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On quantifying the effect of noise in surrogate based stochastic free vibration analysis of laminated composite shallow shells
Abstract
This paper presents the effect of noise on surrogate based stochastic natural frequency analysis of composite laminates. Surrogate based uncertainty quantification has gained immense popularity in recent years due to its computational efficiency. On the other hand, noise is an inevitable factor in every real-life design process and structural response monitoring for any practical system. In this study, a novel algorithm is developed to explore the effect of noise in surrogate based uncertainty quantification approaches. The representative results have been presented for stochastic frequency analysis of spherical composite shallow shells considering Kriging based surrogate model. The finite element formulation for laminated composite shells has been developed based on Mindlin’s theory considering transverse shear deformation. The proposed approach for quantifying the effect of noise is general in nature and therefore, it can be extended to explore other surrogates under the influence of noise.
... Then, the method has been applied to solve uncertainty analysis problems in different areas of engineering e.g., mechanics of solid [16,17], fluid mechanics [18][19][20][21], structural dynamics [22,23], fluid-structure interaction [24,25], etc. A volume of literature is available on the application of SFEM in the uncertainty analysis of laminate composite plates [26][27][28][29][30][31]. ...
... To generate a polynomial chaos series, we refer to Ghanem and Spanos [10]. For ease of understanding, a 2nd order PC expansion using three inputs (see Eqs. (30) and (32)) is shown in expanded form as Eq. (33) and a simplified form as Eq. ...
This article quantifies uncertainty in the dynamic responses of marine cycloidal propeller (MCP) blade by Stochastic Finite Element Method (SFEM) during crash-stop ship maneuver. Stopping a ship in an emergency is known as crash-stop astern maneuver. A MCP contains a horizontal circular metallic disc on which six aerofoil blades are hung vertically. All blades are made up of carbon fiber-reinforced composite laminates. Uncertainty in ship speed and composite material properties are propagated to find out the statistical response of natural frequencies, displacement, and stresses on the blade. SFEM is a combination of Stochastic Response Surface Method (SRSM) and Finite Element Method (FEM). SRSM is a collocation-based non-intrusive polynomial chaos expansion to transform any random distribution into a stochastic response surface equation (surrogate model). During ship maneuvering motion, the whole unit rotates about a vertical axis at the centre. For each rotation of the disc, all blades undergo periodic oscillations. The SRSM for uncertainty propagation is computationally more efficient than the standard Monte Carlo Simulation (MCS) technique without compromising the accuracy of the results. It is observed that uncertainty in vibration responses of MCP blade is significant and worth study.
... The bottom-up surrogate approach for studying the influence of uncertainties on the free vibration behavior of laminates was proposed by Dey et al. [9]. Using a firstorder shear deformation theory-based Kriging surrogate model was proposed by Mukhopadhyay et al. [10] They predicted the influence of noise on the free vibration behavior of shallow laminated composite shells. Dey et al. [11] predicted the stochastic free vibration response of laminated shallow doubly curved shells using the Kriging surrogate model. ...
The present work aims to quantify the influence of uncertainties in the ply orientation of multi-layered bio-inspired helicoidal laminated composite conical, hemispherical, and toroidal shells under thermal conditions. Any change in the ply orientation affects the free vibration behavior of the laminates. The present investigation focuses on the different levels of uncertainties in the ply orientation on the free vibration behavior of the shell. Moreover, the study also focuses on the sensitivity of the uncertainties in ply orientations on the free vibration behavior of the shells, which is also quantified. To quantify the stochastic free vibration behavior of the shells, the Gaussian process regression (GPR) machine learning algorithm-based surrogate model is developed to predict the frequencies of the shells. The surrogate is created in the framework of higher-order shear deformation theory. The uncertainties in the ply orientations are introduced using bootstrapping. The present results are compared with the stochastic frequencies obtained using Monte Carlo simulations (MCS) to determine the model’s accuracy. The study highlights the influence of the temperature, type of shell, and end conditions on the stochastic free vibration behavior of bio-inspired laminated shells.
... Dey et al. (2015aDey et al. ( , b, 2016aDey et al. ( , b, c, d, 2017, Truong et al. (2021) have reviewed the usage of surrogate models for assessing uncertainty in vibration properties of the plates made of composite material. The different surrogate modelling techniques are polynomial regression, kriging (Mukhopadhyay et al. 2015c(Mukhopadhyay et al. , 2016, HDMR, ANN (Dey et al. 2016a, b, c, d), polynomial chaos expansion, support vector regression, moving least square, radial basis function, multivariate adaptive regression splines (Dey et al. 2016a, b, c, d), and support vector machines (Vaishali et al. 2020). They evaluated both computational efficiency and accuracy in handling uncertainty ). ...
Composite materials offer the unique advantage of allowing customization of their properties based on their load. However, the optimization of composite laminate properties can often be challenging, often leading to quasi-isotropic designs or the use of industry guidelines. This paper presents a novel method for optimizing of a composite cylinder under axial compression. It introduces an innovative approach by merging a tailored differential evolution algorithm with a deep neural network. The key modification is in the method of constraint implementation. The initial population and trial vectors are constrained to balanced laminates using a while loop, effectively shrinking the design space and reducing computational requirements. The advantage of the customization is reflected in the faster convergence of the optimization as well as a much more accurate deep neural network model. It also enabled the differential evolution to escape the local maxima. Using the deep neural network to evaluate candidate solutions, further reduces the computational costs. The technique is validated using linear buckling analysis and applied to design an inter-tank truss structure. The optimization resulted in a drop in the mass of the truss structure from 5.28 to 4.87 kg. The study establishes a general optimization method applicable to various composite cylinders, including short and long, thin and thick cylinders, and honeycomb core sandwiched composite structures.
... The use of fiber-reinforced polymer (FRP) bars and geosynthetic geogrids (G) as pile reinforcement materials was discovered to be a promising strategy, due to their high strengthweight ratio, durability, and high ability to anti-corrosion [7] [8]. Composite piles are commonly used as waterfront barriers, fenders, and bearing piles for light structures in ports due to the ongoing creation of novel composite materials [9][10][11][12][13]. They were also used in bridge rehabilitation work as load-bearing substructures [14]. ...
... Meanwhile, there are multiple types of noise, such as uniform noise [5], Gaussian noise [14], and random field noise. Mukhopadhyay et al. [25] has quantified the effect of different types of noise and revealed that the accuracy of the surrogate model is influenced by the type of noise. To evaluate and compare the performance and noise robustness of machine learning models and PIC under different noise types, we conducted experiments as outlined in the Supplementary Materials (Section S3.1). ...
Data-driven discovery of partial differential equations (PDEs) has recently made tremendous progress, and many canonical PDEs have been discovered successfully for proof of concept. However, determining the most proper PDE without prior references remains challenging in terms of practical applications. In this work, a physics-informed information criterion (PIC) is proposed to measure the parsimony and precision of the discovered PDE synthetically. The proposed PIC achieves satisfactory robustness to highly noisy and sparse data on 7 canonical PDEs from different physical scenes, which confirms its ability to handle difficult situations. The PIC is also employed to discover unrevealed macroscale governing equations from microscopic simulation data in an actual physical scene. The results show that the discovered macroscale PDE is precise and parsimonious and satisfies underlying symmetries, which facilitates understanding and simulation of the physical process. The proposition of the PIC enables practical applications of PDE discovery in discovering unrevealed governing equations in broader physical scenes.
... These variations can be interpreted as uncertainties, which may be the cause of variation in their mechanical properties. 3 Generally, uncertainties can be categorised into two types: (i) aleatoric, which arises from inherently random effects and (ii) epistemic, which is due to the lack of information and knowledge in any activity or phase of the modelling process because of ignorance of the environment and system variables. 4 If these are not properly accounted for, unexpected failure might take place. ...
Induced uncertainties during the filament winding process may cause a significant stochastic variation in the mechanical behaviour of composite shells. This paper aims to develop a novel and deep uncertainty quantification (UQ), sensitivity and reliability analyses of filament wound shells considering manufacturing uncertainties. Firstly, a progressive damage analysis is performed to estimate their deterministic burst pressure. Then, a signal-to-noise (SNR) approach is employed using the Taguchi method for sensitivity analysis and screening uncertainties arising from manufacturing. Initial results reveal that the shells are more sensitive to thickness uncertainties for thinner structures. Then, probabilistic and reliability analyses are carried out using the Boosted Decision Trees Regression (BDTR) approach from machine learning algorithms. Despite the complexity and non-linear relationships in the problem, the developed BDTR-based metamodel shows powerful predictive performance. A comparative study shows that ply thickness uncertainty leads to a significant underestimation of failure probability. For expensive and time-consuming models in that only a few runs can be affordable, a modified approximation method for reliability analysis is proposed. Results indicate a high capability at estimating failure probability with high accuracy.
... Recently there has been an increasing trend of coupling Machine learning (ML) with the conventional analysis of composite plates and shells for achieving computational efficiency and more insights [59][60][61][62][63][64][65][66]. ML is a computer algorithm that trains itself through an input-output dataset and subsequently predicts the output corresponding to an unseen set of input parameters. ...
Composite Structures
Available online 2 February 2023, 116756
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Random forest-based surrogates for transforming the behavioral predictions of laminated composite plates and shells from FSDT to Elasticity solutions
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https://doi.org/10.1016/j.compstruct.2023.116756
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Abstract
In the present work, a surrogate model based on the Random Forest (RF) machine learning is employed for transforming the First-order Shear Deformation Theory (FSDT) based solutions to elasticity based solutions. The bending behavior of laminated composite plates and shells is investigated to demonstrate the capability of such surrogate-assisted computational bridging. In the proposed approach, the surrogate model predicts the difference in stress and displacement between the values obtained using FSDT and Elasticity, which are thereby adjusted to the FSDT predictions for obtaining more accurate values. It leads to an accuracy of elasticity solutions, while having the computational expense of FSDT. The number of layers, thickness, the orientation of each layer, material properties, and geometric properties of plates and shells are considered as input variables used for training RF-based surrogate model. The accuracy of the proposed methodology has been determined by comparing the upgraded results with those available in the literature. The RF-based surrogate model can upgrade the FSDT-based governing behavior to more accurate 3D Elasticity based solutions, thus setting a milestone in coupling ML with composite theories to predict the behavior of laminated composite plates and shells more accurately with a low level of computational expenses.
