A Exam Sethu

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Published on September 30, 2007

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Slide1:  Sethuraman Sankaran Admission to Candidacy Exam Presentation Date: April 6, 2007 Sibley School of Mechanical and Aerospace Engineering Cornell University Information-theoretic and collocation based methods for uncertainty modeling and optimization Slide2:  Acknowledgements Special Committee: Prof. Nicholas Zabaras, M&AE, Cornell University Prof. Subrata Mukherjee, T&AM, Cornell University Prof. Tatiyana Apanasovich, ORIE, Cornell University Funding Agencies: ARO, Metallic Materials Program AFOSR Sibley school of Mechanical and Aerospace engineering Cornell Theory Center Slide3:  Brief outline STOCHASTIC MODELING STOCHASTIC OPTIMIZATION MULTISCALE PERSPECTIVE FUTURE WORK Slide4:  Uncertainty in materials Properties Structure Processing Questions of interest Given a structure predict properties? Given desired properties, predict the processing parameters. …. How to deal with such underlying questions when the materials and structures are laced with uncertainties Slide5:  Multiscale Picture Aircraft wing Continuum scale meso scale Micro-scale Mechanics of slip Property prediction at the macro-scale based on structure at the micro-scale Slide6:  Stochastic Modeling Slide7:  The Big Picture Stochastic microscale models Micro Scale Macro Scale Stochastic models (incorporating uncertainty) Provide stochastic models as inputs to other problems at macro scale. Modeling input uncertainties Stochastic design of new processes accounting for possible sources of uncertainties Material response curves-stochastic, deterministic? Slide8:  Why do we need a statistical model? 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 Slide9:  Macro scale – Variation in property statistics Source of variation – topology and crystal orientations in lower scales Big picture Slide10:  Microstructure as a class We compute a microstructural class where there is variability in microstructural features within a class but certain experimentally obtained information is incorporated within the class. MICROSTRUCTURE CLASS TOPOLOGICAL FEATURES TEXTURAL FEATURES A class is defined based on statistics – mean grain size, variance of grain sizes, mean texture etc. A microstructure in the entire specimen is considered to be a sample from this class where a probability is assigned to each microstructure in the class Slide11:  Tool for stochastic modeling Maximum entropy technique The principle of maximum entropy (MAXENT) states that amongst the probability distributions that satisfy our incomplete information about the system, the probability distribution that maximizes entropy is the least-biased estimate that can be made. It agrees with everything that is known but carefully avoids anything that is unknown. P = 0.01 P= 0.005 P = 0.05 Mesoscale MaxEnt for generating stochastic microstructure data Macroscale Upscale Stochastic macroscale model/Stochastic macro scale inputs Homogenization schemes Slide12:  Development of a mathematical model Compute a PDF of microstructures Grain size features Orientation Distribution functions Grain size ODF (a function of 145 random parameters) Assign microstructures to the macro specimen after sampling from the PDF Slide13:  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 DATA GENERATION FEATURE EXTRACTION STOCHASTIC MODELING POST-PROCESSING Generating input microstructures: The phase field model:  Generating input microstructures: The phase field model Define order parameters: where Q is the total number of orientations possible Define free energy function (Allen/Cahn 1979, Fan/Chen 1997) : Non-zero only near grain boundaries Phase field model (contd…):  Phase field model (contd…) Driving force for grain growth: Reduction in free energy: thermodynamic driving force to eliminate grain boundary area (Ginzburg-Landau equations) kinetic rate coefficients related to the mobility of grain boundaries Assumption: Grain boundary mobilties are constant Phase Field – Problem parameters:  Phase Field – Problem parameters Isotropic mobility (L=1) Discretization : problem size : 75x75x75 Order parameters: Q=20 Timesteps = 1000 First nearest neighbor approx. Input microstructural samples:  Input microstructural samples 3D microstructural samples 2D microstructural samples Slide18:  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 Microstructural feature: Grain sizes:  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 Slide20:  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 Slide21:  Distribution of microstructures Grain size ODF Know microstructures at some points Given: Microstructures at some points Obtain: PDF of microstructures MAXENT as a tool for microstructure reconstruction:  MAXENT as a tool for microstructure reconstruction Input: Given average (and lower moments) of grain sizes and ODFs Obtain: microstructures that satisfy the given properties Constraints are viewed