A stochastic approach for modeling incident gust effects on flow quantities

The uncertainty in gust loads on a rigid, flat-plate airfoil at zero angle-of-attack due to imprecise knowledge of the gust parameters is quantified. The loads are computed using the unsteady vortex lattice model, which includes temporal variations in wake vorticity and the associated downwash on the airfoil. The non-intrusive formulation of the polynomial chaos expansion in terms of the multivariate Hermite polynomials is employed to quantify the uncertainty in the predicted unsteady lift. The expansion coefficients were estimated through Latin hypercube sampling of the parameters in the vertical and streamwise gust spectra. The first-order chaos expansion in terms of the uncertain spectral parameters was found to be sufficient for representing the stochastic aerodynamic lift, which was found to be most sensitive to imprecision in the standard deviation of the vertical component of the gust. These conclusions were found to be unaffected by ignoring the effects of gusts on the locations of the shed vortices in the airfoil’s wake.     

Neumann enriched polynomial chaos approach for stochastic finite element problems

An enrichment scheme based upon the Neumann expansion method is proposed to augment the deterministic coefficient vectors associated with the polynomial chaos expansion method. The proposed approach relies upon a split of the random variables into two statistically independent sets. The principal variability of the system is captured by propagating a limited number of random variables through a low-ordered polynomial chaos expansion method. The remaining random variables are propagated by a Neumann expansion method. In turn, the random variables associated with the Neumann expansion method are utilised to enrich the polynomial chaos approach. The effect of this enrichment is explicitly captured in a new augmented definition of the coefficients of the polynomial chaos expansion. This approach allows one to consider a larger number of random variables within the scope of spectral stochastic finite element analysis in a computationally efficient manner. Closed-form expressions for the first two response moments are provided. The proposed enrichment method is used to analyse two numerical examples: the bending of a cantilever beam and the flow through porous media. Both systems contain distributed stochastic properties. The results are compared with those obtained using direct Monte Carlo simulations and using the classical polynomial chaos expansion approach.     

Uncertainty quantification of subcritical bifurcations

Analysing and quantifying parametric uncertainties numerically is a tedious task, even more so when the system exhibits subcritical bifurcations. Here a novel interpolation based approach is presented and applied to two simple models exhibiting subcritical Hopf bifurcation. It is seen that this integrated interpolation scheme is significantly faster than traditional Monte Carlo based simulations. The advantages of using this scheme and the reason for its success compared to other uncertainty quantification schemes like Polynomial Chaos Expansion (PCE) are highlighted. The paper also discusses advantages of using an equi-probable node distribution which is seen to improve the accuracy of the proposed scheme. The probabilities of failure (POF) are defined and plotted for various operating conditions. The possibilities of extending the above scheme to experiments are also discussed.     

Polynomial chaos expansion for sensitivity analysis

In this paper, the computation of Sobol's sensitivity indices from the polynomial chaos expansion of a model output involving uncertain inputs is investigated. It is shown that when the model output is smooth with regards to the inputs, a spectral convergence of the computed sensitivity indices is achieved. However, even for smooth outputs the method is limited to a moderate number of inputs, say 10-20, as it becomes computationally too demanding to reach the convergence domain. Alternative methods (such as sampling strategies) are then more attractive. The method is also challenged when the output is non-smooth even when the number of inputs is limited.     

A probabilistic approach to uncertainty quantification with limited information

Many safety assessments depend upon models that rely on probabilistic characterizations about which there is incomplete knowledge. For example, a system model may depend upon the time to failure of a piece of equipment for which no failures have actually been observed. The analysts in this case are faced with the task of developing a failure model for the equipment in question, having very limited knowledge about either the correct form of the failure distribution or the statistical parameters that characterize the distribution. They may assume that the process conforms to a Weibull or log-normal distribution or that it can be characterized by a particular mean or variance, but those assumptions impart more knowledge to the analysis than is actually available. To address this challenge, we propose a method where random variables comprising equivalence classes constrained by the available information are approximated using polynomial chaos expansions (PCEs). The PCE approximations are based on rigorous mathematical concepts developed from functional analysis and measure theory. The method has been codified in a computational tool, AVOCET, and has been applied successfully to example problems. Results indicate that it should be applicable to a broad range of engineering problems that are characterized by both irreducible andreducible uncertainty.     