... Fibre-reinforced composites usually have complex microstructures and their manufacturing processes are not trivial, which naturally makes them susceptible to variations in their microstructure, geometry and material properties. These variations can be interpreted as uncertainties, which may be the cause of variation in their mechanical properties [1]. ...
... However, the downside of the standard Monte Carlo method is its slow convergence and a large number of realizations (~10 4 ) are required to attain the desired accuracy. To mitigate the computationally expensive nature of MCS, the possible avenues could be parallelization of the Monte Carlo simulations [51] or utilization of surrogate modeling approaches [52,53]. Even though parallelization may be able to reduce the time for Monte Carlo Simulations, it still requires high computational effort. ...
This paper presents an experimental investigation supported by data-driven approaches concerning the influence of critical stochastic effects on the dynamic fracture toughness of glass-filled epoxy composites using a computationally efficient framework of uncertainty quantification. Three different shapes of glass particles are considered including rod, spherical and flaky shapes with coupled stochastic variations in aspect ratio, dynamic elastic modulus and volume fraction. An artificial neural network based surrogate assisted Monte Carlo simulation is carried out here in conjunction with advanced experimental techniques like digital image correlation and scanning electron microscopy to quantify the uncertainty and sensitivity associated with the dynamic fracture toughness of composites in terms of stress intensity factor under dynamic impact. The study reveals that the pre-crack initiation time regime shows the most prominent effect of uncertainty. Additionally, rod shape and the aspect ratio are the most sensitive filler type and input parameter respectively for characterizing dynamic fracture toughness. Here the quantitative results based on large-scale data-driven approaches convincingly demonstrate using a computational mapping between the stochastic input and output parameter spaces that the effect of uncertainty gets pronounced significantly while propagating from the compound source level to the impact responses. Such outcomes based on experimental data essentially bring us to the realization that quantification of uncertainty is of utmost importance for developing a reliable and practically relevant inclusive analysis and design framework for the dynamic fracture of particulate composites. With limited literature available on the determination of fracture toughness considering inertial effects, the present work demonstrates a novel and insightful experimental approach for uncertainty quantification and sensitivity analysis of dynamic fracture toughness of particulate polymer composites based on surrogate modeling.
... The finite element modelling (in the case of computationally intensive simulation of repetitive nature) [55,[95][96][97][98][99][100][101] can be an alternate way where inefficient analytical solutions [92][93][94] are not possible. The surrogate modelling ...
Purpose: This paper presents the sensitivity analysis (SA) of the random natural frequency responses of hybrid multi-functionally graded sandwich (HMGS) shells for establishing a unified measure in the case of multi-objective performances. The
functionally graded materials, laminated composites, and sandwich cores are employed to develop such novel structures to tailor the benefits of each component in a single structure.
Methods: A novel MARS-based sensitivity analysis of these hybrid multi-functionally graded sandwich shells is developed to achieve computational efficiency without compromising the outcome. Such surrogate-assisted FE approaches are
crucial for computationally intensive multi-objective systems. The basic governing equations of random natural frequency are framed based on finite element formulation. The variabilities of major influencing random input parameters (here, geometric
and material properties) are carried out by employing Monte Carlo simulation (MCS). The multivariate adaptive regression spline (MARS) is adopted as a surrogate model to increase computational efficiency.
Results and Conclusion: The results are portrayed to showcase the significant effects of variable input parameters (sensitivity) on random frequency responses of such novel HMGS shells. Hence, it provides the predominant random input parameters and
their relative degree of importance while designing such multi-dimensional structural systems. Thus, the contribution of this article lies in both the development of a computationally efficient sensitivity analysis approach and the insightful numerical
results for hybrid structures presented thereafter. The comprehensive and collective sensitivity quantification considering multi-functional objectives, as presented in this article, would lead to efficient computational modeling of complex structural
systems for more optimized designs and better quality control during manufacturing.
... The ease of handling, lack of requirement for heavy lifting and handling equipment, and corrosion resistance are some other factors that are advantageous in the repair, retrofitting, and rehabilitation of civil engineering structures. Because of continuous research and development on new composite materials [3][4][5][6], the use of such materials is found to be advantageous in terms of weight sensitivity and cost effectiveness. ...
Several approaches were used to explore the characteristics of reinforced concrete (RC) structural elements. Experimental work in the lab was extensively used as a means to examine the structural response and influence of different parameters under shear loads. Also, using numerical analysis to look into these components has been proven effective. This paper focuses on the shear conduct and response of circular beams reinforced with steel bars using the finite element (FE) model by considering the effect of reinforcement type and ratio, shear span-depth ratio (a/d), and member’s size. The FE model results were confirmed with the experimental outcomes of full-scale circular RC specimens tested earlier by scientists. The outcomes from the numerical study displayed that the proposed finite element replica was capable of simulating the characteristics of the beams, tested experimentally in the lab, with credible accuracy. From the FE model, it was found that the concrete shear contribution is best described as a formula that is inversely proportional to the member’s depth and directly proportional to the square root of axial stiffness of the reinforcement.
... They demonstrated that the approach is significantly faster than using ANN-based surrogates, without sacrificing any predictive capabilities. Mukhopadhyay et al. (2016) presented the impact of noise on stochastic natural frequency analyses based on surrogate models of composite laminates. They developed an algorithm for exploring the impact of noise in surrogate-based UQ methods and verified the approach for stochastic frequency analyses of spherical shallow shells using a surrogate model based on Kriging. ...
The utilization of surrogate models to approximate complex systems has recently gained increased popularity. Because of their capability to deal with black-box problems and lower computational requirements, surrogates were successfully utilized by researchers in various engineering and scientific fields. An efficient use of surrogates can bring considerable savings in computational resources and time. Since literature on surrogate modelling encompasses a large variety of approaches, the appropriate choice of a surrogate remains a challenging task. This review discusses significant publications where surrogate modelling for finite element method-based computations was utilized. We familiarize the reader with the subject, explain the function of surrogate modelling, sampling and model validation procedures, and give a description of the different surrogate types. We then discuss main categories where surrogate models are used: prediction, sensitivity analysis, uncertainty quantification, and surrogate-assisted optimization, and give detailed account of recent advances and applications. We review the most widely used and recently developed software tools that are used to apply the discussed techniques with ease. Based on a literature review of 180 papers related to surrogate modelling, we discuss major research trends, gaps, and practical recommendations. As the utilization of surrogate models grows in popularity, this review can function as a guide that makes surrogate modelling more accessible.
... Therefore, reliability analysis along with progressive damage analysis of composite structures is a challenging approach. Recently, surrogate models (also known as metamodels) have been used as a useful tool for enabling uncertainty quantification in presence of a large number of costly model evaluations, as it replaces an expensive high-fidelity model with a cheap whilst trained surrogate model [18][19][20]. Mukhopadhyay et al. [21] have presented a precious and concise review on surrogate-based uncertainty quantification algorithms. In addition to this, for stochastic natural frequency analysis of composite laminates, they carried out a comparative analysis for evaluation of different surrogate models from the viewpoint of computational efficiency and accuracy. ...
Progressive damage analysis of composite structures in the presence of uncertainties is a computationally-expensive and highly-complex process. This work tackles these issues by developing efficient finite element (FE)-based surrogate models that are constructed with artificial neural network (ANN) models and design of experiment (DOE) methods. The proposed framework for building surrogate models (metamodels) can capture various multi-scale uncertainties. In addition, to alleviate the computational burden in non-deterministic analyses, a novel strategy is proposed using the Plackett-Burman method to determine sources of uncertainties that have significant impacts in scattering the response. The response surface methodology (RSM), in cooperation with FE analysis, is used to create datasets. The RSM-based and ANN-based metamodels are constructed by using these datasets. Both are tested on unseen data. Key results show that for complex models, ANN-based metamodels have better accuracy than RSM-based ones in prediction. The high accuracy built surrogate models are used for stochastic, probabilistic and reliability analysis of filament wound composite tubes. As a result, this multi-scale surrogate-based framework significantly decreases the computational time and cost of the analyses. Reliability analysis demonstrates that the statistical correlation between ply properties is significant, and it must be considered for accurate evaluation.
... A significant number of relatively recent works concentrate on the aspect of material and geometric uncertainty of the micro and macro mechanical parameters of composites including the aspect of different forms of defects [46][47][48][49][50][51][52][53][54][55][56]. Machine learning assisted approaches have attracted the attention of researchers lately for achieving computational efficiency and obtaining better insights [57][58][59][60][61][62][63]. The aspect of sensitivity analysis for laminated composites has been addressed by a limited number of studies, albeit considering the performance parameters individually [64][65][66][67]. ...
The present paper proposes a surrogate-assisted moment-independent stochastic sensitivity analysis of laminated composite plates for establishing a unified measure in the case of multi-objective performances. With the advancements in artificially engineered structural systems spanning across different length scales, it has become more common to design composite structures for multi-objective performances like the criteria of deflection, buckling and vibration of multiple modes, different impact parameters, and failure. Normally sensitivity analysis is carried out separately and individually for different such performance parameters. This paradigm is no more suitable for advanced multi-functional structures like composite laminates. In this article, we propose an efficient unified sensitivity analysis approach based on weighted relative importance of different performance parameters by introducing the notion of engineering judgment. A moment-independent sensitivity analysis is proposed here based on finite element modeling of composites in conjunction with the Least Angle Regression assisted Polynomial Chaos Expansion (PCE) to achieve computational efficiency without compromising the outcome. Such surrogate-assisted finite element approaches are particularly crucial for computationally intensive multi-objective systems like composites. The layer-wise unified sensitivity quantification of laminated composites considering multi-functional objectives, as presented here, would lead to more optimized designs and better quality control while manufacturing the complex advanced structural systems.
... There are also some studies belonging to Dey et. al. [12,13] those deal with the uncertainty in composites via a surrogate based approach. Nguyen et. ...
