Materials Process Design and Control Laboratory Advanced computational techniques for materials-by-design Materials Process Design and Control Laboratory Sibley School of Mechanical and Aerospace Engineering 188 Frank H. T. Rhodes Hall Cornell University Ithaca, NY URL: Nicholas Zabaras

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Materials Process Design and Control Laboratory OUTLINE  Multiscale modeling and design Control of microstructure-sensitive material properties  Stochastic deformation modeling GPCE, stochastic Galerkin (non-intrusive, Smolyak collocation, etc.) Robust design (non-intrusive stochastic optimization of SPDE systems)  Interrogation of polycrystal microstructures Homogenization techniques  Computing microstructure PDFs using limited information Maximum entropy methods  Conclusions

Materials Process Design and Control Laboratory MULTISCALE MODELING: Process-property-structure triangle Meso-scale representation Forging Properties Performance Evolving microstructure Process Control process parameters Design properties Improve performance

Materials Process Design and Control Laboratory Process optimization using phenomenological material models Initial Iteration Intermediate Iteration Final Iteration Underfill Flash Preform optimization of a Ti-64 cross shaft Maximize volumetric yield No Flash Final

Materials Process Design and Control Laboratory Crystal/lattice reference frame e1e1 ^ e2e2 ^ Sample reference frame e’ 1 ^ e’ 2 ^ crystal e’ 3 ^ e3e3 ^  Crystallographic orientation  Rotation relating sample and crystal axis  Properties governed by orientation  Discrete aggregate of crystals (Anand et al.)  Comparing & quantifying textures  Continuum representation  Orientation distribution function (ODF)  Handling crystal symmetries  Evolution equation for ODF PHYSICAL APPROACH TO PLASTICITY

Materials Process Design and Control Laboratory ORIENTATION DISTRIBUTION FUNCTION (ODF) Conservation principle Texture can be described, quantified & compared Based on the Taylor hypothesis Eulerian & Lagrangian forms Why continuum approach for ODF? EVOLUTION EQUATION FOR THE ODF (Eulerian) v – re-orientation velocity: how fast are the crystals reorienting r – current orientation of the crystal. A – is the ODF, a scalar field; Constitutive sub-problem  Taylor hypothesis: deformation in each crystal of the polycrystal is the macroscopic deformation.  Compute the reorientation velocity from the spin

MATERIALS PROCESS DESIGN & CONTROL LABORATORY Sibley School of Mechanical & Aerospace Engineering 0 ~ Ω + Ω = Ω (r, t; L+ΔL) r – orientation parameter Ω = Ω (r, t; L) ~ I + (L s ) n+1 F n+1 + F n+1 o x + x = x(X, t; β+Δ β) o B n+1 L + L = L (X, t; β+Δ β) o F n+1 x = x(X, t; β) B n+1 L = L (X, t; β) L = velocity gradient B0B0 L s = design velocity gradient The velocity gradient – depends on a macro design parameter Sensitivity of the velocity gradient – driven by perturbation to the macro design parameter A micro-field – depends on a macro design parameter (and) the velocity gradient as Sensitivity of this micro-field driven by the velocity gradient Sensitivity thermal sub-problem Sensitivity constitutivesub-problem Sensitivity kinematic sub-problem Sensitivity contact & frictionsub-problem MULTI-LENGTH SCALE SENSITIVITY ANALYSIS

Design for the strain rate such that a desired material response is achieved Material: 99.98% pure f.c.c Al Materials Process Design and Control Laboratory MATERIAL POINT SIMULATOR: DESIGN FOR SPECIFIC MATERIAL RESPONSE

Materials Process Design and Control Laboratory Multilength scale process design: control of microstructure-sensitive properties Objective: Design the initial preform such that the die cavity is fully filled and the yield strength is uniform over the external surface (shown in Figure below). Material: FCC Cu + Uniform yield strength desired on this surface Fill cavity Multi-objective optimization Increase Volumetric yield Decrease property variation

