Traditional deterministic methods applied for the analysis and design of aerospace structures and materials serve as essential tools in engineering design. However, it is critical to account for variability in material performance, which arises from both aleatoric and epistemic uncertainty sources such as material defects, manufacturing inconsistencies, environmental conditions, extreme operational loads, as well as variations arising from computational models and model parameters. Such variations may cause deviations in predicted critical properties, potentially causing mechanical performance degradation and even failures in engineering components used in aerospace, for instance, turbine disks and jet engines. [1] To address these challenges, integrating state-of-the-art uncertainty quantification (UQ) methods is crucial for enhancing the characterization, modeling, and design of aerospace materials, ultimately improving the performance and reliability of aerospace systems.
Aerospace engineering is in fact one of the first science fields with documented studies focusing on UQ. The very first examples of UQ for aerospace engineering include the editorial policies published by The American Institute of Aeronautics and Astronautics (AIAA) in 1986 as guidelines for technical papers on numerical convergence analyses. [2] These guidelines are assumed to be equivalent to modern practices of validation and verification, which are studied under the broad theme of UQ. [1,2] Even though UQ studies covered the general spectrum of aerospace engineering, and it has already been identified as one of the pillars of the integrated computational materials engineering paradigm, [3] its particular applications to aerospace materials have not been articulated until more recently.
The range of UQ studies within the context of aerospace materials covers the development of analytical, numerical, and data-driven techniques to quantify the effects of the stochasticity of material features on expected behavior and properties. Microstructural uncertainties arise mainly from the inherent stochasticity introduced by processing and fabrication techniques, which include unavoidable microlevel fluctuations, defects, tolerances, and other imperfections in manufacturing. The uncertainty associated with manufacturing can also lead to unexpected variations in stress and temperature gradients. [1] These aleatoric uncertainties cause variations in properties, within a specimen, and across multiple specimens fabricated with the same manufacturing technique. [4] Other sources for microstructural uncertainties are related to the assumptions and errors in numerical models and parameters used to characterize microstructure features and material behavior. [5] With the incorporation of the microstructural uncertainty, the processing conditions and/or the material properties demonstrate variations that can be represented with a probability distribution as shown in Figure 1. [6] The propagation of microstructural uncertainty is observed to affect the mechanical performance, ranging from the linear elastic properties [1,4,7] to the fatigue/failure behavior [8][9][10][11] for metallic systems, composite materials, and other structures commonly used in the aerospace industry. Therefore, it is important to consider the effects and propagation of the microstructural uncertainty in the design of aerospace materials. As depicted in Figure 1, [6] this design under uncertainty problem has been addressed via forward design and inverse design formulations, respectively. Within the context of investigating the process-microstructure-property linkages of aerospace materials, the forward design problem corresponds to the identification of the optimum thermomechanical processing parameters that minimize/maximize target microstructural features (e.g., crystallographic texture, grain morphology) and/or target mechanical properties, and the identification of the optimum microstructure features to minimize/maximize target mechanical properties. [1] In contrast, the inverse design problem is defined either in the property space or the microstructure space. When defined in the property space, the inverse problem aims to find the underlying microstructural features and/or thermomechanical processing parameters that provide the prescribed/desired properties. When defined in the microstructure space, it aims to identify the underlying thermomechanical processing parameters that lead to prescribed/desired microstructural features. [1] The investigation of the microstructural uncertainty is more common in the forward problem type compared to the inverse design problem. This is anticipated due to the potential curse of dimensionality issues of the inverse UQ problem. [1] This review article will analyze the UQ problem for microstructures of aerospace materials by examining the linkages between process-structure-property (PSP) relationships. First, the types of uncertainties investigated within the context of aerospace materials are presented. Accordingly, the aleatoric uncertainty arising from the inherently stochastic nature of microstructures and experimental measurements is introduced. Next, the variations stemming from the numerical models and parameters, known as epistemic uncertainty, are discussed. Of particular interest for this article is reviewing the state-ofthe-art studies that analyze the propagation of the aleatoric and epistemic uncertainty from process parameters/features to the microstructural features and multiscale mechanical properties. Moreover, the inverse problem, also known as the inverse design, is investigated while accounting for the effects of aleatoric and epistemic uncertainty within PSP relationships. Application problems studied in this article are limited to those materials widely used in aerospace systems, such as lightweight metallic alloys, composites, and new-generation architected systems. The article also covers future topics of importance identified as multiscale topology optimization (MSTO) under uncertainty, utilization of machine learning, particularly physicsinformed neural networks (PINN), to address forward and inverse UQ problems, the exploration of the effects of uncertainty on the extreme mechanical behavior, reliability-based design (RBD) of microstructures, and UQ in additive manufacturing (AM).
The organization of the article is as follows: Section 2 presents the mathematical background behind UQ problems by discussing the types of uncertainties considered for studying aerospace materials as well as the analytical and numerical methods to quantify these variations. Section 3 addresses the forward UQ problem by investigating the process-structure and structure-property problems separately. Section 4 discusses the inverse design problem, also referred to as materials design under uncertainty, through structureprocess (SP) and property-structure (PS) UQ problems. Section 5 explores the future UQ topics and problems for aerospace materials. A summary of the study is presented in Section 6.
Figure 1. Overview of the forward and inverse design for both deterministic and stochastic models. Reproduced with permission. [6] Copyright 2021, The American Institute of Aeronautics and Astronautics.
This section discusses the mathematics behind the UQ problem for the characterization, modeling, and design of microstructures. First, the types of uncertainties arising from different sources, such as fabrication-related and computation-related variations, will be introduced. This will be followed by a detailed explanation of the mathematical formulations of different analytical and numerical UQ algorithms used for microstructure problems.
The different sources of uncertainty in the microstructures of aerospace materials can be categorized as either aleatoric or epistemic uncertainty. The aleatoric uncertainty is defined as the uncertainty that arises from the inherently stochastic nature of microstructures and the variations that stem from experimental measurement techniques. [1,12] In contrast, epistemic uncertainty is defined as the uncertainty arising from the computational methods or the selection of parameters. [1,13] There is a lack of efforts in the literature dedicated to investigating the sole effects of aleatoric or epistemic uncertainty. Most studies focus on analyzing the effects of uncertainty without classifying the uncertainty type. In this section, a few exemplary studies presenting the sole effects of aleatoric and epistemic uncertainty are outlined. A more comprehensive literature review is provided in the following sections with respect to the types of UQ algorithms and problems.
Aleatoric uncertainty-also known as irreducible uncertaintycan originate from a multitude of sources through the fabrication or testing of microstructures. During the fabrication of materials, small-scale defects and imperfections can occur due to a variety of reasons. Some of these reasons include the stress and thermal fluctuations of the material or the manufacturing environment, [14] variations of material quality in AM, [15] and the fluctuations of product porosity in additively manufactured samples. [16] Another source for aleatoric uncertainty is related to the measurement techniques used to capture the microstructures of these materials. [1] When utilizing data samples, such as the measurements of the crystallographic texture of polycrystalline microstructures through electron backscatter diffraction (EBSD) to capture grain orientations, differences in the red-green-blue (RGB) channel values of pixels, even when they demonstrate the same crystallographic orientations, can be caused by temperature/stress fluctuations occurring during processing. [12] Epistemic uncertainty-also known as reducible uncertaintyis related to the variations in the mathematical/numerical models that characterize microstructures and the selection of parameters for those models. Preliminary efforts solely focusing on the identification of the epistemic uncertainty related to the synthetic microstructure reconstruction are presented by Senthilnathan et al. [4,7] in addition to other studies implementing machinelearning-based models to analyze the effects of epistemic uncertainty. [17]
This section explores various algorithms for quantifying and propagating uncertainty, with a particular focus on modeling and designing the microstructures of aerospace materials. While many approaches exist for quantifying and propagating uncertainty across different scales, as classified and illustrated in Figure 2, this section emphasizes the most notable methods that are particularly relevant to the context of this article. Although other UQ methods have been studied in computational mechanics, they are not covered here because they are less commonly applied to the UQ of microstructures in aerospace materials.
The analytical approach is a viable strategy as it can provide exact solutions for UQ problems under some assumptions. However, the applicability of the analytical method highly depends upon well-defined random variables that need to be characterized by explicit probability distribution functions (PDFs). The analytical approach is more commonly applied to study the UQ problem when the random variables are assumed to be Gaussian distributed while it has also been extended to study the propagation of uncertainty arising from random variables defined through non-Gaussian probability distributions. [18] First consider the definition of two Gaussian-distributed random variable vectors, P and A. Assume that they have a correlation coefficient, ρ, and their mean and standard deviation vectors are denoted as (μ P , Σ P ) and (μ A , Σ A ), respectively. If another variable, Z, directly depends on P and A through the relation Z = PA, the mean can be expressed as μ Z ¼ μ P μ A and standard deviation vectors can be represented by the relation as shown in Equation (1), where the square terms represent square of individual matrix elements. [18] Σ 2
The analytical approach can be extended to model any random variable demonstrating Gaussian and non-Gaussian distributions with the use of the transformation of random variables rule. [19] In accordance with this approach, X and Y are defined as input and output parameters, respectively. The relationship between them can be established such that Y is equal to some function of X, denoted as Y ¼ gðX Þ, and this map can be reversed as X ¼ hðYÞ, where h corresponds to the inverse map (g À1 ). Based on this explicit relationship, the methodology calculates a Jacobian, J = h 0 ðYÞ (where J is equal to the derivative of h with respect to Y ), and determines the PDF of the output variable as a product of the Jacobian and the PDF of the input. The probability distribution of the output parameter can be calculated using Equation (2). Here, f Y and f X represent the PDFs of the output and input variables, respectively. An essential advantage of the transformation of random variables rule is that it is applicable to random variables represented with continuous PDFs. The underlying assumption is that the mapping between the input and output parameters should be explicitly accessible and it should be invertible. Given the prior knowledge of the input PDF, f X , and the inverted mapping function, h(Y ), the output PDF can be determined using Equation (2). Subsequently, employing statistical integrals using the output PDF, the variances (var(Y )) and the expected values (E[Y]) of the output parameter matrix can also be evaluated, as demonstrated in Equation ( 3) and (4), respectively, where the minimum and maximum values of the output variable are denoted as Y min and Y max , respectively. [19] f Y ðYÞ ¼ f X ðhðYÞÞjJj
(2)
Acar et al. [6,[20][21][22][23] developed the analytical UQ method and applied it to microstructure design to study the propagation of the microstructural uncertainty over homogenized (mesoscale) material properties. More specifically, an orientation distribution function (ODF) was defined to represent the crystallographic texture of Ti-7Al alloy, a metallic alloy used for jet engine components. The variations of the ODFs were defined as a joint multivariate Gaussian probability distribution using the experimental pole figure data and through the application of the characteristic functions method. [23] The propagation of the microstructural uncertainty on the homogenized elastic modulus was studied using the homogenization formulation and transformation of random variables. In another study, [20] an inverse problem centered on investigating the microstructural stochasticity amid fluctuations in mesoscale material characteristics was studied by proposing an analytical UQ model, AUQLin, which can compute the variations of ODFs given the probability distributions of homogenized properties. Additionally, the analytical UQ method was extended to account for boundedness to represent microstructural variations, even when assuming a Gaussian distribution, with the inclusion of further design constraints for volume normalization and the non-negativity of ODFs defining the upper and lower bounds, respectively. These constraints introduce additional equations related to the mean and covariance that must be considered during the solution process. Further details on this formulation can be found in the referenced articles. [18,23]
The Monte Carlo sampling (MCS) method, widely employed for UQ problems, involves generating multiple samples of the input random variables based on their probability distributions. The output values for each sample are obtained with an available deterministic solution. These output values are then aggregated to statistically evaluate the variability of the output variables. This is typically achieved by generating random values, y 1 , y 2 …, y n , sampled from the distribution of Y, and then calculating the average of Y from Equation ( 5), mean and variance of ŝn from Equation ( 6) and ( 7). [24] The MCS estimator remains unbiased as the anticipated value of ŝn ¼ s. As per the strong law of large numbers, [25] the mean converges to the genuine expected value found from Equation (8). The error of the MCS predictor is proportional with n À1=2 where n represents the total number of samples. σn À1/2 represents the root mean square error of ŝn . One notable advantage of employing the MCS method is its capability to utilize sample values for error estimation. [26] The prevailing practices for estimating σ 2 , the variance, are employed in Equation ( 9) and (10). According to the central limit theorem, the error of ðs À ŝn Þ follows a roughly Gaussian distribution, centered at 0 with the variance value of σ 2 /n. Consequently, confidence intervals can be computed around the estimate, ŝn .