Damage-based modification for fatigue life prediction under non-proportional loadings

This paper proposes a new fatigue model based on virtual strain energy to predict fatigue life under both proportional and non-proportional loadings for different materials including, 1045 Steel, 30CrNiMo8HH, Titanium TC4, and AZ31B magnesium. The results were strongly correlated with experimental results available in the literature. In addition, two damage-based modifications for fatigue life prediction under non-proportional loadings are studied. These modifications are then applied to the fatigue parameters including Smith–Watson–Topper, Fatemi–Socie, maximum shear strain, and the proposed parameter for fatigue life predictions of the studied materials. The results show considering these modifications significantly improves the accuracy of the models.     

Mixed aleatory-epistemic uncertainty quantification with stochastic expansions and optimization-based interval estimation

Uncertainty quantification (UQ) is the process of determining the effect of input uncertainties on response metrics of interest. These input uncertainties may be characterized as either aleatory uncertainties, which are irreducible variabilities inherent in nature, or epistemic uncertainties, which are reducible uncertainties resulting from a lack of knowledge. When both aleatory and epistemic uncertainties are mixed, it is desirable to maintain a segregation between aleatory and epistemic sources such that it is easy to separate and identify their contributions to the total uncertainty. Current production analyses for mixed UQ employ the use of nested sampling, where each sample taken from epistemic distributions at the outer loop results in an inner loop sampling over the aleatory probability distributions. This paper demonstrates new algorithmic capabilities for mixed UQ in which the analysis procedures are more closely tailored to the requirements of aleatory and epistemic propagation. Through the combination of stochastic expansions for computing statistics and interval optimization for computing bounds, interval-valued probability, second-order probability, and Dempster-Shafer evidence theory approaches to mixed UQ are shown to be more accurate and efficient than previously achievable.     

Data-driven probabilistic quantification and assessment of the prediction error model in damage detection applications

Advances in computer hardware and sensor technologies have led to a surge in the use of data-driven modeling and machine learning for structural engineering applications, with Structural Health Monitoring (SHM) being one of them. Despite considerable interest, it remains a research topic due to the difficulty in accurately quantifying aleatoric and epistemic uncertainty in SHM systems. Sources of uncertainty are related to operational and environmental variability, as well as measurement noise and the model prediction error associated with the data used to train damage identification algorithms. In this work, the authors aim to explicitly quantify the statistical structure of model prediction error and assess its influence on the detection performance of strain-based SHM architectures under the existence of aleatoric variability. A structural beam, subjected to probabilistic static loading is used as the reference structure and strain measurements as the damage-sensitive features. Model prediction error is quantified explicitly using robust statistical tools through available laboratory observations and synthetic (Finite Element) data. Monte Carlo simulations enabled the forward propagation of uncertainty to the feature space to generate training data for three binary detectors (Likelihood Ratio Test, Quadratic Discriminant Analysis and Mahalanobis Distance), based on statistical pattern recognition. Detection performance was compared between the explicitly quantified prediction model error and the commonly assumed white Gaussian noise model, showcasing the influence of systematic error (bias) and correlation on the robustness of an SHM system using real-world data.     

