Nicholas Zabaras (PI), Swagato Acharjee, Veera Sundararaghavan NSF Grant Number: DMI Development of a robust computational design simulator for industrial deformation processes Research Objectives: To develop a mathematically and computationally rigorous gradient-based optimization methodology for virtual materials process design that is based on quantified product quality and accounts for process targets and constraints. Equilibrium equation Design derivative of equilibrium equation Material constitutive laws Design derivative of the material constitutive laws Design derivative of assumed kinematics Assumed kinematics Incremental sensitivity constitutive sub-problem Time & space discretized modified weak form Time & space discretized weak form Sensitivity weak form Contact & friction constraints Regularized design derivative of contact & frictional constraints Incremental sensitivity contact sub-problem Conservation of energy Design derivative of energy equation Incremental thermal sensitivity sub-problem Schematic of the continuum sensitivity method (CSM) Continuum problem Design differentiate Discretize 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 Initial 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 Optimal die DIE DESIGN FOR UNIFORM MATERIAL STATE AT EXIT Additional support from AFOSR and ARO. Computing facilities provided by Cornell Theory Center [6] S. Acharjee and N. Zabaras, "Uncertainty propagation in finite deformations -- A spectral stochastic Lagrangian approach", Computer Methods in Applied Mechanics and Engineering, submitted for publication. [1] S. Ganapathysubramanian and N. Zabaras, "Deformation process design for control of microstructure in the presence of dynamic recrystallization and grain growth mechanisms", International Journal for Solids and Structures, Vol. 41/7, pp , 2004 [2] 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, in press [3] V. Sundararaghavan and N. Zabaras, "A dynamic material library for the representation of single phase polyhedral microstructures", Acta Materialia, Vol. 52/14, pp , 2004 [4] S. Acharjee and N. Zabaras "A proper orthogonal decomposition approach to microstructure model reduction in Rodrigues space with applications to the control of material properties", Acta Materialia, Vol. 51/18, pp , 2003 [5] S. Ganapathysubramanian and N. Zabaras, "Design across length scales: A reduced-order model of polycrystal plasticity for the control of microstructure-sensitive material properties", Computer Methods in Applied Mechanics and Engineering, Vol. 193 (45-47), pp , 2004 [7] V. Sundararaghavan and N. Zabaras, "Classification of three-dimensional microstructures using support vector machines", Computational Materials Science, Vol. 32, pp , [8] Velamur Asokan Badri Narayanan and N. Zabaras, "Stochastic inverse heat conduction using a spectral approach", International Journal for Numerical Methods in Engineering, Vol. 60/9, pp , 2004 [9] S. Ganapathysubramanian and N. Zabaras, "Modeling the thermoelastic- viscoplastic response of polycrystals using a continuum representation over the orientation space", International Journal of Plasticity, Vol. 21/1 pp , 2005 [10] V. Sundararaghavan and N. Zabaras, "On the synergy between classification of textures and deformation process sequence selection", Acta Materialia, in press Materials Process Design and Control Laboratory 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 Current capabilities - Thermomechanical deformation process design in the presence of ductile damage -Thermomechanical deformation process design in the presence of dynamic recrystallization -Multi-stage deformation process design -Implementation of 3D continuum sensitivity analysis algorithm. Mathematically rigorous computation of gradients - good convergence observed within few optimization iterations 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 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) Micro problem driven by the velocity gradient F Macro problem driven by the macro-design variable β B n+1 Ω = Ω (r, t; F) ~ Polycrystal plasticity x = x(X, t; β ) F = F (X, t; β ) F = deformation gradient F n+1 B0B0 X Material: % pure polycrystalline f.c.c Aluminum Process: Upset forging Forging rate = 0.01 /s Total deformation = 15% MULTI-LENGTH SCALE FORGING Ongoing efforts Extension to complex multistage forging and extrusion processes -Incorporate remeshing using CUBIT (Sandia) and interface with suitable data transfer schemes -Computational issues – Parallel implementation using PETSC (ANL) -Extension to constitutive modeling and process design of Titanium alloys. -Development of a multiscale version of the design simulator employing a polycrystal plasticity based constitutive model involving a novel two length scale sensitivity analysis for process and materials design Synergistic research activities -Design and analysis of deformation processes in the presence of uncertainty -Statistical learning techniques for process sequence selection -Microstructure classification and reconstruction -Model reduction techniques in multiscale modeling Spectral stochastic simulation: A tension test modeled using a GPCE-based approach. The internal state variable is assumed to be uncertain and derived from an assumed covariance kernel: (a) The initial and mean deformed configuration of the tension specimen (b) The mean load versus displacement curve and a set of embedded sample realizations (c) The standard deviation of the response Recent publications [110] pole figure Feature DATABASE OF ODFs Uniaxial (z-axis) Compression Texture z-axis fiber (BB’) Multi-stage reduced order control of Young’s Modulus Stage: 2 Tension (   = ) Stage: 1 Shear (   = ) Classification Adaptive reduced basis selection Process – 2 Plane strain compression a = Process – 1 Tension a = Initial Conditions: Stage 1 DATABASE Higher dimensional feature Space  x  Design parameter ( 