Nicholas Zabaras

Sparse grid collocation schemes for stochastic natural convection problems

In recent years, there has been an interest in analyzing and quantifying the effects of random inputs in the solution of partial differential equations that describe thermal and fluid flow problems. Spectral stochastic methods and Monte-Carlo based sampling methods are two approaches that have been used to analyze these problems. As the complexity of the problem or the number of random variables involved in describing the input uncertainties increases, these approaches become highly impractical from implementation and convergence points-of-view. This is especially true in the context of realistic thermal flow problems, where uncertainties in the topology of the boundary domain, boundary flux conditions and heterogeneous physical properties usually require high-dimensional random descriptors. The sparse grid collocation method based on the Smolyak algorithm offers a viable alternate method for solving high-dimensional stochastic partial differential equations. An extension of the collocation approach to include adaptive refinement in important stochastic dimensions is utilized to further reduce the numerical effort necessary for simulation. We show case the collocation based approach to efficiently solve natural convection problems involving large stochastic dimensions. Equilibrium jumps occurring due to surface roughness and heterogeneous porosity are captured. Comparison of the present method with the generalized polynomial chaos expansion and Monte-Carlo methods are made.

An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations

In recent years, there has been a growing interest in analyzing and quantifying the effects of random inputs in the solution of ordinary/partial differential equations. To this end, the spectral stochastic finite element method (SSFEM) is the most popular method due to its fast convergence rate. Recently, the stochastic sparse grid collocation method has emerged as an attractive alternative to SSFEM. It approximates the solution in the stochastic space using Lagrange polynomial interpolation. The collocation method requires only repetitive calls to an existing deterministic solver, similar to the Monte Carlo method. However, both the SSFEM and current sparse grid collocation methods utilize global polynomials in the stochastic space. Thus when there are steep gradients or finite discontinuities in the stochastic space, these methods converge very slowly or even fail to converge. In this paper, we develop an adaptive sparse grid collocation strategy using piecewise multi-linear hierarchical basis functions. Hierarchical surplus is used as an error indicator to automatically detect the discontinuity region in the stochastic space and adaptively refine the collocation points in this region. Numerical examples, especially for problems related to long-term integration and stochastic discontinuity, are presented. Comparisons with Monte Carlo and multi-element based random domain decomposition methods are also given to show the efficiency and accuracy of the proposed method.

An adaptive high-dimensional stochastic model representation technique for the solution of stochastic partial differential equations

A computational methodology is developed to address the solution of high-dimensional stochastic problems. It utilizes high-dimensional model representation (HDMR) technique in the stochastic space to represent the model output as a finite hierarchical correlated function expansion in terms of the stochastic inputs starting from lower-order to higher-order component functions. HDMR is efficient at capturing the high-dimensional input-output relationship such that the behavior for many physical systems can be modeled to good accuracy only by the first few lower-order terms. An adaptive version of HDMR is also developed to automatically detect the important dimensions and construct higher-order terms using only the important dimensions. The newly developed adaptive sparse grid collocation (ASGC) method is incorporated into HDMR to solve the resulting sub-problems. By integrating HDMR and ASGC, it is computationally possible to construct a low-dimensional stochastic reduced-order model of the high-dimensional stochastic problem and easily perform various statistic analysis on the output. Several numerical examples involving elementary mathematical functions and fluid mechanics problems are considered to illustrate the proposed method. The cases examined show that the method provides accurate results for stochastic dimensionality as high as 500 even with large-input variability. The efficiency of the proposed method is examined by comparing with Monte Carlo (MC) simulation.

Hierarchical Bayesian models for inverse problems in heat conduction

Stochastic inverse problems in heat conduction with consideration of uncertainties in the measured temperature data, temperature sensor locations and thermophysical properties are addressed using a Bayesian statistical inference method. Both parameter estimation and thermal history reconstruction problems, including boundary heat flux and heat source reconstruction, are studied. Probabilistic specification of the unknown variables is deduced from temperature measurements. Hierarchical Bayesian models are adopted to relax the prior assumptions on the unknowns. The use of a hierarchical Bayesian method for automatic selection of the regularization parameter in the function estimation inverse problem is discussed. In addition, the method explores the length scales in the estimation of thermal variables varying in space and time. Markov chain Monte Carlo (MCMC) simulation is conducted to explore the high dimensional posterior state space. The methodologies presented are general and applicable to a number of data-driven engineering inverse problems.

A sensitivity analysis for the optimal design of metal-forming processes

Abstract A major objective in metal-forming processing is the optimum selection of process conditions in the design stage of processes given the material state and the geometry of the final product, conditions on the initial workpiece and possibly some restrictions on the processes. Since the required process conditions are input to the direct forming analysis, the design of forming processes usually consists of the repeated trial and error use of direct modeling techniques. Here, process conditions refer to die surfaces, die lubrication conditions, preform selection and the selection of the required ram forces and velocities. In this work, a sensitivity analysis for large deformation hyperelastic viscoplastic solids is presented that is consistent with the kinematic analysis and the constitutive integration scheme used in the updated Lagrangian method for solving the direct deformation problem. The method is developed with the problem of die design in metal-forming processes in mind. As such, special attention is given to the modeling of the complex boundary conditions that result in the die-workpiece interface. The effectiveness of the method is demonstrated by solving an extrusion axially symmetric die design problem. In particular, an extrusion die is designed such that the material state in the final product has the least possible standard deviation.