Sparse polynomial chaos expansions of frequency response functions using stochastic frequency transformation

Frequency response functions (FRFs) are important for assessing the behavior of stochastic linear dynamic systems. For large systems, their evaluations are time-consuming even for a single simulation. In such cases, uncertainty quantification by crude Monte-Carlo simulation is not feasible. In this paper, we propose the use of sparse adaptive polynomial chaos expansions (PCE) as a surrogate of the full model. To overcome known limitations of PCE when applied to FRF simulation, we propose a frequency transformation strategy that maximizes the similarity between FRFs prior to the calculation of the PCE surrogate. This strategy results in lower-order PCEs for each frequency. Principal component analysis is then employed to reduce the number of random outputs. The proposed approach is applied to two case studies: a simple 2-DOF system and a 6-DOF system with 16 random inputs. The accuracy assessment of the results indicates that the proposed approach can predict single FRFs accurately. Besides, it is shown that the first two moments of the FRFs obtained by the PCE converge to the reference results faster than with the Monte-Carlo (MC) methods.     

Fatigue damage accumulation of cold expanded hole in aluminum alloys subjected to block loading

Fatigue damage accumulation of cold expanded hole in aluminum alloys used in land transportation components was investigated. Tests were carried out using pre-cracked SENT specimens and inserting an expanded hole at the crack tip. The degree of the cold expansion was chosen equal to 4.3%. Tests were performed in two and four block loading under constant amplitude. Two sequences were compared.The increasing and the decreasing magnitude were compared. The experimental results were compared to the damage calculated by the Miner's rule and a new simple fatigue damage indicator. This comparison shows that the ‘model of the damage stress’, which take into account of the loading history, yields a good estimation of the experimental results. Moreover, the error is minimized in comparison to the Miner's model.     

Identification of missing input distributions with an inverse multi-modal Polynomial Chaos approach based on scarce data

This work presents a framework for predicting the unknown probability distributions of input parameters, starting from scarce experimental measurements of other input parameters and the Quantity of Interest (QoI), as well as a computational model of the system. This problem is relevant to aeronautics, an example being the calculation of the material properties of carbon fibre composites, which are often inferred from experimental measurements of the full-field response. The method presented here builds a probability distribution for the missing inputs with an approach based on probabilistic equivalence. The missing inputs are represented with a multi-modal Polynomial Chaos Expansion (mmPCE), a formulation which enables the algorithm to efficiently handle multi-modal experimental data. The parameters of the mmPCE are found through an optimisation process. The mmPCE is used to produce a dataset for the missing inputs, the input uncertainties are then propagated through the computational model of the system using arbitrary Polynomial Chaos (aPC) in order to produce a probability distribution for the QoI. This is in addition to an estimate of the QoI’s probability distribution arising from experimental measurements. The coefficients of the mmPCE are adjusted such that the statistical distance between the two estimates of the probability distribution of the QoI is minimised. The algorithm has two key aspects: the metric used to quantify the statistical distance between distributions and the aPC formulation used to propagate the input uncertainties. In this work the Kolmogorov–Smirnov (KS) distance was used to quantify the distance between probability distributions for the QoI as it allowed high order statistical moments to be matched and is non-parametric.The framework for back-calculating unknown input distributions was demonstrated using a dataset comprising scarce experimental measurements of the material properties of a batch of carbon fibre coupons. The ability of the algorithm to back-calculate a distribution for the shear and compression strength of the composite, based on limited experimental data, was demonstrated. It was found that it was possible to recover reasonably accurate probability distributions for the missing material properties, even when an extremely scarce data set with a fairly simplistic computational model was used.     