as expectations and lower moments 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. Microstructures are considered as realizations of a random field which comprises of randomness in grain sizes and orientation distribution functions. The MAXENT principle:  The MAXENT principle The principle of maximum entropy (MAXENT) states that amongst the probability distributions that satisfy our incomplete information about the system, the probability distribution that maximizes entropy is the least-biased estimate that can be made. It agrees with everything that is known but carefully avoids anything that is unknown. E.T. Jaynes 1957 MAXENT is a guiding principle to construct PDFs based on limited information There is no proof behind the MAXENT principle. The intuition for choosing distribution with maximum entropy is derived from several diverse natural phenomenon and it works in practice. The missing information in the input data is fit into a probabilistic model such that randomness induced by the missing data is maximized. This step minimizes assumptions about unknown information about the system. Slide24:  MAXENT : a statistical viewpoint MAXENT solution to any problem with set of features is Parameters of the distribution Input features of the microstructure Fit an exponential family with N parameters (N is the number of features given), MAXENT reduces to a parameter estimation problem. Mean provided 1-parameter exponential family (Poisson distribution) Gaussian distribution Mean, variance given No information provided (unconstrained optimiz.) The uniform distribution Commonly seen distributions MAXENT as an optimization problem:  Subject to Lagrange Multiplier optimization Lagrange Multiplier optimization feature constraints features of image I MAXENT as an optimization problem Find Slide26:  Gradient Evaluation Objective function and its gradients: 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) Slide27:  Reconstruction of grain size distributions (for 2D microstructures) Polarized light micrograph of aluminium alloy AA3302 (source Wittridge NJ et al. Mat.Sci.Eng. A, 1999) Input constraints in the form of first two moments. The corresponding MAXENT distribution is shown on the right. Slide28:  Comparison of MaxEnt grain size distribution with the distribution of a phase field microstructure Reconstruction of grain size distributions (for 3D microstructures) Microstructure modeling : the Voronoi structure:  Microstructure modeling : the Voronoi structure {p1,p2,…,pk} : generator points. Cell division of k-dimensional space : Voronoi tessellation of 3d space. Each cell is a microstructural grain. Slide30:  Heuristic algorithm for generating voronoi centers Generate sample points on a uniform grid and each point is associated with a grain size drawn from the given distribution, d. 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. Rcorr(y,d)>0.95? No Yes stop Slide31:  Reconstructed microstructures – 2D voronoi cell tessellations Reconstruction of microstructures based on correlation with the MAXENT grain size distribution. All voronoi tessellations which lead to a size distribution that has correlation coefficient more than 0.9 are accepted. Slide32:  Reconstructing microstructures Computing microstructures using the Sobel sequence method Slide33:  Reconstructing microstructures (contd..) Computing microstructures using the Sobel sequence method Slide34:  Reconstruction of textures Texture is seen as an independent random field similar to grain sizes Grain size provides sizes to individual grains. Textures provide specific crystal properties to each grain. Rodrigues’ representation Fcc fundamental region Cubic crystal ODF: Provides information about volume fraction of crystals with a specific orientation Slide35:  Input ODF Reconstructed samples using MAXENT ODF reconstruction using MAXENT Representation in Frank-Rodrigues space Slide36:  Input ODF Expected property of reconstructed samples of microstructures Ensemble properties Slide37:  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 Homogenization Scheme:  Homogenization Scheme Microstructure is a representation of a material point at a smaller scale Deformation at a macro-scale point can be represented by the motion of the exterior boundary of the microstructure. (Hill, R., 1972) Polycrystal Plasticity models Primary mechanisms: Crystallographic slip and reorientation of individual crystals Veera and Zabaras, IJP, 2006 Slide39:  How many samples are required to predict the mean/standard deviation of the stress-strain curves effectively? Convergence analysis Almost sure convergence: Convergence in probability: Convergence in moments: What we are interested We say that converges to X if Estimates Increase the number of microstructure samples and test if the mean and standard deviation values converge at different locations in the stress-strain curve We use Monte-Carlo techniques to sample microstructures and get statistics of their properties Slide40:  Convergence