Computational Cost Research Articles

Projected Gaussian Markov Improvement Algorithm for High-Dimensional Discrete Optimization via Simulation

This article considers a discrete optimization via simulation (DOvS) problem defined on a graph embedded in the high-dimensional integer grid. Several DOvS algorithms that model the responses at the solutions as a realization of a Gaussian Markov random field (GMRF) have been proposed exploiting its inferential power and computational benefits. However, the computational cost of inference increases exponentially in dimension. We propose the projected Gaussian Markov improvement algorithm (pGMIA), which projects the solution space onto a lower-dimensional space creating the region-layer graph to reduce the cost of inference. Each node on the region-layer graph can be mapped to a set of solutions projected to the node; these solutions form a lower-dimensional solution-layer graph. We define the response at each region-layer node to be the average of the responses within the corresponding solution-layer graph. From this relation, we derive the region-layer GMRF to model the region-layer responses. The pGMIA alternates between the two layers to make a sampling decision at each iteration. It first selects a region-layer node based on the lower-resolution inference provided by the region-layer GMRF, then makes a sampling decision among the solutions within the solution-layer graph of the node based on the higher-resolution inference from the solution-layer GMRF. To solve even higher-dimensional problems (e.g., 100 dimensions), we also propose the pGMIA+: a multi-layer extension of the pGMIA. We show that both pGMIA and pGMIA+ converge to the optimum almost surely asymptotically and empirically demonstrate their competitiveness against state-of-the-art high-dimensional Bayesian optimization algorithms.

Mixed-integer quadratic optimization and iterative clustering techniques for semi-supervised support vector machines

AbstractAmong the most famous algorithms for solving classification problems are support vector machines (SVMs), which find a separating hyperplane for a set of labeled data points. In some applications, however, labels are only available for a subset of points. Furthermore, this subset can be non-representative, e.g., due to self-selection in a survey. Semi-supervised SVMs tackle the setting of labeled and unlabeled data and can often improve the reliability of the results. Moreover, additional information about the size of the classes can be available from undisclosed sources. We propose a mixed-integer quadratic optimization (MIQP) model that covers the setting of labeled and unlabeled data points as well as the overall number of points in each class. Since the MIQP’s solution time rapidly grows as the number of variables increases, we introduce an iterative clustering approach to reduce the model’s size. Moreover, we present an update rule for the required big-M values, prove the correctness of the iterative clustering method as well as derive tailored dimension-reduction and warm-starting techniques. Our numerical results show that our approach leads to a similar accuracy and precision than the MIQP formulation but at much lower computational cost. Thus, we can solve larger problems. With respect to the original SVM formulation, we observe that our approach has even better accuracy and precision for biased samples.

A comparison of computational algorithms for the Bayesian analysis of clinical trials.

Clinical trials are increasingly using Bayesian methods for their design and analysis. Inference in Bayesian trials typically uses simulation-based approaches such as Markov Chain Monte Carlo methods. Markov Chain Monte Carlo has high computational cost and can be complex to implement. The Integrated Nested Laplace Approximations algorithm provides approximate Bayesian inference without the need for computationally complex simulations, making it more efficient than Markov Chain Monte Carlo. The practical properties of Integrated Nested Laplace Approximations compared to Markov Chain Monte Carlo have not been considered for clinical trials. Using data from a published clinical trial, we aim to investigate whether Integrated Nested Laplace Approximations is a feasible and accurate alternative to Markov Chain Monte Carlo and provide practical guidance for trialists interested in Bayesian trial design. Data from an international Bayesian multi-platform adaptive trial that compared therapeutic-dose anticoagulation with heparin to usual care in non-critically ill patients hospitalized for COVID-19 were used to fit Bayesian hierarchical generalized mixed models. Integrated Nested Laplace Approximations was compared to two Markov Chain Monte Carlo algorithms, implemented in the software JAGS and stan, using packages available in the statistical software R. Seven outcomes were analysed: organ-support free days (an ordinal outcome), five binary outcomes related to survival and length of hospital stay, and a time-to-event outcome. The posterior distributions for the treatment and sex effects and the variances for the hierarchical effects of age, site and time period were obtained. We summarized these posteriors by calculating the mean, standard deviations and the 95% equitailed credible intervals and presenting the results graphically. The computation time for each algorithm was recorded. The average overlap of the 95% credible interval for the treatment and sex effects estimated using Integrated Nested Laplace Approximations was 96% and 97.6% compared with stan, respectively. The graphical posterior densities for these effects overlapped for all three algorithms. The posterior mean for the variance of the hierarchical effects of age, site and time estimated using Integrated Nested Laplace Approximations are within the 95% credible interval estimated using Markov Chain Monte Carlo but the average overlap of the credible interval is lower, 77%, 85.6% and 91.3%, respectively, for Integrated Nested Laplace Approximations compared to stan. Integrated Nested Laplace Approximations and stan were easily implemented in clear, well-established packages in R, while JAGS required the direct specification of the model. Integrated Nested Laplace Approximations was between 85 and 269 times faster than stan and 26 and 1852 times faster than JAGS. Integrated Nested Laplace Approximations could reduce the computational complexity of Bayesian analysis in clinical trials as it is easy to implement in R, substantially faster than Markov Chain Monte Carlo methods implemented in JAGS and stan, and provides near identical approximations to the posterior distributions for the treatment effect. Integrated Nested Laplace Approximations was less accurate when estimating the posterior distribution for the variance of hierarchical effects, particularly for the proportional odds model, and future work should determine if the Integrated Nested Laplace Approximations algorithm can be adjusted to improve this estimation.