Comparison of Bayesian methods on parameter identification for a viscoplastic model with damage

The state of materials and accordingly the properties of structures are changing over the period of use, which may influence the reliability and quality of the structure during its life-time. Therefore, identification of the model parameters of the system is a topic which has attracted attention in the content of structural health monitoring. The parameters of a constitutive model are usually identified by minimization of the difference between model response and experimental data. However, the measurement errors and differences in the specimens lead to deviations in the determined parameters. In this article, the focus is on the identification of material parameters of a viscoplastic damaging material using a stochastic simulation technique to generate artificial data which exhibit the same stochastic behavior as experimental data. It is proposed to use Bayesian inverse methods for parameter identification and therefore the model and damage parameters are identified by applying the Transitional Markov Chain Monte Carlo Method (TMCMC) and Gauss–Markov–Kalman filter (GMKF) approach. Identified parameters by using these two Bayesian approaches are compared with the true parameters in the simulation and with each other, and the efficiency of the identification methods is discussed. The aim of this study is to observe which one of the mentioned methods is more suitable and efficient to identify the model and damage parameters of a material model, as a highly non-linear model, using a limited surface displacement measurement vector and see how much information is indeed needed to estimate the parameters accurately.     

Local stress–strain field intensity approach to fatigue life prediction under random cyclic loading

According to the characteristic of the local behavior of fatigue damage, on the basis of stress field intensity approach, a theory of local stress–strain field intensity for fatigue damage at the notch is developed in this paper, which can take account of the effects of the local stress–strain gradient on fatigue damage at the notch. In order to calculate the local stress–strain field intensity parameters, an incremental elastic-plastic finite element analysis under random cyclic loading is used to determine the local stress–strain response. A local stress–strain field intensity approach to fatigue life prediction is proposed by means of elastic-plastic finite element method for notched specimens. This approach is used to predict fatigue crack initiation life, and good correlation was observed with U-shape notched specimens for normalized 45 steel.     

A damage gradient model for fatigue life prediction of notched metallic structures under multiaxial random vibrations

In the present paper, a damage gradient model combing the damage concept with the theory of critical distance (TCD) is established to estimate the fatigue lives of notched metallic structures under multiaxial random vibrations. Firstly, a kind of notched metallic structure is designed, and the biaxial random vibration fatigue tests of the notched metallic structures are carried out under different correlation coefficients and phase differences between two vibration axes. Then, the fatigue lives of the notched metallic structures are evaluated utilizing the proposed model with the numerical simulations. Finally, the proposed model is validated by the experiment results of the biaxial random vibration fatigue tests. The comparison results demonstrate that the proposed model can provide fatigue life estimation with high accuracy.     

Uncertainty of the fatigue damage arising from a stochastic process with multiple frequency modes

Predicting the variance of the fatigue damage due to a stochastic load process is a difficult classical problem that dates back to the 1960s. For many years, the available analytical methods for tackling this problem have been limited to the linear oscillator response under Gaussian white noise excitation. In a recent prior work, the author developed an improved method for calculating the damage variance for a general narrowband Gaussian process. From a fatigue uncertainty perspective, a narrowband process is particularly crucial as the amplitude correlation magnifies the variance. This paper extends the analysis to a multimodal process comprising two or more narrowband components. The proposed method is tractable, involving a single summation for an arbitrary spectral density. Moreover, closed form expressions are available for two special cases, i.e. the components are all linear oscillator responses or the spectral density of each is rectangular. The equations also yield insight on the multilayered correlation mechanisms produced by different narrowband components. The accuracy of the method is verified by rainflow counting of simulated time history stresses.     

Collocation-based stochastic finite element analysis for random field problems

A stochastic response surface method (SRSM) which has been previously proposed for problems dealing only with random variables is extended in this paper for problems in which physical properties exhibit spatial random variation and may be modeled as random fields. The formalism of the extended SRSM is similar to the spectral stochastic finite element method (SSFEM) in the sense that both of them utilize Karhunen–Loeve (K–L) expansion to represent the input, and polynomial chaos expansion to represent the output. However, the coefficients in the polynomial chaos expansion are calculated using a probabilistic collocation approach in SRSM. This strategy helps us to decouple the finite element and stochastic computations, and the finite element code can be treated as a black box, as in the case of a commercial code. The collocation-based SRSM approach is compared in this paper with an existing analytical SSFEM approach, which uses a Galerkin-based weighted residual formulation, and with a black-box SSFEM approach, which uses Latin Hypercube sampling for the design of experiments. Numerical examples are used to illustrate the features of the extended SRSM and to compare its efficiency and accuracy with the existing analytical and black-box versions of SSFEM.     

