Materials Process Design and Control Laboratory

  24. Updated Lagrangian Weak Form for Direct Analysis Kinematic Sub-problem Thermal Sub-problem Constitutive Sub-problem Contact & Friction Sub-problem Remeshing & Data Transfer Sub-problem Updated Lagrangian Weak Form for Sensitivity Analysis Thermal Sub-problem Constitutive Sub-problem Contact & Friction Sub-problem Remeshing & Data Transfer Sub-problem Kinematic Sub-problem DIRECT AND SENSITIVITY PROBLEMS FOR HOT FORMING Materials Process Design and Control Laboratory

  25. DESIGN SENSITIVITY ANALYSIS WITH REMESHING Desired shape Optimization with remeshing Initial solution Optimization without remeshing With remeshing Without remeshing DDM FDM Materials Process Design and Control Laboratory

  26. CLOSED-DIE PREFORM DESIGN PROBLEM Preform design process Objective: Preform design to minimize required force Force reduction Initial preform Optimized preform Force Optimal preform shape Stroke Materials Process Design and Control Laboratory

  27. PREFORM DESIGN FOR POROUS MATERIAL Objective: Minimize the flash and the deviation between the die and the workpiece for a Preforming shape and volume design Material:- 2024-T351Al, 300K, 5% initial void fraction, varying elastic properties (using Budiansky method), co-efficient of friction between die & workpiece = 0.1 Product using guess preform Distribution of shear modulus in product Product using optimal preform Variation of preform shape with optimization iterations Non-dimensional objective Iteration number Materials Process Design and Control Laboratory

  28. Initial extrusion process design Optimal extrusion process design 0 . 9 4 0 . 2 5 0 . 9 3 0 . 2 0 . 9 2 z - axis Final 0 . 1 5 0 . 9 1 Nondimensionalized Objective function Initial 0 . 9 0 . 1 0 . 8 9 0 . 0 5 0 . 8 8 0 1 2 3 4 5 6 7 8 9 0 . 4 8 0 . 4 9 0 . 5 0 . 5 1 Iteration index r - axis EXTRUSION DIE DESIGN FOR CONTROL OF CHEVRON DEFECTS Isothermal frictionless, material with ductile damage area reduction 10.7% 1% initial void fraction Power law model Objective: Design the extrusion die for a fixed reduction of the workpiece s.t. chevron defects are avoided. Initial design has chevron defects, characterized here by the void fraction being > 1%. Materials Process Design and Control Laboratory

  29. ? ? OPTIMAL PREFORM DESIGN EXAMPLE Final product • DESIGN OBJECTIVES • Desired shape • Minimize material utilization • Minimize plastic work/force • Desired microstructure, or • Some of their combinations • CONSTRAINTS • Press force • Equipments • Press temperature • Cost • Material use Continuum shape sensitivity analysis Design problem Optimization Optimum preform shape ? Equivalent stress sensitivity contour (14 remeshing operations) Materials Process Design and Control Laboratory

  30. PREFORM DESIGN – SINGLE STAGE PROCESS Objective: Minimize the flash and the deviation between the die and the workpiece for a Preforming shape design More flash Much more material with a conventional design Unfilled cavity The same material in a conventional design Flash No flash Fully filled cavity The same material with an optimum design Materials Process Design and Control Laboratory

  31. IN A MULTISTAGE DESIGN PROBLEM • Objective: • Minimize the gap between • the finishing die and the • workpiece • in a • two stage forging; • with given finishing die; • unknown die but prescribed • stroke in the preforming • stage. Preforming Stage Finishing Stage Unfilled cavity Initial Design Less unfilled region Iteration No. 2 Fully filled cavity Final Design Materials Process Design and Control Laboratory