analysis A B C D A 0 5 10 15 20 25 30 35 40 45 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 5 10 15 20 25 30 35 40 45 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Number of samples Standard deviation of stress (MPa) Number of samples Standard deviation of stress (MPa) A B Slide41:  0 5 10 15 20 25 30 35 40 45 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 5 10 15 20 25 30 35 40 45 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Number of samples Standard deviation of stress (MPa) Number of samples Standard deviation of stress (MPa) C D A B C D Convergence analysis – contd.. Slide42:  Statistical variation of properties Statistical variation of homogenized stress-strain curves. Aluminium polycrystal with rate-independent strain hardening. Pure tensile test. Slide43:  Motivation Slide44:  Grain size moments (like mean, std, etc) Input Output A class of microstructures satisfying the information Statistical relation Problem statement: We want to predict the class of microstructures which satisfies a given statistic of grain size moment. Input is a 2-tuple and output is a microstructural class. The data is generated using MaxEnt. Moment no. Moment value Method Use MAXENT for certain microstructure classes. Generate MAXENT distribution for certain inputs TRAIN A NON-LINEAR STATISTICAL MAPPER Slide45:  Distribution that is given from the database Distribution from training weights of the network Optimization T: number of microstructure classes that have been pre-computed and stored in a database Kullback-Leibler divergence Non-linear relationship between input moments and parameters of the MaxEnt PDF Input moment Parameters of the MaxEnt PDF Slide46:  Technique-Backpropagation Initialize weights and biases Objective function Gradients Update weights/biases If less than tolerance, terminate Slide47:  20 40 60 80 100 120 140 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Iterations Objective function A B C Information learning – convergence plot Distributions at various iterations Information learning – convergence for 3 input moments Slide48:  Database containing information about two moments Database containing information about three moments Database containing information about four moments Recomputed MaxEnt distributions using information learning Slide49:  Stochastic Optimization Slide50:  Motivation-I (Inverse problems) Heat flux (unknown,random) Uncertain material properties Temperature sensors From the PDF of measurement in temperature sensors, compute the PDF of heat flux. Stochastic inverse problem Slide51:  Motivation-II (Design problems) Uncertain velocity gradient Uncertainty in initial shape Minimize 1. flash (material wastage) 2. stresses/variation in stresses at root of teeth Slide52:  STOCHASTIC DESIGN: Compute the PDF of parameters to minimize/maximize some behavior of material Slide53:  Representation of random process - Karhunen-Loeve, Polynomial Chaos expansions Karhunen-Loèvè expansion Based on the spectral decomposition of the covariance kernel of the stochastic process Random process Mean Set of random variables to be found Eigenpairs of covariance kernel Need to know covariance Converges uniformly to any second order process Set the number of stochastic dimensions, N, based on the eigen-value spectrum Dependence of variables Pose the (N+d) dimensional problem Representing randomness – Karhunen Loeve expansion Slide54:  Representing randomness – Polynomial Chaos expansion e.g.: To represent a normal random variable represents a uniform variable between a and b e.g.: To represent a uniform random variable Slide55:  Generalized Polynomial Chaos expansion Analagous to the Fourier expansion. Each coefficient is interpreted in a spectral space (and is non-physical) The solution is obtained by projecting onto the space occupied by the polynomials themselves – this results in a set of coupled equations for the coefficients, is affected by the curse of dimensionality Slide56:  Solution of SPDE’s using Polynomial chaos Solve for ui Slide57:  Spectral Galerkin method: Spatial domain is approximated using a finite element discretization Stochastic domain is approximated using a spectral element discretization Coupled equations Decoupled equations Collocation method: Spatial domain is approximated using a finite element discretization Stochastic domain is approximated using multidimensional interpolating functions Representing randomness – Stochastic Collocation Scheme Slide58:  Solution of SPDE’s using Stochastic collocation Represent using Lagrange polynomials Slide59:  Smolyak algorithm: reduction in points D = 10 For 2D interpolation using Chebyshev nodes Left: Full tensor product interpolation uses 256 points Right: Sparse grid collocation used 45 points to generate interpolant with comparable accuracy Results in multiple orders of magnitude reduction in the number of points to sample Slide60:  Stochastic Optimization Issues How do you compute stochastic sensitivities (i.e.) gradient of