Materials Process Design and Control Laboratory Sibley School of Mechanical and Aerospace Engineering 169 Frank H. T. Rhodes Hall Cornell University Ithaca, NY URL: NICHOLAS ZABARAS A MULTISCALE APPROACH TO MATERIALS USING STOCHASTIC AND COMPUTATIONAL STATISTICS TECHNIQUES

Materials Process Design and Control Laboratory NEED FOR MULTISCALE ANALYSIS Fingering in porous media Water injector site Porous bed-rock Water displaces the oil layer to the receiving site Water accelerates more in areas with high permeability Fingering reduces quality of the oil received (polluted with water) Oil receiver site Permeability of bed rock is inherently stochastic Statistics like mean permeability, correlation structure are usually constant for a given rock type Stationary probability models can be used Direct simulation of the effect of uncertainty in permeability on the amount of oil received requires enormous computational power Bed rock length scale – typically of order of kms Length scale for permeability variation – typically of order of cms Requirement – blocks for a single dimension (10 12 blocks overall)

Materials Process Design and Control Laboratory NEED FOR MULTISCALE ANALYSIS Transport phenomena in material processes like solidification Engineering component Microstructural features Formation of dendrites, micro- scale flow structure, heat transfer patterns are highly sensitive to perturbations Microstructure is dynamic and evolves with the materials process Uncertainties at the micro- scale are loosely correlated, however macro-scale features like species concentration, temperature, stresses are highly correlated Uncertainty analysis at micro-scale requires considerable computational effort Macro-properties dependant on the dendrite patterns and uncertainty propagation at the micro- scales Uncertainty interactions are no longer satisfy stationarity assumptions – newer probability models based on image analysis and experimentation needed Length scale ~ meters Length scale ~ meters

Materials Process Design and Control Laboratory STOCHASTIC VARIATIONAL MULTISCALE Physical model Statistical variations in properties are significant Discontinuities, loosely correlated structures in properties Large scale system Micro scale features Statistical variations are relatively negligible Discontinuities get smoothed out Process interactions, properties become correlated Uncertainties in boundary and initial conditions Discrete probability distributions to model properties Image analysis to develop the correlation structure Bayesian data analysis interface Green’s functions, RFB type models Explicit subgrid scale model - FEM Experimental, Monte Carlo/ MD models Spectral stochastic/ support-space representation of uncertainty Discretization method like FEM, Spectral, FDM Subgrid scale models Large scale simulation Averaging out the higher statistical features of subgrid scale solutions using Karhunen-loeve/ wavelet filtering Residual Statistical features Large scale solutions obtained from the explicit discretization approach

Materials Process Design and Control Laboratory MULTISCALE TRANSPORT SYSTEMS Flow past an aerofoilAtmospheric flow in Jupiter Solidification process Modeling of dendrites at small scale, fluid flow and transport at large scale Large scale turbulent structures, small scale dissipative eddies, surface irregularities Astro-physical flows, effects of gravitational and magnetic fields Presence of a variety of spatial and time scales - commonality Varied applications – Engineering, Geophysical, Materials Boundary conditions, material properties, small scale behavior inherently are uncertain

Materials Process Design and Control Laboratory IDEA BEHIND VARIATIONAL MULTISCALE - VMS Solidification process Physical model Micro-constitutive laws from experiments, theoretical predictions Subgrid model Resolved model Green’s function Residual free bubbles MsFEM “Hou et al.” TLFEM “Hughes et al.” FEM FDM Spectral Large scale behavior – explicit resolution Small scale behavior – statistical resolution Large scale Residual Subgrid scale solution Where does uncertainty fit in ?

Materials Process Design and Control Laboratory WHY STOCHASTIC MODELING IN VMS ? Model uncertainty Material uncertaintyComputational uncertainty Imprecise knowledge of governing physics Models used from experiments Uncertain boundary conditions Inherent initial perturbations Small scale interactions Surroundings uncertainty Solidification microscale features Material properties fluctuate – only a statistical description possible Uncertainty in codes Machine precision errors Not accounted for in analysis here

Materials Process Design and Control Laboratory SOME PROBABILITY THEORY Probability space – A triplet -- Collection of all basic outcomes of the experiment -- Permutation of the basic outcomes -- Probability associated with the permutations Sample space Real interval Random variable – a function Stochastic process – a random function at each space and time point Notations:

