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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 (28 February 2025) | Viewed by 15196

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Prof. Dr. Joaquim Infante Barbosa

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1. CIMOSM, ISEL—Centro de Investigação em Modelação e Optimização de Sistemas Multifuncionais, Instituto Superior de Engenharia de Lisboa, 1959-007 Lisboa, Portugal
2. IDMEC, IST—Instituto de Engenharia Mecânica, Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisboa, Portugal
Interests: structural mechanics; finite element analysis; structural optimization; composite materials; numerical computation; symbolic computation
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Dr. José Alberto Rodrigues

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CIMA e Departamento de Matemática—Instituto Superior de Engenharia de Lisboa, Rua Conselheiro Emídio Navarro, 1, 1959-007 Lisbon, Portugal
Interests: mathematical analysis; mathematical modeling; finite element method; isogeometric analysis; physics-informed neural networks; image processing
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Many complex problems arising from different areas of science and engineering can be successfully studied by applying mathematical models, demonstrating the multidisciplinary nature of mathematical modeling. In this process, a wide variety of issues are usually addressed in order to obtain robust and accurate numerical schemes.

Thus, this Special Issue aims to bring together modeling approaches that consider mathematical models, serving as a reference to existing structural engineering and mechanics problems.

The key problems addressed in this issue are also the automation and systematization of complex mechanical problems enabling significant time savings, the advantages of merging numerical and computational facets of the problems’ solving process and the development of hybrid analytical-numerical techniques. Topics of interest include (but are not limited to) the following:

Design of robust computational methods and simulations;
Development of parameter fitting methods and inverse problem strategies in ordinary and partial differential equations;
Numerical simulation in structural engineering;
Design optimization of complex structural systems;
Computational methods in mathematics and structural engineering;
Integrated numerical/computational methods.

Prof. Dr. Joaquim Infante Barbosa
Dr. José Alberto Rodrigues
Guest Editors

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Keywords

structural engineering
mechanical problems
fluid-structure interactions
soil-structure interactions
structural reliability
mathematical modeling
numerical schemes
optimization
control methods
ordinary differential equations
partial differential equations

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Related Special Issue

Numerical and Computational Methods in Structural Engineering and Mechanics, 2nd Edition in Mathematics

Published Papers (11 papers)

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Editorial

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3 pages, 152 KiB

Open AccessEditorial

Editorial: Advances in Mathematical Modeling for Structural Engineering and Mechanics

by Joaquim Infante Barbosa and José Alberto Rodrigues

Mathematics 2025, 13(6), 936; https://doi.org/10.3390/math13060936 - 12 Mar 2025

Viewed by 312

Abstract

Mathematical modeling is a cornerstone in addressing complex problems across science and engineering, showcasing its inherently multidisciplinary nature [...] Full article

(This article belongs to the Special Issue Numerical and Computational Methods in Structural Engineering and Mechanics)

Research

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25 pages, 8484 KiB

Open AccessFeature PaperArticle

Extending the Meshless Natural-Neighbour Radial-Point Interpolation Method to the Structural Optimization of an Automotive Part Using a Bi-Evolutionary Bone-Remodelling-Inspired Algorithm

by Carlos Oliveira, Ana Pais and Jorge Belinha

Mathematics 2025, 13(2), 178; https://doi.org/10.3390/math13020178 - 7 Jan 2025

Viewed by 694

Abstract

Topological structural optimization is a powerful computational tool that enhances the structural efficiency of mechanical components. It achieves this by reducing mass without significantly altering stiffness. This study combines the Natural-Neighbour Radial-Point Interpolation Method (NNRPIM) with a bio-inspired bi-evolutionary bone-remodelling algorithm. This combination enables non-linear topological optimization analyses and achieves solutions with optimal stiffness-to-mass ratios. The NNRPIM discretizes the problem using an unstructured nodal distribution. Background integration points are constructed using the Delaunay triangulation concept. Nodal connectivity is then imposed through the natural neighbour concept. To construct shape functions, radial point interpolators are employed, allowing the shape functions to possess the delta Kronecker property. To evaluate the numerical performance of NNRPIM, its solutions are compared with those obtained using the standard Finite Element Method (FEM). The structural optimization process was applied to a practical example: a vehicle’s suspension control arm. This research is divided into two phases. In the first phase, the optimization algorithm is applied to a standard suspension control arm, and the results are closely evaluated. The findings show that NNRPIM produces topologies with suitable truss connections and a higher number of intermediate densities. Both aspects can enhance the mechanical performance of a hypothetical additively manufactured part. In the second phase, four models based on a solution from the optimized topology algorithm are analyzed. These models incorporate established design principles for material removal commonly used in vehicle suspension control arms. Additionally, the same models, along with a solid reference model, undergo linear static analysis under identical loading conditions used in the optimization process. The structural performance of the generated models is analyzed, and the main differences between the solutions obtained with both numerical techniques are identified. Full article

(This article belongs to the Special Issue Numerical and Computational Methods in Structural Engineering and Mechanics)