With a sufficient number of samples, the MCS method can provide a comprehensive depiction of the statistical characteristics of the system. [1] The MCS method finds wide-range applications in the realm of microstructure modeling. Noii et al. [27] established a mathematical framework to model uncertainty by employing a stochastic microstructure analysis for brittleductile failure. They developed the strategy for allocating heterogeneities and generating corresponding meshes in both 2D and 3D scenarios. Subsequently, the Monte Carlo finite-element method (MC-FEM) was utilized to build a stochastic partial differential equation (PDE) model, approximating the expectation and variance of the solution field for brittle-ductile failure through the evaluation of a substantial number of samples. In ref. [28], MCS was employed to explore the propagation of uncertainty within a multiscale model, encompassing nano-scale to mesoscale features of single-walled nanotubes to yield insights into the mesoscale Poisson's ratio and Young's modulus values and their variations across various orientations. Meanwhile, Rouhi et al. [29] utilized the MCS method to derive the effective matrix properties of a nanocomposite material, shedding light on the elastic properties of individual laminae composed of matrices with uncertain elastic parameters. Similarly, Hafiychuk [30] conducted a sensitivity analysis by incorporating uncertainty regarding fiber volume fractions and ply orientations for laminated composite materials through the MCS method, demonstrating the significant influence of initial uncertainty assumptions on macroscopic elastic properties. Monti et al. [31] employed magnetron sputtering parameters, including voltage, pressure, substrate orientation, and working distance, to compute the trajectory of atoms and energy reaching the substrate through MCS simulations conducted with the SiMTRA software. This study accurately depicted how the microstructure varies with deposition conditions and elucidates the correlation between process parameters and typical thin films. The utilization of multilevel sampling techniques, including multi-index Monte Carlo (MIMC) and multilevel Monte Carlo (MLMC), was demonstrated in evaluating the influence of microstructural variations in polycrystalline materials on the forecasts of homogenized material characteristics by Tran et al. [32] Recently, the UQ community has introduced innovative approaches to tackle the computational hurdles associated with MCS-based methods for uncertainty propagation. By expanding upon the notion of control variates, MLMC [33][34][35][36][37] and multifidelity Monte Carlo [32,[38][39][40][41] techniques have been devised. These methods capitalize on the hierarchical structure of models to strike an optimal balance between variance and bias estimates. Their objective is to furnish more precise MC estimations within predefined computational constraints.
The maximum entropy (MaxEnt) principle asserts that the most accurate probability distribution reflecting the knowledge about a system maximizes entropy, given specific prior data or testable information. Typically, the conserved quantities consisted by this prior data, which are mean values of regarding certain moment functions of the PDF under consideration. This application is prevalent in statistical thermodynamics. [42] Alternatively, symmetries of the PDF can be specified. The relationship between symmetry groups and their corresponding conserved quantities suggests a parallel relationship between these two methods of defining testable information within the MaxEnt methodology. [42] The associated entropy for the probability distribution can be expressed as in Equation (11) where P i pðA i Þ ¼ 1 because the sum of all probabilities (p) must be 1 where A denotes all possible states. When addressing complex physical systems, characterized by a vast array of states and consequently a high entropy measured in numerous bits, it becomes practical to scale the summation by the Boltzmann's constant, denoted as k B (1.381 Â 10 À23 J K À1 ). Additionally, employing natural logarithms instead of base 2 logarithms is advantageous. [42] Subsequently, the entropy could be expressed in Equation (12). However, when dealing with a specific quantity within a system, it is crucial to recognize that each state within that system contributes its own distinct amount. The determination of the expected value involves aggregating these state-specific values, weighted by their respective probabilities. [43] For instance, if we denote a quantity G with each state having a value represented by g(A i ), the focus lies on probability distributions where the resultant expected value aligns with G, as represented in Equation ( 13). [42] S ¼ X
A method rooted in information theory was introduced by Sankaran and Zabarasto [44] to measure the impact of uncertainty on properties, particularly at smaller scales. This approach involved the representation of uncertainties through a diverse set of microstructural samples generated via the MaxEnt method. The microstructures were considered stochastic, and the distribution's entropy was maximized under constraints imposed by available microstructural data. The derivation of a MaxEnt fracture model was studied by Chan et al. [45] using the principles of information theory and statistical thermodynamics. Leveraging the MaxEnt principle allows for forecasting the lifespan of a structure amid microstructural variability. This model, characterized by a single parameter, established a correlation between the likelihood of fracture and the accumulated dissipation of entropy at a specific material location. Guilleminot et al. [46] introduced a MaxEnt approach to modeling the apparent elasticity tensor of heterogeneous elastic materials. This methodology enables nonparametric probabilistic modeling while ensuring that classical constraints, such as normalization and invertibility, are satisfied. Additionally, the approach takes into account constraints related to boundedness and material symmetry properties, providing a more comprehensive understanding of the material's behavior. Sobczyk [47] explored the probability distributions of key topological and metric characteristics of random aggregates of grains or cells, focusing on their shape. Drawing from classical statistical mechanics, they derived these distributions using methods akin to those found in Gibbsian ensembles. Employing the method of maximum information entropy, these probability distributions and numerical calculations were conducted for various intriguing scenarios. A novel stochastic micromechanical framework, utilizing MaxEnt principles, was introduced by Zhu et al. [48] This framework aimed to capture the probabilistic nature of effective properties in two-phase composite materials, accounting for interparticle interactions.
A random matrix refers to a matrix where one or more of its elements are random variables, essentially making it a matrixvalued random variable. In 1968, Balian introduced the use of random matrix theory alongside an entropy-based approach in their work on "Random Matrices and Information Theory". [49] The idea of applying random matrix theory to model uncertainty in mechanics was later pioneered by Soize [50][51][52] and was further extended to stochastic topology optimization to enhance the first-order reliability method (FORM) within the domain of reliability-based topology optimization (RBTO) by Vishwanathan and Vio. [53] The given matrix denoted as K demonstrates the uncertainty. Subsequently, the stochastic system is characterized by a mean of the matrix, K o , alongside a mean-normalized standard deviation, σ 2 K , also termed as the dispersion parameter which can be defined by Equation (14). The E(.) and ||.|| F operations represent the mean value and Frobenius norm. The distribution's characteristics remain on two pivotal factors:
where the number of degree of freedom, covariance tensor of K, and the theoretical mean matrix value are denoted by n, Σ and α, respectively. These parameters play a direct role in determining the theoretical properties of both the mean matrix and covariance tensor linked to K. In a comprehensive way, an augmentation in the value of either parameter corresponds to amplified magnitudes within the anticipated mean matrix, denoted as E½K ¼ αΣ, alongside elevated values within the covariance tensor. Consequently, this correlates with heightened fluctuations in the entries of the matrix. [54]
Guilleminot et al. [55] proposed a framework and emphasized the utilization of Huet's partition theorem for modeling mesoscale heterogeneous microstructures. By applying the MaxEnt principle under the constraints of the available data, they introduced two models of random matrices. Furthermore, they showed that reinterpreting the boundedness constraints provides a way to construct a probabilistic model for matrix-valued random fields that are both deterministically bounded and positive definite. The experimental, parametric, and nonparametric errors in aerospace structures analysis were quantified using random matrix theory by Vishwanathan and Vio. [56] The Wishart distribution and random matrix theory were employed for describing the modal uncertainties where the FEM for numerical investigations were used for both an aircraft wing box and a rectangular wing to verify the applicability of random matrix theory and the density function of Marchenko-Pastur. It was demonstrated that the stochastic frequency response shift can be determined by the nonparametric dispersion parameter. This finding proved the capability of random matrix theory, showing that it can effectively quantify the lower bound of unknown sources of error. However, its flexibility is noteworthy as it maintains consistent computational performance and robustness across various correlation lengths and random field variances, as evidenced by numerical experiments. [53,54] Li and Steeves [54] developed a robust structural design model that accounts for uncertain modulus of elasticity. They used random matrix theory and the compliance topology optimization method to calculate the propagated uncertainty in stiffness. Additionally, this method was compared with two well-established approaches: stochastic collocation (SC) and Monte Carlo robust topology optimization. The primary disadvantage of the presented method was that plane-stress bilinear elements yield less accurate results, ultimately producing smaller values compared to established methods. In both material design and structural dynamics, random matrix theory provides a systematic approach to handle randomness and correlations, leading to more informed decision-making, improved designs, and enhanced structural performance. [57] Its importance reclines in its capability for connecting the practical applications and the theoretical modeling in complex systems governed by randomness.
