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Mathematical Modelling, Analysis, and Optimization for Engineering and Mechanics
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Mathematical Modelling, Analysis, and Optimization for Engineering and Mechanics

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

Deadline for manuscript submissions: 10 August 2025 | Viewed by 9448

Special Issue Editors

Dr. Sotos C. Generalis

E-Mail Website
Guest Editor
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
Dr. Philip Trevelyan

E-Mail Website
Guest Editor
School of Engineering and Applied Science, Aston University, Birmingham B4 7ET, UK
Interests: hydrodynamic instabilities; chemical reaction; partial differential equations; fluid dynamics

Special Issue Information

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

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

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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Published Papers (7 papers)

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Research

16 pages, 1242 KiB  
Article
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 317
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 [...] Read more.
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 KiB  
Article
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 488
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 [...] Read more.
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 KiB  
Article
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 690
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 [...] Read more.
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 KiB  
Article
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 1968
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 [...] Read more.
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 v^ ≥ 50 or v^ < 0.01, the pull-off force has nothing to do with the unloading rate, but when 0.01 < 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 KiB  
Article
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 5 | Viewed by 2388
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 [...] Read more.
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 KiB  
Article
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 1641
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 [...] Read more.
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 KiB  
Article
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 1050
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 [...] Read more.
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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