Mathematics

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6 pages, 206 KiB

Open AccessEditorial

Mathematical Modeling and Simulation in Mechanics and Dynamic Systems

by Maria Luminita Scutaru and Catalin-Iulian Pruncu

Mathematics 2022, 10(3), 448; https://doi.org/10.3390/math10030448 - 30 Jan 2022

Cited by 1 | Viewed by 2955

Abstract

Although it has previously been considered difficult to make further contributions in the field of mechanics, the spectacular evolution of technology and numerical calculation techniques has caused this opinion to be reconsidered and to the development of more and more sophisticated models that [...] Read more.

Although it has previously been considered difficult to make further contributions in the field of mechanics, the spectacular evolution of technology and numerical calculation techniques has caused this opinion to be reconsidered and to the development of more and more sophisticated models that describe, as accurately as possible, the phenomena that take place in dynamic systems [...] Full article

(This article belongs to the Special Issue Mathematical Modeling and Simulation in Mechanics and Dynamic Systems)

Research

Jump to: Editorial

26 pages, 8285 KiB

Open AccessArticle

Damage Detection and Isolation from Limited Experimental Data Using Simple Simulations and Knowledge Transfer

by Asif Khan, Jun-Sik Kim and Heung Soo Kim

Mathematics 2022, 10(1), 80; https://doi.org/10.3390/math10010080 - 27 Dec 2021

Cited by 9 | Viewed by 2748

Abstract

A simulation model can provide insight into the characteristic behaviors of different health states of an actual system; however, such a simulation cannot account for all complexities in the system. This work proposes a transfer learning strategy that employs simple computer simulations for [...] Read more.

A simulation model can provide insight into the characteristic behaviors of different health states of an actual system; however, such a simulation cannot account for all complexities in the system. This work proposes a transfer learning strategy that employs simple computer simulations for fault diagnosis in an actual system. A simple shaft-disk system was used to generate a substantial set of source data for three health states of a rotor system, and that data was used to train, validate, and test a customized deep neural network. The deep learning model, pretrained on simulation data, was used as a domain and class invariant generalized feature extractor, and the extracted features were processed with traditional machine learning algorithms. The experimental data sets of an RK4 rotor kit and a machinery fault simulator (MFS) were employed to assess the effectiveness of the proposed approach. The proposed method was also validated by comparing its performance with the pre-existing deep learning models of GoogleNet, VGG16, ResNet18, AlexNet, and SqueezeNet in terms of feature extraction, generalizability, computational cost, and size and parameters of the networks. Full article

(This article belongs to the Special Issue Mathematical Modeling and Simulation in Mechanics and Dynamic Systems)

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15 pages, 18100 KiB

Open AccessArticle

Assessment of Complex System Dynamics via Harmonic Mapping in a Multifractal Paradigm

by Gabriel Gavriluț, Liliana Topliceanu, Manuela Gîrțu, Ana Maria Rotundu, Stefan Andrei Irimiciuc and Maricel Agop

Mathematics 2021, 9(24), 3298; https://doi.org/10.3390/math9243298 - 18 Dec 2021

Cited by 2 | Viewed by 2019

Abstract

In the present paper, nonlinear behaviors of complex system dynamics from a multifractal perspective of motion are analyzed. In the framework of scale relativity theory, by analyzing the dynamics of complex system entities based on continuous but non-differentiable curves (multifractal curves), both the [...] Read more.

In the present paper, nonlinear behaviors of complex system dynamics from a multifractal perspective of motion are analyzed. In the framework of scale relativity theory, by analyzing the dynamics of complex system entities based on continuous but non-differentiable curves (multifractal curves), both the Schrödinger and Madelung scenarios on the holographic implementations of dynamics are functional and complementary. In the Madelung scenario, the holographic implementation of dynamics (i.e., free of any external or internal constraints) has some important consequences explicated by means of various operational procedures. The selected procedures involve synchronous modes through SL (2R) transformation group based on a hidden symmetry, coherence domains through Riemann manifold embedded with a Poincaré metric based on a parallel transport of direction (in a Levi Civita sense). Other procedures used here relate to the stationary-non-stationary dynamics transition through harmonic mapping from the usual space to the hyperbolic one manifested as cellular and channel type self-structuring. Finally, the Madelung scenario on the holographic implementations of dynamics are discussed with respect to laser-produced plasma dynamics. Full article