The Bayesian methodology deals with generating forecasts grounded in preexisting knowledge. [58] Contemporary Bayesian computation has become viable with the advent of the Metropolis-Hastings algorithm, providing the capability to fit Bayesian models without the constraint of relying solely on conjugate priors. [58] The foundation of Bayesian methods rests upon two fundamental principles such as establishing the prior PDF for the random variables, and deriving the posterior PDF for the same variables. [1] The prior probability distribution PðθÞ encapsulates expert/historical insights prior to the collection of any new data. This method facilitates the updating of the earlier PDF of model parameters to a posterior PDF, integrating additional information from the model's application and measurements (D) to the investigated system. The conditional probability definition is utilized by Bayesian update, as depicted in Equation (15). The posterior parameter distribution is determined based on the conditional probability of the investigated data of the parameter vector θ, expressed as PðDjθÞ, and the prior distribution, denoted as PðθÞ. The likelihood function of the model, often referred to in statistical analyses, represents the distribution function. In statistical assessments, it is common to refer to this function, which hypothesizes a normal distribution of residuals between the model and observations. This hypothesis assumes a null average and variance of σ 2 e , establishing a basis for analyzing the statistical relationships within the model. Under this assumption, PðDjθÞ can be expressed multiplicatively as depicted in Equation ( 16), where Y i represents the model responses corresponding to a specific variable's available measurements D i of within a particular cross section of a system. The application of Equation ( 16) relies on the assumptions that residuals exhibit homoscedasticity, independence, and identical distribution over time. [59] PðθjDÞ ¼ PðDjθÞPðθÞ R PðDjθÞPðθÞdθ (15)
In recent times, there has been a notable surge in attention toward Bayesian methods within the domain of UQ concerning the characterization of multiscale materials. [59][60][61] Moreover, Olivier et al. [62] explored techniques for the Bayesian learning of neural networks (NNs), which has offered a comprehensive approach to handling both aleatoric uncertainties, stemming from the inherent randomness in the data generation process, and epistemic uncertainties, arising from limited data availability. The proposed algorithms leverage approximate variational inference and (pseudo-) Bayesian model averaging to strike a balance between precision in uncertainty estimation and computational efficiency. Kim et al. [63] introduced a novel Bayesian learning framework designed to swiftly predict and quantify uncertainties in the physical behavior of multi-material components produced through AM. This Bayesian learning model demonstrated the rapid and accurate beam deflection prediction with any meso-structure when trained with experimental and simulation data from specific multi-material designs. Recently, a physics-informed Bayesian neural network (BNN) has been modeled by Li et al. [64] that combines the physically consistent predictions and fundamental governing law of materials. To assess its efficacy, they demonstrated case studies focused on prognosing the life of steel alloys under the creep loading condition.
The Gaussian process (GP) regression (GPR) has emerged as a prominent approach in tackling forward and inverse UQ problems within materials modeling. Its methodology attracts significant interest owing to its ability to integrate information from various fidelity levels, including estimations, measurements, and simulations. This multifaceted data utilization enables the development of surrogate models that strike a delicate balance between precision and computational speed. Essentially, the essence of this multi-fidelity modeling approach lies in leveraging high-fidelity information to construct an accurate surrogate model while incorporating computationally efficient (yet typically low-fidelity) data. The prime supposition must be that the multifidelity model behaves as a GP with ỹðxÞ∶ Nðμ ỹ, σ 2 ỹ Þ where parameters of the multi-fidelity prediction (ỹ), which are μ ỹ and σ 2 ỹ , represent the mean and variance, respectively. [65] If x and x denote the input data points for the high-fidelity and low-fidelity models, respectively, the covariance expression relies on the kernel function, k. This covariance may be calculated using several kernel definitions, such as squared exponential, exponential, rational quadratic, polynomial, linear, and constant formulations. [65] The objective of employing a multi-fidelity modeling methodology lies in determining the value of Y à ¼ f ðX à Þ, wherein X à represents the intended input for computing the response of the multi-fidelity model, denoted as Y à . [1] The associated covariance matrices can be characterized using Equation ( 17)- (19). It is crucial that the relationship between the variables agrees with the condition: m < r and n ¼ m þ r, where r, m, and n denote the number of low-fidelity, high-fidelity, and all design samples. Consequently, the approximated response of the multi-fidelity model can be expressed through Equation (20). To achieve a comprehensive model representation incorporating multiple levels of fidelity, it is essential to optimize the covariance approximations defined in terms of hyperparameters. A model objective function, as exemplified by Equation ( 21), [66] serves as a guiding framework. This function encompasses a general error definition aimed at refining the hyper-parameters. The problem statement for error computation is adaptable to address specific requirements. Within this case, the objective function aims to minimize disparities between the GP predictions, denoted as Y i à , and the corresponding outputs from high-fidelity models, denoted as Y i , across all samples of available high-fidelity data (where i ranges from 1 to m).
Acar [66] introduced a computational approach built on GPs to facilitate multi-fidelity modeling in crystal plasticity, specifically targeting Ti-7Al alloy, by employing computational methodologies that yield two distinct levels of solution fidelity. The outcomes obtained through the proposed multi-fidelity scheme demonstrated a marked enhancement in the precision of the crystal plasticity modeling of Ti-7Al alloy compared to the findings derived from prior low-fidelity solutions. [66] An experimental investigation was focused on characterizing the microstructures of 27 samples aged at various combinations of temperature and time exposures conducted by Yabansu et al. [67] Among these, 23 samples were specifically utilized to train a GPR model. This model was designed to take the aging temperature and duration as input parameters and forecast the statistical properties of the microstructure as output. Saunders et al. [68] proposed a technique for connecting the morphology of microstructures to their mechanical properties where the functional GP surrogate models within a directed graphical network were utilized. This framework enabled the accurate prediction of mechanical properties in near real time, with errors typically within single-digit magnitudes when forecasting complete stress-strain histories for a specific microstructure.
Often, the physical properties of materials in multiscale modeling remain semi-bounded or bounded, making it mathematically difficult to employ the Gaussian model directly. [69] Even if the models are capable of estimating data to some extent, their accuracy is generally limited to specific scenarios involving single variables or pairs of variables. This limitation necessitates the building of stochastic models for non-Gaussian fields. The intended purpose and specific application determine the choice of model, and then the quality assessment can be done using any appropriate validation metric. A probabilistic approach has been introduced by Guilleminot for multiscale material science and mechanics of materials, which can handle non-Gaussian random fields. [69]
Polynomial chaos expansion (PCE), proposed by Wiener, [70] offers a technique to express a random variable using the polynomial function of random variables. These polynomials are chosen to be orthogonal concerning the joint probability distribution of random variables. The Hermite chaos expansion has demonstrated its efficacy in tackling stochastic differential equations involving both some non-Gaussian and Gaussian input random variables. [71,72] However, while it excels with Gaussian inputs, its convergence rate with non-Gaussian inputs remains elusive. [73] The Wiener-Askey PCE methods were proposed to address this limitation and accommodate a broader range of random input variables with non-Gaussian distributions. [73] This technique serves as the foundation of the widely adopted PCE methodology in various scientific and engineering applications for stochastic systems. The PCE technique endeavors to construct a surrogate model with a predetermined structure comprising a finite series of orthonormal polynomials. These polynomials are characterized by coefficients derived from test or simulation data. [74] This methodology employs a designated set of functions, with the choice of orthonormal functions contingent upon the assumed probability distribution of the stochastic input variables. Ref. [73] provided an overview of the types of orthonormal basis polynomials tailored to various common probability distributions. When opting for foundational functions derived from the PDF of random input variables, the ensuing formulations of Equation ( 22) emerge to depict the inherent uncertainty within the system where the order of the chaos expansion (p) and the number of random variables (n) play pivotal roles in determining the complexity of the orthonormal basis functions utilized within the PCE method. Once the model expression, as outlined in Equation ( 22), is established, the task involves solving for the unknown chaos coefficients (denoted as a i , where i ranges from 1 to p). Commonly employed techniques for solving these coefficients include the least-square algorithms and the least-angle regression. [75,76] Moreover, the PCE method specifies a minimum number of samples necessary to train the algorithm effectively in determining these coefficients, as described by
Here, Hermite polynomials of order n and general secondorder random process are symbolized by H ij and u i , respectively.
The research on investigating the microstructures of aerospace materials has not embraced the PCE techniques extensively. One prominent application includes the study by He et al. [77] investigating the effects of uncertainty on multiscale mechanical behavior using a data-driven model of a concrete material. A generalized PCE framework to analyze uncertainty propagation in laser-based powder bed fusion AM was introduced by Tapia et al. [78] The PCE approach was utilized to evaluate the impact of uncertainties in the input parameters on the distribution of melt pool dimensions. The methodology was validated by comparing its results with those obtained from Monte Carlo simulations and experimental data from a test bed. García-Merino et al. [79] developed a surrogate model using adaptive PCE to efficiently evaluate and analyze the uncertainty of a composite material's elastic properties, derived from 2D and 3D numerical homogenization. To reduce the computational costs and improve the model's robustness, they employed adaptive sparse expansions in conjunction with the least-angle regression algorithm. [80] This method identifies the optimal polynomial order automatically in the PCE by applying an approach of model selection for sparse linear models. Moreover, few studies have been conducted to analyze the effects of macroscale uncertainties on structural behavior using the PCE method. [81][82][83]
Functioning as a mechanism for dimensional reduction, the Karhunen-Loeve (KL) expansion embodies the notion of random fields to depict quantities evolving across space and time. By mitigating the complexities associated with high-dimensional systems, particularly the curse of dimensionality, dimension reduction proves to be a practical and effective tactic for addressing numerical intricacies in analysis. The KL expansion is a type of general orthogonal series expansion. In this expansion, the function f(x) can be expressed using a comprehensive set of normalized and orthogonal basis functions, φ(x), along with corresponding coefficients, b, as depicted in Equation (23). The condition of the uncorrelated factors could be defined by Equation (24). [84] The expression for the eigenvalues (λ i Þ of corresponding eigenfunctions (φ i Þ are given by Equation (25) where μ denotes the mean of the factors b and temporal and the spatial coordinates are represented by x and y, respectively. [72,85] The KL expansion emerges from a sequence of eigenvalues and corresponding eigenfunctions, culminating in the expression provided next in Equation (26) where the uncorrelated random variables and projected field are denoted by ξ i and wðxÞ, respectively. [74] f ðxÞ ¼ b
While the utilization of the KL expansion remains uncommon in addressing UQ challenges within multiscale materials science, a notable application was outlined by Wen et al. [86] They introduced a novel data-driven bi-orthogonal approach to KL expansion aimed at analyzing and capturing the multiscale propagation of uncertainty arising from variability in the initial microstructure of face-centered cubic (FCC) nickel. The study implemented a second-level KL expansion to distinguish between spatial and random coordinates and demonstrated a significant reduction in the complexity of the stochastic problem. Allaix and Carbone [87] developed a method of integration of the KL series expansion with the FEM enabling the discretization of random fields to address reliability concerns, offering a means to forecast the reliability index with heightened precision in discretization. KL method has been implemented by Feng et al. [88] in a new algorithm designed to reconstruct random media, including rocks, composites, and alloys. This algorithm effectively transforms GRFs to align with specific probability distributions and correlation functions without the need for iterative computations, making it suitable for managing large datasets. In another study, [89] the KL method was combined with Hurst and detrended-fluctuation analyses for statistical fluctuation analysis, and with minimal cover and box-counting techniques for fractal analysis. The KL transformation, along with other techniques, such as principal component analysis, NNs, and Gaussian classifiers, was used to process the curves derived from these analyses. Remarkably, the KL transformation, when paired with NNs, achieved nearly 100% accuracy in classifying various microstructures, performing exceptionally well in both training and testing scenarios.
Uncertainty propagation in the context of the forward PSP problems involves quantifying and analyzing how uncertainties in various stages of material processing and characterization propagate through the material system, influencing the final properties. This includes uncertainties in processing parameters, measurement techniques, microstructural features, and model assumptions. The forward problem can be divided into two subproblems: 1) process-structure problem and 2) structureproperty problem.
Understanding uncertainty propagation is crucial for ensuring the reliability and accuracy of predictions on material properties using process and structure data. In this section, state-of-the-art studies addressing the forward UQ problem of predicting stochastic material behavior are explored. The section also unveils the methodologies employed to understand UQ problems, facilitating the design of materials with improved reliability and performance.
