NSF Grant Number: DMI PI: Prof. Nicholas Zabaras Institution: Cornell University Title: 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 including economic aspects. Current capabilities: Development of a general purpose continuum sensitivity method for the design of multi-stage industrial deformation processes Deformation process design for porous materials Design of 3D realistic preforms and dies Extension to polycrystal plasticity based constitutive models with evolution of crystallographic texture Materials Process Design and Control Laboratory, Cornell University (Minimal barreling) Initial guess Optimal preform Macro - continuumMicro-scale Polycrystal plasticity Future research Multiscale metal forming design with reduced order modeling of microstructure Design of formed products with desired directional microstructure dependent properties Probabilistic design using spectral methods with specification of robustness limits in the design variables Design for desired yield stress at a material point Broader Impact: A virtual laboratory for realistic materials process design is developed that will lead to reduction in lead time for process development, trimming the cost of an extensive experimental trial-and-error process development effort, developing processes for tailored material properties and increasing volume/time yield. The design simulator under development provides a robust and handy industrial tool to carry out real-time metal forming design. Normalized yield stress

Deformation Process Design for Tailored Material Properties Difficult Insertion of new materials and processes into production Numerical Simulation Trial-and-error and with no design information Conventional Design Tools Material Modeling Incremental improvements in specific areas Development of designer knowledge base Time consuming and costly Computational Material Process Design Simulator Sensitivity Information points to most influential parameters so as to optimally design the process Virtual Material Process Laboratory Reliability Based Design for material/tool variability & uncertainties in mathematical & physical models Data Mining of Designer Knowledge for rapid solution to complex problems and to further drive use of knowledge Materials Process Design control of microstructure using various length and time scale computational tools Accelerated Insertion of new materials and processes Innovative Processes for traditional materials

Reliability based design Sensitivity information Designer knowledge Materials process design Virtual Materials Process Laboratory Selection of a virtual direct process model Selection of the sequence of processes (stages) and initial process parameter designs Selection of the design variables like die and preform parameterization Continuum multistage process sensitivity analysis consistent with the direct process model Optimization algorithms Interactive optimization environment Virtual Deformation Process Design Simulator

Description of parameter sensitivities: Take F R = I with the design velocity gradient L 0 = 0. Main features:   Gateaux differential referred to the fixed configuration Y   Rigorous definition of sensitivity   Driving force for the sensitivity problem is L R =F R F R -1 o Shape and Parameter Continuum Sensitivity Analysis

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

3D Continuum sensitivity contact sub-problem   Continuum approach for computing traction sensitivities – In line with the continuum sensitivity approach   Accurate computation of traction derivatives using augmented Lagrangian regularization.   Traction derivatives computed without augmentation using oversize penalties Regularization assumptions No slip/stick transition between direct/perturbed problem No admissible/inadmissible region transition between direct/perturbed 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 Parametersensitivityanalysis υ r υ r x = x ( X, t, β s ) B0B0 B’0B’0 BRBR X + X X o x = x ( X + X, t, β s + Δ β s ) ~ o X = X (Y ; β s + Δ β s ) ~ Y X = X (Y ; β s ) ~ ~ x + x B΄B΄ o B Die x Shapesensitivityanalysis τ 2 + τ 2 o τ2τ2 First reported 3D regularized contact sensitivity algorithm

Equivalent stress sensitivity Perturbation of the preform shape parameters CSMFDM Equivalent stress sensitivity Temperature sensitivity Open die forging of a cylindrical billet Validation of 3D thermo-mechanical shape sensitivity analysis

Minimize the flash and the deviation between the die and the workpiece through a preforming shape design Unfilled cavity Flash The same material in a conventional design The same material with an optimum design No flash Fully filled cavity Process design for the manufacture of an engine disk – two possible approaches 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. Initial design Unfilled cavity Optimal design Iteration Number Objective Function (x1.0E-05) Al 1100-O Initially at 673K; Preform and die parameterizati- -on

Objective: Minimize the flash and the deviation between the die and the workpiece for a preforming shape and volume design Material: 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 Product using optimal preform Distribution of the void fraction in product Variation of preform shape with optimization iterations Iteration number Non-dimensional objective Preform design for porous material

3D Preform design to fill die cavity for forging a circular disk 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 for a fixed stroke Material: Al 1100-O at 673 K Iterations Normalized objective

(1) Continuum framework (3) Desired effectiveness in terms of state variables (2) State variable evolution laws Initial configuration B o B F e F p F  F Deformed configuration Intermediate thermal configuration Stress free (relaxed) configuration Phenomenology Polycrystal plasticity Initial configuration B o B F * F p F Deformed configuration Stress free (relaxed) configuration (1) Single crystal plasticity (3) Ability to tune microstructure for desired properties (2) State evolves for each crystal The effectiveness of design for desired product properties is limited by the ability of phenomenological state-variables to capture the dynamics of the underlying microstructural mechanisms Polycrystal plasticity provides us with the ability to capture material properties in terms of the crystal properties. This approach is essential for realistic design leading to desired microstructure- sensitive properties From phenomenology to polycrystal plasticity Need for polycrystalline analysis Challenges in polycrystalline analysis Infinite microstructural degrees of freedom limits the scope of design Solution: Develop microstructure model reduction n0n0 s0s0 s0s0 n0n0 n s