Statistical Moment (SM) based modelling is a quite straightforward approach in stochastic modelling of uncertain structures. However, the method still has deficiencies including determination of SMs of natural frequencies of vibratory structures, and it has not been tested yet for realistic structures. This study aims such verification by employing high-degree statistical moments in stochastic equations. In this respect, SM approach is applied for two different uncertainty cases. In the first case, uncertain parameters are experimentally reproduced from the batch of laminated composite beams. Then, those uncertainties are fed to SM equations used in finite element model to obtain descriptive statistical quantities (mean, variance, skewness, and kurtosis) of stochastic natural frequencies. Next, Pearson model is utilized to obtain probability density function of the natural frequency by using standardized SMs. Beside this, uncertain fundamental natural frequency of fifty samples of composite beams is measured by experimental modal tests. All SM based predictions and modal test results are also compared with numerical Monte Carlo Simulations. The latter case examines composite beams having nonnormal uncertain thickness. Since the results are in good harmony with each other, it is concluded that high-order SM based approach may effectively be used in uncertainty modelling of realistic structures.
... Though it is not possible to completely eliminate these sources of uncertainties, they can be reduced by making the better prediction models by making use of simulated noisy datasets [67]. To create the simulated noisy dataset from traditional design dataset which is used for the formation of surrogate model, a Gaussian white noise is introduced which has a specific variance level (p), in the set of responses [68], while the inputs are kept unchanged. ...
In the present study, to examine the excavation-induced instability in the support system of rock tunnel, an analytical approach namely confinement convergence method (CCM) is employed with two different constitutive models, namely Mohr–coulomb criterion (MC) and Generalized Hoek–Brown criterion (GHB). Probabilistic analysis is performed on a circular tunnel with respect to two limiting functions: criteria regarding the radius of plastic zone and the tunnel convergence. Three surrogate models, namely collocation-based stochastic response surface method (CSRSM), multi-gene genetic programming (MGGP) and multivariate adaptive regression splines (MARS), are used in conjunction with the Monte-Carlo simulation (MCS) to calculate the probability of failure (Pf) of tunnel. The results of each methodology are compared with the traditional MCS, regarding the efficiency and the accuracy. Negative correlation between the shear strength parameters in the MC criterion is constructed using Gaussian copula. A detailed comparison is made among all three surrogates based on the Pf obtained using MCS. To incorporate the effect of the epistemic uncertainty, a Gaussian white noise having a specific variance level (p) is introduced and the noisy datasets are engaged by the MARS and CSRSM surrogate models for the probabilistic analysis.
... Instead, surrogate model approaches, which are constructed based on a limited set of actual input/output data points, are a suitable method when dealing with such complex problems. Several surrogate models have been reported in the literature to evaluate uncertainties in composite laminates, such as Kriging method, 18,19 radial basis function, 20 polynomial chaos expansion (PCE), 15,21,22 and artificial neural network. 23,24 State-of-the-art reviews on the surrogate models for evaluating the uncertainty in structural responses of composite laminates can be found in reference. ...
The natural origin of the fibers combined with random production flaws results in significant uncertainties in the properties of natural fiber-reinforced composites. A probabilistic assessment can help to characterize the uncertainties and evaluate the reliability of natural fiber composites, enabling their use in engineering designs. To this end, this study aims to analyze the uncertainties in the tensile strength and frequency response of a unidirectional flax/epoxy composite due to the actual variability of various input parameters, including the fiber material properties and manufacturing flaws. Based on the available data in the literature, the non-deterministic input variables were divided into normal and uniform variables using a statistical test. A computationally efficient response surface approach based on the polynomial chaos expansion was proposed to conduct the uncertainty analysis with multi-type uncertain variables. Moreover, the results were validated by experimental measurements and compared with the direct Monte Carlo simulation to demonstrate the accuracy and efficiency of the surrogate model.
... Both the methods are found to perform satisfactorily for all the three structures upto a noise level of 1.5%. Subsequently, Mukhopadhyay et al. [402,403] quantified the effect of noise on machine learning based uncertainty quantification of polymer composites. Saseendran et al. [404] analyzed the impact of noise on two ML algorithms namely, polynomial and linear regression with ridge regularization. ...
The superior multi-functional properties of polymer composites have made them an ideal choice for aerospace, automobile, marine, civil, and many other technologically demanding industries. The increasing demand of these composites calls for an extensive investigation of their physical, chemical and mechanical behavior under different exposure conditions. Machine learning (ML) has been recognized as a powerful predictive tool for data-driven multi-physical modeling, leading to unprecedented insights and exploration of the system properties beyond the capability of traditional computational and experimental analyses. Here we aim to abridge the findings of the large volume of relevant literature and highlight the broad spectrum potential of ML in applications like prediction, optimization, feature identification, uncertainty quantification, reliability and sensitivity analysis along with the framework of different ML algorithms concerning polymer composites. Challenges like the curse of dimensionality, overfitting, noise and mixed variable problems are discussed, including the latest advancements in ML that have the potential to be integrated in the field of polymer composites. Based on the extensive literature survey, a few recommendations on the exploitation of various ML algorithms for addressing different critical problems concerning polymer composites are provided along with insightful perspectives on the potential directions of future research.
... Subsequently, Mukhopadhyay et al. [402,403] quantified the effect of noise on machine learning based uncertainty quantification of polymer composites. Saseendran et al. [404] analyzed the impact of noise on two ML algorithms namely, polynomial and linear regression with ridge regularization. ...
The superior multi-functional properties of polymer composites have made them an ideal choice for aerospace, automobile, marine, civil, and many other technologically demanding industries. The increasing demand of these composites calls for an extensive investigation of their physical, chemical and mechanical behavior under different exposure conditions. Machine learning (ML) has been recognized as a powerful predictive tool for data-driven multi-physical modeling, leading to unprecedented insights and exploration of the system properties beyond the capability of traditional computational and experimental analyses. Here we aim to abridge the findings of the large volume of relevant literature and highlight the broad spectrum potential of ML in applications like prediction, optimization, feature identification, uncertainty quantification, reliability and sensitivity analysis along with the framework of different ML algorithms concerning polymer composites. Challenges like the curse of dimensionality, overfitting, noise and mixed variable problems are discussed, including the latest advancements in ML that have the potential to be integrated in the field of polymer composites. Based on the extensive literature survey, a few recommendations on the exploitation of various ML algorithms for addressing different critical problems concerning polymer composites are provided along with insightful perspectives on the potential directions of future research.
... But for safe and economical design of these hybrid FG-sandwich structures, it is very necessary to consider these uncertainties [10,11,16,[64][65][66]. Probabilistic approaches for predicting uncertainty-based dynamic responses in case of complex structures like composite plates and shells have gained extreme attention from the researchers [9,17,51,52]. Uncertainty in the field of dynamic stability of composites was studied [15,[26][27][28][29][30][31]. ...
The present chapter investigates the moment-independent sensitivity analysis for hybrid sandwich structures (having cylindrical shell geometry) subjected to low-velocity impact. These hybrid structures are extensively used in lightweight applications where thermal exposure/resistance is of prime importance. Here, the FG facesheet is placed at the upper layer of core whereas laminated composite facesheet is kept at lower layer so that the structure can sustain high-temperature exposure at reduced weight because of laminated composite facesheet at the inner layer of the core. The probabilistic study is performed for the transient impact response of the structure which in turn utilized to assess the sensitiveness of the parameters. The computational efficiency is achieved by implementing polynomial chaos expansion (PCE) metamodel in conjunction with Monte Carlo simulation (MCS). The results illustrate the parameters which significantly affect the transient impact response of the structure.
... There is nothing as ideal in this world and hence to fully eliminate the epistemic uncertainties is next to impossible. To make the results and simulation more realistic, a part of epistemic uncertainty is incorporated as noise in the performance functions [68,69]. ...
This article presents a computational reliability analysis of reinforced soil-retaining structures (RSRS) under seismic conditions. The internal stability of RSRS is evaluated using the horizontal slice method (HSM) with modified pseudo-dynamic seismic forces. Two different failure modes of RSRS are identified and their reliability indices are computed using the first-order reliability method (FORM). The critical probabilistic failure surface is identified using a three-tier optimization scheme. Reliability index of the system is computed by considering the modes of failure to be connected in series. The tension mode is found to be the most critical mode of failure. The present study identifies that the wall height (H), shear wave velocity of the soil (Vs), and predominant frequency of the input motion (ω) govern the response of RSRS. Reliability indices depend on a parameter termed as the normalized frequency (ωH/Vs) and their values decrease with an increase in the value of ωH/Vs. Increase in the damping ratio of soil, increases the value of reliability indices, especially for ωH/Vs values, which are close to π/2. The FORM suffers from few critical shortcomings such as linear assumption of limit state surface at the most probable point of failure and its ability to consider only the statistical uncertainties excluding the effect of epistemic uncertainties. This calls for sampling-based numerical techniques such as Monte-Carlo simulation (MCS) which gives more comprehensive understanding of the problem under consideration in a probabilistic framework. Thus, a computationally efficient surrogate-assisted MCS is carried out to validate the present formulation and provide numerical insights by capturing the system dynamics over the entire design domain. Adoption of the efficient surrogate-assisted approach allowed us to quantify the epistemic uncertainty associated with the system using Gaussian white noise (GWN). Subsequently, its effects on the system reliability index and probabilistic behavior of the critical parameters are presented. The numerical results clearly indicate that it is imperative to take into account the probabilistic deviations of the critical performance parameters for RSRS to ensure adequate safety and serviceability under operational condition while quantifying the reliability of such systems.
... Recently researchers (Patil et al., 2017;Deierling and Zhupanska, 2018;Gopinath and Batra, 2018;Zhu et al., 2018;Zhou et al., 2020;Mukhopadhyay et al., 2016) investigated the effective properties of composites using methods based on homogenization and RVE/unit cell technique. Patil et al. (2017) predicted the effective properties of heterogeneous materials by applying the extended as well as multiscale FE method (XFEM and MsFEM) in which the particles and voids were incorporated into the matrix. ...