Understanding the mechanical properties of aerospace materials involves navigating the intricate path from the manufacturing processes to the resultant microstructures. This problem, commonly referred to as the "process-to-structure problem," is a critical component in the forward uncertainty propagation paradigm. In this subsection, the challenges and methodologies associated with linking manufacturing processes to the microstructural features that dictate the mechanical behavior of aerospace materials are explored.
The first step in addressing the process-to-structure problem is acknowledging the intricate relationship between manufacturing processes and resulting material microstructures. Aerospace materials, subject to diverse manufacturing techniques such as casting, forging, and AM, undergo complex transformations that significantly influence their final characteristics.
Manufacturing processes often involve the coupling of multiple physical phenomena. Modeling these interactions, such as the interplay between thermal and mechanical aspects during forging, introduces additional complexity. Uncertainties in one aspect can cascade and amplify throughout the process, leading to challenges in predicting microstructural outcomes. Greene et al. [90] introduced a generalized uncertainty propagation criterion derived from benchmark studies, offering a conceptual framework for understanding how uncertainties propagate through microstructured material systems. Their work has established the foundation for subsequent analyses by providing insights into the interdependencies of various manufacturing parameters. Olivier et al. [62] contributed to the literature by emphasizing the role of computational models, particularly BNNs, for UQ in data-driven materials modeling. By integrating machine-learning techniques, they developed a powerful approach to simulate and predict uncertainties in microstructures based on empirical data, enhancing the accuracy of predictions. Spectral methods also play a crucial role in UQ. For instance, Le Matre [91] presented applications of spectral methods to the problems of uncertainty propagation and quantification in model-based computations. They specifically focused on computational and algorithmic features of these methods which are most useful in dealing with models based on PDEs. Tapia et al. [78] focused on uncertainty propagation in computational models for laser powder bed fusion AM. Leveraging PCE, the study provided valuable insights into the uncertainties associated with AM processes. This exploration was crucial for understanding how variations in the manufacturing process manifest as uncertainties in microstructures. The evolution of material properties throughout manufacturing stages is a dynamic process influenced by various factors. Ghasemi et al. [92] contributed to exploring the uncertainties in metamodel-based probabilistic optimization of CNT/polymer composite structures. Their study underscored the significance of stochastic multiscale modeling in understanding uncertainties and optimizing composite materials. They provided valuable information on how uncertainties propagate through different stages of microstructure evolution.
The journey from process to structure in microstructured materials is riddled with challenges, demanding a nuanced approach to address multi-physics interactions and the acquisition of high-quality data for effective data-driven approaches. Attari et al. [93] contributed by exploring uncertainty propagation in a multiscale CALculation of PHAse Diagrams (CALPHAD) reinforced elastochemical phase-field model. This study shed light on the intricacies of integrating CALPHAD and phase-field modeling, addressing the challenges associated with multiphysics interactions. Multi-physics interactions, involving the simultaneous consideration of heat transfer, fluid dynamics, and solid mechanics, demand advanced modeling techniques. [90,93] The complexities arising from these interactions introduce uncertainties that significantly impact the reliability of predictions. Researchers have explored various methodologies to navigate these complexities, providing insights into potential solutions and strategies for improving the accuracy of models in capturing multi-physics phenomena. [94] In addition to multiphysics interactions, the challenge of obtaining high-quality data for data-driven approaches remains a focal point. Data-driven models, particularly those leveraging machine learning, heavily rely on robust and diverse datasets. Significant research has been done in addressing the importance of data quality and suggesting strategies for enhancing data acquisition, thereby strengthening the foundations of data-driven approaches for UQ in microstructured materials. [62,93,95,96] The future of UQ of microstructural features within the context of process-structure forward problem lies in the pursuit of innovative computational techniques, the exploration of novel experimental methodologies, and the seamless integration of these approaches. By addressing current challenges and pushing the boundaries of methodology, researchers can pave the way for more accurate predictions of material properties in aerospace applications. Some researchers have recommended that incorporating advanced numerical methods and simulation algorithms could enhance the fidelity of models, providing more accurate representations of microstructure evolution and its associated uncertainties. [97] The integration of physics-based models with experimental data also emerges as a crucial direction for future research. The synergy between computational predictions and experimental observations can enhance the reliability of UQ. Novel experimental techniques, as highlighted by Honarmandi et al. [97] can provide high-fidelity data for the validation and refinement of computational models, contributing to a more holistic understanding of uncertainties in microstructured materials.
This section addresses the forward problem of uncertainty propagation, with particular emphasis on the impact of microstructural parameter uncertainties on homogenized material properties. The objective is to investigate advanced methodologies for quantifying and propagating these uncertainties through computational models, thereby enhancing the understanding of the complexities between microstructure and mechanical properties in aerospace materials.
The comprehension of material behavior at the microscopic level is important for the optimization of aerospace-grade materials. Polycrystalline microstructures in the aerospace industry are described by features such as textural orientation, grain size and boundaries, phase distribution, defects, and other constituents, which dictate mechanical properties. [98][99][100][101] For composites, microstructural features such as fiber-matrix interactions, fiber orientation, inter-facial bonding, and void content significantly influence mechanical properties such as strength, stiffness, and fatigue resistance. Moreover, the adoption of AM in aerospace faces challenges due to limitation of material options, variability in material properties, and rigorous post-processing and quality control standards, such as Aerospace SAE AS9100 and NASA Standard 6030. [102] These standards are imminent for mitigating risks linked to the uncertainties in AM processes, especially concerning mechanical and fatigue properties. Effective management of these uncertainties requires enhanced quality controls, refined post-processing techniques, and the development of new standards to improve the reliability of AM aerospace components. [102,103] Most UQ studies focusing on the mechanical behavior of microstructures fall within this category. Therefore, this subsection provides an exhaustive review of the relevant literature. One of the prominent studies in the context of the structure-property problem includes Ostoja-Starzewski's research [104] that utilized random field theory to model spatial variability in heterogeneous materials such as composites and polycrystalline metals, where microstructural characteristics, e.g., grain size and phase distribution, significantly influence macroscopic mechanical properties. Through statistical homogenization and effective medium theory, the research demonstrated that macroscopic properties, such as stiffness, are highly sensitive to variations in microstructural features, particularly as the correlation length decreases. These findings demonstrate the critical role of spatial correlations in predicting the overall mechanical behavior of materials, where microstructural heterogeneity directly affects performance. Building on this, Ranganathan and Ostoja-Starzewski [105] explored how disparities in crystal structure affect macroscopic elastic properties, introducing the universal elastic anisotropy index (A U ) index, a measure of elastic anisotropy across crystal symmetries. Their research highlights how variations in crystal symmetry and orientation influence stiffness and shear response, emphasizing the intricate structure-property relationship in complex crystalline materials.
Furthermore, Ostoja-Starzewski [106] employed lattice models to simulate mechanical responses such as elasticity, micropolar elasticity, and crack propagation in composites, metals, ceramics, and polymers. By discretizing the microstructure into spring networks, the study demonstrated how the internal geometry of the microstructure governs mechanical properties, such as Young's modulus and Poisson's ratio. The examination of spatial randomness [107] further identified how local variability in microstructures, captured through statistical volume elements (SVEs), affects stiffness and strength at larger scales, providing critical insights into the link between microstructural randomness and macroscopic material properties. Additionally, the analysis of scale effects in random media, [108] particularly in plasticity, showed how traditional representative volume elements (RVEs) are insufficient to capture small-scale randomness, leading to variability in strain localization and yield strength.
Tootkaboni and Graham-Brady [109] developed a computational framework that links microstructural randomness in composites to their macroscopic mechanical behavior using spectral stochastic methods combined with homogenization theory. This framework models how uncertainties in material properties propagate through the microstructure, affecting elastic properties and overall mechanical performance under varying loading conditions. In related work, Xu and Graham-Brady [110] used stochastic RVEs and emphasized the importance of capturing local stress and strain variations in materials with stochastic elastic properties, demonstrating how these local variations significantly influence macroscopic mechanical responses, e.g., strain localization and failure.
Baxter and Graham's moving-window technique [111] was introduced for characterizing random composites and was found to enhance the predictive accuracy of mechanical models by analyzing local material property variations. This method enables the probabilistic characterization of elastic modulus and yield stress, providing valuable insights into how local stress variations influence mechanical performance.
Grigoriu's research on stochastic frameworks for multiscale material responses highlights the discrepancies between microand macroscale material properties due to microstructural variations. Using stochastic differential equations, Grigoriu demonstrated how local heterogeneity, modeled through random fields, influences global properties such as conductivity and stiffness. [112] The development of finite-dimensional models further refines this approach, showing the need to approximate microstructural randomness to capture essential material features, especially in heterogeneous materials like random fiber systems. [113] Staber and Guilleminot [114] also contributed to stochastic modeling in the context of the structure-property problem by focusing on the behavior of elasticity tensors in heterogeneous fiber-reinforced composites. By incorporating spatial correlation lengths and coefficients of variation, the model captures how local elastic properties propagate to influence macroscopic mechanical behavior under uncertainty. This framework is essential for materials with complex microstructures, where local randomness significantly affects global performance. The researchers conducted another study on the stochastic modeling and identification of hyperelastic constitutive models [115] which highlights how random fluctuations at the microstructural level influence macroscopic mechanical behavior in laminated composites, particularly under complex nonlinear deformation states. Guilleminot's work on bounded elasticity tensors [55] further addressed how local fluctuations in elasticity propagate across scales in polycrystalline materials, offering a more accurate prediction of mechanical responses by integrating the physical bounds of elasticity tensors. Additionally, Chen and Guilleminot's research [116] into nonlinear elastic materials, especially carbon-epoxy laminates, investigated how spatial uncertainties in material parameters impact global mechanical performance under nonlinear deformation. By integrating experimental data into non-GRFs, the research exhibited how randomness in hyperelastic strain energy functions affects the material's mechanical behavior.
Koutsourelakis and Deodatis [117] developed a method for simulating binary random fields applied to two-phase random media, thus capturing the randomness in microstructures such as fiber-reinforced composites and porous materials. This method provided a robust framework for predicting macroscopic properties based on stochastic variations within the microstructure. Agrawal and Koutsourelakis's probabilistic closure models [118] integrated aleatoric and model uncertainty, underscoring the importance of incorporating microstructural variations to accurately predict macroscopic material response.
The study by Senthilnathan and Acar [4] employed the Markov random field (MRF) algorithm to generate large-scale digital representations of metallic microstructures from small-scale experimental data. However, the stochastic nature of the MRF algorithm introduced uncertainty in the microstructural features.
