Computational models for stochastic multiscale systems Materials Process Design and Control Laboratory Nicholas Zabaras Materials Process Design and Control.

Computational models for stochastic multiscale systems Materials Process Design and Control Laboratory Nicholas Zabaras Materials Process Design and Control Laboratory Sibley School of Mechanical and Aerospace Engineering 188 Frank H. T. Rhodes Hall Cornell University Ithaca, NY 14853-3801 Email: zabaras@cornell.edu URL: http://mpdc.mae.cornell.edu/

Materials Process Design and Control Laboratory Outline of presentation 1. Stochastic multiscale modeling of diffusion in random heterogeneous media 2. Some aspects of stochastic multiscale modeling of polycrystal materials x GPCE & support methods for macroscopic models x Modeling mesoscopic uncertainty using maximum entropy methods x Information passing – variability in properties induced by microstructural uncertainties x Robust materials design

Materials Process Design and Control Laboratory MULTISCALE MATERIALS MODELING grain/crystal Inter-grain slip Grain boundary accommodation Twins precipitates atoms Meso Micro Nano Performance Mechanics of slip MD Homogenization Continuum : Process

Materials Process Design and Control Laboratory CLASSIFICATION OF UNCERTAINTY Uncertainty in engineering systems Modeling IntrinsicExtrinsic Parametric  Experimental setup  Sensor errors  Surroundings  Constitutive relations  Assumptions on underlying physics  Surroundings  Boundary conditions  Process parameters

Materials Process Design and Control Laboratory UNCERTAINTY MODELLING TECHNIQUES  Reliability models  Concerned with extreme variations [not in the robust analysis zone]  Analysis beyond second order is extremely complicated  Sensitivity derivatives, Perturbation methods  Linear uncertainty propagation  Cannot address large deviations from mean  Neumann expansions  Are accurate for relatively small fluctuations  Derivation is complicated for higher order uncertainties  Monte Carlo  Most robust of all uncertainty quantification techniques  Extremely computation intensive  Can we combine essential features of one or more of the above [Karhunen-Loeve and Generalized polynomial chaos expansions]

Materials Process Design and Control Laboratory STOCHASTIC PROCESSES AS FUNCTIONS  A probability space is a triple comprising of collection of basis outcomes, permutation of these outcomes and a probability measure  A real-valued random variable is a function that maps the probability space to a real line [regions in go to intervals in the real line]  : Random variable  A space-time stochastic process is can be represented as + other regularity conditions

Materials Process Design and Control Laboratory SERIES REPRESENTATION [CONTD]  Karhunen-Loeve Stochastic process Mean function ON random variables Deterministic functions  The deterministic functions are based on the eigen-values and eigenvectors of the covariance function of the stochastic process.  The orthonormal random variables depend on the kind of probability distribution attributed to the stochastic process.  Any function of the stochastic process (typically the solution of PDE system with as input) is of the form

Materials Process Design and Control Laboratory SERIES REPRESENTATION [CONTD]  Generalized polynomial chaos expansion is used to represent quantities like Stochastic process Askey polynomials in input Deterministic functions Stochastic input  The Askey polynomials depend on the kind of joint PDF of the orthonormal random variables  Typically: Gaussian – Hermite, Uniform – Legendre, Beta -- Jacobi polynomials

Materials Process Design and Control Laboratory MODEL MULTISCALE HEAT EQUATION Permeability of Upper Ness formation Domain Boundary in on  Multiple scale variations in K  K is inherently random  Composites  Diffusion processes  Diffusion processes in crystal microstructure in

Materials Process Design and Control Laboratory STOCHASTIC GOVERNING PDES Denotes a random quantity ASSUMPTIONS  The Dirichlet boundary conditions do not have a multiscale nature (i.e. they can be resolved using the coarse grid)  The correlation function of K decays slowly. Thus only a few random variables are required for its approximation  For steep decays of correlation function, only Monte Carlo methods are viable

Materials Process Design and Control Laboratory STOCHASTIC WEAK FORM Find such that, for all  Basic stochastic function space  Derived function spaces  Stochastic weak form

Materials Process Design and Control Laboratory VARIATIONAL MULTISCALE METHOD h Subgrid scale solution Coarse scale solution Actual solution Hypothesis  Exact solution = Coarse resolved part + Subgrid part [Hughes, 95, CMAME] Induced function space  Solution function space = Coarse function space + Subgrid function space Idea  Model the projection of weak form onto the subgrid function space, calculate an approximate subgrid solution  Use the subgrid solution to solve for coarse solution

