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A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "E2: Control Theory and Mechanics".

Deadline for manuscript submissions: closed (10 August 2025) | Viewed by 13073

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Dr. Sotos C. Generalis

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School of Engineering and Applied Science, Aston University, Birmingham B4 7ET, UK
Interests: turbulence; non-linear fluid dynamics; stability; Floquet theory; bifurcation theory; Navier–Stokes equations; molecular dynamics
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Dr. Philip Trevelyan

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School of Engineering and Applied Science, Aston University, Birmingham B4 7ET, UK
Interests: hydrodynamic instabilities; chemical reaction; partial differential equations; fluid dynamics
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Dear Colleagues,

Mathematical modelling, analysis and further optimization attempts to understand nonlinear phenomena are fundamentally present in nature and form the basis of science, engineering, and consequently, mechanics. Mathematical modelling and optimization form the foundation of our understanding of all engineering applications and are thus present in the solutions/states of the dynamical modelling of mechanics. From spontaneous symmetry breaking in particle physics and the formation of coherent states in ubiquitous flows to analysing pattern and stress formation in physico-chemical processes, mathematical techniques and modelling play a pivotal role in the discovery of the solutions/states preferred by nature. Through the theoretical application and numerical implementation of pioneering theoretical techniques such as Poincaré sections, weakly nonlinear analysis/perturbation methods, Floquet analysis and bifurcation theory within Euclidean or curvilinear space fixed-point/solutions and their stability can be identified, while transient solutions can be followed, offering the possibility to explicitly optimise mechanical states that have both engineering applications and simultaneously present significant theoretical/simulative advances. Mathematical modelling enables us to continuously pioneer new emerging technologies for the benefit of society in general, and further encourage new insights into technology, engineering, and science; explain and organize complex behaviours; and aid in the establishment of the basis of new pathways for further exploitation by future generations.

In order to celebrate the role of “Mathematical Modelling, Analysis, and Optimization for Engineering and Mechanics”, this Special Issue aims to publish original research papers and reviews on the latest advancements in modelling, analysis, and optimization for engineering and mechanics alike.

Dr. Sotos C. Generalis
Dr. Philip Trevelyan
Guest Editors

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Keywords

fluid dynamics
turbulence modelling
nonlinear dynamics
hydrodynamic instabilities
fluids in porous media
thin films
mathematical modelling
nonlinear analysis
numerical methods
partial differential equations
Navier–Stokes systems

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Mathematical Modelling, Analysis, and Optimization for Engineering and Mechanics, 2nd Edition in Mathematics

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Research

19 pages, 1909 KB

Open AccessArticle

Modeling of Soliton Behavior in Nonlinear Transmission Line Systems

by Sadia Medhit, Beenish, Fehaid Salem Alshammari and Isha Bukhar

Mathematics 2025, 13(18), 2997; https://doi.org/10.3390/math13182997 - 16 Sep 2025

Viewed by 207

Abstract

This study focuses on the nonlinear partial differential equation known as the Lonngren wave equation, which plays a significant role in plasma physics, nonlinear wave propagation, and astrophysical research. By applying a suitable wave transformation, the nonlinear model is reduced to an ordinary differential equation. Analytical wave solutions of the Lonngren wave equation are then derived using the extended direct algebraic method. The physical behavior of these solutions is illustrated through 2D, 3D, and contour plots generated in Mathematica. Finally, the stability analysis of the Lonngren wave equation is discussed. Full article

(This article belongs to the Special Issue Mathematical Modelling, Analysis, and Optimization for Engineering and Mechanics)

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32 pages, 1551 KB

Open AccessFeature PaperArticle

Free Vibration Analysis of Porous FGM Plates on Elastic Foundations with Temperature-Dependent Material Properties

by Aleksandar Radaković, Dragan Čukanović, Aleksandar Nešović, Petar Knežević, Milan T. Djordjević and Gordana Bogdanović

Mathematics 2025, 13(18), 2957; https://doi.org/10.3390/math13182957 - 12 Sep 2025