(This article belongs to the Special Issue Mathematical Modeling and Simulation in Mechanics and Dynamic Systems)

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22 pages, 6128 KiB

Open AccessArticle

Fracture Modelling of a Cracked Pressurized Cylindrical Structure by Using Extended Iso-Geometric Analysis (X-IGA)

by Soufiane Montassir, Hassane Moustabchir, Ahmed Elkhalfi, Maria Luminita Scutaru and Sorin Vlase

Mathematics 2021, 9(23), 2990; https://doi.org/10.3390/math9232990 - 23 Nov 2021

Cited by 14 | Viewed by 2509

Abstract

In this study, a NURBS basis function-based extended iso-geometric analysis (X-IGA) has been implemented to simulate a two-dimensional crack in a pipe under uniform pressure using MATLAB code. Heaviside jump and asymptotic crack-tip enrichment functions are used to model the crack’s behaviour. The [...] Read more.

In this study, a NURBS basis function-based extended iso-geometric analysis (X-IGA) has been implemented to simulate a two-dimensional crack in a pipe under uniform pressure using MATLAB code. Heaviside jump and asymptotic crack-tip enrichment functions are used to model the crack’s behaviour. The accuracy of this investigation was ensured with the stress intensity factors (SIFs) and the J-integral. The X-IGA—based SIFs of a 2-D pipe are compared using MATLAB code with the conventional finite element method available in ABAQUS FEA, and the extended finite element method is compared with a user-defined element. Therefore, the results demonstrate the possibility of using this technique as an alternative to other existing approaches to modeling cracked pipelines. Full article

(This article belongs to the Special Issue Mathematical Modeling and Simulation in Mechanics and Dynamic Systems)

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34 pages, 2690 KiB

Open AccessArticle

Lie-Group Modeling and Numerical Simulation of a Helicopter

by Alessandro Tarsi and Simone Fiori

Mathematics 2021, 9(21), 2682; https://doi.org/10.3390/math9212682 - 22 Oct 2021

Cited by 4 | Viewed by 3328

Abstract

Helicopters are extraordinarily complex mechanisms. Such complexity makes it difficult to model, simulate and pilot a helicopter. The present paper proposes a mathematical model of a fantail helicopter type based on Lie-group theory. The present paper first recalls the Lagrange–d’Alembert–Pontryagin principle to describe [...] Read more.

Helicopters are extraordinarily complex mechanisms. Such complexity makes it difficult to model, simulate and pilot a helicopter. The present paper proposes a mathematical model of a fantail helicopter type based on Lie-group theory. The present paper first recalls the Lagrange–d’Alembert–Pontryagin principle to describe the dynamics of a multi-part object, and subsequently applies such principle to describe the motion of a helicopter in space. A good part of the paper is devoted to the numerical simulation of the motion of a helicopter, which was obtained through a dedicated numerical method. Numerical simulation was based on a series of values for the many parameters involved in the mathematical model carefully inferred from the available technical literature. Full article

(This article belongs to the Special Issue Mathematical Modeling and Simulation in Mechanics and Dynamic Systems)

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40 pages, 7858 KiB

Open AccessArticle

Reliability Simulation of Two Component Warm-Standby System with Repair, Switching, and Back-Switching Failures under Three Aging Assumptions

by Kiril Tenekedjiev, Simon Cooley, Boyan Mednikarov, Guixin Fan and Natalia Nikolova

Mathematics 2021, 9(20), 2547; https://doi.org/10.3390/math9202547 - 11 Oct 2021

Cited by 4 | Viewed by 2154

Abstract

We analyze the influence of repair on a two-component warm-standby system with switching and back-switching failures. The repair of the primary component follows a minimal process, i.e., it experiences full aging during the repair. The backup component operates only while the primary component [...] Read more.

