Stochastic Finite Element Model Research Articles

A 3-D numerical study of the random wave-induced response in a spatially heterogenous seabed

The seabed failure poses a significant threat to the stability of marine structures. The actual wave-induced dynamic response in a porous seabed is a complex issue due to the randomness in hydrodynamic pressure and the spatial variability in soil properties. A probabilistic framework was proposed based on a 3-D stochastic finite element model for the investigation of the random wave-induced responses in a spatially heterogenous seabed. The framework integrates the generation of random waves as well as the random soil and the finite element analysis of poroelasticity via a Livelink code. A decomposed Karhunen-Loève expansion method was proposed, making the generation of multi-dimensional random fields with high resolution and large scale possible. Contrastive analyses of different scenarios were conducted to study the effects of the randomness of wave pressure and the spatial variability of the soil shear modulus on the seabed response. Monte Carlo simulations were used for the statistics of seabed response, and then the effects of the correlation length of random fields on the liquefaction depth were investigated. The results of the analysis highlighted the significant impacts of the randomness of ocean waves and the spatial variability in soil properties, which were often ignored in previous studies.

Prediction of biomechanical responses of human lumbar discs - a stochastic finite element model analysis

Background Accurate biomechanical investigation of human intervertebral discs (IVDs) is difficult because of their complicated structural and material features. Aim To investigate probabilistic distributions of the biomechanical responses of the IVD by considering varying nonlinear structural and material properties using a stochastic finite element (FE) model. Methods A FE model of a L3-4 disc was reconstructed, including the nucleus pulposus (NP), annular matrix and fibers. A Monte Carlo method was used to randomly generate 500 sets of the nonlinear material properties and fiber orientations of the disc that were implemented into the FE model. The FE model was analyzed under seven loading conditions: a 500 N compressive force, a 7.5Nm moment simulating flexion, extension, left-right lateral bending, and left-right axial rotation, respectively. The distributions of the ranges of motion (ROMs), intradiscal pressures (IDP), fiber stresses and matrix strains of the disc were analyzed. Results Under the compressive load, the displacement varied between 0.29 mm and 0.76 mm. Under the 7.5Nm moment, the ROMs varied between 3.0° and 6.0° in primary rotations. The IDPs varied within 0.3 MPa under all the loading conditions. The maximal fiber stress (3.22 ± 0.64 MPa) and matrix strain (0.27 ± 0.12%) were observed under the flexion and extension moments, respectively. Conclusion The IVD biomechanics could be dramatically affected by the structural and material parameters used to construct the FE model. The stochastic FE model that includes the probabilistic distributions of the structural and material parameters provides a useful approach to analyze the statistical ranges of the biomechanical responses of the IVDs.

Modal properties of macaw palm fruit-rachilla system: An approach by the stochastic finite element method (SFEM)

In the last few years, the trend towards the use of fossil fuels has been enormously discouraged, driving research and development into new energy sources, joining efforts from both the scientific community and industry. The global trend to increase the use of renewable energy sources is essential to develop appropriate technologies for the cultivation and management of crops in order to obtain production on an industrial scale, associated with the least environmental impact. The Macaw palm (Acrocomia aculeata), a palm tree native to Tropical America, has great potential as a raw material for the biodiesel production. Macaw palm fruits can be harvested by mechanical vibrations and the modal properties identification, such as natural frequencies and modes of vibration, of the fruit-rachilla system is fundamental for the development of harvesting machines and devices that use this principle. This paper presents a systematic approach to identify the modal properties of macaw palm fruit-rachilla system, when the inherent variability of the biological systems are involved. The inherent variability of the specific mass and elasticity modulus of the fruit-rachilla systems of macaw palm in different plantation sites, as well as the influence of the fruit maturation stage, were considered in the analysis, as random variables. Using the stochastic finite element model it was possible to determine the spectral response of the natural frequencies of interest, as well as its statistical parameters in order to serve as base of knowledge for design harvesting machines. The analysis and discussions had contributed to establish three possible operating frequency ranges to design macaw palm harvesting machines.