A dual fusion recognition model for sleep posture based on air mattress pressure detection

In order to solve the difficult portability problem of traditional non-invasive sleeping posture recognition algorithms arising from the production cost and computational cost, this paper proposes a sleeping posture recognition model focusing on human body structural feature extraction and integration of feature space and algorithms based on a specific air-spring mattress structure, called SPR-DE (SPR-DE is the Sleep Posture Recognition-Data Ensemble acronym form). The model combines SMR (SMR stands for Principle of Spearman Maximal Relevance) with horizontal and vertical division based on the barometric pressure signals in the human body’s backbone region to reconstruct the raw pressure data into strongly correlated non-image features of the sleep postures in different parts and directions and construct the feature set. Finally, the recognit-ion of the two sleep postures is accomplished using the AdaBoost-SVM integrated classifier. SPR-DE is compared with the base and integrated classifiers to verify its performance. The experimental results show that the amount of significant features helps the algorithm to classify different sleeping patterns more accurately, and the f1 score of the SPR-DE model determined by the comparison experiments is 0.998, and the accuracy can reach 99.9%. Compared with other models, the accuracy is improved by 2.9% ~ 7.7%, and the f1-score is improved by 0.029 ~ 0.076. Therefore, it is concluded that the SMR feature extraction strategy in the SPR-DE model and the AdaBoost-SVM can achieve high accuracy and strong robustness in the task of sleep posture recognition in a small area, low-density air-pressure mattress, taking into account the comfort of the mattress structural design and the sleep posture recognition, integrated with the mattress adaptive adjustment system.

A Comprehensive Review of Methods for Hydrological Forecasting Based on Deep Learning

Artificial intelligence has undergone rapid development in the last thirty years and has been widely used in the fields of materials, new energy, medicine, and engineering. Similarly, a growing area of research is the use of deep learning (DL) methods in connection with hydrological time series to better comprehend and expose the changing rules in these time series. Consequently, we provide a review of the latest advancements in employing DL techniques for hydrological forecasting. First, we examine the application of convolutional neural networks (CNNs) and recurrent neural networks (RNNs) in hydrological forecasting, along with a comparison between them. Second, a comparison is made between the basic and enhanced long short-term memory (LSTM) methods for hydrological forecasting, analyzing their improvements, prediction accuracies, and computational costs. Third, the performance of GRUs, along with other models including generative adversarial networks (GANs), residual networks (ResNets), and graph neural networks (GNNs), is estimated for hydrological forecasting. Finally, this paper discusses the benefits and challenges associated with hydrological forecasting using DL techniques, including CNN, RNN, LSTM, GAN, ResNet, and GNN models. Additionally, it outlines the key issues that need to be addressed in the future.

An extended isogeometric collocation method for fracture analysis

AbstractA collocation method is developed for discrete fracture models in the context of the partition‐of‐unity method. Spline technologies used in isogeometric analysis (IGA) are exploited to provide a smooth inter‐element transition of gradients, thus allowing to get rid of extra flux terms at element boundaries which are generated by Lagrange polynomials. Bézier extraction is utilised to formulate IGA commensurate with a standard finite element data‐structure. The efficacy of the proposed approach is examined through different numerical examples and is compared with other discrete methods for fracture analysis. The proposed approach is competitive in terms of accuracy with the least computational cost, rendering it a suitable candidate for superseding available collocation approaches for fracture simulation. Moreover, the approach naturally assesses the possibility of physics informed neural networks for fracture simulation, to which collocation is central.

