Advanced Computational Techniques for Materials-by-design

Modeling of uncertainty propagation in multi-scale models of deformation is extremely complex considering the nonlinear coupled phenomena that need to be accounted for. The ongoing work addresses key mathematical and computational issues related to robust control of deformation processes. Our research accomplishments for this year include development of new mathematical models based on spectral polynomial chaos, support space, and entropy maximization techniques for modeling sources of uncertainties in material deformation processes. These models, in conjunction with multi-scale models, allow simulations of the effect of microstructural variability on the reliability of macroscale systems. We have developed the first stochastic variational multi-scale simulator with explicit sub-grid modeling, and a robust deformation process simulator for simulating uncertainties in metal forming processes. The non-intrusive stochastic Galerkin method developed as a part of the deformation simulator provides highly accurate estimates of the statistical quantities of interest within a fraction of time required using existing Monte-Carlo methods, and with minimal modification of existing deterministic software. The technique has also been applied to enable stochastic optimization of deformation processes. Finally, an information theoretic framework to capture microstructural uncertainties and its effect on macro-scale properties is summarized. 1 Status of effort Substantial progress has been made in the achievement of this AFOSR-Comp Math project objectives in the 3 year of this project. Key developments are listed below: • Development of non-intrusive stochastic Galerkin method for robust modeling of deformation processes[1][2] • Development of continuum stochastic sensitivity method (CSSM) for robust optimization of forming processes[2] • Development of stochastic variational multi-scale model with explicit subgrid modeling for solving multi-scale partial differential equations (PDEs) in random heterogeneous microstructures [3][4] • Development of maximum entropy techniques for modeling topological uncertainties in polycrystalline metallic microstructures and its influence of homogenized properties [5] Particular contributions are briefly summarized below with more details given in the provided references. 1.1 Development of a stochastic framework for analysis of metal forming processes Two distinct approaches are being followed towards modeling uncertainties in deformation processes. In the first technique, Spectral Stochastic Finite Element Method (SSFEM), a spectral expansion of the current configuration of a deforming body is proposed using Legendre chaos expansions to compute the stochastic deformation gradient which is in turn used to compute statistics of several critical fields in large deformation analysis. A stochastic large deformation analysis following this approach is presented in our work in [6]. The second algorithm is based on a finite element representation of the support space of the random variables. This method is particularly useful for capturing instabilities and bifurcations in physical phenomena as demonstrated in [7]. The support space representation can lead to a non-intrusive decoupled as well as intrusive coupled formulation for evaluating the stochastic process. The highlight of the decoupled approach, called Non-Intrusive Stochastic Galerkin (NISG) method, is that it can be directly applied to presently available deterministic codes with minimal effort needed to compute the complete probability density function (PDF) of a stochastic process. In NISG method, the stochastic process is represented over the support space using piecewise continuous orthogonal polynomials in multi-dimensional random variables. The polynomials we choose are locally supported element shape functions used for representing functions in the finite element method. The h and p convergence characteristics of the discretized stochastic domain are identical to spatial finite elements. In Fig. 1, we show a simulation of an isothermal, extrusion operation with a die of average diameter reduction of 14% [1]. The objective of the problem is to ascertain the effect of uncertain die geometry on the steady state distribution of the state variable at the exit. A 9x9 grid is used to discretize the support space. The mean and standard deviations of the state variable distribution at the die exit are plotted in Fig. 1(b&c) respectively. For this problem, NISG method provides highly accurate estimates of the statistical quantities of interest within a fraction of the time required using existing Monte Carlo methods. Our current effort focuses on adaptive methods based on sparse grid interpolants (variants of Smolyak algorithm) to efficiently represent stochastic quantities in large dimensional spaces. This development will allow inclusion of microstructural uncertainties during design of processes and allow reliability assessment in multi-scale models. (a) (b) (c) Fig. 1 (a) Initial and final mean configurations for the isothermal extrusion problem (b) Mean and (c) standard deviation of the state variable at die exit [1] State variable: s (MPa)