Materials Process Design and Control Laboratory Multi-scale design – OFHC Copper closed die forging: Iteration Underfill Yield strength (MPa) Optimal yield strength Optimal fill

Materials Process Design and Control Laboratory Extrusion design problem Objective: Design the extrusion die for a fixed reduction such that the deviation in the Young’s Modulus at the exit cross section is minimized Material: FCC Cu Minimize Youngs Modulus variation across cross-section Young's Modulus Distribution Material point sub-problem - ~ In sample coordinates In crystal coordinates Polycrystal average in sample coodinates Young’s modulus (along sample x-axis) ODF Die design for improved properties

Materials Process Design and Control Laboratory Multiscale Extrusion – Control of Youngs Modulus: Iteration Youngs Modulus (GPa) Optimal solution

Materials Process Design and Control Laboratory Two way flow of statistical information 11e21e41e61e9 Engineering Length Scales ( ) Physics Chemistry Materials 0 A Information flow Statistical filter Electronic Nanoscale Microscale Mesoscale Continuum Material information – inherently statistical in nature. Atomic scale – Kinetic theory, Maxwell’s distribution etc. Microstructural features – correlation functions, descriptors etc. Information flow across scales Material heterogeneity STOCHASTIC MATERIALS MODELING: MOTIVATION

Materials Process Design and Control Laboratory State variable based power law model. State variable – Measure of deformation resistance- mesoscale property Material heterogeneity in the state variable assumed to be a second order random process with an exponential covariance kernel. Eigen decomposition of the kernel using KLE. Eigenvectors Initial and mean deformed config. UNCERTAINTY DUE TO MATERIAL HETEROGENEITY: GPCE approach

Materials Process Design and Control Laboratory Load vs Displacement SD Load vs Displacement Dominant effect of material heterogeneity on response statistics UNCERTAINTY DUE TO MATERIAL HETEROGENEITY: GPCE approach

Materials Process Design and Control Laboratory Mean InitialFinal Using 6x6 uniform support space grid Uniform 0.02 Material heterogeneity induced by random distribution of micro-voids modeled using KLE and an exponential kernel. Gurson type model for damage evolution Effect of random voids on material response: Support space approach

Materials Process Design and Control Laboratory Load displacement curves Effect of random voids on material behavior: Suppose space approach

Materials Process Design and Control Laboratory Axisymmetric cylinder upsetting – 60% height reduction (Initial height 1.5 mm) Random initial radius – 10% variation about mean (1 mm)– uniformly distributed Random die workpiece friction U[0.1,0.5] Power law constitutive model Using 10x10 support space grid Random ? Shape Random ? friction PROCESS UNCERTAINTY

Materials Process Design and Control Laboratory Force SD Force PROCESS STATISTICS: Support space approach

Materials Process Design and Control Laboratory ParameterMonte Carlo (20000 LHS samples) Support space 10x10 Mean2.2859e e6 SD m e e6 m41.850e e10 Final force statisticsConvergence study PROCESS STATISTICS Relative Error

Materials Process Design and Control Laboratory Design Objective Probability Constraint Norm Constraint SPDE Constraint Augmented Objective ROBUST DESIGN OF DEFORMATION PROCESSES

Materials Process Design and Control Laboratory CSSM problem decomposed into a set of CSM problems Compute sensitivities of parameters with respect to stochastic design variables by defining perturbations to the PDF of the design variables. Decomposition based on the fact that perturbations to the PDF are local in nature A CONTINUUM STOCHASTIC SENSITIVITY SCHEME (CSSM)

Materials Process Design and Control Laboratory Design Objective – unconstrained case Set of Nel E *n objective functions NISG APPROXIMATION FOR OBJECTIVE FUNCTION