Materials Process Design and Control Laboratory VMS – A MATHEMATICAL INTRODUCTION Find such that, for all  Stochastic weak form  VMS hypothesis Exact solution Coarse solution Subgrid solution  Induced function spaces Solution function space Trial function space

Materials Process Design and Control Laboratory SCALE PROJECTION OF WEAK FORM Find such that, for all and  Eliminate the subgrid solution in the coarse weak form using a modeled subgrid solution obtained from the subgrid weak form  Projection of weak form onto coarse function space  Projection of weak form onto subgrid function space  Idea

Materials Process Design and Control Laboratory INTERPRETATION OF SUBGRID SOLUTION  Projection of weak form onto subgrid function space  Assumptions Subgrid solution Homogenous part Affine correction  Incorporates all coarse scale information that affects the subgrid solution  Is that part of subgrid solution that has no coarse scale dependence Find such that, for all and

Materials Process Design and Control Laboratory SPLITTING THE SUBGRID SCALE WEAK FORM  Subgrid homogeneous weak form  Subgrid affine correction weak form  The homogenous subgrid solution is also denoted as the C2S map (coarse-to-subgrid)  We design multiscale basis functions to determine the C2S map  The affine correction is modeled explicitly Find such that, for all and

Materials Process Design and Control Laboratory EXAMINING DYNAMICS ũCũC ūCūC ûFûF 1 1 Coarse solution field at start of time step Coarse solution field at end of time step  Time scale for exact solution =  Time scale for coarse solution =  Local time coordinate =  In each element, we use a truncated GPCE representation for the coarse solution

Materials Process Design and Control Laboratory A CLOSER LOOK AT THE COARSE SOLUTION  Coarse solution is entirely specified by the coefficients  Any coarse scale information that is passed on to the subgrid solution can only be through this coefficient field  Without loss of generality, we assume the following  Multiscale basis functions

Materials Process Design and Control Laboratory DYNAMICS OF  After sufficient algebraic manipulation, we get ũCũC ūCūC ûFûF 1 1 Coarse solution field at start of time step Coarse solution field at end of time step  Without loss of generality, inside a coarse time step, we assume

Materials Process Design and Control Laboratory MORE RESULTS  The subgrid homogenous solution can be written as  With some involved derivations, we can show Askey polynomial FEM shape function Multiscale basis function Diffusion coefficient with multiple scales

Materials Process Design and Control Laboratory COMPLETE SPECIFICATION OF C2S MAP  The above evolution equation requires the specification of an initial condition (in each coarse element) and boundary conditions (on each coarse element boundary).  Before that, we introduce a new variable  This reduces the C2S map governing equation as follows

Materials Process Design and Control Laboratory BOUNDARY CONDITIONS FOR C2S MAP  On each coarse element, we have Coarse element boundary   Where, we have on the boundary

Materials Process Design and Control Laboratory THE AFFINE CORRECTION TERM  On each coarse element, we have  The affine correction term originates from 2 sources  Effect of source, sink terms at subgrid level [A]  Effect of the subgrid component of the true initial conditions [B]  The [B] effect is global and is not resolved in our implementation  Since [B] effect is decaying in time, we choose a time cut-off after which the subgrid solutions are accurate and can be used. This is also called the burn-in time

Materials Process Design and Control Laboratory BOUNDARY CONDITIONS  The affine correction term has no coarse-scale dependence, we can assume it goes to zero on coarse-element boundaries  If we need to avoid complications due to burn-in time and the effect of above assumption, we can use a quasistatic subgrid assumption

Materials Process Design and Control Laboratory MODIFIED COARSE SCALE FORMULATION  We can substitute the subgrid results in the coarse scale variational formulation to obtain the following  We notice that the affine correction term appears as an anti- diffusive correction  Often, the last term involves computations at fine scale time steps and hence is ignored  Time-derivatives of subgrid quantities are approximated using difference formulas (EBF)

Materials Process Design and Control Laboratory COMPUTATIONAL ISSUES  Based on the indices in the C2S map and the affine correction, we need to solve (P+1)(nbf) problems in each coarse element  At a closer look we can find that  This implies, we just need to solve (nbf) problems in each coarse element (one for each index s)