We analyze the influence of repair on a two-component warm-standby system with switching and back-switching failures. The repair of the primary component follows a minimal process, i.e., it experiences full aging during the repair. The backup component operates only while the primary component is being repaired, but it can also fail in standby, in which case there will be no repair for the backup component (as there is no indication of the failure). Four types of system failures are investigated: both components fail to operate in a different order or one of two types of switching failures occur. The reliability behavior of the system is investigated under three different aging assumptions for the backup component during warm-standby: full aging, no aging, and partial aging. Four failure and repair distributions determine the reliability behavior of the system. We analyzed two cases—in the First Case, we utilized constant failure rate distributions. In the Second Case, we applied the more realistic time-dependent failure rates. We used three methods to identify the reliability characteristics of the system: analytical, numerical, and simulational. The analytical approach is limited and only viable for constant failure rate distributions i.e., the First Case. The numerical method integrates simultaneous Algebraic Differential Equations. It produces a solution in the First Case under any type of aging, and in the Second Case but only under the assumption of full aging in warm-standby. On the other hand, the developed simulation algorithms produce solutions for any set of distributions (i.e., the First Case and the Second Case) under any of the three aging assumptions for the backup component in standby. The simulation solution is quantitively verified by comparison with the other two methods, and qualitatively verified by comparing the solutions under the three aging assumptions. It is numerically proven that the full aging and no aging solutions could serve as bounds of the partial aging case even when the precise mechanism of partial aging is unknown. Full article

(This article belongs to the Special Issue Mathematical Modeling and Simulation in Mechanics and Dynamic Systems)

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20 pages, 3570 KiB

Open AccessArticle

“Holographic Implementations” in the Complex Fluid Dynamics through a Fractal Paradigm

by Alexandra Saviuc, Manuela Gîrțu, Liliana Topliceanu, Tudor-Cristian Petrescu and Maricel Agop

Mathematics 2021, 9(18), 2273; https://doi.org/10.3390/math9182273 - 16 Sep 2021

Cited by 5 | Viewed by 1548

Abstract

Assimilating a complex fluid with a fractal object, non-differentiable behaviors in its dynamics are analyzed. Complex fluid dynamics in the form of hydrodynamic-type fractal regimes imply “holographic implementations” through velocity fields at non-differentiable scale resolution, via fractal solitons, fractal solitons–fractal kinks, and fractal [...] Read more.

Assimilating a complex fluid with a fractal object, non-differentiable behaviors in its dynamics are analyzed. Complex fluid dynamics in the form of hydrodynamic-type fractal regimes imply “holographic implementations” through velocity fields at non-differentiable scale resolution, via fractal solitons, fractal solitons–fractal kinks, and fractal minimal vortices. Complex fluid dynamics in the form of Schrödinger type fractal regimes imply “holographic implementations”, through the formalism of Airy functions of fractal type. Then, the in-phase coherence of the dynamics of the complex fluid structural units induces various operational procedures in the description of such dynamics: special cubics with SL(2R)-type group invariance, special differential geometry of Riemann type associated to such cubics, special apolar transport of cubics, special harmonic mapping principle, etc. In such a manner, a possible scenario toward chaos (a period-doubling scenario), without concluding in chaos (nonmanifest chaos), can be mimed. Full article

(This article belongs to the Special Issue Mathematical Modeling and Simulation in Mechanics and Dynamic Systems)

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18 pages, 1315 KiB

Open AccessArticle

Machine Learning Approach for Modeling and Control of a Commercial Heliocentris FC50 PEM Fuel Cell System

by Mohamed Derbeli, Cristian Napole and Oscar Barambones

Mathematics 2021, 9(17), 2068; https://doi.org/10.3390/math9172068 - 26 Aug 2021

Cited by 14 | Viewed by 3040

Abstract

In recent years, machine learning (ML) has received growing attention and it has been used in a wide range of applications. However, the ML application in renewable energies systems such as fuel cells is still limited. In this paper, a prognostic framework based [...] Read more.