Viewed by 274

Abstract

This study investigates the free vibration behaviors of functionally graded (FGM) plates with a porous structure, resting on a Kerr-type elastic foundation, while accounting for thermal effects and complex material property distributions. Within the framework of higher-order shear deformation theory (HSDT), two novel shape functions are introduced to accurately model transverse shear deformation across the plate thickness without employing shear correction factors. These functions are constructed to satisfy shear stress boundary conditions and capture nonlinear effects induced by material gradation and porosity. A variational formulation is developed to describe the dynamic response of FGM plates in a thermo-mechanical environment, incorporating temperature-dependent material properties and three porosity distributions: uniform, linear, and trigonometric. Numerical solutions are obtained using in-house MATLAB codes, allowing complete control over the formulation and interpretation of the results. The model is validated through detailed comparisons with existing literature, demonstrating high accuracy. The findings reveal that the porosity distribution pattern and gradient intensity significantly influence natural frequencies and mode shapes. The trigonometric porosity distribution exhibits favorable dynamic performance due to preserved stiffness in the surface regions. Additionally, the Kerr-type elastic foundation enables fine tuning of the dynamic response, depending on its specific parameters. The proposed approach provides a reliable and efficient tool for analyzing FGM structures under complex loading conditions and lays the groundwork for future extensions involving nonlinear, time-dependent, and multiphysics analyses. Full article

(This article belongs to the Special Issue Mathematical Modelling, Analysis, and Optimization for Engineering and Mechanics)

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29 pages, 2543 KB

Open AccessArticle

A Finite Element–Finite Volume Code Coupling for Optimal Control Problems in Fluid Heat Transfer for Incompressible Navier–Stokes Equations

by Samuele Baldini, Giacomo Barbi, Giorgio Bornia, Antonio Cervone, Federico Giangolini, Sandro Manservisi and Lucia Sirotti

Mathematics 2025, 13(11), 1701; https://doi.org/10.3390/math13111701 - 22 May 2025

Viewed by 792

Abstract

In this work, we present a numerical approach for solving optimal control problems for fluid heat transfer applications with a mixed optimality system: an FEM code to solve the adjoint solution over a precise restricted admissible solution set and an open-source well-known code for solving the state problem defined over a different one. In this way, we are able to decouple the optimality system and use well-established and validated numerical tools for the physical modeling. Specifically, two different CFD codes, OpenFOAM (finite volume-based) and FEMuS (finite element-based), have been used to solve the optimality system, while the data transfer between them is managed by the external library MEDCOUPLING. The state equations are solved in the finite volume code, while the adjoint and the control are solved in the finite element code. Two examples taken from the literature are implemented in order to validate the numerical algorithm: the first one considers a natural convection cavity resulting from a Rayleigh–Bénard configuration, and the second one is a conjugate heat transfer problem between a fluid and a solid region. Full article

(This article belongs to the Special Issue Mathematical Modelling, Analysis, and Optimization for Engineering and Mechanics)

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16 pages, 1242 KB

Open AccessArticle

A Note on the Dynamics of Modified rf-SQUIDs: Simulations and Possible Control over Oscillations

by Nikolay Kyurkchiev, Tsvetelin Zaevski, Anton Iliev and Todor Branzov

Mathematics 2025, 13(5), 722; https://doi.org/10.3390/math13050722 - 24 Feb 2025

Viewed by 560

Abstract

The so-call SQUIDs (abbreviated from superconducting quantum interference device) are very sensitive apparatuses especially built for metering very low magnetic fields. These systems have applications in various practical fields—biology, geology, medicine, different engineering areas, etc. Their features are mainly based on superconductors and the Josephson effect. They can be differentiated into two main groups—direct current (DC) and radio frequency (RF) SQUIDs. Both of them were constructed in the 1960s at Ford Research Labs. The main difference between them is that the second ones use only one superconducting tunnel junction. This reduces their sensitivity, but makes them significantly cheaper. We investigate namely the rf-SQUIDs in the present work. A number of authors devote their research to the rf-SQUIDs driven by an oscillating external flux. We aim to enlarge the theoretical base of these systems by adding new factors in their dynamics. Several particular cases are explored and simulated. We demonstrate also some specialized modules for investigating the proposed model. One application for possible control over oscillations is also discussed. It is based on the Fourier transform and, as a consequence, on the characteristic function of some probability distributions. Full article

(This article belongs to the Special Issue Mathematical Modelling, Analysis, and Optimization for Engineering and Mechanics)