Materials Process Design and Control Laboratory NUMERICAL EXAMPLES  Stochastic investigations  Example 1: Decay of a sine hill in a medium with random diffusion coefficient  The diffusion coefficient has scale separation and periodicity  Example 2: Planar diffusion in microstructures  The diffusion coefficient is computed from a microstructure image  The properties of microstructure phases are not known precisely [source of uncertainty]  Future issues

Materials Process Design and Control Laboratory EXAMPLE 1

Materials Process Design and Control Laboratory EXAMPLE 1 [RESULTS AT T=0.05]

Materials Process Design and Control Laboratory EXAMPLE 1 [ERROR PLOTS] Quasistatic subgrid Dynamic subgrid  Note that the L2 error in upscaling is larger in the case of the dynamic subgrid assumption? Why?

Materials Process Design and Control Laboratory QUASISTATIC SEEMS BETTER  There are two important modeling considerations that were neglected for the dynamic subgrid model  Effect of the subgrid component of the initial conditions on the evolution of the reconstructed fine scale solution  Better models for the initial condition specified for the C2S map (currently, at time zero, the C2S map is identically equal to zero implying a completely coarse scale formulation)  In order to avoid the effects of C2S map, we only store the subgrid basis functions beyond a particular time cut-off (referred to herein as the burn-in time)  These modeling issues need to be resolved for increasing the accuracy of the dynamic subgrid model

Materials Process Design and Control Laboratory EXAMPLE 2  The thermal conductivities of the individual consituents is not known  We use a mixture model

Materials Process Design and Control Laboratory DESCRIPTION OF THE MIXTURE MODEL  Assume the pure (  ) and (  ) phases have the following thermal conductivities  The following mixture model is used for describing the microstructure thermal conductivity  The following initial conditions and boundary conditions are specified

Modeling uncertainty propagation in large deformations Materials Process Design and Control Laboratory

MOTIVATION:UNCERTAINTY IN FINITE DEFORMATION PROBLEMS Metal forming Composites – fiber orientation, fiber spacing, constitutive model Biomechanics – material properties, constitutive model, fibers in tissues Initial preform shape Material properties/models Forging velocity Texture, grain sizes Die/workpiece friction Die shape Small change in preform shape could lead to underfill

Materials Process Design and Control Laboratory WHY UNCERTAINTY AND MULTISCALING ? Macro Meso Micro  Uncertainties introduced across various length scales have a non-trivial interaction  Current sophistications – resolve macro uncertainties  Use micro averaged models for resolving physical scales  Imprecise boundary conditions  Initial perturbations  Physical properties, structure follow a statistical description

Materials Process Design and Control Laboratory UNCERTAINTY ANALYSIS USING SSFEM Key features Total Lagrangian formulation – (assumed deterministic initial configuration) Spectral decomposition of the current configuration leading to a stochastic deformation gradient B n+1 (θ) x n+1 (θ)=x(X,t n+1, θ, ) B0B0 X x n+1 (θ) F(θ)F(θ)

Materials Process Design and Control Laboratory TOOLBOX FOR ELEMENTARY OPERATIONS ON RANDOM VARIABLES Scalar operations Matrix\Vector operations 1.Addition/Subtraction 2.Multiplication 3.Inverse 1.Addition/Subtraction 2.Multiplication 3.Inverse 4.Trace 5. Transpose Non-polynomial function evaluations 1.Square root 2.Exponential 3.Higher powers Use precomputed expectations of basis functions and direct manipulation of basis coefficients Use direct integration over support space Matrix Inverse Compute B(θ) = A -1 (θ) Galerkin projection Formulate and solve linear system for B j (PC expansion)

Materials Process Design and Control Laboratory UNCERTAINTY ANALYSIS USING SSFEM Linearized PVW On integration (space) and further simplification Galerkin projection Inner product

Materials Process Design and Control Laboratory UNCERTAINTY DUE TO MATERIAL HETEROGENEITY 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.