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23 pages, 10731 KiB

Open AccessArticle

A New Fault Diagnosis Method for Rolling Bearings with the Basis of Swin Transformer and Generalized S Transform

by Jin Yan, Xu Zhu, Xin Wang and Dapeng Zhang

Mathematics 2025, 13(1), 45; https://doi.org/10.3390/math13010045 - 26 Dec 2024

Viewed by 801

Abstract

In view of the rolling bearing fault signal non-stationarity, strong noise can lead to low fault diagnosis accuracy. A Swin Transformer and generalized S Transform fault diagnosis method is proposed to solve the problems of difficult signal feature extraction and low diagnostic accuracy. Generalized S transform is used to improve the resolution of bearing fault signals, the Swin Transformer model is used to master the shallow weight required for identifying rolling bearing faults for highly fault characteristic expression signals, and the deep weight is obtained by backpropagation training. Finally, the extracted features are input into the improved Softmax classifier for fault classification. The various signal processing methods for the bearing signal processing ability are compared, and this model’s diagnosis ability and the ability to resist noise are verified. The experimental results show that the method has a remarkable ability and an accuracy of above 90% in the anti-noise test and also has a good robustness. Full article

(This article belongs to the Special Issue Numerical and Computational Methods in Structural Engineering and Mechanics)

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26 pages, 1793 KiB

Open AccessArticle

Numerical Simulation of Shock Wave in Gas–Water Interaction Based on Nonlinear Shock Wave Velocity Curve

by Zongduo Wu, Dapeng Zhang, Jin Yan, Jianhua Pang and Yifang Sun

Mathematics 2024, 12(20), 3268; https://doi.org/10.3390/math12203268 - 18 Oct 2024

Viewed by 747

Abstract

In a gas–water interaction problem, the nonlinear relationship between shock wave velocity is introduced into a Hugoniot curve, and a Mie–Grüneisen Equation of state (EOS) is established by setting the Hugoiot curve as the reference state. Unlike other simple EOS based on the thermodynamics laws of gas (such as the Tait EOS), the Mie–Grüneisen EOS uses reference states to cover an adiabatic impact relationship and considers the thermodynamics law separately. However, the expression of the EOS becomes complex, and it is not adaptive to many methods. A multicomponent Mie–Grüneisen mixture model is employed in this study to conquer the difficulty of the complex form of an EOS. In this model, some coefficients in the Mie–Grüneisen EOS are regarded as variables and solved using newly constructed equations. The performance of the Mie–Grüneisen mixture model in the gas–water problem is tested by low-compression cases and high-compression cases. According to these two tests, it is found that the numerical solutions of the shock wave under the Mie–Grüneisen EOS agrees with empirical data. When compared to other simple-form EOSs, it is seen that the Mie–Grüneisen EOS has slight advantages in the low-compression case, but it plays an important role in the high-compression case. The comparison results show that the solution of the simple-form EOS clearly disagrees with the empirical data. A further study shows that the gap between the Mie–Grüneisen EOS and other simple-form EOSs becomes larger as the initial pressure and particle velocity increase. The impact effects on the pressure, density and particle velocity are studied. Moreover, the gas–water interaction in a spherical coordinate plane and a two-dimensional coordinate is a significant part of our work. Full article

(This article belongs to the Special Issue Numerical and Computational Methods in Structural Engineering and Mechanics)

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18 pages, 7899 KiB

Open AccessArticle

Research on Auxetic Lattice Structure for Impact Absorption in Machines and Mechanisms

by Levente Széles, Richárd Horváth and Livija Cveticanin

Mathematics 2024, 12(13), 1983; https://doi.org/10.3390/math12131983 - 27 Jun 2024

Cited by 2 | Viewed by 1200

Abstract

In this paper, a new type of filled doubly re-entrant auxetic lattice structure for application in damping and energy absorption devices is considered. The structure is modeled to give protection for machines and mechanisms of intensive impact. The suggested structure is the modified version of the auxetic one with silicone fillings. The unit of the structure is assumed as a re-entrant hexagon with four quadrangular absorbers. For the assumed model of unit, the deformation properties and the Poisson’s ratio were computed. The obtained results were experimentally tested. Specimens of filled and unfilled structures were investigated under quasi-static compression. The measured results show that the energy dissipation is more than two times higher for filled structure than for unfilled ones. In the filled structure, the absorber’s rigidity has the crucial role. If the rigidity is small, the absorber, inside the unit, continues to deform from rectangle into rhomboid. Otherwise, if the rigidity is high, units with absorbers form a beam-like structure that buckles and shows high energy absorption effect. The experimentally obtained results are in good agreement with the theoretical ones. Full article

(This article belongs to the Special Issue Numerical and Computational Methods in Structural Engineering and Mechanics)

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16 pages, 13482 KiB

Open AccessArticle

A p-Refinement Method Based on a Library of Transition Elements for 3D Finite Element Applications

by Adnan Shahriar and Ahmed Jenan Mostafa

Mathematics 2023, 11(24), 4954; https://doi.org/10.3390/math11244954 - 14 Dec 2023