The research investigated the impact of this uncertainty on homogenized mechanical properties using a shape descriptor based on moment invariants to analyze variations in microstructural features. Synthetic microstructure data for Ti-7Al alloy was generated using MRF, and uncertainty in the reconstructed samples was assessed using graphical methods and statistical metrics. Subsequently, GPR was applied to explore how microstructural uncertainty propagated to homogenized properties. Principal eigenvalue moments (PEM) based on Hu moments were assigned as shape descriptors, enabling both graphical and statistical evaluations of variation. GPR, particularly through the kriging method, predicted mean values and covariance of mechanical properties based on microstructural descriptors provided by PEM. The study has demonstrated the accuracy of the GP model through significant overlap between predictions and test data, as depicted in Figure 3a,b. [4] Additionally, Figure 3c,d illustrates probability distributions of 2D additively manufactured microstructures for Young's modulus and yield strength, highlighting the variations in predicted and actual values due to microstructural variations. [4] Another study [23] addressed the aleatoric uncertainty arising during the forging process of the Ti-7Al alloy and quantified the resulting variations in the microstructure level in terms of the variations of the ODF values (Figure 4), representing volume densities of crystallographic orientations. In addition, the resulting variations of the homogenized compliance parameters, S 11 and S 66 , demonstrating the material's normal and shear compliance, as well as Young's modulus (E 1 ) and shear modulus (G 12 ), were computed as shown in Figure 5. The study utilized EBSD data of Ti-7Al microstructures fabricated with the same processing technique and parameters to observe the effects of the variations. The bars in Figure 4 and 5 demonstrate computations performed by the MCS method to find the variations of the ODFs and homogenized properties while the red curves indicate the analytical model predictions. In particular, the analytical UQ algorithm used in this study assumed a multivariate Gaussian representation for the ODF parameters, and the distribution was maintained for the homogenized properties due to the linear relationship between the homogenized properties and the ODF values. [23] Grigoriu [119] pioneered the development of probabilistic models for linear elastic random microstructures, also specifically targeting the compliance tensor. Building on this work, a novel method was created to estimate the distributions of extreme material responses using a remarkably small number of response samples. This innovative approach leverages extreme value theory to deliver accurate and efficient predictions, outperforming traditional Monte Carlo simulations.
One study utilizing MCS for UQ was performed by Zhang [11] and employed SVEs with features described by probability distributions sampled via MCS to evaluate fatigue nucleation life prediction using the fatigue indicator parameter (FIP). A reduced-order homogenization model predicted the mechanical response under fatigue loading for each SVE, treating FIP and critical FIP as random variables with probability density functions. The study validated the framework against experimental data on titanium alloy Ti-6242, demonstrating its capability to capture fatigue life variability induced by microstructural variations through comparisons between predicted and experimental data at different stress levels and cycle ranges. Additionally, as c,d) probability distributions of 2D additively manufactured microstructures for Young's modulus and yield strength, respectively. Reproduced with permission. [4] Copyright 2022, Elsevier.
Figure 4. Probability histograms of the ODFs. Reproduced with permission. [23] Copyright 2017, Elsevier.
shown in Figure 6a, [11] the predicted probability of fatigue nucleation was compared against experimental data over a range of cycles, with the red curve representing the calibrated model and black dots depicting experimental data points. The calibration aimed to minimize the disparity between model predictions and experimental results. Figure 6b [11] shows a histogram comparing predicted and experimental data for the number of cycles until fatigue nucleation, specifically at a stress level equivalent to 91.5% of the yield stress. This histogram illustrates the distribution of cycles predicted for fatigue nucleation, alongside experimental data. Figure 6c [11] compares predictions with experimental data at a stress level of 95.5% of yield stress, serving as a validation at a higher stress level.
Korshunovaa et al. [15] integrated the finite cell method (FCM) with the MLMC method to process computed tomography (CT) images for material characterization, focusing on square and octet-truss lattices. Variability in microstructure, primarily induced by AM-caused defects, was captured using binary random fields from CT images to represent porosity. The effective Young's modulus was computed via FEM for each binary random field result by applying uniaxial displacement and calculating the stress-strain ratio. The MCS method estimated the mean and variance of the effective Young's modulus across multiple results of perturbed geometrical input, leading to mechanical property variability. The conclusions of this study indicated that low porosity variance indicates uniformity in sample stiffness, while potential wall thickness underestimation may underscore structural stability or reduced local stiffness. The study by Tran [32] presented a novel UQ approach, using the crystal plasticity (CP) FEM (CPFEM) framework to analyze the influence of Reproduced with permission. [23] Copyright 2017, Elsevier. microstructural variation on homogenized material properties. The methodology relied on multilevel sampling techniques such as MLMC and MIMC to consider different fidelity parameters simultaneously, including mesh resolutions, integration time steps, and constitutive models. By leveraging a hierarchy of approximations with varying fidelity levels, the framework effectively reduced the number of microstructures required to achieve the desired level of accuracy. Moreover, the approach was extended to a multi-fidelity framework, offering the flexibility to choose between plasticity and dislocation-density-based models. Another study conducted by Elleithy et al. [120] investigated the impact of uncertainty on the homogenized stress-strain behavior of Ti-6Al-4 V alloy. They modeled the deviations from nominal temperature and crystallographic orientations using MCS to evaluate variations in the stress-strain response. Results highlighted a nonlinear relationship between plastic behavior and temperature, with minimal effect on homogenized elastic modulus and yield strength. Furthermore, perturbations in polycrystalline orientations exhibited minimal deviation in the stress-strain relationship, indicating that mechanical properties such as elastic modulus and yield strength are not linearly correlated with orientation changes. In a subsequent study, Elleithy and Acar [121] used MCS for UQ in the exploration of the multiscale plasticity behavior of Ti-6Al-4V alloy. Through CPFEM modeling and calibration, the study accurately determined local solutions for the CP parameters, including slip-system specific parameters, through error minimization fitting with experimental data. The uncertainty associated with these CP parameters was then quantified using MCS, revealing a considerable impact on mechanical properties such as Young's modulus, yield strength, and plastic-region slopes. Sensitivity analysis highlighted the critical role of initial slip resistance in both elastic and plastic behavior. Figure 7 [121] illustrates the effects of a 10% variation in CP parameters on the alloy's mechanical behavior, showcasing Gaussian distribution patterns observed in histogram plots for the mechanical properties.
A different class of the structure-property problem where UQ is of interest involves the modeling of mechanical failure and fatigue life. A significant portion of mechanical failures occur due to fatigue crack formation in structures. This fact prompts the need to design a fatigue model that accounts for uncertainty to effectively model the microstructures of materials. Several notable studies have explored fatigue modeling under uncertainty. [10,[122][123][124][125] Wang et al. [124] proved that utilizing digital twin (DT) technology emerges as a potential strategy in tackling the aforementioned issue of fatigue crack formation in structures. The authors took the lead in establishing a framework driven by DT to formulate a universal approach ensuring precise forecasting of structural fatigue life. The innovative DT-driven framework facilitated the seamless integration of both the virtual and physical models of the structure through the utilization of dynamic Bayesian network inference. Yeratapally et al. [10] proposed a model for predicting fatigue life, which was concerned with microstructure and deformation mechanisms. They validated its efficacy by employing the persistent slip band firmness on a fatigue crack initiation basis. Rigorous sensitivity and uncertainty analyses verified its reliability. Three categories of uncertainties in model parameters were delineated: physically measurable, physically unmeasurable, and empirically or semiempirically derived parameters. Uncertainty propagation employed the MCS method, where parameter uncertainties toward whole posterior distributions permeate between the model, enabling the calculation of fatigue life predictions amid uncertainties.
Another important microstructure modeling application in the context of UQ has focused on creating phase diagrams, which stands as a cornerstone in alloy design, particularly within the Figure 7. Maximum and minimum stress-strain variations in response to 10% perturbation within CP parameters. Reproduced with permission. [121] Copyright 2024, The American Institute of Aeronautics and Astronautics. c) prediction at 95.5% of yield stress. Reproduced with permission. [11] Copyright 2022, Frontiers in Materials.
realm of multiscale material modeling connecting from micro-to macroscale. [126][127][128][129][130] To ensure originality and accuracy, a comprehensive probabilistic evaluation of CALPHAD model parameters of Hf-Si binary case study was conducted by Honarmandi et al. [131] by employing a Markov chain Monte Carlo (MCMC) sampling techniques. This meticulous process yielded not only the most probable parameter values but also their associated uncertainties. [131] In this study, a fusion of an error correlationbased model fusion and Bayesian model averaging (BMA) technique were employed to integrate the various model outcomes, each serving its distinct purpose. Within BMA, the probability of each model's accuracy was considered, and the resulting fused estimate was a weighted mean across all models. Alternatively, employing an information fusion technique that leveraged correlations among model deviations was found to improve accuracy and reduce uncertainties, surpassing outcomes derived from individual models.
Another numerical UQ methodology finding applications in the structure-property problem is the SC method, which is employed through CPFEM. For instance, Tran et al. [132] used SC to assess the uncertainty of common constitutive models in CPFEM across different crystal structures, including FCC copper, body-centered-cubic tungsten, and hexagonal close-packed magnesium. This method interpolates the system's response at collocation points within the stochastic parameter space, which consequently facilitates for an accurate quantification of how uncertainties in crystal plasticity parameters, such as critical resolved shear stresses and hardening parameters, impact material behavior. These parameters were treated as stochastic variables to capture inherent variability. Variance-based global sensitivity analysis using Sobol indices reveals the contribution of these parameters to yield strength and initial yield strain variance. Figure 8 [132] illustrates simulation stress-strain curves, demonstrating the impact of constitutive models on initial yield behavior and modulus of elasticity across different crystal structures, thereby providing valuable insights for robust constitutive model calibration.
Behnam et al. [133] conducted UQ on a microstructural FE model for creep in Grade 91 steel. The sensitivity analyses identified six important parameters, including slip resistances, power law exponent, diffusional creep parameter, viscosity, diffusivity, and nucleation rate constant. These parameters were found to be crucial for understanding the material response to creep and its microstructural evolution. Reduced-order models focusing on these key parameters were generated to allow for efficient analysis. The generalized PCE (gPCE) method was then utilized to propagate uncertainty on the minimum creep rate. Results demonstrated that limiting variation in these parameters can effectively control uncertainty in the creep behavior, showcasing the significance of the gPCE method in efficiently capturing and managing uncertainty in predictions. Additionally, the comparison with MCS confirmed that the gPCE method provided similar mean and standard deviation values with significantly fewer simulations, highlighting its efficiency and accuracy in UQ.
A study by Kumar et al. [134] employs an adaptive sparse PCE (SPCE) method for uncertainty and sensitivity analyses in stochastic composite structures. Findings suggest that uncertainties in laminate stacking sequence orientation angles have minimal impact on output variables, guiding future exploration involving diverse sampling schemes and optimization under uncertainties for manufacturing cost and structural strength. In another work by Kumar, [135] SPCE was applied to model and quantify uncertainties in the design of a composite leaf spring, considering variables from material properties, layer stacking sequence, and macroscale. The findings demonstrated that incorporating uncertainties using SPCE in the optimization process led to the identification of design solutions that balance cost and stiffness while adhering to mass constraints. Furthermore, Galbincea et al. [136] employed PCE to systematically quantify the effects of material variability on the energy absorption capacity of an aerospace component during impact scenarios. By integrating PCE with stochastic finite-element analysis (FEA), the study enhanced the predictive modeling of aerospace structures, enabling the determination of design parameters optimizing energy dissipation efficiency under uncertain material properties.