Stochastic finite element model assessing length effect for unreinforced masonry walls subjected to one-way vertical bending under out-of-plane loading

The strength of unreinforced masonry (URM) walls subjected to one-way vertical bending under out-of-plane loading (no pre-compression) is known to be affected by the tensile bond strength. Factors such as batching, workmanship, and environmental exposure alter the strength of this bond, resulting in spatial variability for any URM assembly. In narrow wall panels a single weak joint may dictate the failure load of a masonry wall, whereas for longer walls there is higher potential for weak joints to occur and load redistribution. This paper focuses on a stochastic assessment of clay brick URM walls with spatially variable tensile bond strength subjected to uniformly distributed out-of-plane loads in one-way vertical bending and assessing the effect of wall length on the ultimate failure load. Stochastic computational modelling combining 3D non-linear Finite Element Analysis (FEA) and Monte Carlo Simulation (MCS) is used to account for bond strength variability when estimating the walls ultimate failure loads. For this assessment FEA MCS has been applied to a set of existing test data for walls 1, 2, 4, and 10 units long, by ten different masons. Models were also developed to consider walls in the intermediate length range, 7 units long, and walls outside of this range, 15 units long. For each set of simulations the peak pressure and load–displacement data was extracted and analysed, showing agreement with the results of wall test data. The panel strength is shown to increase with wall length from 1 to 4 units, then stabilize with further length increase. The variability of the failure load is shown to decrease with increasing wall length.

Dynamic Response of Angle Ply Laminates with Uncertainties Using MARS, ANN-PSO, GPR and ANFIS.

In the present work, for the first time, free vibration response of angle ply laminates with uncertainties is attempted using Multivariate Adaptive Regression Spline (MARS), Artificial Neural Network-Particle Swarm Optimization (ANN-PSO), Gaussian Process Regression (GPR), and Adaptive Network Fuzzy Inference System (ANFIS). The present approach employed 2D C0 stochastic finite element (FE) model based on the Third Order Shear Deformation Theory (TSDT) in conjunction with MARS, ANN-PSO, GPR, and ANFIS. The TSDT model used eliminates the requirement of shear correction factor owing to the consideration of the actual parabolic distribution of transverse shear stress. Zero transverse shear stress at the top and bottom of the plate is enforced to compute higher-order unknowns. C0 FE model makes it commercially viable. Stochastic FE analysis done with Monte Carlo Simulation (MCS) FORTRAN inhouse code, selection of design points using a random variable framework, and soft computing with MARS, ANN-PSO, GPR, and ANFIS is implemented using MATLAB in-house code. Following the random variable frame, design points were selected from the input data generated through Monte Carlo Simulation. A total of four-mode shapes are analyzed in the present study. The comparison study was done to compare present work with results in the literature and they were found in good agreement. The stochastic parameters are Young’s elastic modulus, shear modulus, and the Poisson ratio. Lognormal distribution of properties is assumed in the present work. The current soft computation models shrink the number of trials and were found computationally efficient as the MCS-based FE modelling. The paper presents a comparison of MARS, ANN-PSO, GPR, and ANFIS algorithm performance with the stochastic FE model based on TSDT.

On generative models as the basis for digital twins

AbstractA framework is proposed for generative models as a basis fordigital twinsormirrorsof structures. The proposal is based on the premise that deterministic models cannot account for the uncertainty present in most structural modeling applications. Two different types of generative models are considered here. The first is a physics-based model based on the stochastic finite element (SFE) method, which is widely used when modeling structures that have material and loading uncertainties imposed. Such models can be calibrated according to data from the structure and would be expected to outperform any other model if the modeling accurately captures the true underlying physics of the structure. The potential use of SFE models as digital mirrors is illustrated via application to a linear structure with stochastic material properties. For situations where the physical formulation of such models does not suffice, a data-driven framework is proposed, using machine learning and conditional generative adversarial networks (cGANs). The latter algorithm is used to learn the distribution of the quantity of interest in a structure with material nonlinearities and uncertainties. For the examples considered in this work, the data-driven cGANs model outperforms the physics-based approach. Finally, an example is shown where the two methods are coupled such that a hybrid model approach is demonstrated.