In recent years, machine learning (ML) has received growing attention and it has been used in a wide range of applications. However, the ML application in renewable energies systems such as fuel cells is still limited. In this paper, a prognostic framework based on artificial neural network (ANN) is designed to predict the performance of proton exchange membrane (PEM) fuel cell system, aiming to investigate the effect of temperature and humidity on the stack characteristics and on tracking control improvements. A large part of the experimental database for various operating conditions has been used in the training operation to achieve an accurate model. Extensive tests with various ANN parameters such as number of neurons, number of hidden layers, selection of training dataset, etc., are performed to obtain the best fit in terms of prediction accuracy. The effect of temperature and humidity based on the predicted model are investigated and compared to the ones obtained from real-time experiments. The control design based on the predicted model is performed to keep the stack operating point at an adequate power stage with high-performance tracking. Experimental results have demonstrated the effectiveness of the proposed model for performance improvements of PEM fuel cell system. Full article

(This article belongs to the Special Issue Mathematical Modeling and Simulation in Mechanics and Dynamic Systems)

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22 pages, 14254 KiB

Open AccessArticle

Alternative Artificial Neural Network Structures for Turbulent Flow Velocity Field Prediction

by Koldo Portal-Porras, Unai Fernandez-Gamiz, Ainara Ugarte-Anero, Ekaitz Zulueta and Asier Zulueta

Mathematics 2021, 9(16), 1939; https://doi.org/10.3390/math9161939 - 14 Aug 2021

Cited by 16 | Viewed by 3580

Abstract

Turbulence in fluids has been a popular research topic for many years due to its influence on a wide range of applications. Computational Fluid Dynamics (CFD) tools are able to provide plenty of information about this phenomenon, but their computational cost often makes [...] Read more.

Turbulence in fluids has been a popular research topic for many years due to its influence on a wide range of applications. Computational Fluid Dynamics (CFD) tools are able to provide plenty of information about this phenomenon, but their computational cost often makes the use of these tools unfeasible. For that reason, in recent years, turbulence modelling using Artificial Neural Networks (ANNs) is becoming increasingly popular. These networks typically calculate directly the desired magnitude, having input information about the computational domain. In this paper, a Convolutional Neural Network (CNN) for predicting different magnitudes of turbulent flows around different geometries by approximating the equations of the Reynolds-Averaged Navier-Stokes (RANS)-based realizable k-ε two-layer turbulence model is proposed. Using that CNN, alternative network structures are proposed to predict the velocity fields of a turbulent flow around different geometries on a rectangular channel, with a preliminary stage to predict pressure and vorticity fields before calculating the velocity fields, and the obtained results are compared with the ones obtained with the basic structure. The results demonstrate that the proposed structures clearly outperform the basic one, especially when the flow becomes uncertain. In addition, considering the results, the best network configuration is proposed. That network is tested with a domain with multiple geometries and a domain with a narrowing of the channel, which are domains with different conditions from the training ones, showing fairly accurate predictions. Full article

(This article belongs to the Special Issue Mathematical Modeling and Simulation in Mechanics and Dynamic Systems)

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24 pages, 819 KiB

Open AccessArticle

Thermal Scaling of Transient Heat Transfer in a Round Cladded Rod with Modern Dimensional Analysis

by Botond-Pál Gálfi, Ioan Száva, Daniela Șova and Sorin Vlase

Mathematics 2021, 9(16), 1875; https://doi.org/10.3390/math9161875 - 6 Aug 2021

Cited by 14 | Viewed by 1892

Abstract

Heat transfer analysis can be studied efficiently with the help of so-called modern dimensional analysis (MDA), which offers a uniform and easy approach, without requiring in-depth knowledge of the phenomenon by only taking into account variables that may have some influence. After a [...] Read more.

Heat transfer analysis can be studied efficiently with the help of so-called modern dimensional analysis (MDA), which offers a uniform and easy approach, without requiring in-depth knowledge of the phenomenon by only taking into account variables that may have some influence. After a brief presentation of the advantages of this method (MDA), the authors applied it to the study of heat transfer in straight bars of solid circular section, protected but not thermally protected with layers of intumescent paints. Two cases (two sets of independent variables) were considered, which could be easily tracked by experimental measurements. The main advantages of the model law obtained are presented, being characterized by flexibility, accuracy, and simplicity. Additionally, this law and the MDA approach allow us to obtain much more advantageous models from an experimental point of view, with the geometric analogy of the model with the prototype not being a necessary condition. To the best knowledge of the present authors there are no studies reporting the application of the MDA method as it was used in this paper to heat transfer. Full article