  32. MULTISTAGE DEFORMATION PROCESS • Objective: • Minimize the gap between • the finishing die and the • workpiece • in a • two stage forging; • with given finishing die; • unknown die but prescribed • stroke in the preforming • stage. Preforming Stage Finishing Stage Unfilled cavity Initial Design Less unfilled region Iteration No. 3 Fully filled cavity Final Design Materials Process Design and Control Laboratory

  33. 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 ) Design Objective Objective Function In MPa Initial Optimal Average state 50.2 52.3 Iteration number Deviation 3.73 1.88 PREFORMING DIE DESIGN FOR CONTROL OF MICROSTRUCTURE Preforming stage Finishing stage Preforming Stage Finishing Stage 1100-Al workpiece Initial temperature 673 K Axisymmetric problem Standard ambient conditions 2 pre-defined stages - preforming & finishing Initial design Height (mm) Optimal design Objective: Radius (mm) Design the preforming die for a fixed volume of the workpiece such that the variation in state in the product is minimum Materials Process Design and Control Laboratory

  34. CURRENT CAPABILITIES & FUTURE RESEARCH PLANS CURRENT CAPABILITY FUTURE RESEARCH 2D forming process design • Thermo-mechanical analyses for materials with ductile damage • Design objectives Shape optimization Force minimization Material utilization rates • Forming process design considering thermal effects in the die Remeshing & data transfer • Effective remeshing based on geometric criteria • Accurate data transfer techniques • Assumed strain sensitivity methods Other important features • Very accurate and efficient computation of sensitivity fields (gradient calculation) • Innovative OOP for multistage design • Accurate kinematics & contact modeling • State variable-based constitutive modeling • Testing and further developments for single-stage designs with complex 2D geometries • Design of processes for microstructure and damage control • Multi-length scale design models • Sensitivity analysis for texture-sensitive design • Modeling and design of grain growth • Simultaneous thermal & mechanical design Multi-stage forming design • Optimization framework with multiple constraints and competing objectives • Coupling with ideal forming & microstructure evolution paths based initial designs • Reduced order models for design & control Robust materials process design Development of a 3D forming design simulator • Use most features of the 2D simulator • Remeshing & contact algorithms • Industrial design applications Materials Process Design and Control Laboratory

  35. S. Ganapathysubramanian and N. Zabaras, "Computational design of deformation processes for materials with ductile damage", Computer Methods in Applied Mechanics and Engineering, in press N. Zabaras, S. Ganapathysubramanian and Q. Li, "A continuum sensitivity method for the design of multi-stage metal forming processes", International Journal of Mechanical Sciences, submitted for publication REFERENCES CONTACT VIA http://www.mae.cornell.edu/zabaras/ S. Ganapathysubramanian and N. Zabaras, "A continuum sensitivity method for finite thermo-inelastic deformations with applications to the design of hot forming processes", International Journal for Numerical Methods in Engineering, Vol. 55, pp. 1391--1437, 2002 N. Zabaras, S. Ganapathysubramanian and Q. Li, "A continuum sensitivity method for the design of multi-stage metal forming processes", International Journal of Mechanical Sciences, submitted for publication S. Ganapathysubramanian and N. Zabaras, "Computational design of deformation processes for materials with ductile damage", Computer Methods in Applied Mechanics and Engineering, in press Materials Process Design and Control Laboratory

  36. PREFORM DESIGN FOR POROUS MATERIAL Materials Process Design and Control Laboratory

  37. EXTRUSION DIE DESIGN FOR CONTROL OF CHEVRON DEFECTS Materials Process Design and Control Laboratory

  38. PREFORM DESIGN – SINGLE STAGE PROCESS Materials Process Design and Control Laboratory

  39. MULTISTAGE DEFORMATION PROCESS Materials Process Design and Control Laboratory

  40. MULTISTAGE DEFORMATION PROCESS Materials Process Design and Control Laboratory

  41. PREFORMING DIE DESIGN FOR CONTROL OF MICROSTRUCTURE Materials Process Design and Control Laboratory

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