Robust Design Optimization of Car-Door Structures with Spatially Varied Material Uncertainties

This paper presents an effective approach for robust design optimization of car-door structures with spatially varied material properties. This spatially varied material property causes structural response quantities; for example, the natural frequency and the lateral stiffness coefficient become random variables. In this regard, the Karhunen-Loève expansion is first used to represent the elastic modulus and the mass density random fields as a series of random variables. Then, a stochastic finite-element model is formulated for uncertainty quantification of the car-door structure. Combined with a polynomial-based response surface model to mimic the true performance indicator, this allows one to efficiently evaluate probability constraints for the robust design optimization of the uncertain car-door structure. In numerical simulations, design variables of the uncertain car-door structure are defined as thickness values of the tailor rolled blank structure at various regions, whereas multiple design objectives are formulated via the structural weight, the first-order natural frequency, and the lateral stiffness coefficient. Results have shown that the mean value of performance indicators can be generally improved, whereas the response variance is further minimized to archive the robust design objective. The probability-based constraint is significant to relate the Pareto optimum set to the targeted structural safety level. The proposed approach is simple, suggesting an attractive tool for the robust design optimization of car-door structures with spatially varied material uncertainties.

Three-Dimensional Stochastic Modelling of Wavy Carbon Nanotube Reinforced Epoxy Nanocomposites

Carbon nanotubes (CNTs) are frequently used as the nanoscale filler materials. It has been observed that when they are added to the polymer matrix by a small fraction, they are able to form a strong yet light-weight multifunctional composite materials. Previous studies found these properties to be significantly influenced by the morphology of CNTs embedded in the matrix. In general, long CNTs become wavy and randomly oriented when dispersed in the matrix material. Collectively, they form an interconnected network. Over the past several decades, many theoretical and computational studies have been carried out to capture the mechanics of CNT-reinforced nanocomposites. In most of these models, CNTs are modeled as straight fibers. In addition, a perfect interphase between CNT and epoxy is generally assumed. As such, these models may not capture the realistic morphology of CNT reinforced composites. In the current study, we have developed a method to construct stochastic three-dimensional finite element models of wavy CNT reinforced epoxy nanocomposites. Both aligned type and randomly distributed type are considered. Uniform random distributions are assumed for both waviness and orientation angles of the dispersed CNTs. To study the effect of waviness on the elastic properties, finite element models with maximum CNT waviness angles of 0° (straight), 10° , 20° , and 30° are generated for CNT volume fractions ranging from 0.1 to 1%. A significant decrease in the properties is observed as the maximum allowable waviness angle (θmax) of the embedded CNTs in the epoxy is increased. The obtained values are compared with the theoretical models available in the literature. The three-phase model including CNTs, CNT/epoxy interphase and matrix is also studied to evaluate the effect of ‘soft’ (Ei = 10–75% Em) as well as ‘hard’ interphase (Ei = 110–175% Em) on the composite modulus. These results suggest that due to ineffective load transfer between CNT and epoxy in the presence of CNT waviness, the elastic modulus of the CNT composites is compromised by 4–8%. The modulus is further decreased by nearly 2.5% as the CNT/epoxy interphase modulus decreases by 90%. The ‘hard’ interphase model confirms an efficient load transfer from matrix to CNT fibers, leading to the increase in composite stiffness. It is found that the composite modulus is more sensitive to the ‘soft’ interphase.

Stochastic normal mode frequency analysis of hybrid angle ply laminated composite skew plate with opening using a novel approach

In the present work, for the first time, stochastic normal mode frequency analysis of hybrid angle ply laminated composite skew plate with an opening is attempted by a novel approach. The present novel approach employed 2 D C0 stochastic finite element (FE) model based on higher-order shear deformation theory (HSDT) in conjunction with artificial neural network (ANN) and artificial neural network-particle swarm optimization (ANN-PSO). HSDT model used eliminates the requirement of shear correction factor owing to consideration of the actual parabolic distribution of transverse shear stress. Zero transverse shear stress at the top and bottom of the plate is enforced to compute higher-order unknowns. C0 FE model makes it commercially viable. Stochastic vibration using FE analysis with Monte Carlo simulation, design of experiment (DOE), and soft computing with ANN, ANN-PSO is implemented. With the DOE method, a total of 128 design points are generated. Since there are no results for the present problem in the literature is available. Hence suitable comparisons have been made and found in the good agreement. The stochastic parameters are Young’s elastic modulus, shear modulus, and the Poisson ratio. Lognormal distribution of properties is assumed in the present work. The present soft computation models shrink the trial and are found computationally in agreement with the Monte Carlo simulation-based finite element modeling (MCS-FEM). Current work reveals that ANN-PSO is a fast and straightforward method with result accuracy close to MCS-FEM, and the future researcher in this area may use ANN-PSO.