Materials Process Design and Control Laboratory SPECTRAL STOCHASTIC EXPANSIONS Covariance kernel required – known only for inputs Best possible representation in mean-square sense Series representation of stochastic processes with finite second moments - Mean of the stochastic process - Coefficient dependant on the eigen-pairs of the covariance kernel of the stochastic process - Orthogonal random variables Karhunen-Loeve expansion Generalized polynomial chaos expansion - Coefficients dependant on chaos-polynomials chosen - Chaos polynomials chosen from Askey-series (Legendre – uniform, Jacobi – beta)

Materials Process Design and Control Laboratory SUPPORT-SPACE/STOCHASTIC GALERKIN - Joint probability density function of the inputs - The input support-space denotes the regions where input joint PDF is strictly positive Triangulation of the support- space Any function can be represented as a piecewise polynomial on the triangulated support-space - Function to be approximated - Piecewise polynomial approximation over support-space L 2 convergence – (mean-square) h = mesh diameter for the support-space discretization q = Order of interpolation Error in approximation is penalized severely in high input joint PDF regions. We use importance based refinement of grid to avoid this

Materials Process Design and Control Laboratory BOUSSINESQ NATURAL CONVECTION Temperature gradients are small Constant fluid properties except in the force term viscous dissipation negligible Momentum equation boundary conditions Energy equation boundary conditions

Materials Process Design and Control Laboratory DEFINITION OF FUNCTION SPACES - Spatial domain - Time interval of simulation [0,t max ] Function spaces for deterministic quantities Function spaces for stochastic quantities

Materials Process Design and Control Laboratory DERIVED FUNCTION SPACES Velocity function spaces Uncertainty is incorporated in the function space definition Solution velocity, temperature and pressure are in general multiscale quantities (as Rayleigh number increases) the computational grid capture less and less information Pressure function spaces Temperature function spaces

Materials Process Design and Control Laboratory WEAK FORMULATION – BOUSSINESQ EQNS Find such that for all, the following holds Energy equation – weak form Momentum and continuity equations – weak form

Materials Process Design and Control Laboratory VARIATIONAL MULTISCALE DECOMPOSITION Bar denotes large scale/resolved quantity Prime denotes subgrid scale/ unresolved quantity Induced multiscale decomposition for function spaces Interpretation Large scale function spaces correspond to finite element spaces – piecewise polynomial and hence are finite dimensional Small scale function spaces are infinite dimensional

Materials Process Design and Control Laboratory SCALE DESOMPOSED WEAK FORM - ENERGY Find and such that for all and, the following holds Small scale strong form of equations Time discretization rule

Materials Process Design and Control Laboratory ELEMENT FOURIER TRANSFORM Spatial domain Subgrid scale solution denotes unresolved part of the solution, hence dominated by large wave number modes!! Spatial derivative approximation Other techniques to solve for an approximate subgrid solution include: - Residual-free bubbles, Green’s function approach - Two-level finite element method – explicit evaluation - Multiscale FEM – Incorporates subgrid features in large scale weighting function

Materials Process Design and Control Laboratory ALGEBRAIC SUBGRID SCALE MODEL Time discretization Element Fourier transform Parseval’s theorem Mean value theorem

Materials Process Design and Control Laboratory STABILIZED FINITE ELEMENT EQUATIONS Strong regularity conditions Stabilized weak formulation where Time integration has a role to play in the stabilization (Codina et al.) Stochastic intrinsic time scale (subgrid scale solution has a stochastic model)

Materials Process Design and Control Laboratory CONSIDERATIONS FOR MOMENTUM EQUATION Picard’s linearization Fairly accurate for laminar up to transition (moderate Reynolds number flows) For high Reynolds number flows, the term assumes importance since small scales act as momentum dissipaters Small scale strong form of equations

Materials Process Design and Control Laboratory SUBGRID VELOICTY AND PRESSURE Element Fourier transform Simultaneous solve Parseval’s theorem Mean-value theorem

Materials Process Design and Control Laboratory STABILIZED FINITE ELEMENT EQUATIONS Strong regularity conditions Stabilized weak formulation where Momentum equation Continuity equation