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18 pages, 3702 KB

Open AccessArticle

Effect of Rotation in Radial Microwave Irradiation: A Numerical Approach

by María Cruz Navarro and Damián Castaño

Mathematics 2025, 13(3), 357; https://doi.org/10.3390/math13030357 - 23 Jan 2025

Viewed by 652

Abstract

In this paper, we study numerically the effect of rotation within a solvent in a cylindrical container subject to radial microwave irradiation. Two solvents with different dielectric and thermophysical properties are used: water and ethylene glycol. The samples are irradiated at a frequency of 2.45 GHz and a power of 80 W. The higher the rotation rate is, the faster the state becomes fully 3D. For water, the bifurcation occurs earlier in time due to its lower viscosity. For ethylene glycol, more susceptible to microwaves than water but with a higher viscosity, the flow remains axisymmetric for a long time and it becomes 3D when it has almost reached a stationary homogeneous maximum temperature all along the cell. We use a 3D temporal model coupling heat and momentum equations and the Maxwell equations based on spectral methods to perform the simulations. Full article

(This article belongs to the Special Issue Mathematical Modelling, Analysis, and Optimization for Engineering and Mechanics)

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18 pages, 1082 KB

Open AccessArticle

Analytical Investigation of Time-Dependent Two-Dimensional Non-Newtonian Boundary Layer Equations

by Imre Ferenc Barna, Laszló Mátyás, Krisztián Hriczó and Gabriella Bognár

Mathematics 2024, 12(23), 3863; https://doi.org/10.3390/math12233863 - 9 Dec 2024

Viewed by 959

Abstract

In this study, five different time-dependent incompressible non-Newtonian boundary layer models in two dimensions are investigated with the self-similar Ansatz, including external magnetic field effects. The power-law, the Casson fluid, the Oldroyd-B model, the Walter fluid B model, and the Williamson fluid are analyzed. For the first two models, analytical results are given for the velocity and pressure distributions, which can be expressed by different types of hypergeometric functions. Depending on the parameters involved in the analytical solutions of the nonlinear ordinary differential equation obtained by the similarity transformation, a vast range of solution types is presented. It turned out that the last three models lack self-similar symmetry; therefore, no analytic solutions can be derived. Full article

(This article belongs to the Special Issue Mathematical Modelling, Analysis, and Optimization for Engineering and Mechanics)

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17 pages, 5307 KB

Open AccessArticle

Research on Adhesion Pull-Off Behavior of Rigid Flat Punch and Viscoelastic Substrate

by Tao Zhang, Yunqi Zhang and Kai Jiang

Mathematics 2024, 12(22), 3454; https://doi.org/10.3390/math12223454 - 5 Nov 2024

Cited by 1 | Viewed by 2346

Abstract

Interfacial adhesion is one of the key factors affecting the reliability of micro–nano systems. The adhesion contact mechanism is still unclear as the time-dependent viscoelasticity of soft materials. To clarify the adhesion interaction, the pull-off detachment between the rigid flat punch and viscoelastic substrate is explored considering the viscoelasticity of soft materials and rate-dependent adhesion. Taking the Lennard-Jones (L-J) potential characterizing interfacial adhesion and the Prony series defining the viscoelasticity of materials as references, the bilinear cohesion zone model (CZM) and standard Maxwell model are employed, and an adhesion analysis framework is established by combining finite element technology. The influence laws of the loading and unloading rates, material relaxation coefficients and size effect on adhesion pull-off behavior are revealed. The results show that the pull-off force is independent of the material relaxation effect and related to the unloading rate. When

\hat{v}

≥ 50 or

\hat{v}

< 0.01, the pull-off force has nothing to do with the unloading rate, but when 0.01 <

\hat{v}

< 50, the pull-off force increases with the increasing unloading rate. Also, it is controlled by the size effect, and the changing trend conforms to the MD-n model proposed by Jiang. The energy required for interfacial separation (i.e., effective adhesion work) is a result of the comprehensive influence of unloading rates, material properties and the relaxation effect, which is consistent with Papangelo1’s research results. In addition, we derive the critical contact radius of the transition from the Kendall solution to the strength control solution. This work not only provides a detailed solution for the interfacial adhesion behavior but also provides guidance for the application of adhesion in Micro-Electro-Mechanical Systems (MEMSs). Full article

(This article belongs to the Special Issue Mathematical Modelling, Analysis, and Optimization for Engineering and Mechanics)

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18 pages, 5375 KB

Open AccessArticle

Kinematics Parameter Calibration of Serial Industrial Robots Based on Partial Pose Measurement

by Tiewu Xiang, Xinyi Jiang, Guifang Qiao, Chunhui Gao and Hongfu Zuo

Mathematics 2023, 11(23), 4802; https://doi.org/10.3390/math11234802 - 28 Nov 2023