This article presents different micromechanical modelling techniques based on analytical and numerical approaches to determine the effective elastic and piezoelectric (piezoelastic) properties of graphene-based composite materials. Different types, orientations and shapes as well as different geometrical parameters of fiber reinforcement are considered for estimating the effective properties. The effective properties of composite are predicted with and without considering the strong covalent bond which provides interaction and in-plane stability of 2Dcrystalline graphene or strong van der Wall forces formed between graphene layers and the matrix. It is revealed that the axial, transverse and shear effective piezoelastic properties of graphene reinforced piezoelectric composite (GRPC) are significantly enriched due to the incorporation of graphene into the epoxy matrix. The importance of incorporating graphene as nanofillers/interphase into the conventional epoxy matrix to form an advanced composite and its effective properties are illustrated while these results show excellent agreement with previously reported experimental estimates. These results reveal that due to incorporation of graphene nanofillers, there is a significant enhancement in effective properties of composite. The results would also help to recognize the most important material properties with respect to different shapes and orientation of reinforcements which influences the performance of system significantly. To confirm safety, robustness and sustainability of the structure, it is the most prior requirement to determine the effective properties of composites considering different parameters for the different static and structural analyses.
... The ease of handling, lack of requirement for heavy lifting and handling equipment, and corrosion resistance are some other factors that are advantageous in the repair, retrofitting, and rehabilitation of civil engineering structures. Because of continuous research and development on new composite materials [3][4][5][6], the use of such materials is found to be advantageous in terms of weight sensitivity and cost effectiveness. ...
A numerical analysis investigation, using finite element model (FEM) and modified compression field theory (MCFT), was conducted to evaluate the shear capacity and behavior of circular concrete piles reinforced with steel and FRP bars by considering shear behavior, shear strength, and deflection shape. The FEM and MCFT models were verified against the experimental results of full-scale circular concrete speciemns previously tested by the authors. Load-deformation response curve, load-strain for the concrete and reinforced rebars were predicted using finite element analysis were compared to experimental results. The FEM outcomes showed that the model was able to capture the behavior of the specimens with good accuracy. While, the modified compression field theory using Response 2000 (R2K) software provided good predictions of the shear strength with an average value of V exp /V Response2000 for specimens is 1.17. Subsequently, a parameric study was performed to study the effect of member's depth (300, 400, and 600 mm), longitudinal reinforcement ratio (1.5, 2.5, and 3.5%), and reinforcing bars material (Steel, Glass-FRP, Carbon-FRP) on the behaviour of circular concrete piles.
... The ease of handling, lack of requirement for heavy lifting and handling equipment, and corrosion resistance are some other factors that are advantageous in the repair, retrofitting, and rehabilitation of civil engineering structures. Because of continuous research and development on new composite materials [3][4][5][6], the use of such materials is found to be advantageous in terms of weight sensitivity and cost effectiveness. ...
A numerical analysis investigation, using finite element model (FEM) and modified compression field theory (MCFT), was conducted to evaluate the shear capacity and behavior of circular concrete piles reinforced with steel and FRP bars by considering shear behavior, shear strength, and deflection shape. The FEM and MCFT models were verified against the experimental results of full-scale circular concrete speciemns previously tested by the authors. Load-deformation response curve, load-strain for the concrete and reinforced rebars were predicted using finite element analysis were compared to experimental results. The FEM outcomes showed that the model was able to capture the behavior of the specimens with good accuracy. While, the modified compression field theory using Response 2000 (R2K) software provided good predictions of the shear strength with an average value of V exp /V Response2000 for specimens is 1.17. Subsequently, a parameric study was performed to study the effect of member's depth (300, 400, and 600 mm), longitudinal reinforcement ratio (1.5, 2.5, and 3.5%), and reinforcing bars material (Steel, Glass-FRP, Carbon-FRP) on the behaviour of circular concrete piles.
... The ease of handling, lack of requirement for heavy lifting and handling equipment, and corrosion resistance are some other factors that are advantageous in the repair, retrofitting, and rehabilitation of civil engineering structures. Because of continuous research and development on new composite materials [3][4][5][6], the use of such materials is found to be advantageous in terms of weight sensitivity and cost effectiveness. ...
A numerical analysis investigation, using finite element model (FEM) and modified compression field theory (MCFT), was conducted to evaluate the shear capacity and behavior of circular concrete piles reinforced with steel and FRP bars by considering shear behavior, shear strength, and deflection shape. The FEM and MCFT models were verified against the experimental results of full-scale circular concrete speciemns previously tested by the authors. Load-deformation response curve, load-strain for the concrete and reinforced rebars were predicted using finite element analysis were compared to experimental results. The FEM outcomes showed that the model was able to capture the behavior of the specimens with good accuracy. While, the modified compression field theory using Response 2000 (R2K) software provided good predictions of the shear strength with an average value of V exp /V Response2000 for specimens is 1.17. Subsequently, a parameric study was performed to study the effect of member's depth (300, 400, and 600 mm), longitudinal reinforcement ratio (1.5, 2.5, and 3.5%), and reinforcing bars material (Steel, Glass-FRP, Carbon-FRP) on the behaviour of circular concrete piles.
... The ease of handling, lack of requirement for heavy lifting and handling equipment, and corrosion resistance are some other factors that are advantageous in the repair, retrofitting, and rehabilitation of civil engineering structures. Because of continuous research and development on new composite materials [3][4][5][6], the use of such materials is found to be advantageous in terms of weight sensitivity and cost effectiveness. ...
... Le prédicteur fourni par le krigeage est donc associé à une estimation de l'erreur locale (variance) de la prédiction. Le KRI a servi dans des problèmes d'optimisation [120][121], et dans de nombreux problèmes de sciences de l'ingénieur comme en mécanique [122] mais aussi en CEM [123]. Il existe de nombreuses variantes : adaptative [67], le Co-krigeage [124][125] [126], Poisson-krigeage [127], krigeage bayésien [128], en association avec la méthode fiabiliste de subset simulation [129]. ...
Une Interférence Electromagnétique Intentionnelle (IEMI) agressant un équipement électronique peut provoquer une défaillance de ce dernier. L’étude des effets d’une IEMI commence par l’évaluation du risque de défaillance afin de mettre en place les protections adéquates. Malheureusement, une prédiction déterministe d’une défaillance est impossible car les caractéristiques de l’équipement et de l’agression sont très incertaines. La stratégie proposée consiste à modéliser la contrainte générée par l’agression, ainsi que la susceptibilité de l’équipement, comme des variables aléatoires. Ensuite, trois étapes sont nécessaires. La première concerne l’estimation de la distribution probabiliste de la variable aléatoire de susceptibilité, la seconde porte sur celle de la contrainte, pour enfin, dans une troisième étape, en déduire la probabilité de défaillance. Pour la première étape, nous utilisons des méthodes d’inférence statistique sur un petit échantillon de seuils de susceptibilités mesurés. Nous comparons deux types d’inférence paramétrique : bayésienne et celle du maximum de vraisemblance. Nous concluons qu’une approche pertinente pour l'analyse du risque CEM consiste à utiliser les intervalles de confiance ou de crédibilité des estimations des paramètres pour encadrer la probabilité de défaillance, quelle que soit la méthode d’inférence choisie. Pour la deuxième étape, nous explorons les techniques de recherche de valeurs extrêmes tout en réduisant le nombre de simulations nécessaires. En particulier, nous proposons la technique de la stratification contrôlée par un métamodèle de krigeage. Nous montrons que cette technique améliore drastiquement les performances par rapport à l’approche classique (simulation Monte Carlo). De plus, nous proposons une implémentation particulière de cette technique afin de maitriser le coût de calcul. Enfin, la troisième étape est la plus simple une fois les deux premières franchies puisque, par définition, une défaillance survient lorsque la contrainte est supérieure à la susceptibilité. A partir d’un cas test final comportant la simulation de l’agression d’un équipement et de données de susceptibilité de cet équipement, nous calculons un encadrement de la probabilité de défaillance en recourant aux méthodes développées pendant cette thèse. En particulier, nous montrons que l’utilisation conjointe de la stratification contrôlée par le krigeage et de l’inférence de la distribution de la susceptibilité permet effectivement d’encadrer l’estimation de la vraie probabilité de défaillance.
... Stochastic natural frequency of noise-induced support vector regression (SVR) model for laminated composite curved panels is analysed by Dey et al. [49]. The effect of noise in surrogate model is quantified by Mukhopadhyay et al. [50], whereas Dey et al. [44] presented the effect of noise using polynomial neural network approach in uncertainty quantification of natural frequency, and Chakraborty [51] predicted delamination in laminated composite using ANN approach. Lee et al. [52] determined the equivalent material properties of glass/epoxy composite by applying the homogenization method, and stochasticity in the equivalent material properties are assessed by MCS and found that variation in individual material properties has a significant effect on the equivalent material properties. ...
The present paper portrays the mapping for stochasticity in low-velocity impact responses of functionally graded material (FGM) plates by employing multivariate adaptive regression splines (MARS) surrogate model in conjunction with finite element (FE) approach. The unavoidable stochastic variabilities (caused due to numerous errors involved in manufacturing processes) in material properties of FGM plates are considered in order to map the effect of elemental variabilities on global response of the structure. The material properties of FGM plates are considered to follow the rule of mixture in conjunction to power law. The Newmark’s time integration scheme and modified Hertzian contact law are employed to solve the time-dependent equation. The present FE formulation is based on an eight noded isoparametric element in which each element has five degrees of freedom. The effects of variability in temperature and power-law exponent on stochastic low-velocity impact responses are also portrayed. The maximum contact force, plate and impactor displacement are considered as the response parameters. The present MARS model is coupled with the finite element to achieve the higher efficiency with adequate accuracy as compared to the FE-based full-scale Monte Carlo simulation. The statistical results illustrate that the stochasticity in material properties significantly influences the low-velocity impact responses of FGM plates.
... Owing to its numerous advantages, metamodels have recently received significant attention in structural engineering applications ranging from prediction [1,2] and uncertainty quantification [3][4][5][6][7][8] to single-objective [9,10] and multi-objective optimization [11,12]. Among all such metamodels, polynomial regression (PR) is perhaps the most widely used approach. ...