The fatigue life prediction for structure with surface scratch considering cutting residual stress,initial plasticity damage and fatigue damage

In this paper, a continuum damage mechanics based fatigue model is used to evaluate the effect of surface scratches resulting from accidental scrapes on the fatigue life of structures. First, a dynamic analysis is conducted to simulate scratch generation. Second, the initial damage field caused by plastic deformation in the scraping process is calculated. Third, for structures with scratches under fatigue loading, Chaudonneret’s damage model for multiaxial fatigue is applied and the finite element implementation is presented. At last, this method is applied to life calculation for scratched specimens and for a scratched fixed plate. The theoretical calculation tallies with the experimental results.     

Cycle distribution and fatigue damage assessment in broad-band non-Gaussian random processes

The aim of the paper is to propose a method to assess the cycle distribution and the fatigue damage in stationary broad-band non-Gaussian processes; the method is a further development of an existing procedure proposed for Gaussian processes [Int J Fatigue 2002; 24(11)]. By introducing a suitable transformation, we link a non-Gaussian process to an underlying Gaussian one, for which we can estimate the cumulative distribution of counted cycles; the corresponding joint density for the non-Gaussian process is then derived. The analysis of time histories measured on Mountain-bikes in off-road tracks shows that the new method is able to correctly assess the distribution of ‘rainflow’ counted cycles taking into account the non-normality of the load.     

Fatigue life prediction under variable loading based on a new damage model

To examine the performance of nonlinear models proposed in the estimation of fatigue damage and fatigue life of components under random loading, a batch of specimens made of 6082 T 6 aluminium alloy has been studied and some of the results are reported in the present paper. The paper describes an algorithm and suggests a fatigue cumulative damage model, especially when random loading is considered. This paper contains the results of mono-axial random load fatigue tests with different mean and amplitude values performed on 6082 T 6 aluminium alloy specimens. Cycles were counted with rainflow algorithm and damage was cumulated with a new model proposed in this paper and with the Palmgren–Miner model. The proposed model has been formulated to take into account the damage evolution at different load levels and it allows the effect of the loading sequence to be included by means of a recurrence formula derived for multilevel loading, considering complex load sequences. It is concluded that a ‘damaged stress interaction damage rule’ proposed here allows a better fatigue damage prediction than the widely used Palmgren–Miner rule, and a formula derived in random fatigue could be used to predict the fatigue damage and fatigue lifetime very easily. The results obtained by the model are compared with the experimental results and those calculated by the most fatigue damage model used in fatigue (Miner’s model). The comparison shows that the proposed model, presents a good estimation of the experimental results. Moreover, the error is minimized in comparison to the Miner’s model.     

Bearing damage evolution of a pinned joint in CFRP laminates under repeated tensile loading

A mechanical pinned joint in the CFRP laminates such as [0/±45/90]_3S, [90/±45/0]_3S, [0/±45/90]_2S and [90/±45/0]_2s is loaded statically and cyclically to finally obtain the critical condition for fatigue. It is derived that in the static loading, the critical damage that yields shear matrix crack is kink and the critical condition to the final failure is the appearance of kink in every inner 0° layer and that in the fatigue loading within the moderate load, the critical damage that yields shear matrix crack is almost always kink-like damage along the collapse front and at high load it is rather kink. Next, the non-elastic elongation of a joint at the maximum load subtracted by the one at 10th cycle is focused on and its capability is figured out for various stacking sequences. The critical value U_NE,F^* for the elongation rate change to the final fatigue failure is around 50–65 μm in the present material. The critical condition to the final fatigue failure and corresponding to U_NE,F^* is roughly the appearance of mostly kink-like damage in every inner 0° layer.