Materials Process Design and Control Laboratory IMPLEMENTATION ISSUES - GPCE Spatial domain Generic function Random coefficient Galerkin shape function GPCE expansion for random coefficients Random coefficient Askey polynomial Each node has P+1 degrees of freedom for each scalar stochastic process Interpolation is accomplished by tensor-product basis functions (P+1) times larger than deterministic problems Assume the inputs have been represented in Karhunen-Loeve expansion such that the input uncertainty is summarized by few random variables

Materials Process Design and Control Laboratory IMPLEMENTATION ISSUES – SUPPORT SPACE Spatial domain A stochastic process can be interpreted as a random variable at each spatial point Two-level grid approach Spatial grid Support-space grid Mesh dense in regions of high input joint PDF Element There is finite element interpolation at both spatial and random levels Each spatial location handles an underlying support-space grid Highly OOP structure

Materials Process Design and Control Laboratory NUMERICAL EXAMPLES Flow past a circular cylinder with uncertain inlet velocity – Transient behavior RB convection in square cavity with adiabatic body at the center – uncertainty in the hot wall temperature (simulation away from critical points) - Transient behavior - Simulation using GPCE, validation using deterministic simulation RB convection in square cavity – uncertainty in Rayleigh number (simulation about a critical point) - Failure of the GPCE approach - Analysis support-space method - Comparison of prediction by support-space method with deterministic simulations In the last example, temperature contours do not convey useful information and hence are ignored

Materials Process Design and Control Laboratory FLOW PAST A CIRCULAR CYLINDER Computational details – 2000 bilinear elements for spatial grid, third order Legendre chaos expansion for velocity and pressure, preconditioned parallel GMRES solver Time of simulation – 180 non- dimensional units Inlet velocity – Uniform random variable between 0.9 and 1.1 Kinematic viscosity 0.01 Time stepping – 0.03 non- dimensional units Inlet Traction free outlet No-slip Investigations Onset of vortex shedding Shedding near wake regions, flow statistics

Materials Process Design and Control Laboratory ONSET OF VORTEX SHEDDING Mean pressure at t = 79.2 Vortex shedding is just initiated Not in the periodic shedding mode First order term in Legendre chaos expansion of pressure at t = 79.2 Vortex shedding is predominant Periodic shedding behavior noticed

Materials Process Design and Control Laboratory FULLY DEVELOPED VORTEX SHEDDING Mean pressure contours First order term in LCE of pressure contours Second order term in LCE of pressure contours

Materials Process Design and Control Laboratory VORTEX SHEDDING - CONTD The FFT of the mean velocity shows a broad spectrum with peak at frequency The spectrum is broad in comparison to deterministic results wherein a sharp shedding frequency is obtained Mean velocity has superimposed frequencies Mean velocity has comparatively lower magnitude than the deterministic velocity (Y- velocities compared at near wake region)

Materials Process Design and Control Laboratory RB CONVECTION - CENTRAL ADIABATIC BODY Computational details – 2048 bilinear elements for spatial grid, third order Legendre chaos expansion for velocity, pressure and temperature, preconditioned parallel GMRES solver Time of simulation – 1.5 non-dimensional units Rayleigh number Prandtl number – 0.7 Time stepping – non-dimensional units Transient behavior of temperature statistics ( Flow results in paper ) Cold wall Hot wall Insulated Adiabatic body

Materials Process Design and Control Laboratory TRANSIENT BEHAVIOR – TEMPERATURE Mean temperature contours Steady conduction like state not reached Second order term in the Legendre chaos expansion of temperature First order term in the Legendre chaos expansion of temperature

Materials Process Design and Control Laboratory CAPTURING UNSTABLE EQUILIBRIUM Computational details – 1600 bilinear elements for spatial grid Time of simulation – 1.5 non- dimensional units Rayleigh number – uniformly distributed random variable between 1530 and 1870 (10% fluctuation about 1700) Prandtl number – 6.95 Time stepping – non- dimensional units Support-space grid – One- dimensional with ten linear elements Simulation about the critical Rayleigh number – conduction below, convection above Both GPCE and support-space methods are used separately for addressing the problem Failure of Generalized polynomial chaos approach Support-space method – evaluation and results against a deterministic simulation Cold wall Hot wall Insulated

Materials Process Design and Control Laboratory FAILURE OF THE GPCE X-vel Y-vel Mean X- and Y- velocities determined by GPCE yields extremely low values !! (Gibbs effect) X- and Y- velocities obtained from a deterministic simulation with Ra = 1870 (the upper limit)