(This article belongs to the Special Issue Mathematical Modeling and Simulation in Mechanics and Dynamic Systems)

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14 pages, 3925 KiB

Open AccessArticle

Thermodynamic Optimization of a High Temperature Proton Exchange Membrane Fuel Cell for Fuel Cell Vehicle Applications

by Bing Xu, Dongxu Li, Zheshu Ma, Meng Zheng and Yanju Li

Mathematics 2021, 9(15), 1792; https://doi.org/10.3390/math9151792 - 28 Jul 2021

Cited by 16 | Viewed by 2634

Abstract

In this paper, a finite time thermodynamic model of high temperature proton exchange membrane fuel cell (HT-PEMFC) is established, in which the irreversible losses of polarization and leakage current during the cell operation are considered. The influences of operating temperature, membrane thickness, phosphoric [...] Read more.

In this paper, a finite time thermodynamic model of high temperature proton exchange membrane fuel cell (HT-PEMFC) is established, in which the irreversible losses of polarization and leakage current during the cell operation are considered. The influences of operating temperature, membrane thickness, phosphoric acid doping level, hydrogen and oxygen intake pressure on the maximum output power density

P_{m a x}

and the maximum output efficiency

η_{m a x}

are studied. As the temperature rises,

P_{m a x}

and

η_{m a x}

will increase. The decrease of membrane thickness will increase

P_{m a x}

, but has little influence on the

η_{m a x}

. The increase of phosphoric acid doping level can increase

P_{m a x}

, but it has little effect on the

η_{m a x}

. With the increase of hydrogen and oxygen intake pressure,

P_{m a x}

and

η_{m a x}

will be improved. This article also obtains the optimization relationship between power density and thermodynamic efficiency, and the optimization range interval of HT-PEMFC which will provide guidance for applicable use of HT-PEMFCs. Full article

(This article belongs to the Special Issue Mathematical Modeling and Simulation in Mechanics and Dynamic Systems)

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18 pages, 1228 KiB

Open AccessArticle

Power Spectral Density Analysis of Nanowire-Anchored Fluctuating Microbead Reveals a Double Lorentzian Distribution

by Gregor Bánó, Jana Kubacková, Andrej Hovan, Alena Strejčková, Gergely T. Iványi, Gaszton Vizsnyiczai, Lóránd Kelemen, Gabriel Žoldák, Zoltán Tomori and Denis Horvath

Mathematics 2021, 9(15), 1748; https://doi.org/10.3390/math9151748 - 24 Jul 2021

Cited by 3 | Viewed by 2708

Abstract

In this work, we investigate the properties of a stochastic model, in which two coupled degrees of freedom are subordinated to viscous, elastic, and also additive random forces. Our model, which builds on previous progress in Brownian motion theory, is designed to describe [...] Read more.

In this work, we investigate the properties of a stochastic model, in which two coupled degrees of freedom are subordinated to viscous, elastic, and also additive random forces. Our model, which builds on previous progress in Brownian motion theory, is designed to describe water-immersed microparticles connected to a cantilever nanowire prepared by polymerization using two-photon direct laser writing (TPP-DLW). The model focuses on insights into nanowires exhibiting viscoelastic behavior, which defines the specific conditions of the microbead. The nanowire bending is described by a three-parameter linear model. The theoretical model is studied from the point of view of the power spectrum density of Brownian fluctuations. Our approach also focuses on the potential energy equipartition, which determines random forcing parametrization. Analytical calculations are provided that result in a double-Lorentzian power density spectrum with two corner frequencies. The proposed model explained our preliminary experimental findings as a result of the use of regression analysis. Furthermore, an a posteriori form of regression efficiency evaluation was designed and applied to three typical spectral regions. The agreement of respective moments obtained by integration of regressed dependences as well as by summing experimental data was confirmed. Full article

(This article belongs to the Special Issue Mathematical Modeling and Simulation in Mechanics and Dynamic Systems)

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17 pages, 4184 KiB

Open AccessArticle

SARS-CoV-2 Spread Forecast Dynamic Model Validation through Digital Twin Approach, Catalonia Case Study

by Pau Fonseca i Casas, Joan Garcia i Subirana, Víctor García i Carrasco and Xavier Pi i Palomés

Mathematics 2021, 9(14), 1660; https://doi.org/10.3390/math9141660 - 14 Jul 2021

Cited by 10 | Viewed by 3047

Abstract

The spread of the SARS-CoV-2 modeling is a challenging problem because of its complex nature and lack of information regarding certain aspects. In this paper, we explore a Digital Twin approach to model the pandemic situation in Catalonia. The Digital Twin is composed [...] Read more.