High-fidelity multi-scale design optimization of many real-life applications in structural engineering still remains largely intractable due to the computationally intensive nature of numerical solvers like finite element method. Thus, in this paper, an alternate route of metamodel based design optimization methodology is proposed in multi-scale framework based on a symbolic regression implemented using genetic programming (GP) coupled with D-optimal design. This approach drastically cuts the computational costs by replacing the finite element module with appropriately constructed robust and efficient metamodels. Resulting models are compact, have good interpretability and assume a free-form expression capable of capturing the non-linearly, complexity and vastness of the design space. Two robust nature-inspired optimization algorithms, viz. multi-objective genetic algorithm (MOGA) and multi-objective particle swarm optimization (MOPSO), are used to generate Pareto optimal solutions for several test problems with varying complexity. TOPSIS, a multi-criteria decision-making approach is then applied to choose the best alternative among the Pareto optimal sets. Finally, the applicability of GP in efficiently tackling multi-scale optimization problems of composites is investigated, where a real-life scenario is explored by varying fractions of pertinent engineering materials to bring about property changes in the final composite structure across two different scales. The study reveals that a micro-scale optimization leads to better optimized solutions, demonstrating the advantage of carrying out a multi-scale optimization without any additional computational burden.
... investigated on the stochasticity in structural integrity and material distribution in mechanical properties of metamaterials, while in high stochastic dimension and modelling error, probabilistic model of random field is constructed, identified and presented for computational structural dynamics by Batou and Soize (2013). The effect of noise in a surrogate model is quantified by Mukhopadhyay et al. (2016), whereas Dey et al. (2016b) presented the effect of noise using polynomial neural network approach in uncertainty quantification of natural frequency and Chakraborty (2005) predicted delamination in laminated composite using artificial neural network approach. Few researchers used modal stability procedure-based Monte Carlo simulation (MCS) approach for stochastic analysis (Druesne et al., 2016;Yin et al., 2018), whereas some researchers used surrogate-based MCS approach for stochastic analysis (Karsh et al., 2018a(Karsh et al., , 2018b(Karsh et al., , 2018c(Karsh et al., , 2019Kumar et al., 2019;Mukhopadhyay et al., , 2018 to reduce the computational time and cost. ...
Purpose – The 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/approach – The 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.
Findings – Results 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/value – The 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.
... The uncertainties incurred during the layup process are due to the misalignment of ply-orientation. Typical uncertainties incurred from the curing process are intralaminate voids, incomplete curing of resin, excess resin between plies, excess matrix voids and porosity and variation in ply thickness [65][66][67][68][69]. ...
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.
... The aspect of considering uncertainty in the analysis and design of composite structures are becoming increasingly recognised in last few years. Probabilistic approaches are found to be predominantly used to quantify the effect of uncertainty in composite structures following intrusive [Lal and Singh (2010), Scarth and Adhikari (2017), Zhou et al. (2017)] as well as non-intrusive [Shaw et al. (2010), Dey et al. (2016aDey et al. ( , 2018, Umesh and Ganguli (2013), , Karsh et al. (2018), Sakata et al. (2008), , 2017c, 2018, Mukhopadhyay and Adkhikari (2016b); Mukhopadhyay et al. (2016c)] methods. The prerequisite of carrying out a probabilistic analysis is to 3 Fig. 1 Fuzzy multi-scale analysis of composite laminates considering the coupled effect of material and geometric uncertainty (a) Typical distribution of a typical material property y (such as E 1 ) along a cross-sectional view (X-Z plane) of two laminae for a particular realization in case of the conventional approach where spatial variation of y is ignored (b) Typical distribution of a material property y along a cross-sectional view (X-Z plane) of two laminae for a particular realization in case of the present approach considering spatial variation of material properties (c) Typical representation of a fuzzy based analysis have the statistical distribution of stochastic input parameters available. ...
This article presents a non-probabilistic fuzzy based multi-scale uncertainty propagation framework for studying the dynamic and stability characteristics of composite laminates with spatially varying system properties. Most of the studies concerning the uncertainty quantification of composites rely on probabilistic analyses, where the prerequisite is to have the statistical distribution of stochastic input parameters. In many engineering problems, these statistical distributions remain unavailable due to the restriction of performing large number of experiments. In such situations, a fuzzy-based approach could be appropriate to characterize the effect of uncertainty. A novel concept of fuzzy representative volume element (FRVE) is developed here for accounting the spatially varying non-probabilistic source-uncertainties at the input level. Such approach of uncertainty modelling is physically more relevant than the prevalent way of modelling non-probabilistic uncertainty without considering the ply-level spatial variability. An efficient radial basis function based stochastic algorithm coupled with the fuzzy finite element model of composites is developed for the multi-scale uncertainty propagation involving multi-synchronous triggering parameters. The concept of a fuzzy factor of safety (FFoS) is discussed in this paper for evaluation of safety factor in the non-probabilistic regime. The results reveal that the present physically relevant approach of modelling fuzzy uncertainty considering ply-level spatial variability obtains significantly lower fuzzy bounds of the global responses compared to the conventional approach of non-probabilistic modelling neglecting the spatially varying attributes.
In this work, the free vibration characteristics of flexbeam like structures, typically used in helicopter main or tail rotor blades, are investigated. The tapering effect in these laminated composite structures is introduced by terminating plies, which act as potential delamination sites. Delamination and uncertainties in material properties at various scales influence their dynamic behavior. The variational asymptotic method (VAM) is used as a mathematical framework to develop a model of a tapered composite beam with delamination. Subsequently, the modal characteristics are obtained by coupling the VAM model with the reduced 1D finite element beam model. Stochastic natural frequencies are investigated by combining the source uncertainties in the delaminated structure. Following a detailed validation study, the model capability is demonstrated on flexbeams typically used in helicopter rotor blades.
This paper proposes a stochastic approach for estimating the natural frequencies of reducible composite strips. The framework accounts for multi-scale uncertainty at micro, meso, and component levels; additionally, it accounts for damages due to matrix cracking and delamination in a pretwisted composite strip. An asymptotically accurate Variational Asymptotic Method (VAM) based model for cross-sectional stiffness is coupled with a one-dimensional Finite Element Method (FEM) approach for the determination of natural frequencies of the pretwisted strips. The proposed model is validated by comparing the deterministic natural frequencies of composite strips with experimental and numerical results reported in the literature. The quantification of uncertainty is carried out by determining the probabilistic variations in the first three natural frequencies of the strips and analyzing the local and global sensitivity measures. The study identifies the uncertainties in material attributes, addressed at micro and macro levels, influencing the free vibration response of the strips.
Composite laminates have found wide-ranging applications in various areas of structural, marine and aerospace industries. Their design and optimization is a challenging task due to involvement of a large number of design variables. Because of high accuracy of the laminate modeling theories and presence of numerous design variables, laminate design and optimization is primarily carried out in silico. Integratin the high accuracy of these laminate modeling theories using numerical solvers, like finite element method, boundary element method etc. with the iterative improvement capability of different optimization algorithms is a well-established approach and can be broadly referred to as high-fidelity optimization. However, in recent times with the advent of machine learning and statistical approaches, metamodel-based optimization has gained significant prominence, primarily due to its less computational time and effort. In this review paper, the essence of nearly 300 research articles (about 26% and 50% of them are from last 5 and 10 years respectively.) on high-fidelity and metamodel-based optimization of composite laminates is comprehensively assessed and presented. Special emphasis is provided on the discussion of various metamodels. The methodology and key outputs of each research article are concisely presented in this paper, which would make it an asset for the future researchers and design engineers.
The wire electric discharge machining (WEDM) is a potential alternative over the conventional machining methods, in terms of accuracy and ease in producing intricate shapes. However, the WEDM process parameters are exposed to unavoidable and unknown sources of uncertainties, following their inevitable influence over the process performance features. Thus, in the present work, we quantified the role of parametric uncertainty on the performance of the WEDM process. To this end, we used the practically relevant noisy experimental dataset to construct the four different machine learning (ML) models (linear regression, regression trees, support vector machines, and Gaussian process regression) and compared their goodness of fit based on the corresponding R ² and RMSE values. We further validated the prediction capability of the tested models by performing the error analysis. The model with the highest computational efficiency among the tested models is then used to perform data-driven uncertainty quantification and sensitivity analysis. The findings of the present article suggest that the pulse on time ( T on ) and peak current (IP) are the most sensitive parameters that influence the performance measures of the WEDM process. In this way, the current study achieves two goals: first, it proposes a predictive framework for determining the performance features of WEDM for unknown design points, and second, it reports data-driven uncertainty analysis in the light of parametric perturbations. The observations reported in the present article provide comprehensive computational insights into the performance characteristics of the WEDM process.
Thin-walled cylindrical shells are commonly used structures in explosion containment vessels. When subjected to internal blast loading, these cylindrical shells are prone to dynamic buckling failure. In this work the dynamic stability of cylindrical fiber composite shells with metal liner subjected to uniform internal pressure pulse was investigated through both finite element simulations and theoretical modelling. In particular, pulse buckling of the inner metal liner and vibrational buckling of the outer fiber composite shell were observed and studied. Dynamic, Implicit analysis in ABAQUS Standard considering the initial geometric imperfections were conducted to study the mechanisms for the dynamic buckling of the inner metal liner and outer fiber composite shell. The simulated buckled shape using a simplified two-dimensional model could successfully reproduce the previous experimental results. The dynamic pulse buckling of the metal liner was found to occur during the plastic compression process. The effect of the buckling amplitude of the inner metal liner on the dynamic stability of the outer fiber composite shell was also revealed using numerical simulations. Instead of preventing the outer fiber composite shell from buckling instability, severely buckled metal liner with large buckling amplitude may even accelerate the growth of unstable mode of the fiber composite shell. The theoretical solution for the radial deformation of the metal liner was established by modelling the growth of the buckling mode as the amplification of the initial geometric or velocity imperfection. The most amplified mode and corresponding amplification were determined by the derived amplification function. The solutions for the radial deformation of the outer fiber composite shell were represented by a set of Mathieu differential equations. The dynamic instability of the fiber composite shells was determined from the Mathieu stability charts. The dependence of the dynamic instability of the bilayer composite shells on various parameters such as composite layup, thickness-to-radius ratio and impulse of pressure pulse loading was theoretically predicted and compared with the finite element simulation results.