Materials Process Design and Control Laboratory PREDICTION BY SUPPORT-SPACE METHOD X-vel Y-vel Mean X- and Y- velocities determined by support-space method at a realization Ra=1870 X- and Y- velocities obtained from a deterministic simulation with Ra = 1870 (the upper limit)

Bayesian computation and Multiscale simulation Materials Process Design and Control Laboratory base-level: multiscale prior distribution modeling explore time and spatial length scales of random unknown fields model temporal and spatial dependence (GRF, MRF, DAMRF etc.) mid-level: multiscale direct simulation (likelihood computation) build meta models reduce computation cost top level: uncertainty quantification of multiscale simulation Y = F(θ, ω s ) + ω m A two-way interaction stochastic upscaling --- using data (simulation results on fine scale) to estimate the distribution of upscaling coefficient on simulation scale Estimation of random equivalent parameter --- estimating the unknown equivalent parameter to account for dispersion effect Calibration to deterministic upscaling roles of multiscale simulation in Bayesian computation framework Stochastic upscaling via Bayesian computation

Uncertainty quantification in multiscale simulation via Bayesian Materials Process Design and Control Laboratory fine grid (DNS model) two scale model three scale model *up-scaling *subgrid modeling *averaging equation *up-scaling *subgrid modeling *averaging equation resolve all length scales accurate high computation cost solve system eqns on macroscales model energy dissipation on medium and micro length scales inaccurate low computation cost solve system eqns on macro and medium length scales model energy dissipation on microscale accuracy and computation cost in between DNS and two scale model A key step in solving inverse problems --- how to quantify errors brought by multiscale simulation Y = F(θ, ω s ) DNS data multiscale solver solution errors p( ω s |Y )  p(Y| ω s )p( ω s ) e.g. Y = F(θ) + ω s assuming deviation is large than error ),( ~ N  ωsωs Cov

Stochastic upscaling via Bayesian Materials Process Design and Control Laboratory fine scale simulation coarse grid simulation with upscaling coarse grid simulation without upscaling measurement (fine) scale grid numerical simulation scale grid compute distribution of equivalent parameters) what is stochastic upscaling why stochastic upscaling unsolved heterogeneity unsolved physics macro image of permeability micro-structure of the porous medium viscous fingering phenomenon

Stochastic upscaling via Bayesian Materials Process Design and Control Laboratory Why Bayesian approach Existing methods: upscaling through local average, homogenization or renormalization assumes the knowledge of thermal properties are accurate on the fine scale (not true!) An inverse approach can use both simulation results on fine scale and the real temperature data to obtain the equivalent thermal dispersion coefficient Bayesian can quantify the measurement error and numerical error Bayesian provides distribution of the effective parameter Bayesian uses the information on fine scale (the distribution from renormalization group method as prior) Bayesian provides self-calibration

Stochastic upscaling via Bayesian (cont.) Materials Process Design and Control Laboratory Bayesian upscaling framework microscale simulation and modeling statistical modeling of random porosity, permeability, diffusivity … on microscale simulation on microscale parallel computation domain decomposition reduced order modeling … macroscale simulation and modeling probability distribution upscaling and renormalization simulation on coarse grid data collection on microscale re-estimate distribution of upscaling coefficients on coarse grid )()|(),|()|,(    ppYpYp  hierarchical Bayesian model equivalent parameters to be re-estimated data from microscale simulation and experiment likelihood from coarse grid simulation prior from probability upscaling of fine grid distribution

Stochastic upscaling via Bayesian (cont.) Materials Process Design and Control Laboratory A example problem: K q top q bottom T in T out Assumptions: - isotropic and heterogeneous porous material - no radiation, no viscous dissipation - work done by pressure is neglected - local thermal equilibrium - surface porosity = porosity - no heat generation - forced convection velocity obeys Darcy’s Law and known a priori on the coarse grid P K v    fine grid equation: )()( )( TkTvC t T C mf m       )(  kk m  is a function of porosity and varies significant over length scales coarse grid equation: )()()( * TkTvC t T C mfm      effective thermal dispersion coefficient