The spread of the SARS-CoV-2 modeling is a challenging problem because of its complex nature and lack of information regarding certain aspects. In this paper, we explore a Digital Twin approach to model the pandemic situation in Catalonia. The Digital Twin is composed of three different dynamic models used to perform the validations by a Model Comparison approach. We detail how we use this approach to obtain knowledge regarding the effects of the nonpharmaceutical interventions and the problems we faced during the modeling process. We use Specification and Description Language (SDL) to represent the compartmental forecasting model for the SARS-CoV-2. Its graphical notation simplifies the different specialists’ understanding of the model hypotheses, which must be validated continuously following a Solution Validation approach. This model allows the successful forecasting of different scenarios for Catalonia. We present some formalization details, discuss the validation process and present some results obtained from the validation model discussion, which becomes a digital twin of the pandemic in Catalonia. Full article

(This article belongs to the Special Issue Mathematical Modeling and Simulation in Mechanics and Dynamic Systems)

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15 pages, 2005 KiB

Open AccessArticle

Ecological Performance Optimization of a High Temperature Proton Exchange Membrane Fuel Cell

by Dongxu Li, Siwei Li, Zheshu Ma, Bing Xu, Zhanghao Lu, Yanju Li and Meng Zheng

Mathematics 2021, 9(12), 1332; https://doi.org/10.3390/math9121332 - 9 Jun 2021

Cited by 19 | Viewed by 2556

Abstract

According to finite-time thermodynamics, an irreversible high temperature proton exchange membrane fuel cell (HT-PEMFC) model is established, and the mathematical expressions of the output power, energy efficiency, exergy efficiency and ecological coefficient of performance (ECOP) of HT-PEMFC are deduced. The ECOP is a [...] Read more.

According to finite-time thermodynamics, an irreversible high temperature proton exchange membrane fuel cell (HT-PEMFC) model is established, and the mathematical expressions of the output power, energy efficiency, exergy efficiency and ecological coefficient of performance (ECOP) of HT-PEMFC are deduced. The ECOP is a step forward in optimizing the relationship between power and power dissipation, which is more in line with the principle of ecology. Based on the established HT-PEMFC model, the maximum power density is obtained under different parameters that include operating temperature, operating pressure, phosphoric acid doping level and relative humidity. At the same time, the energy efficiency, exergy efficiency and ECOP corresponding to the maximum power density are acquired so as to determine the optimal value of each index under the maximum power density. The results show that the higher the operating temperature and the doping level, the better the performance of HT-PEMFC is. However, the increase of operating pressure and relative humidity has little effect on HT-PEMFC performance. Full article

(This article belongs to the Special Issue Mathematical Modeling and Simulation in Mechanics and Dynamic Systems)

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14 pages, 3236 KiB

Open AccessArticle

Advanced Models for Modulus and Strength of Carbon-Nanotube-Filled Polymer Systems Assuming the Networks of Carbon Nanotubes and Interphase Section

by Yasser Zare and Kyongyop Rhee

Mathematics 2021, 9(9), 990; https://doi.org/10.3390/math9090990 - 28 Apr 2021

Cited by 3 | Viewed by 1982

Abstract

This study focuses on the simultaneous stiffening and percolating characteristics of the interphase section in polymer carbon nanotubes (CNTs) systems (PCNTs) using two advanced models of tensile modulus and strength. The interphase, as a third part around the nanoparticles, influences the mechanical features [...] Read more.