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

Materials Process Design and Control Laboratory EFFECT OF UNCERTAIN FIBER ORIENTATION Aircraft nozzle flap – composite material, subjected to pressure on the free end Orthotropic hyperelastic material model with uncertain angle of orthotropy modeled using KL expansion with exponential covariance and uniform random variables Two independent random variables with order 4 PCE (Legendre Chaos)

Materials Process Design and Control Laboratory SUPPORT SPACE METHOD - INTRODUCTION Finite element representation of the support space. Inherits properties of FEM – piece wise representations, allows discontinuous functions, quadrature based integration rules, local support. Provides complete response statistics. Convergence rate identical to usual finite elements, depends on order of interpolation, mesh size (h, p versions). Easily extend to updated Lagrangian formulations. Constitutive problem fully deterministic – deterministic evaluation at quadrature points – trivially extend to damage problems. True PDF Interpolant FE Grid

Materials Process Design and Control Laboratory SUPPORT SPACE METHOD – SOLUTION SCHEME Linearized PVW Galerkin projection GPCE Support space

Materials Process Design and Control Laboratory EFFECT OF RANDOM VOIDS ON MATERIAL BEHAVIOR Mean InitialFinal Using 6x6 uniform support space grid Uniform 0.02

Materials Process Design and Control Laboratory PROBLEM 2: EFFECT OF RANDOM VOIDS ON MATERIAL BEHAVIOR Load displacement curves

Materials Process Design and Control Laboratory FURTHER VALIDATION Comparison of statistical parameters ParameterMonte Carlo (1000 LHS samples) Support space 6x6 uniform grid Support space 7x7 uniform grid Mean6.11756.11766.1175 SD0.7991250.7987060.799071 m30.0831688 0.08114570.0831609 m40.9362120.9242770.936017 Final load values

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

Materials Process Design and Control Laboratory PROCESS STATISTICS Force SD Force ParameterMonte Carlo (7000 LHS samples) Support space 10x10 Mean2.2859e32.2863e6 SD297.912299.59 m3-8.156e6 - -9.545e6 m41.850e101.979e10 Final force statistics

Materials Process Design and Control Laboratory Sethuraman Sankaran and Nicholas Zabaras Materials Process Design and Control Laboratory Sibley School of Mechanical and Aerospace Engineering 188 Frank H. T. Rhodes Hall Cornell University Ithaca, NY 14853-3801 Email: ss524@cornell.edu, zabaras@cornell.edu URL: http://mpdc.mae.cornell.edu/ An Information-theoretic Tool for Property Prediction Of Random Microstructures

Materials Process Design and Control Laboratory Idea Behind Information Theoretic Approach Statistical Mechanics Information Theory Rigorously quantifying and modeling uncertainty, linking scales using criterion derived from information theory, and use information theoretic tools to predict parameters in the face of incomplete Information etc Linkage? Information Theory Basic Questions: 1. Microstructures are realizations of a random field. Is there a principle by which the underlying pdf itself can be obtained. 2. If so, how can the known information about microstructure be incorporated in the solution. 3. How do we obtain actual statistics of properties of the microstructure characterized at macro scale.

Materials Process Design and Control Laboratory Information Theoretic Scheme: the MAXENT principle Input: Given statistical correlation or lineal path functions Obtain: microstructures that satisfy the given properties Constraints are viewed as expectations of features over a random field. Problem is viewed as finding that distribution whose ensemble properties match those that are given. Since, problem is ill-posed, we choose the distribution that has the maximum entropy. Additional statistical information is available using this scheme.

Materials Process Design and Control Laboratory Find Subject to Lagrange Multiplier optimization feature constraints features of image I MAXENT as an optimization problem Partition Function

Materials Process Design and Control Laboratory MAXENT FRAMEWORK MAXENT: A way to generate the complete probabilistic characterization of a quantity based on limited measurements The algorithm is solely based on the information contained in the data (the reconstruction is statistical) [comparisons with kriging? ] Sampling from the PDF Component Macro scale Experimental microstructure images Process data Grain size using Heyn intercept method Obtain PDF of grain sizes using MAXENT

Materials Process Design and Control Laboratory PARALLEL GIBBS SAMPLING Reconstruction of microstructures from grain size PDFs typically involve sampling from a large dimensional random space Need parallel sampling procedures [but how for Gibbs samplers] Improper PDF Choose a sample microstructure image Do domain decomposition for grains Choose a random grain Sample properties for the grain conditioned on other grains Pre process At the level of individual processors