Artificial Neural Network Modeling in the Presence of Uncertainty for Predicting Hydrogenation Degree in Continuous Nitrile Butadiene Rubber Processing

The transition from batch to continuous production in the catalytic hydrogenation of nitrile butadiene rubber (NBR) into hydrogenated NBR (HNBR) marks a significant advance for applications under demanding conditions. This study introduces a continuous process utilizing a static mixer (SM) reactor, which notably achieves a hydrogenation conversion rate exceeding 97%. We thoroughly review a mechanistic model of the SM reactor to elucidate the internal dynamics governing the hydrogenation process and address the inherent uncertainties in key parameters such as the Peclet number (Pe), dimensionless time (θτ), reaction coefficient (R), and flow rate coefficient (q). A comprehensive dataset generated from varied parameter values serves as the basis for training an artificial neural network (ANN), which is then compared against traditional models including linear regression, decision tree, and random forest in terms of efficacy. Our results clearly demonstrate the ANN’s superiority in predicting the degree of hydrogenation, achieving the lowest root mean squared error (RMSE) of 3.69 compared to 21.90 for linear regression, 4.94 for decision tree, and 7.51 for random forest. The ANN’s robust capability for modeling complex nonlinear relationships and dynamics significantly enhances decision-making, planning, and optimization of the reactor, reducing computational demands and operational costs. In other words, this approach allows users to rely on a single ML-based model instead of multiple mechanistic models for reflecting the effects of possible uncertainties. Additionally, a feature importance study validates the critical impact of time and element number on the hydrogenation process, further supporting the ANN’s predictive accuracy. These findings underscore the potential of ML-based models in streamlining and enhancing the efficiency of chemical production processes.

High-order finite-element forward-modeling algorithm for the 3D borehole-to-surface electromagnetic method using octree meshes

The borehole-to-surface electromagnetic (BSEM) method complements surface and airborne electromagnetic (EM) methods. The borehole is typically filled with conductive media, which significantly impacts the responses. To accurately simulate the responses, the borehole needs to be modeled. However, the size of the borehole is very small compared with the entire computational domain, making it challenging to discretize the borehole accurately. We develop a high-accuracy 3D finite-element forward-modeling algorithm for the BSEM method using octree meshes. The basic idea of octree meshes is to locally refine the mesh by dividing the hexahedral element into eight smaller elements. This approach has the advantage of being able to accurately discretize small geometric features while using coarse meshes in other areas, thus reducing the computational cost. In addition, high-order basis functions are used to improve the accuracy of numerical solutions. We develop a 3D forward-modeling code for BSEM using the C++ programming language. We verify the correctness and accuracy of our implementation by comparing the numerical responses with analytical solutions. Furthermore, we determine the influence of the borehole on the EM responses, showing that the borehole will cause a decrease of several orders of magnitude in the responses when the borehole is filled with highly conductive media. We also use two 3D models to demonstrate the efficacy of octree meshes in the discretization of complex structures and to discuss the effectiveness and sensitivity of BSEM data in the exploration of mineral and oil-gas resources. The code is open source under the Massachusetts Institute of Technology license and can be used as an accurate modeling tool for BSEM responses, providing useful guidance for field exploration. The code is designed with a modular architecture, ensuring ease of maintenance and the ability to extend to other EM methods and serve as the foundation for 3D inversion.

Task-oriented machine learning surrogates for tipping points of agent-based models

We present a machine learning framework bridging manifold learning, neural networks, Gaussian processes, and Equation-Free multiscale approach, for the construction of different types of effective reduced order models from detailed agent-based simulators and the systematic multiscale numerical analysis of their emergent dynamics. The specific tasks of interest here include the detection of tipping points, and the uncertainty quantification of rare events near them. Our illustrative examples are an event-driven, stochastic financial market model describing the mimetic behavior of traders, and a compartmental stochastic epidemic model on an Erdös-Rényi network. We contrast the pros and cons of the different types of surrogate models and the effort involved in learning them. Importantly, the proposed framework reveals that, around the tipping points, the emergent dynamics of both benchmark examples can be effectively described by a one-dimensional stochastic differential equation, thus revealing the intrinsic dimensionality of the normal form of the specific type of the tipping point. This allows a significant reduction in the computational cost of the tasks of interest.