High-fidelity structural analysis using numerical techniques, such as finite element method (FEM), has become an essential step in design of laminated composite structures. Despite its high accuracy, the computational intensiveness of FEM is its serious drawback. Once trained properly, the metamodels developed with even a small training set of FEM data can be employed to replace the original FEM model. In this paper, an attempt is put forward to investigate the utility of radial basis function (RBF) metamodels in the predictive modelling of laminated composites. The effectiveness of various RBF basis functions is assessed. The role of problem dimensionality on the RBF metamodels is studied while considering a low-dimensional (2-variable) and a high-dimensional (16-variable) problem. The effect of uniformity of training sample points on the performance of RBF metamodels is also explored while considering three different sampling methods, i.e., random sampling, Latin hypercube sampling and Hammersley sampling. It is observed that relying only on the performance metrics, such as cross-validation error that essentially reuses training samples to assess the performance of the metamodels, may lead to ill-informed decisions. The performance of metamodels should also be assessed based on independent test data. It is further revealed that uniformity in training samples would lead towards better trained metamodels.
In the present chapter, multivariate adaptive regression splines (MARS) is explored as a surrogate model in conjunction to Monte Carlo simulation (MCS) to analyse the random first-ply failure loads of graphite–epoxy laminated composite plates. The five failure criteria, namely maximum strain theory, maximum stress theory, Tsai–Hill theory, Tsai–Wu theory, and Hoffman theory, are considered. The numerical validation of deterministic failure load is presented first. Thereafter, a concise investigation is carried out to examine the capability of MARS model for efficiently predicting the first-ply failure loads. Comparative results are presented using scatter plots and probability density function plots to access the prediction capability with respect to direct MCS. The current results portray the successful application of MARS as the surrogate model to achieve computational efficiency and analyse the first-ply failure loads.
In this study, the influences of spatially varying stochastic properties on free vibration analysis of composite plates were investigated via development of a new approach named the deterministic-stochastic Galerkin-based semi-analytical method. The material properties including tensile modulus, shear modulus, and density of the plate were assumed to be spatially varying and uncertain. Gaussian fields with first-order Markov kernels were utilized to define the aforementioned material properties. The stochastic fields were decomposed via application of the Karhunen-Loeve theorem. A first-order shear deformation theory was assumed, following which the displacement field was defined using admissible trigonometric modes to derive the potential and kinetic energies. The stochastic equations of motion of the plate were obtained using the variational principle. The deterministic-stochastic Galerkin-based method was utilized to find the probability space of natural frequencies, and the corresponding mode shapes of the plate were determined using a polynomial chaos approach. The proposed method significantly reduced the size of the mathematical models of the structure, which is very useful for enhancing the computational efficiency of stochastic simulations. The methodology was verified using a stochastic finite element method and the available results in literature. The sensitivity of natural frequencies and corresponding mode shapes due to the uncertainty of material properties was investigated, and the results indicated that the higher-order modes are more sensitive to uncertainty propagation in spatially varying properties.
Due to the absence of adequate control at different stages of complex manufacturing process, material and geometric properties of composite structures are often uncertain. For a secure and safe design, tracking the impact of these uncertainties on the structural responses is of utmost significance. Composite materials, commonly adopted in various modern aerospace, marine, automobile and civil structures, are often susceptible to low-velocity impact caused by various external agents. Here, along with a critical review, we present machine learning based probabilistic and non-probabilistic (fuzzy) low-velocity impact analyses of composite laminates including a detailed deterministic characterization to systematically investigate the consequences of source-uncertainty. While probabilistic analysis can be performed only when complete statistical description about the input variables are available, the non-probabilistic analysis can be executed even in the presence of incomplete statistical input descriptions with sparse data. In this study, the stochastic effects of stacking sequence, twist angle, oblique impact, plate thickness, velocity of impactor and density of impactor are investigated on the crucial impact response parameters such as contact force, plate displacement, and impactor displacement. For efficient and accurate computation, a hybrid polynomial chaos based Kriging (PC-Kriging) approach is coupled with in-house finite element codes for uncertainty propagation in both the probabilistic and non-probabilistic analyses. The essence of this paper is a critical review on the hybrid machine learning algorithms followed by detailed numerical investigation in the probabilistic and non-probabilistic regimes to access the performance of such hybrid algorithms in comparison to individual algorithms from the viewpoint of accuracy and computational efficiency.
The paper presents an extension of the Certain Generalized Stresses Method (CGSM) for the static finite element analysis of homogeneous and laminated shells with variability. The basic assumption is that the generalized stresses do not depend on input parameters perturbation. The CGSM is a non-intrusive method that requires only one finite element analysis with some load cases to calculate the variability of mechanical quantities of interest. The uncertain input parameters are material and physical properties. Uniform random parameters as well as random fields are considered. The displacements statistical results: mean value, standard deviation and distribution are obtained by Monte Carlo simulations, using a semi-analytical formula. Two examples are treated: the Scordelis–Lo shell roof and an automotive windscreen. The method provides results of good quality and is very economical from a computational time point of view.
This paper quantifies the influence of uncertainty in the low-velocity impact responses of sandwich plates with composite face sheets considering the effects of obliqueness in impact angle and twist in the plate geometry. The stochastic impact analysis is conducted by using finite element (FE) modelling based on an eight nodded isoparametric quadratic plate bending element coupled with multivariate adaptive regression spline (MARS) in order to achieve computational efficiency. The modified Hertzian contact law is employed to model contact force and other impact parameters. Newmark's time integration scheme is used to solve the time-dependent equations. Comprehensive deterministic as well as probabilistic results are presented by considering the effects of location of impact, ply orientation angle, impactor velocity, impact angle, face-sheet material property, twist angle, plate thickness and mass of impactor. The relative importance of various input parameters is determined by conducting a sensitivity analysis. The results presented in this paper reveal that the impact responses of sandwich plates are significantly affected by the effect of source-uncertainty that in turn establishes the importance of adopting an inclusive stochastic design approach for impact modelling in sandwich plates.
The initiation and propagation of uncertainties in structural behaviour of complex anisotropic sandwich structure has significant computational challenges. Due to limitation of experimental data, probabilistic descriptions of uncertain parameters are not practically feasible to expedite. This paper presents the prediction capability of surrogate model (polynomial neural network) to estimate the stochastic buckling behaviour of sandwich plates. The polynomial neural network (PNN) is utilized to construct the surrogate. The computational time and cost is significantly reduced by using the proposed model in conjunction to finite element model employing higher order zigzag theory (HOZT). Both material and geometric uncertainties are considered to obtain the statistical quantity of interest (QoI). The computational efficacy of PNN is validated by means of scatter plot and probability density function plot. The constructed PNN-model is found to be convergent with the results obtained by direct Monte Carlo simulation (MCS) techniques. The proposed model can be used for more complex structures in future.
This paper deals with the stochastic sensitivity analysis of functionally graded material (FGM) plates subjected to free vibration and low-velocity impact to identify the most influential parameters in the respective analyses. A hybrid moment-independent sensitivity analysis is proposed coupled with the least angle regression based adaptive sparse polynomial chaos expansion. Here the surrogate model is integrated in the sensitivity analysis framework to achieve computational efficiency. The current paper is concentrated on the relative sensitivity of material properties in the free vibration (first three natural frequencies) and low-velocity impact responses of FGM plates. Typical functionally graded materials are made of two different components, where a continuous and inhomogeneous mixture of these materials is distributed across the thickness of the plate based on certain distribution laws. Thus, besides the overall sensitivity analysis of the material properties, a unique spatial sensitivity analysis is also presented here along the thickness of the plate to provide a comprehensive view. The results presented in this paper would help to identify the most important material properties along with their depth-wise spatial sensitivity for low-frequency vibration and low-velocity impact analysis of FGM plates. This is the first attempt to carry out an efficient adaptive sparse PCE based moment-independent sensitivity analysis (depth-wise and collectively) of FGM plates under the simultaneous susceptibility of vibration and impact. Such simultaneous multi-objective sensitivity analysis can identify the important system parameters and their relative degree of importance in advanced multi-functional structural systems.
This paper deals with portraying the stochastic natural frequencies of cantilever plates made up of functionally graded materials (FGMs) by employing the radial basis function (RBF)-based finite element (FE) approach. The material modeling of FGM plates is carried out by employing three different distribution laws, namely power law, sigmoid law, and exponential law. A generalized algorithm is developed for uncertainty quantification of natural frequencies of the FGM structures due to stochastic variation in the material properties and temperature. The deterministic FE code is validated with the previous literature, whereas convergence study is carried out in between stochastic results obtained from full scale direct Monte Carlo Simulation (MCS) and MCS results obtained from RBF surrogate model of different sample sizes. The percentage of error present in the RBF model is also determined. The influence of crucial parameters such as distribution law, degree of stochasticity, power law index and temperature are determined for natural frequencies analysis of FGMs plates. The results illustrate the input parameters considered in the present study have significant effects on the first three stochastic natural frequencies of cantilever FGM plates.
This article presents a concise overview on condition monitoring and retrofitting/ strengthening of structures including a practical case study of strengthening for an existing historical building. Condition assessment of an existing structure is required mainly to check serviceability and safety requirements of the structure after short term events like earthquake or long term degradation of the structure with time. It is carried out to assess the ability of a structure to perform its intended operations under changed loading conditions with time or modification in its structural system as per newly imposed requirements. The condition assessment and strengthening may also be required for integrated extension of an existing structure. After assessing the condition of the structure, either it is retrofitted (or strengthened) or it is demolished according to the severity of the damage. In this article, such a critical condition assessment for an existing historical masonry building is presented and appropriate strengthening schemes are suggested by following two separate measures (concrete jacketing and fiber reinforced polymer strengthening). Subsequently, the relative advantages and disadvantages of the strengthening measures are discussed from a practical engineering perspective. Aim of this article is not to propose any new method for condition assessment and strengthening of structures, rather we take a systematic approach to demonstrate our experience. Critical case studies on condition assessment and strengthening of historical buildings with adequate technical insights are very scarce to find in scientific literature. This article would serve as a valuable reference for the practicing engineers and the concerned scientific community.
Present study aims to investigate the influence of material uncertainties on vibration and bending behaviour of skewed sandwich FGM plates. Reddy's higher order shear deformation theory has been employed to model the displacement field. Variational approach has been used to derive the governing differential equations. Effect of material uncertainties in the formulation have been incorporated using first order perturbation technique (FOPT). An efficient stochastic finite element formulation (SFEM) have been used for the calculation of first and second order statics of natural frequency and transverse deflection. Validation of the results have been performed with the help of available literature and separately developed Monte Carlo formulation (MCS) algorithm. A large number of examples have been solved to quantify the effect of uncertainties on the vibration and bending characteristics of functionally graded skew sandwich plates.