Effect of the platelet size on the fracturing behavior and size effect of discontinuous fiber composite structures

This study investigates the mode I intra-laminar fracture and size effect in Discontinuous Fiber Composites (DFCs). Towards this goal, the results of fracture tests on geometrically-scaled Single Edge Notch Tension (SENT) specimens are presented and critically discussed for three platelet sizes.The results clearly show a decrease in nominal strength as the specimen size increases. It is found that, when the specimen is sufficiently large, the structural strength scales according to Linear Elastic Fracture Mechanics (LEFM) and the failure occurs in a very brittle way. In contrast, small specimens exhibit a more pronounced pseudo-ductility with a limited scaling effect and a significant deviation from LEFM.To characterize the fracture energy and the effective length of the fracture process zone, an approach combining equivalent fracture mechanics and stochastic finite element modeling is proposed. The model accounts for the complex random mesostructure of the material by modeling the platelets explicitly. Thanks to this theoretical framework, the mode I fracture energy of DFCs is estimated for the first time and it is shown to depend significantly on the platelet size. In particular, the fracture energy is shown to increase linearly with the platelet size in the range investigated in this work.Another importance of this work is that, compared to traditional unidirectional composites, DFC structures exhibit higher pseudo-ductility and their strength is, by far, less sensitive to notches, defects and cracks. This aspect can be used advantageously in structural design only upon the condition that proper certification guidelines acknowledging the more pronounced quasibrittleness of DFCs is formulated. The size effect analysis presented in this work represents a first step in this direction as it allows the assessment of the severity of a defect or notch in DFCs.

Effect of the thickness on the fracturing behavior of discontinuous fiber composite structures

In this study, we investigate experimentally and numerically the mode I intra-laminar fracture and size effect of Discontinuous Fiber Composites (DFCs) as a function of the structure thickness.Fracture tests on geometrically-scaled Single Edge Notch Tension (SENT) specimens show a notable size effect on the nominal strength of DFCs. By integrating equivalent fracture mechanics and stochastic finite element modeling we estimate, for the first time, the fracture energy and the effective size of the Fracture Process Zone (FPZ) as a function of the thickness of the DFC structure.From the integrated analysis, it is found that the fracture energy depends significantly on the structure thickness. It is shown to increase gradually with increasing thickness, t, and to saturate for t⩾3 mm to a value of 57.77 N/mm, which is 4.81 times larger than a typical aluminum alloy. For thicknesses lower than 1.1 mm, corresponding to an average number of platelets through the thickness of roughly 8, it was found that the fracturing process can often be triggered by weak spots in the structure rather than the stress-free notch.

Stochastic natural frequency analysis of skewed sandwich plates

PurposeThe purpose of this paper is to investigate the first three stochastic natural frequencies of skewed sandwich plates, considering uncertain system parameters. To conduct the sensitivity analysis for checking the criticality of input parameters.Design/methodology/approachThe theoretical formulation is developed based on higher-order-zigzag theory in accordance with the radial basis function (RBF) and stochastic finite element (FE) model. A cubic function is considered for in-plane displacement over thickness while a quadratic function is considered for transverse displacement within the core and remains constant in the facesheet. RBF is used as a surrogate model to achieve computational efficiency and accuracy. In the present study, the individual and combined effect of ply-orientation angle, skew angle, number of lamina, core thickness and material properties are considered for natural frequency analysis of sandwich plates.FindingsResults presented in this paper illustrates that the skewness in the sandwich plate significantly affects the global dynamic behaviour of the structure. RBF surrogate model coupled with stochastic FE approach significantly reduced the computational time (more than 1/18 times) compared to direct Monte Carlo simulation approach.Originality/valueThe stochastic results for dynamic stability of sandwich plates show that the inevitable source uncertainties present in the input parameters result in significant variation from the deterministic value demonstrates the need for inclusive design paradigm considering stochastic effects. The present paper comprehensively establishes a generalized new RBF-based FE approach for efficient stochastic analysis, which can be applicable to other complex structures too.