Streaming prediction-error filters

Prediction-error filters (PEFs) are essential in seismic deconvolution and other geophysical estimation problems. We show that non-stationary multidimensional PEFs can be computed in a “streaming” manner, where the filter gets updated incrementally by accepting one new data point at a time. The computational cost of estimating a streaming PEF reduces to the cost of a single convolution. In other words, the cost of PEF design while filtering equals the cost of applying the filter. Moreover, the non-linear operation of finding and applying a streaming PEF is invertible at a similar cost, which enables a fast approach to missing data interpolation.

Atmospheric Chemistry Surrogate Modeling With Sparse Identification of Nonlinear Dynamics

AbstractModeling atmospheric chemistry is computationally expensive and limits the widespread use of chemical transport models. This computational cost arises from solving high‐dimensional systems of stiff differential equations. Previous work has demonstrated the promise of machine learning (ML) to accelerate air quality model simulations but has suffered from numerical instability during long‐term simulations. This may be because previous ML‐based efforts have relied on explicit Euler time integration—which is known to be unstable for stiff systems—and have used neural networks which are prone to overfitting. We hypothesize that parsimonious models combined with modern numerical integration techniques can overcome this limitation. Using a small‐scale mechanism to explore the potential of these methods, we have created a machine‐learned surrogate by (a) reducing dimensionality using singular value decomposition to create an interpretably‐compressed low‐dimensional latent space and (b) using Sparse Identification of Nonlinear Dynamics (SINDy) to create a differential‐equation‐based representation of the underlying dynamics in the compressed latent space with reduced stiffness. The root mean square error of ML model prediction for ozone concentration over 9 days is 37.8% of the root mean square concentration across all simulations in our testing data set. The surrogate model is 10× faster with 12× fewer integration timesteps compared to reference model and is numerically stable in all tested simulations. Overall, we find that SINDy can be used to create fast, stable, and accurate surrogates of a simple photochemical mechanism. In future work, we will explore the application of this method to more detailed mechanisms and their use in large‐scale simulations.

Group sparse-based Taylor expansion method for liver pharmacokinetic parameters imaging of dynamic fluorescence molecular tomography

Objective. Pharmacokinetic parametric images obtained through dynamic fluorescence molecular tomography (DFMT) has ability of capturing dynamic changes in fluorescence concentration, thereby providing three-dimensional metabolic information for applications in biological research and drug development. However, data processing of DFMT is time-consuming, involves a vast amount of data, and the problem itself is ill-posed, which significantly limits the application of pharmacokinetic parametric images reconstruction. In this study, group sparse-based Taylor expansion method is proposed to address these problems. Approach. Firstly, Taylor expansion framework is introduced to reduce time and computational cost. Secondly, group sparsity based on structural prior is introduced to improve reconstruction accuracy. Thirdly, alternating iterative solution based on accelerated gradient descent algorithm is introduced to solve the problem. Main results. Numerical simulation and in vivo experimental results demonstrate that, in comparison to existing methods, the proposed approach significantly enhances reconstruction speed without a degradation of quality, particularly when confronted with background fluorescence interference from other organs. Significance. Our research greatly reduces time and computational cost, providing strong support for real-time monitoring of liver metabolism.

Combining simulation and experimental data via surrogate modelling of continuum dislocation dynamics simulations

Abstract Several computational models have been introduced in recent years to
yield comprehensive insights into microstructural evolution analyses. However, the
identification of the correct input parameters to a simulation that corresponds to a
certain experimental result is a major challenge on this length scale. To complement
simulation results with experimental data (and vice versa) is not trivial since, e.g.,
simulation model parameters might lack a physical understanding or uncertainties
in the experimental data are neglected. Computational costs are another challenge
mesoscale models always have to face, so comprehensive parameter studies can be
costly. In this paper, we introduce a surrogate model to circumvent continuum
dislocation dynamics simulation by a data-driven linkage between well-defined input
parameters and output data and vice versa. We present meaningful results for a
forward surrogate formulation that predicts simulation output based on the input
parameter space, as well as for the inverse approach that derives the input parameter
space based on simulation as well as experimental output quantities. This enables, e.g.,
a direct derivation of the input parameter space of a continuum dislocation dynamics
simulation based on experimentally provided stress-strain data.