This study focuses on the simultaneous stiffening and percolating characteristics of the interphase section in polymer carbon nanotubes (CNTs) systems (PCNTs) using two advanced models of tensile modulus and strength. The interphase, as a third part around the nanoparticles, influences the mechanical features of such systems. The forecasts agree well with the tentative results, thus validating the advanced models. A CNT radius of >40 nm and CNT length of <5 μm marginally improve the modulus by 70%, while the highest modulus development of 350% is achieved with the thinnest nanoparticles. Furthermore, the highest improvement in nanocomposite’s strength (350%) is achieved with the CNT length of 12 μm and interfacial shear strength of 8 MPa. Generally, the highest ranges of the CNT length, interphase thickness, interphase modulus and interfacial shear strength lead to the most desirable mechanical features. Full article

(This article belongs to the Special Issue Mathematical Modeling and Simulation in Mechanics and Dynamic Systems)

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13 pages, 1926 KiB

Open AccessFeature PaperArticle

A Computational Approach to Solve a System of Transcendental Equations with Multi-Functions and Multi-Variables

by Chukwuma Ogbonnaya, Chamil Abeykoon, Adel Nasser and Ali Turan

Mathematics 2021, 9(9), 920; https://doi.org/10.3390/math9090920 - 21 Apr 2021

Cited by 8 | Viewed by 3859

Abstract

A system of transcendental equations (SoTE) is a set of simultaneous equations containing at least a transcendental function. Solutions involving transcendental equations are often problematic, particularly in the form of a system of equations. This challenge has limited the number of equations, with [...] Read more.

A system of transcendental equations (SoTE) is a set of simultaneous equations containing at least a transcendental function. Solutions involving transcendental equations are often problematic, particularly in the form of a system of equations. This challenge has limited the number of equations, with inter-related multi-functions and multi-variables, often included in the mathematical modelling of physical systems during problem formulation. Here, we presented detailed steps for using a code-based modelling approach for solving SoTEs that may be encountered in science and engineering problems. A SoTE comprising six functions, including Sine-Gordon wave functions, was used to illustrate the steps. Parametric studies were performed to visualize how a change in the variables affected the superposition of the waves as the independent variable varies from

x_{1}

= 1:0.0005:100 to

x_{1}

= 1:5:100. The application of the proposed approach in modelling and simulation of photovoltaic and thermophotovoltaic systems were also highlighted. Overall, solutions to SoTEs present new opportunities for including more functions and variables in numerical models of systems, which will ultimately lead to a more robust representation of physical systems. Full article

(This article belongs to the Special Issue Mathematical Modeling and Simulation in Mechanics and Dynamic Systems)

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10 pages, 3290 KiB

Open AccessArticle

An Extended Finite Element Method (XFEM) Study on the Elastic T-Stress Evaluations for a Notch in a Pipe Steel Exposed to Internal Pressure

by Khadija Yakoubi, Soufiane Montassir, Hassane Moustabchir, Ahmed Elkhalfi, Catalin Iulian Pruncu, Jamal Arbaoui and Muhammad Umar Farooq

Mathematics 2021, 9(5), 507; https://doi.org/10.3390/math9050507 - 2 Mar 2021

Cited by 16 | Viewed by 2680

Abstract

The work investigates the importance of the K-T approach in the modelling of pressure cracked structures. T-stress is the constant in the second term of the Williams expression; it is often negligible, but recent literature has shown that there are cases where T-stress [...] Read more.

The work investigates the importance of the K-T approach in the modelling of pressure cracked structures. T-stress is the constant in the second term of the Williams expression; it is often negligible, but recent literature has shown that there are cases where T-stress plays the role of opening the crack, also T-stress improves elastic modeling at the point of crack. In this research study, the most important effects of the T-stress are collected and analyzed. A numerical analysis was carried out by the extended finite element method (X-FEM) to analyze T-stress in an arc with external notch under internal pressure. The different stress method (SDM) is employed to calculate T-stress. Moreover, the influence of the geometry of the notch on the biaxiality is also examined. The biaxiality gave us a view on the initiation of the crack. The results are extended with a comparison to previous literature to validate the promising investigations. Full article

(This article belongs to the Special Issue Mathematical Modeling and Simulation in Mechanics and Dynamic Systems)

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