Reduction and interpretation of matrices of frequency response functions by Bayesian independent component analysis

This work seeks an effective data reduction method for matrices of Frequency Response Functions (FRF) in a way that preserves, as much as possible, the physical interpretation of FRFs in the full targeted frequency range. Also, this reduction method is wished able to cope with the different sources of uncertainties linked to the definition of the mechanical system whose FRFs are processed. It is shown that a Bayesian formulation of Independent Component Analysis (ICA) serves this purpose. It is used here to decompose a FRF matrix as a sum of frequency independent matrices multiplied by a frequency dependent scalar component. On the one hand, the independence property of this processing allows the scalar component to be concentrated in a narrow frequency range, on the other hand the chosen Bayesian approach presents itself as the most natural way to take into account uncertainties in the input FRFs whether they are due to measurement errors or structural uncertainties. Moreover, the probabilistic framework is shown to provide credible intervals on the estimation of the decomposition factors, thus allowing some considerations on the reliability of the processing and the development of a straightforward thresholding method to enhance the data reduction. A first application on measured automotive vibro-acoustic transfer functions shows the reduction performance of the approach and its interest when trying to analyse the measurements. A second application on non-parametric random FRFs computed through a stochastic finite element model illustrates the capacity of the proposed approach to take into account the uncertainty of the FRFs data and to propagate it to the factors of the decomposition.

Probability sensitivity estimation of linear stochastic finite element models applying Line Sampling

This paper presents a framework for probability sensitivity estimation of a class of problems involving linear stochastic finite element models. The sensitivity measure consists of the derivative of the failure probability with respect to the statistics of the underlying random field associated with the model. The framework is formulated as a post-processing step of Line Sampling and it is implemented considering two different approaches. The performance of these two approaches is studied by means of numerical examples. It is concluded that both offer effective means for estimating the sought sensitivity measure. Furthermore, it is observed that the correlation length associated with the random field controls the magnitude of the sensitivity measure.

Calculation of second order statistics of uncertain linear systems applying reduced order models

This contribution investigates the application of reduced order models for approximating the response of a class of uncertain linear systems. The basis associated with the reduced model is generated based on the results of a single analysis of the system plus a sensitivity analysis. The aforementioned strategy is applied in combination with control variates, thus allowing to estimate the second order statistics of the system’s response. In this way, accurate estimates of these statistics can be generated with reduced numerical efforts, as a large number of samples of the system’s response is evaluated with the reduced order model while only a small number of samples of the full model is required. Numerical examples comprising stochastic finite element models suggest that the proposed approach can produce estimates of the second order statistics with reduced variability.

Stochastic Finite-Element Modeling and Dynamic Characteristics Analysis of Pipe-Conveying Fluid

Randomness of fluid and structure properties inherently exists in actual pipes. The waves of parameters may play a significant role in the pipeline dynamic characteristics. However, the vibration analysis of pipe-conveying fluid is usually done by considering the deterministic parameters. To overcome the limitation, a stochastic dynamic model is proposed for the dynamic characteristics analysis of pipe-conveying fluid. Considering the uncertain fluid and structure properties described as one-dimensional stochastic field function, a stochastic dynamic pipeline model based on the stochastic finite element is presented for three-dimensional dynamic characteristics analysis in this paper. A stochastic analytical formula is deduced to obtain the statistical results of the structural dynamic characteristics and responses of pipeline with random properties. The statistics obtained from the presented model are compared with the prior experimental data and Monte Carlo simulation so as to validate the presented model and analysis method, and good agreement is obtained. Effects of structure and fluid randomness are studied in linear and nonlinear pipeline systems, respectively. It is found that the effects of random properties on pipeline eigenvalue and dynamic response are significant, especially the influences of the random elastic modulus, density and flow velocity.