Over the last few decades, uncertainty quantification in composite materials and structures has gained a lot of attention from the research community as a result of industrial requirements. This book presents computationally efficient uncertainty quantification schemes following meta-model-based approaches for stochasticity in material and geometric parameters of laminated composite structures. Several metamodels have been studied and comparative results have been presented for different static and dynamic responses. Results for sensitivity analyses are provided for a comprehensive coverage of the relative importance of different material and geometric parameters in the global structural responses.
Structural damage identification and quantification of damage using non-destructive methods are important aspects for any civil, mechanical and aerospace engineering structures. In this study, a novel damage identification algorithm has been developed using random sampling-high dimensional model representation approach. A global sensitivity analysis based on random sampling-high dimensional model representation is adopted for important parameter screening purpose. Three different structures (spring mass damper system, simply supported beam and fibre-reinforced polymer composite bridge deck) have been used for various single and multiple damage conditions to validate the proposed algorithm. The performance of this method is found to be quite satisfactory in the realm of damage detection in structures. The random sampling-high dimensional model representation-based approach for meta-model formation is particularly useful in damage identification as it works well when large numbers of input parameters are involved. In this study, two different optimization methods have been used and their relative capability to identify damage has been discussed. Performance of this damage identification algorithm under the influence of noise has also been addressed in this article.
A secant function based shear deformable finite element model is developed for the flexural behavior of laminated composite and sandwich plates with various conditions. The structural kinematics of the plate is expressed by means of secant function based shear deformation theory newly developed by the authors. The theory possesses non-linear shear deformation and also satisfies the zero transverse shear conditions on top and bottom surfaces of the plate. The field variables are elegantly utilized in order to ensure C0 continuity requirement. Penalty parameter is implemented to secure the constraints arising due to independent field variables. A biquadratic quadrilateral element with eight nodes and 56 degrees of freedom is employed to discretize the domain. Extensive numerical tests for the flexural behavior of laminated composite and sandwich plates are conducted to affirm the validity of the present finite element model in conjunction with the improved structural kinematics. Influences of boundary conditions, loading conditions, lamination sequences, aspect ratio, span-thickness ratio, etc on the flexural behavior are investigated specifically and compared with the existing results in order to indicate the performance of the present mathematical treatment.
This paper presents the quantification of rotational and ply level uncertainty of random natural frequency for laminated composite conical shells by using surrogate modelling approach. The stochastic eigenvalue problem is solved by using QR iteration algorithm. Sensitivity analysis is carried out to address the influence of different input parameters on the output natural frequencies. The sampling size and computational cost is reduced by employing the present approach compared to direct Monte Carlo simulation. The stochastic mode shapes are also depicted for a typical laminate configuration. Statistical analysis is presented to illustrate the results and its performance.
The propagation of thermal uncertainty in composite structures has significant computational challenges. This paper presents the thermal, ply-level and material uncertainty propagation in frequency responses of laminated composite plates by employing surrogate model which is capable of dealing with both correlated and uncorrelated input parameters. The present approach introduces the generalized high dimensional model representation (GHDMR) wherein diffeomorphic modulation under observable response preserving homotopy (D-MORPH) regression is utilized to ensure the hierarchical orthogonality of high dimensional model representation component functions. The stochastic range of thermal field includes elevated temperatures up to 375 K and sub-zero temperatures up to cryogenic range of 125 K. Statistical analysis of the first three natural frequencies is presented to illustrate the results and its performance.
Non-destructive structural damage identification (SDI) and quantification of damage are important issues for any engineering structure. In this study, a comparative assessment of the damage identification capability of different design of experiment (DOE) methods (such as, 2 k factorial design, central composite design, Box-Behnken design, D-optimal design and Taguchi’s OA design) used in response surface methodology (RSM) has been carried out. Three different structures (simply supported beam, spring mass damper system and fibre reinforced polymer composite bridge deck) have been used for various single and multiple damage conditions to access the comparative ability of the aforementioned methods in identifying damage addressing two critically important criteria: accuracy and computational efficiency. The study reveals that central composite design and D-optimal design are most recommendable among the five considered DOE methods for SDI. Two different input parameter screening methods (sensitivity analysis using RSM utilizing 2 k factorial design and D-optimal design, general sensitivity analysis) have been explored in this study, and their comparative performance is also discussed. It is found that both the methods used in sensitivity analysis for the purpose of input parameter screening in the damage identification process work satisfactorily. Performance of RSM-based damage identification algorithm for different DOE methods under the influence of noise has also been addressed in this paper.
This paper presents the Kriging model approach for stochastic free vibration analysis of composite shallow doubly curved shells. The finite element formulation is carried out considering rotary inertia and transverse shear deformation based on Mindlin’s theory. The stochastic natural frequencies are expressed in terms of Kriging surrogate models. The influence of random variation of different input parameters on the output natural frequencies is addressed. The sampling size and computational cost is reduced by employing the present method compared to direct Monte Carlo simulation. The convergence studies and error analysis are carried out to ensure the accuracy of present approach. The stochastic mode shapes and frequency response function are also depicted for a typical laminate configuration. Statistical analysis is presented to illustrate the results using Kriging model and its performance.
This paper presents a generic random sampling-high dimensional model representations (RS-HDMR) approach for free vibration analysis of angle-ply composite plates. A metamodel is developed to express stochastic natural frequencies of the system. A global sensitivity analysis is carried out to address the influence of input random parameters on output natural frequencies. Three different types of input variables (fiber-orientation angle, elastic modulus and mass density) are varied to validate the proposed algorithm. The present approach is efficiently employed to reduce the sampling effort and computational cost when large number of input parameters is involved. The stochastic finite element approach is coupled with rotary inertia and transverse shear deformation based on Mindlin’s theory. Statistical analysis is carried out to illustrate the features of the RS-HDMR and to compare its performance with full-scale Monte Carlo simulation results. The stochastic mode shapes are also depicted for a typical laminate configuration. Based on the numerical results, some new physical insights are drawn on the dynamic behavior of the system.
This paper is devoted to the construction and to the identification of a probabilistic model of random fields in the presence of modeling errors, in high stochastic dimension and presented in the context of computational structural dynamics. Due to the high stochastic dimension of the random quantities which have to be identified using statistical inverse methods (challenging problem), a complete methodology is proposed and validated. The parametric–nonparametric (generalized) probabilistic approach of uncertainties is used to perform the prior stochastic models: (1) system-parameters uncertainties induced by the variabilities of the material properties are described by random fields for which their statistical reductions are still in high stochastic dimension and (2) model uncertainties induced by the modeling errors are taken into account with the nonparametric probabilistic approach in high stochastic dimension. For these two sources of uncertainties, the methodology consists in introducing prior stochastic models described with a small number of parameters which are simultaneously identified using the maximum likelihood method and experimental responses. The steps of the methodology are explained and illustrated through an application.
The use of kriging models for approximation and metamodel-based design and optimization has been steadily on the rise in the past decade. The widespread use of kriging models appears to be hampered by 1) computationally efficient algorithms for accurately estimating the model's parameters, 2) an effective method to assess the resulting model's quality, and 3) the lack of guidance in selecting the appropriate form of the kriging model. We attempt to address these issues by comparing 1) maximum likelihood estimation and cross validation parameter estimation methods for selecting a kriging model's parameters given its form and 2) an R2 of prediction and the corrected Akaike information criterion assessment methods for quantifying the quality of the created kriging model. These methods are demonstrated with six test problems. Finally, different forms of kriging models are examined to determine if more complex forms are more accurate and easier to fit than simple forms of kriging models for approximating computer models.
A computer experiment generates observations by running a computer model at inputs x and recording the output (response) Y. Prediction of the response Y to an untried input is treated by modeling the systematic departure of Y from a linear model as a realization of a stochastic process. For given data (selected inputs and the computed responses), best linear prediction is used. The design problem is to select the inputs to predict efficiently. The issues of choice of stochastic-process model and computation of efficient designs are addressed, and applications are made to some chemical kinetics problems.
Two types of sampling plans are examined as alternatives to simple random sampling in Monte Carlo studies. These plans are shown to be improvements over simple random sampling with respect to variance for a class of estimators that includes the sample mean and the empirical distribution function. 6 figures.
Using the Stochastic Finite Element Method (SFEM) to perform reliability analysis of the free vibration of composite plates
with material and fabrication uncertainties has received much attention lately. In this work the stochastic analysis is performed
using the First-Order Reliability Method (FORM-method 2) and the Second-Order Reliability Method (SORM). The basic random
variables include laminae stiffness properties and material density, as well as the randomness in ply orientation angles.
Modeling of the composite behavior utilizes a nine-noded isoparametric Lagrangian element based on the third-order shear deformation
theory. Calculating the eigenvectors at the mean values of the variables proves to be a reasonable simplification which significantly
increases solution speed. The stochastic finite element code is validated using comparisons with results of Monte Carlo simulation
technique with importance sampling. Results show that SORM is an excellent rapid tool in the stochastic analysis of free vibration
of composite plates, when compared to the slower Monte Carlo simulation techniques.
A structural damage detection algorithm using static test data is presented in this paper. Changes in the static response of a structure are characterized as a set of non-linear simultaneous equations that relate the changes in the static response to the location and severity of damage. Damage is considered as a reduction in the structural stiffness parameter (axial and/or flexural). An optimality criterion is introduced to solve these equations by minimizing the difference between the load vector of the damaged and the undamaged structure. The overall formulation leads to a non-linear optimization problem with non-linear equality and linear inequality constraints. A method based on stored strain energy in elements is presented for selecting the loading location. Measurement locations are selected based on the Fisher Information Matrix. Numerical results for a plane truss demonstrate the ability of this method in detecting damage in a given structure with the presence of noise in the measurements.
A Monte Carlo procedure was developed for optimizing symmetric fiber reinforced composite laminates such that the weight is minimum and the Tsai-Wu strength failure criterion is satisfied in each ply. The laminate may consist of several materials in cluding an idealized core, and may be subjected to several sets of combined in-plane and bending loads. The procedure yields the number of plies, the fiber orientation, and the material of each ply and the material and thickness of the core. A user friendly computer code was written for performing the numerical calculations. Laminates optimized by the code were compared to laminates resulting from existing optimization methods. These comparisons showed that the present Monte Carlo procedure is a useful and efficient tool for the design of composite laminates.