Grain sizes: Heyn’s intercept method PDF of grain sizes Materials Process Design and Control Laboratory MAXENT DISTRIBUTION OF GRAIN SIZES Given: Experimental image of Al alloy, material properties of individual components, mean orientation of the grains Find the class of microstructures of which the current image is a member

Materials Process Design and Control Laboratory RECONSTRUCTION OF 3D MICROSTRUCTURES Input PDF (grain size distribution) Microstructure samples from the PDF

Materials Process Design and Control Laboratory RECONSTRUCTION OF ODFs Input ODF (expectation value) Statistical samples of ODF Average of ODF computed from samples Orientation distribution function: The probability distribution of orientation of individual grains in a microstructure

Materials Process Design and Control Laboratory Implementation of homogenization scheme Largedef formulation for macro scale Update macro displacements Boundary value problem for microstructure Solve for deformation field, determine average stress Consistent tangent formulation (macro) Integration of constitutive equations Continuum slip theory Consistent tangent formulation (meso) Macro-deformation gradient Homogenized (macro) stress, Consistent tangent meso stress, consistent tangent meso deformation gradient Macro Meso Micro

Materials Process Design and Control Laboratory Study properties of real microstructures X Y Z (a) (c) (b) (d)

Materials Process Design and Control Laboratory Study property variability in a material FORM Approximation Design iterations Input PDF (grain size distribution) Microstructure samples (Voronoi models) from the PDF MAXENT algorithm Homogenization Statistical variability in material property (uniaxial stress- strain curve) Gibbs sampler Grain size lower order statistics (average grain sizes, shape data) from manufacturer Property statistics

Materials Process Design and Control Laboratory ROBUST DESIGN AND OPTIMIZATION WITH UNCERTAINTY  Extending functional optimization methods from the deterministic world  Non-intrusive optimization methods (based on the support method)

Materials Process Design and Control Laboratory EXAMPLE – SIHCP 00 hh  II  Thermal conductivity and heat capacity are stochastic processes  Need to find the unknown flux with variability limits such that the temperature solution is matched with the sensor readings on the internal boundary

Materials Process Design and Control Laboratory DESIGNING THE OBJECTIVE  Need to match extra temperature readings at the boundary  I Stochastic temperature solution at the inner boundary  I, given a guess for the unknown heat flux q 0 Temperature sensor readings with specified variability at the inner boundary  I  Try to match above in mean-square sense

Materials Process Design and Control Laboratory SOLVING THE OPTIMIZATION PROBLEM  Obtain the gradient of the objective function in a distributional sense  We have the definition of continuum stochastic sensitivity embedded in the definition of the gradient  But what is the physical reasoning behind a stochastic sensitivity ?

Materials Process Design and Control Laboratory CONTINUUM STOCHASTIC SENSITIVITY Temperature at a point Perturbation in PDF  Generic definition: Change in output for an infinitesimal change in the design variable  Here: Change in output PDF for an infinitesimal change in PDF of the design variable

Materials Process Design and Control Laboratory GRADIENT DEFINITION VIZ ADJOINT  The definition of the gradient is implicit in the following equality  We use the adjoint based approach for defining the gradient in an indirect manner  Simplifying the above equation leads to an adjoint problem using which the gradient can be obtained

Materials Process Design and Control Laboratory INCORPORATING LOSS FUNCTION  In particular, we obtain the following residual term that we equate to the loss function [ difference in temperature solution and sensor readings at the internal boundary ] 00 hh  II Loss function is used as a flux boundary condition for the adjoint problem

Materials Process Design and Control Laboratory ADJOINT EQUATIONS  The final adjoint equation is obtained as follows  The above unstable backward-diffusion equation is converted into a stable diffusion equation using the transformation  The gradient of the objective function is

Materials Process Design and Control Laboratory TRIANGLE FLUX PROBLEM Insulated X = 0  Triangle flux estimation: Beck J.V and Blackwell. B X = 1 X = d Termperature sensor  Data generation  Flux applied to left end following the profile [see Fig]  Sensor readings [polluted with noise] collected at location x = d

Materials Process Design and Control Laboratory DATA SIMULATING REAL EXPERIMENT Sensor readings Large noise level Small noise level  Sensor accuracy Vs estimation results  Estimation of lower moments like mean

Computational models for stochastic multiscale systems Materials Process Design and Control Laboratory Nicholas Zabaras Materials Process Design and Control.

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