Stochastic buckling analysis of sandwich plates: The importance of higher order modes

The stochastic buckling behaviour of sandwich plates is presented considering uncertain system parameters (material and geometric uncertainty). The higher-order-zigzag theory (HOZT) coupled with stochastic finite element model is employed to evaluate the random first three buckling loads. A cubic in-plane displacement variation is considered for both face sheets and core while quadratic transverse displacement is considered within the core and assumed constant in the faces beyond the core. The global stiffness matrix is stored in a single array by using skyline technique and stochastic buckling equation is solved by simultaneous iteration technique. The individual as well as compound stochastic effect of ply-orientation angle, core thickness, face sheets thickness and material properties (both core and laminate) of sandwich plates are considered in this study. A significant level of computational efficiency is achieved by using artificial neural network (ANN) based surrogate model coupled with the finite element approach. Statistical analyses are carried out to illustrate the results of stochastic buckling behaviour. Normally in case of various engineering applications, the critical buckling load with the least Eigen value is deemed to be useful. However, the results presented in this paper demonstrate the importance of considering higher order buckling modes in case of a realistic stochastic analysis. Besides that, the probabilistic results for global stability behaviour of sandwich structures show that a significant level of variation with respect to the deterministic values could occur due to the presence of inevitable source-uncertainty in the input parameters demonstrating the requirement of an inclusive design paradigm considering stochastic effects.

A stochastic hybrid blade tip timing approach for the identification and classification of turbomachine blade damage

Abstract Blade Tip Timing (BTT) has been in existence for many decades as an attractive vibration based condition monitoring technique for turbomachine blades. The technique is non-intrusive and online monitoring is possible. For these reasons, BTT may be regarded as a feasible technique to track the condition of turbomachine blades, thus preventing unexpected and catastrophic failures. The processing of BTT data to find the associated vibration characteristics is however non-trivial. In addition, these vibration characteristics are difficult to validate, therefore resulting in great uncertainty of the reliability of BTT techniques. This article therefore proposes a hybrid approach comprising a stochastic Finite Element Model (FEM) based modal analysis and Bayesian Linear Regression (BLR) based BTT technique. The use of this stochastic hybrid approach is demonstrated for the identification and classification of turbomachine blade damage. For the purposes of this demonstration, discrete damage is incrementally introduced to a simplified test blade of an experimental rotor setup. The damage identification and classification processes are further used to determine whether a damage threshold has been reached, therefore providing sufficient evidence to schedule a turbomachine outage. It is shown that the proposed stochastic hybrid approach may offer many short- and long-term benefits for practical implementation.

Incorporating spatial variability of pavement foundation layers stiffness in reliability-based mechanistic-empirical pavement performance prediction

Abstract Adequate pavement quality and performance are critical for road users’ safety, ride comfort, vehicle operation and travel delay costs, and vehicle durability. An accurate and robust pavement design is essential for realistic life cycle cost analysis, as well as overall management of the infrastructure. Compared to deterministic design methods, probabilistic methods are more realistic and can capture the inherent uncertainty in pavement and foundation materials; and loading conditions. In this study, spatial variability and systematic measurement errors in foundation layers’ (including the base and subgrade layers) stiffness are incorporated in reliability-based mechanistic-empirical (ME) pavement performance models. Geospatial models are used to characterize both, the spatial variability and systematic measurement errors. To predict the long term pavement performance, the geospatial models were used to construct stochastic finite element (FE) models, which were then used to predict the performance based on the mechanistic-empirical pavement design guide equations (MEPDG). It is found that the typical covariance functions, also known as semivariograms or variograms, should be handled carefully when used in probabilistic performance modeling. Separating the inherent spatial variability from other uncertainties is necessary for performing risk and reliability analysis. Moreover, incorporating the inherent spatial variability in the stochastic FE models can alter the location of the critical response as described in the MEPDG.

Multi-Objective Optimization of Rail Pre-Grinding Profile in Straight Line for High Speed Railway

In order to modify the rail pre-grinding profile smoothly, non-uniform rational B-spline (NURBS) curve with weight factors is used to establish a parameterized model of the profile. A wheel-rail contact stochastic finite element model (FEM) is constructed by the Latin hypercube sampling method and 3D elasto-plastic FEM, in which the wheelset’s lateral displacement quantity is regarded as a random variable. The maximum values of nodal accumulated contact stress (NACS) and nodal mean contact stress (NMCS) in different pre-grinding profiles with differential weight factors are calculated and taken as the training samples to establish two Kriging models. A multi-objective optimization model of pre-grinding profile is established, in which the objective functions are the NACS and NMCS Kriging models. The optimum weight factors are sought using a non-dominated sorting genetic algorithm II (NSGA-II), and the corresponding optimum pre-grinding profile is obtained. The contact stress calculation before and after optimization indicates that the maximum values of NACS and NMCS decline significantly.