In this present article, large amplitude free vibration behaviour of doubly curved composite shell panels have been analysed using the nonlinear finite element method. The nonlinear mathematical model is derived using Green Lagrange type geometric nonlinearity in the framework of higher order shear deformation theory. In addition to that all the nonlinear higher order terms are included in the mathematical model to achieve more general case. The nonlinear governing equation of free vibrated curved panel is derived based on Hamilton׳s principle and solved numerically by using the direct iterative method. The developed mathematical model has been validated by comparing the responses with those available numerical results. Finally, some new numerical experimentation (orthotropicity ratio, stacking sequence, thickness ratio, amplitude ratio and support conditions) have been carried out to show the significance and the efficacy of the proposed mathematical model.
In this paper, the modified Fourier series method is applied to study the vibration behavior of composite laminated doubly-curved shells of revolution with elastic restraints. The theoretical formulation is based on the first order shear deformation shell theory considering the effects of the rotary inertia and initial curvature. In summary, the shell energy functional, written in terms of stress resultants and mid-surface strains, is expressed as a function of five displacement components by using the constitutive and kinematic relationships. Each displacement of the shell is then expanded as a superposition of the standard cosine Fourier series and several auxiliary functions introduced to remove any potential discontinuous of the original displacement and its derivatives at the ends. The desired solutions are obtained by using the variational operation. The convergence and accuracy of the presented solutions are validated, with good agreement observed. A systematic parametric study is also performed regarding the effects of the boundary conditions, lamination schemes, material and geometrical parameters. Finally, a variety of new vibration results including frequencies and mode shapes for circular toroidal, elliptical, paraboloidal and hyperbolical shells with classical and elastic boundary conditions as well as different geometric and material parameters are also presented, which may serve as benchmark solutions for the future researches.
In the present study, stochastic perturbation-based finite element for buckling statistics of functionally graded plates (FGM) with uncertain material properties in thermal environments is investigated. The effective material properties of the gradient plates are assumed to be temperature-dependent and vary in the thickness direction only according to the power-law distribution of the volume fractions of the constituents. An improved structural kinematics proposed earlier by author's is employed which accounts parabolic variations for the transverse shear strains with stress free boundary conditions at the top and bottom faces of the plate. An efficient C stochastic finite element based on the first-order perturbation technique (FOPT) is proposed, and the fundamental equations are obtained using variational approach. Convergence and comparison studies have been performed to describe the efficiency of the present formulation. The numerical results are highlighted with different system parameters and boundary conditions.
In the present paper a generalization of the Refined Zigzag Theory (RZT) to doubly-curved multilayered structures is proposed. The displacement field characteristic of Naghdi’s shell model is enriched with RZT kinematics and a four-node shell finite element is formulated. Assumed Natural Strain (ANS) strategy is employed to overcome shear locking and Enhanced Assumed Strain (EAS) technique is applied to alleviate membrane locking and bending locking. For efficiency purpose, a one-point quadrature rule is used for the in-plane integration and hourglass stabilization is introduced. Finally, several numerical examples, involving static analysis of thick as well as thin shells, are performed to demonstrate the efficiency and accuracy of the proposed shell finite element.
An energy harvesting concept has been proposed comprising a piezoelectric patch on a vertical cantilever beam with a tip mass. The cantilever beam is excited in the
transverse direction at its base. This device is highly nonlinear with two potential wells for large tip masses, when the beam is buckled. For the pre-buckled case considered here, the stiffness is low and hence the displacement response is large, leading to multiple solutions to harmonic excitation that are exploited in the
harvesting device. To maximise the energy harvested in systems with multiple solutions the higher amplitude response should be preferred. This paper investigates the amplitude of random noise excitation where the harvester is unable to sustain the high amplitude solution, and at some point will jump to the low amplitude solution. The investigation is performed on a validated model of the harvester and the effect is demonstrated experimentally.
A computationally efficient C0 finite element model is developed for laminated composite and sandwich plates by implementing the inverse hyperbolic shear deformation theory recently developed by the authors. This model is used to determine responses of general laminates subjected to various combinations of boundary conditions. The present formulation has been generalized for all existing shear deformation theories involving shear strain function. An eight noded serendipity element with 56 degrees of freedom is used to discretize the plate domain. Influences of lamination sequence (cross ply and angle ply), span to thickness ratio, and boundary conditions are investigated for the flexural behavior of laminated composite and sandwich plates. Further, the stability behavior of plates subjected to in-plane loads (uni-axial and bi-axial) is investigated for a variety of examples. Effects of boundary conditions and applied loads on the critical buckling loads and buckling mode shapes are also assessed for a class of laminates in order to show the efficacy of the present mathematical technique to predict the buckling mode shapes.
Thermal post-buckled vibration of laminated composite doubly curved panel embedded with shape memory alloy (SMA) fiber is investigated and presented in this article. The geometry matrix and the nonlinear stiffness matrices are derived using Green–Lagrange type nonlinear kinematics in the framework of higher order shear deformation theory. In addition to that, material nonlinearity in shape memory alloy due to thermal load is incorporated by the marching technique. The developed mathematical model is discretized using a nonlinear finite element model and the sets of nonlinear governing equations are obtained using Hamilton’s principle. The equations are solved using the direct iterative method. The effect of nonlinearity both in geometric and material have been studied using the developed model and compared with those published literature. Effect of various geometric parameters such as thickness ratio, amplitude ratio, lamination scheme, support condition, prestrains of SMA, and volume fractions of SMA on the nonlinear free vibration behavior of thermally post-buckled composite flat/curved panel been studied in detail and reported.
This paper presents the effect of rotational speeds on free vibration characteristics of delaminated twisted graphite–epoxy cross-ply composite conical shells employing finite element method. Theoretical formulation is based on Mindlin's theory considering an eight noded isoparametric plate bending element. A generalized dynamic equilibrium equation is derived from Lagrange's equation of motion neglecting the Coriolis effect for moderate rotational speeds. The multi-point constraint algorithm is utilized to ensure the compatibility of deformation and equilibrium of resultant forces and moments at the delamination crack front. The QR iteration algorithm is used for solution of standard eigenvalue problem. Finite element codes are developed to obtain the numerical results concerning the combined effects of twist angle and rotational speed on the natural frequencies of cross-ply composite shallow conical shells. The mode shapes for a typical laminate configuration are also depicted. Numerical results obtained for cross-ply laminates with delamination are the first known non-dimensional frequencies for the type of analyses carried out here.
In this paper a high precision composite plate bending element has been presented and its application to the analysis of isotropic and composite folded plates has been shown to study the performance of the element. In the present element the effect of shear deformation has been considered. Numerical examples have been solved by the proposed element and the results obtained in the form of bending moment, in-plane force and deflection have been compared with the published results (where available) to show the potentiality of the element. Number of new results have been presented, which are expected to be useful in future research.
The first known solution of the title problem is presented. The Ritz method is used, with algebraic polynomials forming the set of trial functions. The condition that all three components of displacement be zero at the four corners is straightforwardly enforced. Numerical studies show that the convergence is relatively slow, requiring more terms than for shells which are completely free. The class of problems studied includes independent, constant curvature in each of the directions parallel to the edges, yielding vibration modes which fall into one of four symmetry classes, with symmetry or antisymmetry of the displacements existing with respect to each of the two symmetry axes of the problem. Detailed results are given for the frequencies and mode shapes of the first two modes of each symmetry class for shells having square planform and circular cylindrical, spherical and hyperbolic paraboloidal curvatures. Accuracy of the results is partially established by comparison with other previously published, accurate results for the corner supported flat square plate.
This paper describes a novel method for a stochastic analysis of a multiscale homogenization problem. An inhomogeneous material such as a composite material has a complex microstructure, and sometimes it has an uncertainty in geometry or a material property of a microstructure. This microscopic uncertainty will cause dispersion of a macroscopic homogenized material property of a composite material. In order to analyze this problem, an approximation-based stochastic analysis approach is developed.The proposed method uses a flexible approximation technique and variable transformation of a probabilistic density function. In this paper, the Kriging method is used for approximation and integral estimation of a probabilistic density function.As a numerical example, a stochastic response analysis for a homogenized elastic tensor and homogenized elastic constants of a unidirectional fiber reinforced composite material is performed using the Monte-Carlo simulation, the perturbation-based homogenization method and the proposed method. Uncertainties in material properties and geometry of a microstructure of a unidirectional fiber reinforced composite plate are taken into account. From the numerical results, validity and effectiveness of the proposed method are shown.
A finite element analysis for studying the free vibration behaviour of generalized doubly curved laminated composite shells is presented using eight-noded curved quadrilateral isoparametric finite elements. The formulation assumes first-order shear deformation theory for thin and shallow shells, and also considers the two principal radii of curvature and the radius of cross-curvature. Some of the results obtained are compared with those present in the existing literature. Several other numerical results are presented by varying fibre orientations, lamination schemes, support spacings, aspect ratio, lower height to higher height ratio (for conoids), the thickness to radius ratio and radii of curvature ratio (for elliptic and hyperbolic paraboloids), which are relevant to the doubly curved shells.
The Ritz method with algebraic polynomial displacement functions is used to analyze free vibrations of thin cantilevered laminated plates and shallow shells having rectangular planforms. Convergence studies are made which demonstrate that accurate results are obtained using 192 displacement terms for spherical, circular cylindrical, hyperbolic paraboloidal shallow shells and 64 terms for plates. Results are compared with those obtained experimentally and by finite element methods. It is found that the present method needs considerably less degrees of freedom than the finite element method to obtain the same accuracy and compares well with results from experiment. The effect of various parameters (material, number of layers, fiber orientation, curvature) upon the frequencies is studied.
Finite Element Procedures in Engineering Analysis, PHI, New Delhi. [27] Meirovitch L., (1992) Dynamics and Control of Structures Use of Kriging models to approximate deterministic computer models
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Structural damage identification: a RS-HDMR approach
- T Mukhopadhyay
- R Chowdhury
- A Chakrabarti
Mukhopadhyay T., Chowdhury R., Chakrabarti A. (2016) Structural damage identification: a
RS-HDMR approach, Advances in Structural Engineering (In Press)