The solutions to the UQ problem have also benefited from the use of machine-learning and deep-learning strategies. Recently, BNNs have been utilized to quantify both aleatoric and epistemic uncertainties in data-driven materials modeling, offering a robust method to predict mechanical responses while accounting for inherent stochastic variations in material microstructure. [62] In addition, Koutsourelakis' development of a machine-learning framework for upscaling microstructural randomness in solid mechanics ensured that microscale variations are accurately reflected in macroscopic predictions, emphasizing the significance of microstructural heterogeneity in determining global material behavior. [137] A methodology was proposed by Terayama et al. [129] for the efficient construction of phase diagrams using machine learning. The active learning technique of uncertainty sampling was employed to extensively sample regions surrounding phase boundaries. Demonstrations of this approach included the construction of three well-known experimental phase diagrams. Introducing the microstructure-guided deep material network, Huang et al. [138] developed an innovative deep-learning methodology involving a robust framework for modeling multiphase materials by considering plasticity to predict mechanical responses across diverse microstructure characterization and loading paths. They exemplified the methodology by conducting UQ with the modeling of short fiber-reinforced polymer composites. The findings revealed that merely four foundational DMNs were essential to forecast the nonlinear Figure 8. Equivalent ε VM -σ VM plots for FCC Cu. Reproduced with permission. [132] Copyright 2022, Frontiers in Materials.
stress reactions across various loading scenarios, and orientation states, encompassing different constituent properties and fiber volume fractions, as long as the combination falls within the microstructure space outlined by the base models. In Figure 9, [138] they illustrated the coefficients of variations stemming from the upper-left 3 Â 3 sub-matrix of the stiffness matrix alongside the remaining entries on its principal diagonal. The significance of the unrepresented values lay in their trivial nature, typically registering 1000 times lower than those depicted in the heatmaps.
Microstructure design of the materials under uncertainty has been taken into the forefront by material scientists and researchers with the development of modern computational simulation tools and studied over decades. [139,140] The prediction methods based on computational (i.e., numerical, analytical, and semianalytical) approaches are preferred to prevent expensive experimental trial and errors. [141][142][143] Most of the existing research efforts have been carried out to extract material property related to process and structure, which is called as "forward" design method where the purpose is to determine material property according to the computationally achieved optimal design considering manufacturing process physics and constraints in addition to the material physics and design constraints. [144][145][146] This forward design/optimization approach is not only utilized for microscale but also for multiscale including micro-and macroscales together. [147][148][149][150][151] In contrast, data-driven models enable to capture desired microstructure using process parameters/conditions and properties called as "inverse" design method, which is cost-effective in comparison to forward design approach. [152] In the last decade, inverse design of material microstructures has gained a revival of interest with the development of data-driven methods by scientists due to its efficiency and practicality over traditional forward design approach. [153] For this respect, the recent research efforts on the inverse design of the microstructures under uncertainty are reviewed in terms of SP and PS perspective in this section.
Material microstructures exhibit inherent probabilistic behavior, which brings uncertainties due to the complex physics of manufacturing processes. These aleatoric uncertainties require to be quantified as the microstructure exerts a substantial effect on engineering design product properties. [23,[154][155][156][157][158] The inverse approach provides the required process planning for aimed final design by employing desired properties of the material while taking uncertainties arising from the stochastic nature of the process into account. [97,159] The inverse method can utilize the Figure 9. Heatmaps of coefficient of variation of the stiffness matrices S ij (%). Reproduced with permission. [138] Copyright 2022, Elsevier.
set of inputs and outputs obtained from simulations and experiments in a forward fashion to relate SP and PS, as visualized in Figure 10. The desired/prescribed mechanical response can be employed to determine microstructure design and process parameters under uncertainty with the merits of common numerical methods such as UQ, deep learning, machine learning, and numerical optimization.
In the late 90's, PSP relations and purposive inverse approach were discussed by Olson [140] for the first time in the literature. Later on, Horstemeyer [159,160] extended this linkage through modeling and application cases. Johnson and Arróyave [161] developed an inverse framework to predict and optimize the heat treatment process of nickel-based shape memory alloy according to the required microstructure (i.e., grain size and distribution). Nellippallil et al. [162] developed an inverse design approach for the multiple-stage hot rod rolling process to achieve the proper manufacturing stages and parameters according to the desired macro-and microstructure design of the rod element. For this respect, they considered interstaged flow information called horizontal integration and formulated a compromise decision support problem using test and simulation data considering uncertainties of the complicated manufacturing system composed of various stages. A year later, they [163] published their new investigation, which has been improved by including a multiscale design method (i.e., vertical integration) mostly focusing on the microstructure of the product manufactured through the hot-rolling process and the inverse design of hot-rolling stages. The importance of these vertical and horizontal processstructure integrations has been discussed by Tennyson et al. [164] previously. Tran et al. [165] conducted calibration of the welding process parameters to achieve optimal target microstructure. In this regard, they developed a model that combines temperaturebased kinetic Monte Carlo (kMC) and Bayesian optimization which determines target microstructures from a pool of candidate microstructures. Their optimization convergence plot is given in Figure 11, [165] where the objective y is minimized. The objective is minimized as the optimization process advances. Reproduced with permission. [165] Copyright 2020, Elsevier.
In recent work, Noguchi and Inoue [166] inversely determined cold rolling of low-carbon steel process parameters using pixel convolutional neural networks (PixelCNN) that relate microstructural features to material properties and manufacturing parameters. They modeled uncertainty propagation related to stochastic relations between process parameters such as cooling rate and microstructure features such as grain size and volume fraction of the ferrite.
In addition to the inverse problem in terms of SP relations, the inverse PS linkage, which provides goal-oriented microstructure design in accordance with the desired properties is in high demand. The existing research effort on the inverse PS problem with UQ is reviewed in the following subsection.
The inverse design approach aims to find the optimal desired microstructure texture that meets foreknown functionality efficiently. [167][168][169][170] As an early research effort, Sigmund [171] proposed an inverse design approach for 2D lattice microstructures for prescribed homogenized elastic material features such as modulus of elasticity and Poisson's ratio. In this study, topology optimization was solved using optimality criteria minimizing the volume fraction of the lattice, which is demanded for aerospace applications due to low-weight requirements. [172] Recently, Thillaithevan et al. [173] conducted a similar inverse design approach using deformation-based homogenization for 2D periodic microstructures and targeting nonlinear stress-strain properties.
In recent years, PS linkage and inverse microstructure design under uncertainty have gained popularity with the development of efficient data-driven methods such as machine learning and deep learning inverting the input (i.e., microstructure) and output (i.e., property) relations. [174,175] Furthermore, recent advancements in forward microstructure design tools, such as the Bayesian approach for microstructure property estimation from images, investigated by Wade and Graham-Brady, [176] enable the achievement of inverse design methods. The epistemic uncertainties due to simplifications or lack of information in microstructure modeling might be challenging to be quantified. To overcome this, Choi et al. [144] presented a goal-oriented inductive design exploration methodology, which is a multistage inverse design approach that also facilitates the quantification of different types of propagated uncertainties. Attari et al. [177] conducted an inverse design framework for two-phase microstructures targeting the thermal conductivity property. Furthermore, PS has been correlated through the deep variational auto encoder (VAE) method. Pei et al. [175] utilized VAE to inversely predict phase distribution of 9% Cr martensite/ ferrite steels for desired mechanical properties such as yield stress and creep life. In another study, VAE has been conducted to perform the multiscale inverse design of metamaterials using the generated microstructure database by Wang et al. [178] Their overall framework for the inverse design of nonhomogeneous graded metamaterial composed of a pool of microstructures is depicted in Figure 12. [178] Mahdavi et al. [153] inversely modeled dual phase (α and β) of the Ti-6Al-4V alloy aiming volume fractions and Corson's coefficients for required strength. Tan et al. [179] presented a deep convolutional generative adversarial network, which uses images of the microstructures as inputs combined with the convolutional neural network (CNN) to invert PS relation of the microstructure targeting the compliance tensor of the microstructure. Zang and Koutsourelakis [180] proposed a method, called PSP-GEN, based on deep learning to link whole PSP relationships and perform inverse microstructure design while considering the inherent stochasticity of the problem.
Truss-based lattices inspired by material crystal structures are commonly used as mechanical metamaterials due to their high strength-to-weight ratio, which makes them attractive for Figure 12. Overview of the proposed framework. Reproduced with permission. [178] Copyright 2020, Elsevier. aerospace applications. Their homogenized properties can be inversely modeled with the help of deep-learning techniques such as NNs. [181,182] Vant't Sant et al. [183] presented a data-driven approach for the design of growth-based periodic naturally inspired microstructures based on anisotropic homogenized elastic properties of the microstructures. Later on, materialproperty spaces extracted through the genetic algorithm were utilized for the inverse design and optimization of the naturally inspired microstructures by Liu and Acar. [184] In addition, the MCS technique was utilized to quantify uncertainties due to the surface distortion. In their study, they considered 2D microstructures as periodic RVEs consisting of periodic and nonperiodic microstructures.
Currently, the inverse design of a novel material called spinodoid shapes inspired by a phase separation process (i.e., spinodal decomposition) have become prominent due to their smooth topologies preventing high-stress concentrations and providing enhanced mechanical performance maintaining low-weight requirements of aerospace applications. [185][186][187] In these materials, uncertainties due to AM need to be quantified. [188] These materials were modeled using a non-iterative GRF method and proper shapes were determined by linking homogenized elastic properties and the underlying structure with the help of FEA and conditional generative adversarial networks by Liu and Acar. [189] Their inverse design approach for 2D spinodoids is demonstrated in Figure 13. [189] 5. Future Topics for Uncertainty UQ This section highlights key future directions in UQ for aerospace materials, with a focus on the role of microstructures. AM has revolutionized the ability to design materials with customized microstructures, creating new opportunities to improve the performance of lightweight aerospace components. However, these advances also introduce challenges, such as uncertainties in material properties, process conditions, and resulting microstructures, which can significantly affect mechanical performance. Addressing these uncertainties is essential to ensure the reliability, safety, and efficiency of additively manufactured components, particularly given the stringent mechanical and safety standards in aerospace. Thus, UQ is crucial for driving further innovation in this field. MSTO and RBD optimization (RBDO) provide a structured approach to tackle these challenges, offering pathways to balance performance, weight, and structural integrity in the design process, which is especially important as materials are pushed to their physical limits. As aerospace components are exposed to extreme mechanical environments resulting in high-stress concentrations, fatigue, and failure, uncertainties in microstructure can significantly influence material response, potentially compromising the integrity of the component. Therefore, integrating UQ into MSTO and RBDO frameworks is necessary for predicting and ensuring performance under such extreme conditions. Looking ahead, PINNs offer a versatile solution, efficiently incorporating physical laws and real-world data into predictive models. The application of PINNs can be extended to optimizing AM processing, RBDO of microstructures, and UQ for extreme mechanical behavior. These advancements have the potential to significantly enhance the future of UQ for microstructures in aerospace, addressing both current challenges and future needs.