R value Normalized objective function Desired value: α = {1.2,0,0,0,0} T, Initial guess: α = {0.5,0,0,0,0} T Converged reduced order solution: α = {1.19,0.05,0.001,0,0} T Design problem: Ξ = {F,G,H,N} T (from Hill’s anisotropic yield criterion) Design for microstructure sensitive property – R value Iterations

Desired value: α = {1,0,0,0,0} T Initial guess: α = {0.5,0,0,0,0} T Converged solution: α = {0.987,0.011,0,0,0} T Angle from rolling direction Initial Intermediate Optimal Desired h Normalized hysteresis loss Normalized objective function Design Problem Hysteresis loss Crystal direction. Easy direction of magnetization – zero power loss External magnetization direction Materials by design Design of microstructure and deformation for minimal hysteresis loss Iterations

Process design for tailored material properties The guess die shape resulted in large grains along the exit cross- section of the extruded product. Using the optimal die shape, presence of such large grains is eliminated. Optimal solution Guess solution Design the extrusion die for a fixed reduction so that the variation in grain size (at the exit) is minimized Material: 0.2%C steel, friction coefficient of 0.01 Phenomenological approach Polycrystal approach Design for the strain rate such that a desired material response is achieved Material: 99.98% pure f.c.c Al

Hi – peformance computing USER INTERFACE Robust product specifications Control and reduced order modeling Stochastic optimization, Spectral/Bayesian framework Design database, simulations and experiments User update Output design Input Modifications in objectives Starting with robust product specifications, you compute not only the full statistics of the design variables but also the acceptable variability in the system parameters Directly incorporate uncertainties in the system into the design analysis Experimentation and testing driven by product design specifications Improve overall design performance Robust design simulator

Suppose we had a collection of data (from experiments or simulations) for the ODF: such that it is optimal for the ODF represented as Is it possible to identify a basis POD technique – Proper orthogonal decomposition Solve the optimization problem Method of snapshots where Applications of microstructure model reduction GE 90 Boeing 747 Modern aircraft engine design and materials selection is an extremely challenging area. Desired directional properties include:   strength at high temperatures, R- values   elastic, fatigue, fracture properties   thermal expansion, corrosion resistance, machinability properties Developing advanced materials for gas turbine engines is expensive – Is it possible to control material properties and product performance through deformation processes?

Further developments for multi-stage designs – 3D geometries [Ref. 1-5]   Simultaneous thermal & mechanical design   Sensitivity analysis for multi-body deformations Design across length scales [Ref. 3-6]   Coupled length scale analysis with control of grain size, phase distribution and orientation   Microstructure development through stochastic processes   Generate universal snap shots for reduced order modeling   Develop algorithms for real-time microstructural reduced order model mining - Ref. [6] Robust design algorithms [Ref. 7-8]   Can we design a process with desired robustness limits in the objective?   Work includes a spectral method for the design of thermal systems – Ref. [7]   Develop a spectral stochastic FE approach towards robust deformation process design   Introduce Bayesian material parameter estimation – Ref. [8] Couple materials process design with required materials testing selection   Develop an integrated approach to materials process design and materials testing selection: Materials testing driven by design objectives!   With given robustness limits on the desired product attributes, a virtual design simulator can point to the required materials testing that can obtain material properties with the needed level of accuracy. Forthcoming research efforts

ACKNOWLEDGEMENTS The work presented here was funded by NSF grant DMI with additional support from AFOSR and AFRL. [6] S. Ganapathysubramanian and N. Zabaras, "Modeling the thermoelastic-viscoplastic response of polycrystals using a continuum representation over the orientation space", International Journal of Plasticity, submitted for publication. References [1] S. Ganapathysubramanian and N. Zabaras, "Computational design of deformation processes for materials with ductile damage", Computer Methods in Applied Mechanics and Engineering, Vol. 192, pp , [2] 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, in press.. [3] 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, submitted for publication. [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 , [5] V. Sundararaghavan and N. Zabaras, "A dynamic material library for the representation of single phase polyhedral microstructures", Acta Materialia, submitted for publication. [7] Velamur Asokan Badri Narayanan and N. Zabaras, "Stochastic inverse heat conduction using a spectral approach", International Journal for Numerical Methods in Engineering, in press. [8] Jingbo Wang and N. Zabaras, "A Bayesian inference approach to the stochastic inverse heat conduction problem", International Journal of Heat and Mass Transfer, accepted for publication.