Materials Process Design and Control Laboratory BENCHMARK APPLICATION Case 1 – Deterministic problem Case 2 – 1 random variable (uniformly distributed) – friction – 66% variation about mean (0.3) (10x1 grid) – 1D problem Case 3 – 2 random variables (uniformly distributed) – friction (66%) and desired shape (10% about mean) (10x10 grid) - 2D problem Flat die upsetting of a cylinder

Materials Process Design and Control Laboratory Deterministic problem - optimal solution Deterministic problem 1D problem 2D problem OBJECTIVE FUNCTION

Materials Process Design and Control Laboratory DESIGN PARAMETERS Deterministic problem 2D problem 1D problem Initial guess parameters Mean SD Mean SD

Materials Process Design and Control Laboratory OBJECTIVE FUNCTION

Materials Process Design and Control Laboratory FINAL FREE SURFACE SHAPE CHARACTERISTICS Mean SD

Materials Process Design and Control Laboratory Fine scale heterogeneities Coarse scale heterogeneities Nature of randomness differs significantly between scales, though not fully uncorrelated. Need a multiscale evaluation of the correlation kernels Present method Assume correlation between macro points Decompose using KLE grain size, texture, dislocations macro-cracks, phase distributions MULTISCALE NATURE OF MATERIAL HETEROGENEITIES

Materials Process Design and Control Laboratory Motivation for Maximum Entropy Approach When a specimen is manufactured, the microstructures at a sample point will not be the same always. How do we compute the class of microstructures based on some limited information? Different statistical samples of the manufactured specimen

Materials Process Design and Control Laboratory PDF of microstructures (topology) and its features Grain size ODF (a function of 145 random parameters) Know microstructures at some points Given: Microstructures at some points Obtain: PDF of microstructures

Materials Process Design and Control Laboratory The main idea Extract features of the microstructure Geometrical: grain size Texture: ODFs Phase field simulations Experimental microstructures Compute a PDF of microstructures MAXENT Compute bounds on macro properties

Materials Process Design and Control Laboratory Generating input microstructures: Phase field method Isotropic mobility (L=1) Isotropic mobility (L=1) Discretization : Discretization : problem size : 75x75x75 Order parameters: Q=20 Timesteps = 1000 Timesteps = 1000 First nearest neighbor approx. First nearest neighbor approx.

Materials Process Design and Control Laboratory Input microstructural samples 3D microstructural samples 2D microstructural samples

Materials Process Design and Control Laboratory The main idea Extract features of the microstructure Geometrical: grain size Texture: ODFs Phase field simulations Experimental microstructures Compute a PDF of microstructures MAXENT Compute bounds on macro properties

Materials Process Design and Control Laboratory Microstructural feature: Grain sizes Grain size obtained by using a series of equidistant, parallel lines on a given microstructure at different angles. In 3D, the size of a grain is chosen as the number of voxels (proportional to volume) inside a particular grain. 2D microstructures 3D microstructures Grain size is computed from the volumes of individual grains

Materials Process Design and Control Laboratory The main idea Extract features of the microstructure Geometrical: grain size Texture: ODFs Phase field simulations Experimental microstructures Compute a PDF of microstructures MAXENT Tool for microstructure modeling Compute bounds on macro properties

Materials Process Design and Control Laboratory Subject to Lagrange multiplier approach Lagrange multiplier optimization feature constraints features of image I MAXENT as an optimization problem ( E.T.Jaynes 1957) Partition function Find

Materials Process Design and Control Laboratory Gradient Evaluation Objective function and its gradients: Objective function and its gradients: Infeasible to compute at all points in one conjugate gradient iteration Infeasible to compute at all points in one conjugate gradient iteration Use sampling techniques to sample from the distribution evaluated at the previous point (Gibbs sampler) Use sampling techniques to sample from the distribution evaluated at the previous point (Gibbs sampler)

Materials Process Design and Control Laboratory The main idea Extract features of the microstructure Geometrical: grain size Texture: ODFs Phase field simulations Experimental microstructures Compute a PDF of microstructures MAXENT Compute bounds on macroscopic properties Tool for microstructure modeling

Materials Process Design and Control Laboratory Microstructure modeling : the Voronoi structure Division of n-D space into non-overlapping regions such that the union of all regions fills the entire space. Voronoi cell tessellation : {p 1,p 2,…,p k } : generator points. Division of into subdivisions so that for each point, p i there is an associated convex cell, Cell division of k-dimensional space : Voronoi tessellation of 3d space. Each cell is a microstructural grain.