With advances in modern fabrication techniques, AM provides a promising way to fabricate advanced materials with outstanding features, such as low density, negative Poisson's ratio, high energy absorption capability, and high strength and stiffness. The rapid development in manufacturing techniques promotes the characteristic sizes of engineered structures being reduced. The overall performance of a macroscopic engineering structure not only depends on its macroscopic configuration but also on the corresponding microstructural features. Therefore, to optimize the macroscopic structure, a hierarchy of its model and simulation must be linked to create a multiscale description of material properties. Many recent works have been devoted to the study of multiscale modeling. The studies involve determining the effective constitutive law of materials [190][191][192][193] with spatially varying properties. It helps to achieve effective properties of the materials at the macroscale. Other studies involve developing nonconcurrent [194][195][196][197] and concurrent [198][199][200][201] multiscale optimization frameworks. The concurrent design framework shown in Figure 14 optimizes the structure at two scales simultaneously. [148] It allows users to explore a vast design space and optimize the structure at the macroscale as well as at the microscale.
However, these optimized structures are sensitive to defects associated with manufacturing processes. For example, the cooling rate in the melting process of AM can affect the pore and Figure 13. Flowchart of the proposed framework. Reproduced with permission. [189] Copyright 2024, The American Institute of Aeronautics and Astronautics.
grain size of fabricated materials. [202,203] Therefore, to achieve a reliable multiscale design, the uncertainty in the microscopic level of materials should be taken into account. The challenge of multiscale modeling under uncertainty is to understand how errors are introduced at each scale and to alleviate the computational burden of UQ. In recent years, various uncertainties have been considered in stochastic optimization problems. One of the uncertainties is related to the geometric dimensions of microstructures. For materials that are periodically distributed in the design domain, variations in their microstructures are commonly characterized by known probability distributions. [139,184,204] For heterogeneous materials, microscale uncertainties in materials can be presented by the second-order explicit mixture random field model, [4,205,206] which is compatible with sensitivity analysis in topology optimization. Another type of uncertainty is related to the stochastic loading conditions at the macroscale. The uncertainty in the external load can be described by an ellipsoid model, a known distribution within bounds, or other random fields. [207][208][209] In multiscale modeling problems, it is difficult to address robust topology optimization with a high number of random parameters. The high dimensionality can be caused by representing microstructures in the pixels/voxels level and defining complex external loading conditions. In that case, many efficient surrogate-based reliability analysis methods are no longer feasible. [210] In addition, computational cost can be increased by a high number of random parameters involved in UQ. [207] To overcome this issue, Gao et al. [206] transferred the reliability constraint to an equivalent and simplified one using the reliability index. The accuracy of the simplified constraint was guaranteed by the most probable point importance sampling. [211] De et al. [212] addressed this issue by constructing stochastic estimates of the objective, constraints, and gradients. In the optimization process, the stochastic estimates are solved by either adaptive moment estimation (Adam) or the globally convergent method of moving asymptotes. [212] In addition, to increase the computational efficiency, the dimensionality can be reduced by the truncated KL expansion. [205,207] Another solution is to reformulate the problem into a PCE representation while maintaining the high dimension. [205]
PINNs, as one of the modern computational methodologies, combine ML with the fundamental principles of physics, mainly to solve problems involving PDEs. [213] This combination of datadriven learning and domain-specific knowledge makes PINNs Figure 14. Design framework of concurrent topology optimization of a multiscale mechanical system. Reproduced with permission. [148] Copyright 2019, Elsevier.
particularly capable of capturing patterns in diverse applications, ranging from fluid dynamics and material science to optimization problems. It also mitigates one of the major challenges in deep learning which is the dependency on large datasets since the simulations in most of the engineering domains are highly time-consuming. PINN utilizes the NNs as the universal function approximator and attempts to minimize the composite loss function provided as [213] ℒ
where ℒ PDE and ℒ Data represent the loss function for the ordinary differential equation/PDE solution and predicted versus actual data, respectively. The formulation could also incorporate terms such as the difference between the boundary conditions ℒ BC and initial conditions ℒ IC . The input provided for the network is defined as spatiotemporal, and the output is then used for the loss function. As NNs have the capability of automatic differentiation (AD), the gradient terms in the loss functions can be calculated without requiring excessive computing times.
Figure 15 summarizes the process. Since the NN is used as the function approximator, the selection of the network is flexible and can be modified for the specific problem. If the problem is defined such that the input is in a matrix or image form, then the CNNs are used together with the physics-informed loss. [214][215][216][217][218][219] Another approach in the CNN-PINN context is to replace the AD with an approximation of a differentiation operator to decrease the computation time, with guaranteed convergent rate. [220] Furthermore, instead of a composite loss function, using Legendre multiwavelets as the basis function [221] or Hermite spline kernels [222] for continuous interpolation of grid-based states, it is possible to set the loss function involving only the physical constraint, meaning there is no need for precomputed training data. If the problem requires prediction of temporal features, or there is a history dependence, then long short-term memory (LSTM) is used as the NN for the PINN architecture. [223][224][225][226][227][228] Additionally, a combination of aforementioned networks (Conv-LSTM) can be employed to solve periodic PDEs. [229] Recently emerging graph neural networks (GNNs) are also utilized in that context where the problem conforms to graph structure, [230][231][232][233] as well as the graph convolutional networks. [234] Domain-wise, PINN is used for several engineering fields, fluid dynamics, [235][236][237][238] solid mechanics, [239][240][241][242] heat transfer, [243,244] and control systems. [245][246][247] It is particularly efficient in optimization, solving inverse problems, and reduced-order modeling. In the material science context, particularly in aerospace applications, PINN is found to be useful for nondestructive evaluation of 15. Schematic of a physics-informed neural network (PINN). A fully connected neural network is given as a representation, where Σ and σ stand for summation of the neuron output and activation function, respectively, which can be replaced by different architectures such as CNN, LSTM, and GNN, based on the application. Time and space coordinates ðt, xÞ are provided as inputs to the network that generates output u. The derivatives of u with respect to the inputs using automatic differentiation (AD) and the output itself are used to construct the loss function for the partial differential equation. The other loss functions involve data, the initial condition, and the boundary condition that use either the output or the boundary-related differential. These loss functions then constitute the total loss function with weights ω 1 , ω 2 , ω 3 , ω 4 that can be specified for the related problem. If the loss function is sufficiently minimized, then the loop ends, if not then the network parameters are updated to minimize the function in the next step. polycrystalline nickel, by solving the inverse problem to find the material properties such as stiffness tensor, [248] as well as finding specific heat, thermal conductivity, and convective heat-transfer coefficient of different metals. [210] Zhou et al. [249] proposed combining the smoothed FEM (S-FEM) with PINN first to train the model with a small amount of data, only to use afterward as transfer learning for the actual model with more data generated by S-FEM, to solve the elastic-plastic inverse problem with increased accuracy and efficiency. The method's efficiency was presented in molecular dynamics particularly generating results with high accuracy compared to the DFT calculations to predict the properties of Aluminum. [250] The applications further involve composite materials focusing on fracture analysis, [251] damage characterization, [252] mechanical response, [253] elastic properties, [254] and thermomechanical features [255,256] and find many examples in AM particularly on fatigue life prediction. [257][258][259] Despite recent advances in applying PINNs to materials science, their use in UQ of microstructures remains a critical and open area of research for aerospace materials. Traditional data-driven models for UQ of microstructures, whether in forward or inverse problems, often require large datasets for high accuracy, making them computationally expensive. Moreover, these models tend to lack interpretability, complicating efforts to understand the underlying physics of microstructural behavior. In contrast, PINNs present a promising alternative to conventional methods, offering potential advantages for various UQ applications. In addition, traditional UQ methods require explicit knowledge of the underlying mathematical model, which may be challenging or even unknown in complex systems. [260] Furthermore, methods such as MCS, which are used to quantify uncertainty, require the generation of many samples. This can result in a numerically intractable solution due to model complexity. PINNs, in contrast, allow the integration of known physical laws into the NN architecture, enabling the model to learn the inherent physics of the system while simultaneously learning from available data. This dual capability of PINNs facilitates a more flexible approach when exploring UQ. Compared with the solution of time-consuming FEM simulations, the accuracy of results related to the uncertainty of output parameters is sufficiently high. [261] In general, the framework to quantify uncertainty with PINNs falls into two categories: [262] deterministic and probabilistic. In the deterministic approach, several different surrogate models are employed that are supposed to generate a single output for a certain input. These models constitute the ensemble structure, and they are trained separately, each with different parameters. [263][264][265][266] The other ensemble approaches involve training the same network with different datasets, [267] or searching for different local minimums by adjusting the learning rate. [268] To further improve the capability to capture uncertainty, particularly in cases where the data and governing equations do not match due to simplification of the physics, the notion of applying an extra layer of NNs is proposed. [269] In contrast, the probabilistic approach involves learning the posterior probability distribution directly, which is achieved through Bayesian techniques. For this purpose, BNNs are used as function approximators, with sampling performed iteratively using the MCMC method. [270,271] The sampling after a certain time converges to the desired distribution, achieved by the Metropolis algorithm.
Alternatively, instead of directly sampling from the posterior distribution, the variational inference method tries to approximate the complex distribution by simpler ones. [272] The advancements highlighted for addressing UQ with PINNs can be further leveraged in the future to study forward and inverse UQ problems in the microstructures of aerospace materials.
As materials engineering and research reach new heights, it becomes increasingly critical to understand and measure the uncertainty associated with severe mechanical behavior in aerospace materials. This is crucial for ensuring the reliability and safety of aircraft components. [273] Recent advancements in UQ methodologies have provided valuable insights into the complexities impacting critical mechanical properties. This section explores key studies that have contributed to the understanding of uncertainty in extreme mechanical behavior like fatigue, failure, and aeroelastic instabilities and lays out a motivation for future work focusing on complex problems related to extreme mechanics. One significant problem of extreme mechanical behavior is fatigue crack initiation, which poses a substantial risk to the structural integrity of aerospace components. [274] The study by Yeratapally et al. [10] presented a microstructuresensitive model validated using Bayesian UQ to systematically quantify uncertainties in model parameters and their propagation on the fatigue life prediction, enhancing the understanding of fatigue crack initiation mechanisms in polycrystalline materials. Similarly, predicting the remaining useful life of aerospace components is essential for maintenance planning and risk assessment. Sankararaman et al. [275] investigated the use of reliability methods for quantifying uncertainty in remaining useful life prediction, providing insights into efficient methods for risk assessment and decision-making in aerospace maintenance operations. Furthermore, Giannella et al. [8] presented a stochastic framework for fatigue crack-growth predictions, considering uncertainties such as material properties, loading conditions, and environmental factors. Their sensitivity analysis and UQ enhance the understanding of factors influencing fatigue crack growth in aerospace materials, guiding future research directions for improving prediction accuracy and reliability. Moreover, DT technology offers promising capabilities for predicting fatigue life and managing the health of aerospace components. [276] Leser et al. [277] explored the feasibility of DT for fatigue life prediction, coupling in situ diagnostics and prognostics in a probabilistic framework. Through Monte Carlo methods and high-fidelity FEM, they demonstrated the potential of DT for predicting fatigue life with decreasing uncertainty overtime.