An efficient Bayesian inference approach to inverse problems based on an adaptive sparse grid collocation method

A new approach to modeling inverse problems using a Bayesian inference method is introduced. The Bayesian approach considers the unknown parameters as random variables and seeks the probabilistic distribution of the unknowns. By introducing the concept of the stochastic prior state space to the Bayesian formulation, we reformulate the deterministic forward problem as a stochastic one. The adaptive hierarchical sparse grid collocation (ASGC) method is used for constructing an interpolant to the solution of the forward model in this prior space which is large enough to capture all the variability/uncertainty in the posterior distribution of the unknown parameters. This solution can be considered as a function of the random unknowns and serves as a stochastic surrogate model for the likelihood calculation. Hierarchical Bayesian formulation is used to derive the posterior probability density function (PPDF). The spatial model is represented as a convolution of a smooth kernel and a Markov random field. The state space of the PPDF is explored using Markov chain Monte Carlo algorithms to obtain statistics of the unknowns. The likelihood calculation is performed by directly sampling the approximate stochastic solution obtained through the ASGC method. The technique is assessed on two nonlinear inverse problems: source inversion and permeability estimation in flow through porous media.

Shape optimization and preform design in metal forming processes

Abstract A continuum sensitivity analysis is presented for the computation of the shape sensitivity of finite hyperelastic–viscoplastic deformations involving contact with friction using a direct differentiation method. Weak shape sensitivity equations are developed that are consistent with the kinematic analysis, constitutive sub-problem as well as the analysis of the contact/friction sub-problem used in the solution of the direct deformation problem. The shape sensitivities are defined in a rigorous sense and the linear sensitivity analysis is performed in an infinite-dimensional continuum framework. The direct deformation and the sensitivity deformation problems are implemented using the finite element method. The shape sensitivity analysis is validated by a comparison of the results with those obtained from the solution of the perturbed direct deformation problem (i.e. using finite differences). Finite-dimensional gradients of objective functions are then computed using the results of the shape sensitivity analysis for the purpose of preform design and shape optimization in metal forming. The effectiveness of the proposed methodology is demonstrated by solving practical shape optimal design problems in forging processing.

DESIGN OF TWO-DIMENSIONAL STEFAN PROCESSES WITH DESIRED FREEZING FRONT MOTIONS

This paper presents a methodology for the design of two-dimensional solidification processes, It is aimed particularly at the calculation of the boundary flux or temperature for a body that solidifies with a desired motion of the freezing front. Such problems are of particular technological significance, considering that the freeing front motion is directly related to the quality of casting. Solution of these problems can be used to achieve a desired east structure, to optimize the time of casting, to prevent choking of the liquid flow, or to control liquid feeding to the contracting front so that cavities and partial solidification in complex castings are avoided. Spatial smoothing and a modification of Becks future information method are used to solve this ill-posed design problem. A moving and deforming finite-element formulation is employed. The accuracy of the method is demonstrated through several two-dimensional numerical examples.

Kernel principal component analysis for stochastic input model generation

Abstract : Stochastic analysis of random heterogeneous media provides useful information only if realistic input models of the material property variations are used. These input models are often constructed from a set of experimental samples of the underlying random field. To this end, the Karhunen-Loeve (K-L) expansion, also known as principal component analysis (PCA), is the most popular model reduction method due to its uniform mean-square convergence. However, it only projects the samples onto an optimal linear subspace, which results in an unreasonable representation of the original data if they are non-linearly related to each other. In other words, it only preserves the second-order statistics (covariance) of a random field, which is insufficient for reproducing complex structures. This paper applies kernel principal component analysis (KPCA) to construct a reduced-order stochastic input model for the material property variation in heterogeneous media. KPCA can be considered as a nonlinear version of PCA. Through use of kernel functions, KPCA further enables the preservation of high-order statistics of the random field, instead of just two-point statistics as in the standard Karhunen-Lo`eve (K-L) expansion. Thus, this method can model non-Gaussian, non-stationary random fields. In addition, polynomial chaos (PC) expansion is used to represent the random coefficients in KPCA which provides a parametric stochastic input model. Thus, realizations, which are consistent statistically with the experimental data, can be generated in an efficient way. We showcase the methodology by constructing a low-dimensional stochastic input model to represent channelized permeability in porous media.

AN ADJOINT METHOD FOR THE INVERSE DESIGN OF SOLIDIFICATION PROCESSES WITH NATURAL CONVECTION

This paper presents a finite element algorithm based on the adjoint method for the design of a certain class of solidification processes. In particular, the paper addresses the design of directional solidification processes for pure materials such that a desired freezing front heat flux and growth velocity are achieved. This is the first time that an infinite-dimensional continuum adjoint formulation is obtained and implemented for the solution of such inverse/design problems with moving boundaries and Boussinesq incompressible flow. The present design problem belongs to a category of inverse problems in which one is looking for the unknown conditions in part of the boundary, while overspecified boundary conditions are supplied in another part of the boundary (here the freezing interface). The solidification design problem is mathematically posed as a whole time-domain optimization problem. The gradient of the cost functional is calculated using the solution of an appropriately defined continuous adjoint problem. The minimization process is realized by the conjugate gradient method via the solutions of the direct, adjoint and sensitivity sub-problems. The proposed methodology is demonstrated with the solidification of an initially superheated liquid aluminum confined in a square mold. The non-uniformity in the casting product in the direction of gravity due to the existence of natural convection in the melt is emphasized. The inverse design problem is then posed as finding the appropriate spatial-temporal variations of the boundary heat flux on the vertical mold walls that can eliminate or reduce the effects of convection on the freezing interface heat fluxes and growth velocity. The numerical example demonstrates the accuracy and convergence of the adjoint formulation. Finally, open related research design problems are discussed.