Materials Process Design and Control Laboratory Stochastic modeling of microstructures Sampling using grain size distribution Sampling using mean grain size Match the PDF of a microstructure with PDF of grain sizes computed from MaxEnt Each microstructure is referred to by its mean value Mean Grain size Probability Weakly consistent scheme Grain size Probability Strongly consistent scheme

Materials Process Design and Control Laboratory Algorithm for generating voronoi centers Generate sample points on a uniform grid from Sobel sequence Mean Grain size Probability Forcing function Objective is to minimize norm (F). Update the voronoi centers based on F Construct a voronoi diagram based on these centers. Let the grain size distribution be y. R corr (y,d)>0.95? No Yes stop Given: grain size distribution Construct: a microstructure which matches the given distribution

Materials Process Design and Control Laboratory Input ODF Reconstructed samples using MaxEnt PDF of textures ODF reconstruction using MAXENT Representation in Frank- Rodrigues space

Materials Process Design and Control Laboratory Input ODF Expected property of reconstructed samples of microstructures Ensemble properties

Materials Process Design and Control Laboratory The main idea Extract features of the microstructure Geometrical: grain size Texture: ODFs Phase field simulations Experimental microstructures Compute a PDF of microstructures MAXENT Compute bounds on macroscopic properties Tool for microstructure modeling

Materials Process Design and Control Laboratory Input constraints: macro grain size observable. First four grain size moments, expected value of the ODF are given as constraints. Output: Entire variability (PDF) of grain size and ODFs in the microstructure is obtained. MAXENT tool 3D random microstructures – evaluation of property statistics Given some known information on lower-order microstructure moments (limited samples), compute the grain size and ODF probability distributions using the MaxEnt technique as well as the PDF of the homogenized material properties.

Materials Process Design and Control Laboratory Grain volume (voxels) Probability mass function Grain volume distribution using phase field simulations pmf reconstructed using MaxEnt K.L.Divergence= nats Grain size distribution computed using MaxEnt Comparison of MaxEnt grain size distribution with the distribution of a phase field microstructure

Materials Process Design and Control Laboratory Grain volume (voxels) Probability mass function R corr = KL= Reconstructing strongly consistent microstructures Computing microstructures using the Sobel sequence method

Materials Process Design and Control Laboratory Grain volume (voxels) Probability mass function R corr = KL=0.05 Reconstructing strongly consistent microstructures (contd..) Computing microstructures using the Sobel sequence method

Materials Process Design and Control Laboratory (First order) homogenization scheme 1.Microstructure is a representation of a material point at a smaller scale 2.Deformation at a macro-scale point can be represented by the motion of the exterior boundary of the microstructure (Hill 1972)Hill

Materials Process Design and Control Laboratory Computing the variability in the strength of a polycrystal induced from microstructural uncertainty Statistical variation of homogenized stress- strain response Aluminium polycrystal with rate-independent strain hardening. Pure tensile test. COMPUTING THE PDF OF HOMOGENIZED PROPERTIES

Materials Process Design and Control Laboratory Many important applications of MaxEnt  Stochastic data-driven simulations  Account for the propagation of the information (uncertainty) not captured in the data  Stochastic multiscale modeling and design Use as a downscaling tool Quantify statistical assumptions across length-scales  how (statistical) information propagates across scales?  stochastic homogenization  Model processes on distributions of microstructures rather than on particular realizations With one simulation capture both heterogeneities and randomness Diffusion, advection, deformation, fracture