Composite materials play a vital role in aerospace applications, and accurately predicting their ultimate strength is essential for ensuring structural reliability. [278,279] Bhattacharyya et al. [280] introduced a methodology for addressing the effects of epistemic uncertainty on the ultimate strength prediction of composite laminates. Through calibration and validation of a multiscale mechanical model, they quantified and reduced uncertainties in constitutive parameters, offering valuable insights into the reliability of composite structures under extreme mechanical loading conditions. Additionally, Shang [281] addressed the challenges of failure simulation in composite materials under extreme loading conditions through multiscale modeling, employing surrogate modeling and Bayesian inference to estimate the effects of uncertainty on the strength of composite materials.
Aeroelastic interactions significantly impact the performance and safety of aircraft and thus aeroelastic instabilities can be considered as a part of extreme mechanical behavior. Therefore, addressing the effects of uncertainties in this domain is crucial for aerospace design and certification. The study by Beran et al. [282] reviews traditional analytical techniques and emerging computational methods for UQ with applications to aeroelasticity. They identified critical parameters such as flutter and limit cycle oscillations, demonstrating how uncertainties in these parameters impact aeroelastic responses. By addressing challenges such as discontinuous responses and high computational costs, the study provides valuable insights into improving system robustness and safety in aerospace applications.
The outlined studies in this section are prominent examples of addressing the effects of uncertainties on extreme mechanical behavior, particularly related to fatigue/failure properties and aeroelastic instabilities. However, the literature in this field is not sufficiently extensive to provide a comprehensive understanding of the effects of microstructural uncertainties on extreme mechanical behavior, thereby offering a future topic of importance within the context of UQ for aerospace materials.
RBD has emerged as an essential tool in modern structural design, particularly in aerospace engineering. While advanced manufacturing processes necessitate accounting for uncertainties in material properties and microstructural features, traditional design optimization assumes deterministic conditions. However, this fails to consider the inevitable variations observed in material microstructures and/or external conditions. Therefore, integrating UQ into the optimization process becomes vital for ensuring robust and reliable designs.
One traditional approach in RBD is to consider worst-case scenarios, aiming for designs with zero probability of failure. While this conservative approach can prevent failures, it often results in inefficient, costly designs. A more practical approach involves RBDO, which seeks to balance performance and reliability by considering the likelihood of failure without overconservatism. In recent years, significant strides have been made in this area, particularly within the realm of continuum structure topology optimization. For instance, Liu et al. [283] introduced an RBTO methodology that decouples the expensive reliability analysis and topology optimization processes using the topology description function and FORM. This method enhances computational efficiency, though further improvements in accuracy and layout smoothness are still needed.
Further developments by Liu et al. [284] include methods that address uncertainties in loading directions, transforming uncertain topology optimization problems into deterministic equivalents with multiple load cases. This was efficiently handled using the soft-kill bidirectional evolutionary structural optimization (BESO) method, which reduced computational costs compared to the traditional MCS method. Additionally, the cloud model introduced by Liu et al. [285] combined probabilistic and fuzzy uncertainties to further refine the RBTO process.
One key challenge in practical applications of RBTO is avoiding infeasible designs, such as those resulting from zero resultant forces. This issue was addressed by Liu et al. [286] through a modified BESO approach that incorporates load direction uncertainties, ensuring practical, feasible designs under real-world conditions. These advancements highlight the importance of considering various types of uncertainties-probabilistic, fuzzy, and interval-based-within topology optimization to achieve robust and reliable structural design.
A notable challenge in applying traditional RBD techniques to microstructures arises from the complexity of high-fidelity material models, especially those analyzing multiscale behavior. Precise derivative calculations of constraint functions with respect to random parameters, as required by methods such as the FORM and second-order reliability method, become computationally cumbersome in such cases. However, integrating analytical UQ algorithms into RBD was shown to allow for modeling both Gaussian and non-Gaussian distributions of random parameters, thereby maintaining a high level of reliability in the optimization process. [141] For instance, RBD was successfully applied to optimize the microstructural texture of Galfenol alloy, achieving significant improvements in mechanical properties under the effects of microstructural uncertainty modeled with the analytical approach. [141] Despite these advancements, most studies on multiscale structural topology optimization (MSTO) still operate under deterministic assumptions, neglecting uncertainties such as material variability, loading fluctuations, and deviations from design criteria. [287][288][289][290][291][292] Recent work by Wang et al. [293] addressed this gap by developing a multiscale RBTO (MRBTO) method that considers unknown-but-bounded uncertainties in truss-like microstructures. This method represents a significant step toward incorporating uncertainty into multiscale designs, though challenges remain in addressing the "curse of dimensionality" inherent in high-dimensional UQ problems for microstructures.
Furthermore, most existing RBTO approaches consider aleatoric and epistemic uncertainties separately. [294][295][296][297] Meng et al. [298] proposed a hybrid RBTO method that simultaneously accounts for both types of uncertainties using nested triple-loops based on FORM and the level-cut method. Similarly, Zaman and Mahadevan [299] developed a computationally efficient method that decouples design from uncertainty analysis, simplifying the optimization process for multidisciplinary systems.
RBTO is particularly crucial for small aerospace systems, such as microelectromechanical and micro-opto-electromechanical devices, where uncertainties in material microstructures can significantly affect performance. However, the high dimensionality of these systems is challenging for many surrogate model-based reliability analysis methods, such as efficient global reliability analysis [300] and active Kriging MCS. [301] Existing RBTO methods often simplify reliability constraints using approximations, e.g., using the reliability index or minimum performance target point; [302,303] however, these approximations can fall short in complex nonlinear problems. Gao et al. [206] addressed this issue with the adjoint important sampling method, which efficiently computes reliability and sensitivity using gradient-based optimization.
In addition, Du and Sun [304] proposed a reliability-based vibroacoustic MSTO model that considers uncertainties in loading direction, excitation frequency, or both. This approach balances reliability and performance, further advancing the field of RBD for microstructures. Despite significant advancements in RBD, as outlined in this subsection, challenges and opportunities remain in addressing forward, and especially inverse, design problems while accounting for microstructural uncertainty in aerospace systems.
AM has revolutionized the fabrication of complex geometries with minimal material waste, yet the process remains prone to variability, such as material defects, manufacturing inconsistencies, and other imperfections. Modeling the AM process is crucial for optimizing process parameters and minimizing energy and material waste. However, the presence of uncertainties in the process-from material properties to operating conditions-necessitates the incorporation of UQ to ensure reliable, high-accuracy predictions. [14,305] While most studies on AM modeling focus on predicting and analyzing melt-pool geometry and assessing how materials and process parameters affect the melting process, [306][307][308] there is growing interest in developing robust UQ frameworks that can quantify the variations of underlying microstructural features.
Several multiscale, multi-physics models have explored the evolution of microstructures during the AM processes. At the macroscale, numerical methods such as the FEM or finitedifference methods (FDMs) are commonly used to capture the physics of layer deposition with a moving heat source, which helps estimate melt pool dynamics. Meanwhile, microscale models using Monte Carlo methods, phase-field models, and cellular automata simulate the grain structure evolution during solidification. [309] The coupling of these macro-and microscale models allows for a more comprehensive simulation of the AM process, enabling the prediction of the resulting microstructure. [309,310] However, most of these studies leverage deterministic simulations, often overlooking how combinations of various uncertainties can influence the final material properties. Nath et al. [311] addressed this gap by developing a framework that couples a macroscale melt pool model with a microscale solidification model to simulate the microstructure evolution during AM. Their analysis investigated the grain size distribution and its effects on macroscale properties, incorporating multiple sources of uncertainty, such as variability in material properties and uncertainties in grain nucleation and growth models. These uncertainties were aggregated using a multilevel UQ approach. In a similar vein, Wang et al. [312] proposed a generalized UQ framework that explores uncertainty propagation from process parameters to material microstructure and eventually to macrolevel mechanical properties in metallic AM processes. This framework selects and validates a multiscale physics-based model, generates surrogate models from simulation data, and uses these models to quantify uncertainty and analyze its propagation. Additionally, sensitivity analyses were conducted to identify the impact of individual uncertainty sources on product quality.
In recent years, AM has also enabled the production of intricate lattice microstructures, particularly in metals. These structures are increasingly utilized due to their excellent mechanical performance and lightweight properties. [313] However, additively manufactured lattices often exhibit geometric deviations from their nominal designs, leading to significant discrepancies in mechanical behavior. [314] Incorporating as-manufactured geometries into numerical analyses is therefore essential for accurately predicting lattice performance. While this approach is computationally expensive, it provides a more realistic assessment of the mechanical behavior. For example, Liu et al. [315] highlighted the importance of accounting for geometric imperfections in their analysis of elastic properties, and Du et al. [316] examined the effects of geometric variability on the mechanical response of AM lattices. Lei et al. [317] further evaluated the correlation between geometric deviations and material performance, emphasizing the need for accurate models.
To efficiently address the computational challenges, Korshunova et al. [15] developed a binary random field model to generate CT images of as-manufactured lattices. This model captures defects from material scans, offering a practical solution to evaluate the impact of microstructural variability on mechanical behavior. The model, combined with the FCM and MLMC simulations, is found to reduce computational costs significantly while maintaining accuracy. These advancements demonstrate the importance of incorporating UQ into AM processes, particularly in scenarios where geometric deviations and material inconsistencies are critical to the structural integrity of the final product. However, there exists many challenges and opportunities in this field in the future given the complex thermomechanical interactions causing microstructural uncertainty during AM processes and the potential of fabricating many complex microstructural geometries and/or lattice microstructures with more advanced AM techniques. Interested readers are referred to several studies reviewing the state-of-the-art literature for UQ in AM [14,96,318] for further details on this topic.
In this work, we have discussed the uncertainties inherent in the characterization, modeling, and design of microstructures for aerospace materials, emphasizing both forward and inverse problems. The study provides a comprehensive review for understanding and addressing uncertainties arising from fabrication processes, computational modeling, and measurement techniques. The forward problem focuses on the uncertainties in processing parameters, microstructural features, and model assumptions that significantly affect the prediction of mechanical properties. This emphasizes the need for robust uncertainty propagation methods that can reliably capture the effects of the variations on the expected material performance. The inverse problem highlights the importance of designing processing parameters and/or microstructures under uncertainty to achieve desired mechanical properties. The review of SP and PS problems underscores the necessity of leveraging advanced computational tools to guide the design process, minimizing reliance on costly experimental trials.
The article also identifies the topics that need further investigation to study the UQ problem in the characterization, development, modeling, and design of the microstructures of aerospace materials. These topics of future interest include the following: 1) MSTO of aerospace materials while accounting for the effects of the aleatoric and epistemic uncertainty; 2) utilization of physics-informed machine-learning models to create predictive models to address the computationally expensive forward and inverse UQ problems while maintaining sufficient prediction accuracy and improving the interpretability of predictions; 3) investigation of the potentially detrimental effects of uncertainty on the extreme mechanical behavior of microstructures; 4) RBD of microstructures; and 5) UQ of microstructures fabricated with AM techniques.
Overall, this review article provides a foundational perspective on the UQ problem for microstructure modeling and design, offering insights into current methodologies and future directions for enhancing the predictions of mechanical properties of aerospace materials through rigorous UQ.
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