objective function with respect to PDF of parameters? How do you perturb the PDF? GPCE expansion – perturbation in spectral space Stochastic collocation-perturbation at specific collocation points Uncoupled set of equations for computing stochastic sensitivities Coupled equations. Unwieldy to solve in the event of large stochastic dimensions Stochastic optimization problem is converted into a large dimensional deterministic optimization problem Applications IHCP in rolling specimens induced by uncertainties in BC’s, material properties, measurement errors Contaminant source identification in porous media Slide61:  Stochastic Sensitivity analysis – using GPCE expansions This is used to compute stochastic gradients – How much does the Objective function change based on a small perturbation to the PDF of parameters index modal coefficients How do the modal coefficients vary with respect to a perturbation in the PDF. Each coefficient is dependent on the whole PDF. Sensitivity of first modal coefficient. Each coefficient is dependent on the entire PDF of q Slide62:  q q+ Deterministic perturbation Stochastic Sensitivity analysis – using Stochastic Collocation Stochastic sensitivities can be computed by solving a set of decoupled deterministic equations at stochastic collocation points. The computational expense is improved further by employing a sparse grid scheme in collocation space Slide63:  Stochastic Optimization Issues How do you compute stochastic sensitivities (i.e.) gradient of objective function with respect to PDF of parameters? How do you perturb the PDF? GPCE expansion – perturbation in spectral space Stochastic collocation-perturbation at specific collocation points Uncoupled set of equations for computing stochastic sensitivities Coupled equations. Unwieldy to solve in the event of large stochastic dimensions Stochastic optimization problem is converted into a large dimensional deterministic optimization problem Applications IHCP in rolling specimens induced by uncertainties in BC’s, material properties, measurement errors Contaminant source identification in heterogeneous porous media Slide64:  0.0005 0.0003 0.0000 0.0000 0.0118 0.0118 -0.0001 0.0000 d=3, t=20 0.0027 0.0027 0.0000 0.0000 0.0916 0.0916 -0.0007 -0.0007 d=3, t=16 0.00272 0.00271 0.0000 0.0000 0.0915 0.0915 0.0000 -0.0000 d=3, t=7 (*10-1) 0.00012 0.00012 0.0000 0.0000 0.0055 0.0055 0.0000 -0.0000 d=2, t=8 0.0027 0.0027 0.0000 0.0000 0.0916 0.0916 -0.0007 -0.0007 d=2, t=4 0.0068 0.0068 0.0236 0.0236 0.0824 0.0824 0.2870 0.2870 d=2, t=1 0.0006 0.0006 0.0013 0.0013 0.0164 0.0164 0.0000 0.0000 d=1, t=3 0.0013 0.0013 0.0000 0.0000 0.0274 0.0272 0.0000 0.0000 d=1, t=2 0.0068 0.0068 0.0236 0.0236 0.0824 0.0824 0.2870 0.2870 d=1, t=1 Case Stochastic Sensitivity analysis – numerical verification Slide65:  To illustrate this method, we choose a Stochastic Inverse Heat Conduction Problem Temperature sensors Stochastic Optimization - Illustration From the PDF of temperature measured at the sensors, compute the PDF of heat flux at the boundaries. This problem is posed as a stochastic optimization problem: Find: such that Slide66:  Algorithm Step 4: Update k=k+1. where Go to step 2. Slide67:  Unknown random time varying heat flux One-dimensional stochastic inverse heat conduction problem Two temperature sensors were employed. The stochastic optimization technique was employed to re-compute the heat flux. Statistical moments of heat flux Estimated heat flux at the right boundary Slide68:  Estimated heat flux at the left boundary One-dimensional stochastic inverse heat conduction problem Slide69:  Non-intrusiveness in design The technique needs just a deterministic direct solver and a deterministic sensitivity solver, (i.e.), it can be applied on top of a deterministic solver to perform stochastic optimization. Showcase this by coupling the scheme with ANSYS Infinite solid A semi-infinite solid is initially at temperature To. The solid is then suddenly exposed to an environment having a temperature Te and a surface convection coefficient h. h is uncertain. Stochastic sensitivity – How does temperature profile at certain nodal locations vary for small perturbations in the PDF of h Design – Based on noisy temperature measurements at some point in the solid, infer the PDF of convection coefficient. Slide70:  Input: Time history of temperature measurements at some point within the body Start from a random guess for convection coefficient Automated extraction of data files and postprocessing Executes the Ansys input file Method Slide71:  Results Uni-modal convection coefficient Bi-modal convection coefficient Multi-modal convection coefficient Results of the measurement problem and the stochastic optimization match very well with each other Slide72:  IHCP on a rolling cylinder Problem Definition: There is a rolling cylinder with four sensor locations. Based on temperature PDF’s at these points, it is desired to reconstruct the heat flux. Unknown heat flux Thermal conductivity is a random