Cited by 8 | Viewed by 2997

Abstract

The kinematics parameter error is the main error factor that affects the absolute accuracy of industrial robots. The absolute accuracy of industrial robots can be effectively improved through kinematics calibration. The error model-based method is one of the main methods for calibrating the kinematics parameter error. This paper presents a kinematics parameter calibration method for serial industrial robots based on partial pose measurement. Firstly, the kinematics and the pose error models have been established based on the modified Denavit–Hartenberg (MDH) model. By introducing the concept of error sensitivity, the average significance index is proposed to quantitatively analyze the effects of the kinematics parameter error on the pose error of a robot. The results show that there is no need to measure the full pose error of the robot. Secondly, a partial pose measurement device and method have been presented. The proposed device can measure the position error and the attitude error on the x-axis or y-axis. Finally, the full pose error model, the NP-type partial pose error model, and the OP-type partial pose error model have been applied for calibrating the kinematics parameter errors. The experimental results show that the effectiveness of the OP-type partial pose error model is consistent with the full pose error model. Full article

(This article belongs to the Special Issue Mathematical Modelling, Analysis, and Optimization for Engineering and Mechanics)

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16 pages, 3564 KB

Open AccessFeature PaperArticle

A Decoupling Method for Successive Robot Rotation Based on Time Domain Instantaneous Euler Angle

by Xin Zhou and Jianxu Zhu

Mathematics 2023, 11(18), 3882; https://doi.org/10.3390/math11183882 - 12 Sep 2023

Cited by 1 | Viewed by 1872

Abstract

In the present study, a novel time domain decoupling method was proposed for the multiple successive rotations of different kinds of robots. This is achieved through the utilization of instantaneous Euler angles. For a general parallel mechanism, the Plücker coordinates of the intersection line of the before and after rotation plane are determined through the reciprocal product principle of screw theory. Additionally, the angle between these two rotation planes is defined as the instantaneous Euler angle. The analysis of the general parallel mechanism was used as an example to illustrate the solution method of the instantaneous Euler angle. To investigate the intrinsic relationship between the instantaneous Euler angle and the conventional Euler angle, the mathematical mapping relationship and the difference between the instantaneous Euler angle and the two kinds of Euler angles (Z-Y-X and Z-Y-Z) were explored, respectively. Simulations of a 3-sps-s parallel mechanism and a robotic arm were employed to illustrate the superiority of the instantaneous Euler angle. The findings showed that the instantaneous Euler angle exhibited enhanced temporal consistency compared to the conventional Euler angle. Further, it is better suited for accurately describing the decoupled rotation of robotic systems. The proposed approach is also generally applicable to robot performance evaluation, mechanism design, and other relevant fields. Full article

(This article belongs to the Special Issue Mathematical Modelling, Analysis, and Optimization for Engineering and Mechanics)

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26 pages, 7410 KB

Open AccessArticle

Improved Power Series Solution of Transversely Loaded Hollow Annular Membranes: Simultaneous Modification of Out-of-Plane Equilibrium Equation and Radial Geometric Equation

by Xiao-Ting He, Fei-Yan Li and Jun-Yi Sun

Mathematics 2023, 11(18), 3836; https://doi.org/10.3390/math11183836 - 7 Sep 2023

Cited by 1 | Viewed by 1159

Abstract

The ability to accurately predict the shape of a transversely loaded hollow annular membrane is essential to the design of bending-free hollow annular shells of revolution, which requires a further improvement in the hollow annular membrane solution to meet the needs of this accurate prediction. In this paper, the large deflection problem of a transversely loaded hollow annular membrane is reformulated by simultaneously modifying the out-of-plane equilibrium equation and radial geometric equation, and a newer and more refined power series solution is derived. The reason why the classical radial geometry equation induces errors is revealed. The convergence and asymptotic behavior of the power series solution obtained is analyzed numerically. The newly derived solution is compared with the two previously derived solutions graphically, showing that the newly derived solution performs basically as well as expected. In addition, the anticipated use of the hollow and not-hollow annular membrane solutions for the design application of bending-free annular shells of revolution is discussed. Full article

(This article belongs to the Special Issue Mathematical Modelling, Analysis, and Optimization for Engineering and Mechanics)

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