Materials Process Design and Control Laboratory Uncertainty analysis in finite deformation problems using SSFEM Some finite deformation problems and sources of uncertainty Metal forming – process parameters, initial shape, friction coefficient Composites – fiber orientation, fiber spacing, constitutive model Biomechanics – material properties, constitutive model, fibers in tissues Key features Spectral decomposition of the current configuration leading to a stochastic deformation gradient Toolbox for elementary operations of spectral expansions of stochastic quantities B n+1 (θ) F(θ) B0B0 Effect of uncertain fiber orientation on an aircraft nozzle flap

Materials Process Design and Control Laboratory MICROSTRUCTURE RECONSTRUCTION & CLASSIFICATION WITH APPLICATIONS IN 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: / Veeraraghavan Sundararaghavan and Nicholas Zabaras

Materials Process Design and Control Laboratory MATERIALS DESIGN FRAMEWORK Machine learning schemes Microstructure Information library Accelerated Insertion of new materials Optimization of existing materials Tailored application specific material properties Virtual process simulations to evaluate alternate designs Computational process design simulator Virtual materials design framework

Materials Process Design and Control Laboratory DESIGNING MATERIALS WITH TAILORED PROPERTIES Micro problem driven by the velocity gradient L Macro problem driven by the macro-design variable β B n+1 Ω = Ω (r, t; L) ~ Polycrystal plasticity x = x(X, t; β ) L = L (X, t; β ) L = velocity gradient F n+1 B0B0 Reduced Order Modeling Data mining techniques Database Multi-scale Computation Design variables (β) are macro design variables Processing sequence/parameters Design objectives are micro-scale averaged material/process properties

Materials Process Design and Control Laboratory DATABASE FOR POLYCRYSTAL MATERIALS Meso-scale database for polycrystalline materials Machine Learning Database Feature Extraction Data Organization Reduced order basis generation Youngs Modulus RD TD Multi-scale microstructure evolution models Process design for desired properties RD R-value TD ODF Pole Figures

Materials Process Design and Control Laboratory MOTIVATION 1.Creation of 3D microstructure models for property analysis from 2D images 2. 3D imaging requires time and effort. Need to address real–time methodologies for generating 3D realizations. 3. Make intelligent use of available information from computational models and experiments. vision Database Pattern recognition Microstructure Analysis 2D Imaging techniques

Materials Process Design and Control Laboratory PATTERN RECOGNITION (PR) STEPS DATABASE CREATION FEATURE EXTRACTION TRAINING PREDICTION Datasets: microstructures from experiments or physical models Extraction of statistical features from the database Creation of a microstructure class hierarchy: Classification methods Prediction of 3D reconstruction, process paths, etc. PATTERN RECOGNITION : A DATA-DRIVEN OPTIMIZATION TOOL Feature matching for reconstruction of 3D microstructures Microstructure representation Texture(ODF) classification for process path selection Real-time

Materials Process Design and Control Laboratory 3D MICROSTRUCTURE RECOGNITION: A TWO-CLASS PROBLEM Training FeaturesClass(y) Feature Vector (x) – single feature type (Grain size feature) Match lower order features using PR New Feature (From a 2D image) – To which 3D class does this belong? Heyn intercept histogram of a 2D cross-section Feature Extraction

Materials Process Design and Control Laboratory MULTIPLE CLASSES Class-A Class-B Class-C A C B A B C Given a new planar microstructure with its ‘s’ features given by find the class of 3D microstructure (y ) to which it is most likely to belong. p = 3 One Against One Method: Step 1: Pair-wise classification, for a p class problem Step 2: Given a data point, select class with maximum votes out of

Materials Process Design and Control Laboratory TWO PHASE MICROSTRUCTURE: CLASS HIERARCHY Class - 1 3D Microstructures Feature vector : Three point probability function 3D Microstructures Class - 2 Feature: Autocorrelation function LEVEL - 1 LEVEL - 2 r  m 

Materials Process Design and Control Laboratory STATISTICAL CORRELATION MEASURES MC Sampling: Computing the three point probability function of a 3D microstructure(40x40x40 mic) S 3 (r,s,t), r = s = t = 2, 5000 initial points, 4 samples at each initial point. Rotationally invariant probability functions (S i N ) can be interpreted as the probability of finding the N vertices of a polyhedron separated by relative distances x 1, x 2,..,x N in phase i when tossed, without regard to orientation, in the microstructure.