Multifidelity topology design of a maritime survey operation with UUVs

Advances in autonomous systems, maritime communications, and sensing technologies lead to increasing applications of unmanned underwater vehicles (UUVs). In this paper, we study a maritime survey operation topology design problem with UUVs that traverse an ocean environment and collect data from prespecified sensors or locations. This maritime scenario is analyzed via several models and simulation for the purpose of topology design for mission planning. We use a multifidelity approach to examine the trade-offs between different potential topology configurations of assigning UUVs to data collection sensors or locations. We develop three low-fidelity models that make simplifying assumptions. These models provide insight into the design characteristics and allow for sensitivity analysis with low computational cost. They are used to down-select potential configuration designs for further evaluation using a high-fidelity simulation model. A high-fidelity simulation model removes many simplifying assumptions and predicts how a topology design would perform under more realistic conditions. It gathers detailed performance metrics at the expense of higher computational cost. Our study uses this multifidelity approach to demonstrate key trade-offs for topology design. The optimal design of UUVs depends on mission-specific goals.

Benchmark Investigation of SCC-DFTB against Standard and Hybrid DFT to Model Electronic Properties in Two-Dimensional MOFs for Thermoelectric Applications.

Recent studies have shown that metal-organic frameworks (MOFs) have potential as thermoelectric materials, and the topic has received increasing attention. The main motivation for this project is to further our knowledge of thermoelectric properties in MOFs and find which available self-consistent-charge density functional tight binding (SCC-DFTB) method can best predict (at least trends in) the electronic properties of MOFs at a lower computational cost than standard density functional theory (DFT). In this work, the electronic properties of monolayer, serrated, AA-stacked, and/or AB-stacked Zn3C6O6, Cd3C6O6, Zn-NH-MOF─for which no previous calculations of thermoelectric performance exist─and Ni3(HITP)2 MOFs are modeled with DFT-PBE, DFT-HSE06, GFN1-xTB, GFN2-xTB, and DFTB-3ob/mio. The band structures, density of states, and their relative orbital contributions, as well as the electrical conductivity, Seebeck coefficient, and power factor, are compared across methods and geometries. Our results suggest that GFN-xTB is adequate to predict the MOFs' band structure shape and density of states but not band gap. Our calculations further indicate that Zn3C6O6, Cd3C6O6, and Zn-NH-MOF have higher power factor values than Ni3(HITP)2, one of the highest performing synthesized MOFs, and are therefore promising for thermoelectric applications.

LoRa Radio Frequency Fingerprinting with Residual of Variational Mode Decomposition and Hybrid Machine-Learning/Deep-Learning Optimization

RFF refers to the technique for identifying and classifying wireless devices on the basis of their physical characteristics, which appear in the digital signal transmitted in space. Small differences in the radio frequency front-end of the wireless devices are generated across the same wireless device model during the implementation and manufacturing process. These differences create small variations in the transmitted signal, even if the wireless device is still compliant with the wireless standard. By using data analysis and machine-learning algorithms, it is possible to classify different electronic devices on the basis of these variations. This technique has been well proven in the literature, but research is continuing to improve the classification performance, robustness to noise, and computing efficiency. Recently, DL has been applied to RFF with considerable success. In particular, the combination of time-frequency representations and CNN has been particularly effective, but this comes at a great computational cost because of the size of the time-frequency representation and the computing time of CNN. This problem is particularly challenging for wireless standards, where the data to be analyzed is extensive (e.g., long preambles) as in the case of the LoRa (Long Range) wireless standard. This paper proposes a novel approach where two pre-processing steps are adopted to (1) improve the classification performance and (2) to decrease the computing time. The steps are based on the application of VMD where (in opposition to the known literature) the residual of the VMD application is used instead of the extracted modes. The concept is to remove the modes, which are common among the LoRa devices, and keep with the residuals the unique intrinsic features, which are related to the fingerprints. Then, the spectrogram is applied to the residual component. Even after this step, the computing complexity of applying CNN to the spectrogram is high. This paper proposes a novel step where only segments of the spectrogram are used as input to CNN. The segments are selected using a machine-learning approach applied to the features extracted from the spectrogram using the LBP. The approach is applied to a recent LoRa radio frequency fingerprinting public data set, where it is shown to significantly outperform the baseline approach based on the full use of the spectrogram of the original signal in terms of both classification performance and computing complexity.

