Recent progress of uncertainty quantification in small-scale materials science

Abstract

This work addresses a comprehensive review of the recent efforts for uncertainty quantification in small-scale materials science. Experimental and computational studies for analyzing and designing materials in small length-scales, such as atomistic, molecular, and meso levels, have emerged substantially over the last decade. With the advancement in computational resources, uncertainty quantification has started to garner interest in the community. The effects of uncertainties have been found to be critical in numerous studies as they lead to significant deviations on the expected material response and alter the component performance. In the field of small-scale materials science, typical resources of the uncertainties are classified as: (i) inherent material stochasticity (aleatoric uncertainty) associated with processing; (ii) modeling and algorithmic variations (epistemic uncertainty) that arise from the lack of knowledge on the systems/models. The present work reviews the recent efforts in the field and categorize according to various aspects: (i) types of uncertainties, (ii) types of uncertainty quantification problems, (iii) algorithms that are used to study the uncertainties, and (iv) length-scales in different applications. The extensive discussion covers the state-of-the-art and promising future techniques and applications, including the integration of the uncertainty quantification, design, optimization and reliability methods, and uncertainty quantification in advanced manufacturing.

Introduction

Uncertainty Quantification (UQ) is a significant research area for all science and engineering fields as it involves many critical aspects, including the uncertainty representation, uncertainty propagation across different length and time scales, sensitivity analysis, validation and verification for predictive science, and visualization of the uncertainty in high-dimensional design spaces [1], [2], [3], [4], [5], [6], [7]. Accordingly, the UQ has been used as an umbrella term, which refers to the diverse analysis methods and tools that are appropriate for critical assessment of measurements, models, and simulations [8].

The field of UQ has a long history. Early examples of the development include the long-term U.S. Department of Energy program, which aimed to explore the UQ for assessing critical simulations in support of the stewardship of the U.S. nuclear weapons stockpile. Here, the goal was to support the computational models and simulations for the nuclear arsenal by the UQ analysis [9], [10]. The early examples for publications in the field of UQ started in 1986 and included the studies by the American Institute of Aeronautics and Astronautics (AIAA) [11] and American Society of Mechanical Engineers (ASME) [12]. Accordingly, the AIAA and ASME first published editorial policies to report the guidelines for performing convergence analyses on the numerical results of the submitted technical papers [9]. In fact, these guidelines are equivalent to the modern practice on the verification, which is accepted as a cornerstone of UQ [9]. Even though early studies addressed the relevant topics of UQ in general science and engineering fields, the practice of UQ was not broadly articulated within the materials science and engineering community.

In the field of materials science and engineering, the interest towards the UQ has been amplified with the introduction of the Materials Genome Initiative (MGI) by the U.S. Government [13] and Integrated Computational Materials Engineering (ICME) paradigm [14]. Later, the UQ has been identified as a critical research area to support the improvement in the state-of-the-art experimental and computational efforts in the material communities. As a result, the UQ for small-scale (i.e., atomistic, nano, micro, meso) features and volume-averaged properties of materials has been recently recognized as an important research area despite the limited progress in this field [15], [16]. The main objective of the present study is to provide the audience with a comprehensive review of these efforts on the UQ in small-scale materials science, with a particular focus on the types of the uncertainties, algorithms, and problems in different length-scales.

The atomistic-scale applications are extensive and mostly centered around analyzing the uncertainty that affects the material properties of the electronic structure [17], [18], [19]. In particular, most efforts have addressed the Bayesian inference techniques to solve the UQ and propagation problems through the density functional theory (DFT) calculations [17], [20], [21], [22], [23]. The atomistic-level applications have dominantly addressed the effects of the model uncertainties [24], [25], [26], [27], while the analysis of the inherent material stochasticity was also of interest in different works [17], [19].

The molecular-level studies mostly addressed the influence of the parametric uncertainties in predictive simulations [28], [29], [30], [31]. In the field of the molecular dynamics (MD), many previous efforts employed the Bayesian statistics to investigate the variations in the interatomic potentials [32], [33], [34], [35]. Instead of performing the computationally-expensive MD simulations using all targeted combinations of the uncertain variables, one common practice has been determined as the utilization of the corresponding analytical expressions to surpass at least some of these MD simulations [34]. However, this is not possible in complex molecular systems and thus a large number of computationally demanding operations is needed to study the uncertainty problem using numerical algorithms. Consequently, the computationally-expensive nature of the UQ problem has remained as the main cause of the limitations on observing the quantification and propagation of the uncertainty in molecular-scale applications.

The meso-scale UQ problems are mostly concentrated on quantifying the uncertainty in the micro-level (i.e., crystallographic texture, grain topology) and volume-averaged features of the multi-phase and polycrystalline materials. In the case of the UQ of the polycrystalline microstructures, the previous studies were fundamentally based on the stochastic computations to determine the variations in homogenized properties that occur due to the crystallographic texture uncertainty [36], [37], [38], [39], [40], [41]. This material-point problem was solved with traditional UQ techniques, including Monte Carlo Simulation (MCS) [37], [38], [39], [41], [42], [43], stochastic collocation [36], spectral decomposition [40], Kriging [43] and polynomial chaos expansion [23], [44]. Other approaches in this domain include the explicit analytical UQ formulation that was developed by the author [45], [46], [47], [48], [49], [50], [51] to quantify the processing uncertainty and model its propagation on the meso-scale mechanical response by computing the variations in the microstructural texture.

In fact, the aleatoric uncertainty in polycrystalline materials is associated with the unanticipated fluctuations that occur in the stress and thermal fields during thermo-mechanical processing. Computing the variations in microstructural features arising from the thermo-mechanical processing uncertainty is an under-studied problem. It requires a multi-scale modeling strategy that must analyze the effects of the microstructural variations on the material response. However, the multi-scale UQ is still an open research problem when the uncertainties in all orientational (i.e., crystallographic texture, sub-texturing) and morphological (i.e., grain size, grain shape, grain boundaries) features are considered at the same time. With the exponentially increasing problem complexity in the multi-scale domain, the robust numerical techniques, such as MCS, become intractable due to computational time requirements. The application of the computationally-expensive numerical UQ algorithms may still be possible with the use of the simplified multi-scale analysis that does not utilize high-fidelity physical models. Nevertheless, it remains as a challenge for the concurrent multi-scale modeling that requires a two-way communication between different length-scales. Therefore, the implementation of the computationally-efficient UQ techniques and strategies is highly desired to address the concurrent multi-scale problem.

The present work aims to provide an extensive review of the recent progress of UQ in small-scale materials science, by discussing the problem in many different aspects, as explained in Fig. 1 and following:

• Types of the uncertainties (aleatoric and epistemic) that are studied in the field (Section 2);

• Types of the UQ problems (forward and inverse problems) generally addressed in the field (Section 3);

• Types of the UQ algorithms (computational and analytical methods) (Section 4);

• Application problems in different length-scales (atomistic, molecular, micro and meso scales, as well as multi-scale problems) (Section 5).

After the comprehensive discussion of the progress of UQ in small-scale materials science, a discussion on the state-of-the-art techniques and problems, potential future topics and perspective is presented in Section 6. A brief summary of the present work is provided in Section 7.