Materials Process Design and Control Laboratory 3D RECONSTRUCTION Ag-W composite (Umekawa 1969) A reconstructed 3D microstructure 3 point probability function Autocorrelation function

Materials Process Design and Control Laboratory ELASTIC PROPERTIES: YOUNGS MODULUS 3D image derived through pattern recognition Experimental image

Materials Process Design and Control Laboratory MICROSTRUCTURE REPRESENTATION USING SVM & PCA COMMON-BASIS FOR MICROSTRUCTURE REPRESENTATION Does not decay to zero A DYNAMIC LIBRARY APPROACH Classify microstructures based on lower order descriptors. Create a common basis for representing images in each class at the last level in the class hierarchy. Represent 3D microstructures as coefficients over a reduced basis in the base classes. Dynamically update the basis and the representation for new microstructures

Materials Process Design and Control Laboratory PCA MICROSTRUCTURE RECONSTRUCTION Pixel value round-off Basis Components X 5.89 X Project onto basis Reconstruct using two basis components Representation using just 2 coefficients (5.89,14.86)

Materials Process Design and Control Laboratory ORIENTATION FIBERS: LOWER ORDER FEATURES OF AN ODF Points (r) of a (h,y) fiber in the fundamental region angle Crystal Axis = h Sample Axis = y Rotation (R) required to align h with y (invariant to, ) Fibers: h{1,2,3}, y || [1,0,1] {1,2,3} Pole Figure Point y (1,0,1) Integration is performed over all fibers corresponding to crystal direction h and sample direction y For a particular (h), the pole figure takes values P(h,y) at locations y on a unit sphere.

Materials Process Design and Control Laboratory LIBRARY FOR TEXTURES [100] pole figure [110] pole figure Parameter Feature Vector DATABASE OF ODFs

Materials Process Design and Control Laboratory SUPERVISED CLASSIFICATION USING SUPPORT VECTOR MACHINES Given ODF/texture Tension (T) Stage 1 LEVEL – 2 CLASSIFICATION Plane strain compression T+P LEVEL – I CLASSIFICATION Tension identified Stage 2 Stage 3 Multi-stage classification with each class affiliated with a unique process Identifies a unique processing sequence: Fails to capture the non-uniqueness in the solution

Materials Process Design and Control Laboratory UNSUPERVISED CLASSIFICATION Find the cluster centers {C 1,C 2,…,C k } such that the sum of the 2-norm distance squared between each feature x i, i = 1,..,n and its nearest cluster center C h is minimized. Identify clusters Clusters DATABASE OF ODFs Feature Space Cost function Each class is affiliated with multiple processes

Materials Process Design and Control Laboratory PROCESS PARAMETERS LEADING TO DESIRED PROPERTIES Young’s Modulus (GPa) Angle from rolling direction CLASSIFICATION BASED ON PROPERTIES Class - 1 Class - 2 Class - 3 Class - 4 Velocity Gradient Different processes, Similar properties Database for ODFs Property Extraction ODF Classification Identify multiple solutions

Materials Process Design and Control Laboratory A TWO-STAGE PROBLEM Process – 2 Plane strain compression  = Process – 1 Tension  = Initial Conditions: Stage 1 Sensitivity of material property Initial Conditions- stage 2 DATABASE Reduced Basis  (1)  (2) Direct problem Sensitivity problem

Materials Process Design and Control Laboratory MULTIPLE PROCESS ROUTES Desired Young’s Modulus distribution Magnetic hysteresis loss distribution Stage: 1 Shear-1  = Stage: 2 Plane strain compression (  = ) Stage: 1 Shear -1  = Stage: 2 Rotation-1 (  = ) Stage 1: Tension  = Stage 2: Shear-1  = Stage 1: Tension  = Stage 2: Rotation-1  = Classification

Materials Process Design and Control Laboratory DESIGN FOR DESIRED ODF: A MULTI STAGE PROBLEM Initial guess,   = 0.65,   = -0.1 Desired ODFOptimal- Reduced order control Full order ODF based on reduced order control parameters Stage: 1 Plane strain compression (   = ) Stage: 2 Compression (   = )

Materials Process Design and Control Laboratory DESIGN FOR DESIRED MAGNETIC PROPERTY h Crystal direction. Easy direction of magnetization – zero power loss External magnetization direction Stage: 1 Shear – 1 (   = ) Stage: 2 Tension (   = )