field with an exponential correlation kernel KLE realizations Slide73:  Results Mean temperature statistic Second moment of temperature Slide74:  Third moment of temperature Fourth moment of temperature Results Slide75:  qin qout Given: the PDF of concentrations at sensor locations at some time instant Compute: PDF of initial concentrations of the resident fluid Water is pumped through a porous Medium. It is desired to estimate the PDF of initial concentrations of water given its concentration PDF at a later time (given a source and a sink). Inverse Problem-Concentration reconstruction Slide76:  Stochastic Optimization problem Direct governing PDE’s Sensitivity governing PDE’s B.C. for Direct problem B.C. for Sensitivity problem Parameters Slide77:  Stochastic Optimization problem Convergence of objective function Slide78:  Recomputed concentration statistics First and second cumulants of (a) actual and (b) recomputed concentrations at t=0.1 Third and fourth cumulants of (a) actual and (b) recomputed concentrations at t=0.1 Slide79:  Recomputed concentration statistics at t=0.8 Comparison of (a) actual and (b) recomputed mean concentrations Comparison of (a) actual and (b) recomputed second cumulant concentrations Comparison of (a) actual and (b) recomputed third cumulant concentrations Comparison of (a) actual and (b) recomputed fourth cumulant of concentrations Slide80:  Comparison with Bayesian technique Bayesian is another oft-used technique for parameter estimations. Drawback of Bayesian: Assumes a prior distribution on the parameters. Hence, statistics higher than the mean may not be reliably estimated. Advantage: If the prior distribution is very close to the actual distribution, time taken for computation of parameters is less compared to the current stochastic optimization scheme The collocation based stochastic optimization technique is a generalization of Bayesian scheme in the sense that no assumption on the prior is made Slide81:  Multiscale Picture Slide82:  Where do all of these fit in? Properties Structure Processing Questions of interest Given a structure predict properties? Given desired properties, predict the processing parameters. …. How to deal with such underlying questions when the materials and properties are laced with uncertainties Slide83:  Uncertainty in constitutive relation Uncertainties in multiple scales Slide84:  Multilength scale picture Stochastic models at this scale Characterizing stochastic behavior and computing optimal design parameters for specified behavior Upscaling between these scales Main Drawback Multiscaling is interwoven into the materials-process-property triangle. To couple the tools developed, a multiscale solver is required. Future work:  Materials Process Design and Control Laboratory Future work Slide86:  Stochastic deformation process design Velocity gradient Friction coefficient: Unif [fl fu] (at die surface) Significance Several processes have uncontrollable sources of uncertainties. How to take them into account during design? Control parameters (such as ram velocities) may be uncertain. How to account for them? Ability to implement The methodology is in place (stochastic optimization). We need to get in place some solvers for re-meshing during deformation processing. Slide87:  Big Picture A suite of tools with emphasis on (a) stochastics and (b) multiscaling that can enhance the current capabilities at MPDC. MaxEnt tool for stochastic models at the meso-scale and generating statistics of material properties at the macro scale. Stochastic optimization tools for design in the presence of uncertainties at the macro scale. Wavelet tools for adaptive multiscaling. Slide88:  Coupling stochastic models at the macro and micro scales Develop a tool for multiscaling that can be used in conjunction with stochastic tools already developed. Questions of interest How does the microstructure evolve as you apply a deformation on the macro-specimen? How will uncertainties in ram velocity or material property affect microstructure evolution? Is there a computationally efficient model for describing microstructure evolution? Slide89:  Each dyadic level is a particular scale Details at a particular scale Details at lower scales Multiscale model xjk=x2kj+1 Microstructure Invoke higher scales near grain boundaries Other multiscale applications of wavelets O.Vasilyev et. al. Vortex merging Significance: Strong potential for solving phase field models of microstructure evolution – can a reduced wavelet basis be useful? Serves as a useful tool for other multiscale applications (such as contamination source identification etc.) Slide90:  October 2007: Complete the deformation process based stochastic design using the collocation technique March 2008: August 2008: Develop adaptive wavelet based solvers using wavelet collocation technique Applications of the adaptive wavelet solver to microstructure evolution. Specific Time Plan Slide91:  Thank You

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