Simulation of Sloped-Bed Tuned Liquid Dampers Using a Nonlinear Shallow Water Model

Simulation of Sloped-Bed Tuned Liquid Dampers Using a Nonlinear Shallow Water Model

Exploring the Mechanisms behind Non-aromatic Fluorescence with the Density Functional Tight Binding Method.

Recent experimental findings reveal nonconventional fluorescence emission in biological systems devoid of conjugated bonds or aromatic compounds, termed non-aromatic fluorescence (NAF). This phenomenon is exclusive to aggregated or solid states and remains absent in monomeric solutions. Previous studies focused on small model systems in vacuum show that the carbonyl stretching mode along with strong interaction of short hydrogen bonds (SHBs) remains the primary vibrational mode explaining NAF in these systems. In order to simulate larger model systems taking into account the effects of the surrounding environment, in this work we propose using the density functional tight-binding (DFTB) method in combination with non-adiabatic molecular dynamics (NAMD) and the mixed quantum/molecular mechanics (QM/MM) approach. We investigate the mechanism behind NAF in the crystal structure of l-pyroglutamine-ammonium, comparing it with the related nonfluorescent amino acid l-glutamine. Our results extend our previous findings to more realistic systems, demonstrating the efficiency and robustness of the proposed DFTB method in the context of NAMD in biological systems. Furthermore, due to its inherent low computational cost, this method allows for a better sampling of the nonradiative events at the conical intersection which is crucial for a complete understanding of this phenomenon. Beyond contributing to the ongoing exploration of NAF, this work paves the way for future application of this method in more complex biological systems such as amyloid aggregates, biomaterials, and non-aromatic proteins.

A differential evolution framework based on the fluid model for feature selection

Feature selection in machine learning is a crucial step to effectively address the issue of feature redundancy in classification problems. Numerous feature selection algorithms have been developed to minimize the number of features, reduce computational cost, and improve classification accuracy. Differential evolution algorithms have the advantage of being simple in structure, robust, fast in convergence, and frequently used to solve feature selection problems. However, it is worth noting that differential evolution algorithms are susceptible to local optimum and stagnation issues, particularly when applied to high-dimensional data. To address this issue, in this study, we propose a differential evolution framework based on the fluid model, named DEF-FM, for feature selection. DEF-FM has the capability to speed up the convergence of differential evolution algorithms and alleviate the effects of local optima. The proposed framework is validated and compared against eight popular differential evolution algorithms using 12 publicly available benchmark datasets and experimental results unequivocally demonstrate the superiority of the proposed framework.

Free vibration analysis of laminated anisotropic doubly-curved shell structures reinforced with three-phase polymer/CNT/fiber material

The present work investigates the vibrational response of laminated anisotropic doubly-curved shell structures reinforced with Carbon Nanotubes (CNTs) short fibers. The fundamental equations are derived from a curvilinear reference system of principal coordinates, employing the Equivalent Single Layer (ESL) methodology. Furthermore, a general variation in laminate thickness is considered. The unknown field variable is described using higher order theories through the unified formulation, taking into account a general interpolation of the unknown variables with Lagrange polynomials across the physical domain. The Hamiltonian Principle is adopted for the derivation of the dynamic equilibrium equations, which arediscretized numerically by using the Generalized Differential Quadrature (GDQ) method. In addition, an efficient isogeometric NURBS-based mapping is adopted for the distortion of the physical domain. The boundary conditions of the differential problem are modelled through a general distribution of linear elastic springs along the lateral surfaces of the three-dimensional doubly-curved solid. The theory is then applied to study the vibrational modes of structures with different curvatures and lamination schemes, taking into account a general distribution of CNTs along the thickness direction. The homogenized properties of CNTs layers are obtained from the Mori-Tanaka procedure, which accounts for the agglomeration effects of the dispersed nanofibers within the matrix. Unlike previous studies focusing on doubly-curved shells reinforced with composite materials and CNTs, the present formulation enables to derive the vibration characteristics of structures made of generally anisotropic materials with general orientations. Furthermore, the calibration of the parameters for the linear elastic springs’ distribution leads to general external constraints within a single element, thus reducing the computational cost of the problem.