Cited by 1 | Viewed by 1482

Abstract

Wave propagation or acoustic emission waves caused by impact load can be simulated using the finite element (FE) method with a refined high-fidelity mesh near the impact location. This paper presents a method to refine a 3D finite element mesh by increasing the polynomial order near the impact location. Transition elements are required for such a refinement operation. Three protocols are defined to implement the transition elements within the low-order FE mesh. Due to the difficulty of formulating shape functions and verification, there are no transition elements beyond order two in the current literature for 3D elements. This paper develops a complete set of transition elements that facilitate the transition from first- to fourth-order Lagrangian elements, which facilitates mesh refinement following the protocols. The shape functions are computed and verified, and the interelement compatibility conditions are checked for each element case. The integration quadratures and shape function derivative matrices are also computed and made readily available for FE users. Finally, two examples are presented to illustrate the applicability of this method. Full article

(This article belongs to the Special Issue Numerical and Computational Methods in Structural Engineering and Mechanics)

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24 pages, 2887 KiB

Open AccessArticle

Seismo-VLAB: An Open-Source Software for Soil–Structure Interaction Analyses

by Danilo S. Kusanovic, Elnaz E. Seylabi, Peyman Ayoubi, Kien T. Nguyen, Joaquin Garcia-Suarez, Albert R. Kottke and Domniki Asimaki

Mathematics 2023, 11(21), 4530; https://doi.org/10.3390/math11214530 - 3 Nov 2023

Cited by 2 | Viewed by 2934

Abstract

In the fields of structural and geotechnical engineering, improving the understanding of soil–structure interaction (SSI) effects is critical for earthquake-resistant design. Engineers and practitioners often resort to finite element (FE) software to advance this objective. Unfortunately, the availability of software equipped with boundary representation for absorbing scattered waves and ensuring consistent input ground motion prescriptions, which is necessary for accurately representing SSI effects, is currently limited. To address such limitations, the authors developed Seismo-VLAB (SVL v1.0-stable) an open-source software designed to perform SSI simulations. The methodology considers the integration of advanced techniques, including the domain decomposition method (DDM), perfectly matched layers (PMLs), and domain reduction method (DRM), in addition to parallel computing capabilities to accelerate the solution of large-scale problems. In this work, the authors provide a detailed description of the implementation for addressing SSI modeling, validate some of the SVL’s features needed for such purpose, and demonstrate that the coupled DRM–PML technique is a necessary condition for accurately solving SSI problems. It is expected that SVL provides a significant contribution to the SSI research community, offering a self-contained and versatile alternative. The software’s practical application in analyzing SSI and directionality effects on 3D structures under seismic loading demonstrates its capability to model real-world earthquake responses in structural engineering. Full article

(This article belongs to the Special Issue Numerical and Computational Methods in Structural Engineering and Mechanics)

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18 pages, 575 KiB

Open AccessArticle

Free Vibration Analyses of Stiffened Functionally Graded Graphene-Reinforced Composite Multilayer Cylindrical Panel

by Yuhua Zhou, Yanhu Zhang, Brighton Nyasha Chirukam, Jianwei Li, Congshan Lu, Masoud Babaei and Kamran Asemi

Mathematics 2023, 11(17), 3662; https://doi.org/10.3390/math11173662 - 24 Aug 2023

Cited by 6 | Viewed by 1762

Abstract

In this paper, the free vibration response of a stiffened functionally graded graphene nanoplatelet (GPL)-reinforced composite multilayer cylindrical shell panel is studied for the first time. The shell is stiffened by both stringers and rings. Additionally, the effect of reinforcing the shell panel, ring and stinger with GPLs is independently studied. Halpin–Tsai relations are employed to evaluate the mechanical properties of the shell panel, rings and stringers. The first-order shear deformation shell theory, accompanied by the Lekhnitsky smeared stiffener model, using the numerical finite element method and Hamilton principle, is employed to develop the governing motion equations of the shell panel. Four different types of GPL patterns, including FG-A, FG-X, FG-O and UD, are assumed across the thickness of the shell panel, rings and stringers. The effects of different factors, including various weight fractions and patterns of GPLs nanofillers, the geometry of the shell panel and stiffeners and two displacement boundary conditions, on the natural frequencies of the shell panel, have been studied. Full article

(This article belongs to the Special Issue Numerical and Computational Methods in Structural Engineering and Mechanics)

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16 pages, 10872 KiB

Open AccessArticle

Understanding In-Line Connections Behavior from Experimental and Numerical Analyses on Rectangular and Circular Hollow Section Elements

by Calin-Ioan Birdean, Ioan Both, Ioan Mărginean and Anghel Cernescu

Mathematics 2023, 11(15), 3416; https://doi.org/10.3390/math11153416 - 5 Aug 2023

Cited by 1 | Viewed by 1197

Abstract

Depending on the connection type, especially semi-rigid connections, the analyses of building structures offer accurate results function of the rigidity and ductility. The present paper analyzes the in-line connection of rectangular and circular hollow sections, categorized as semi-rigid connections, suitable for an architectural design of invisible joints. For such connection the standards do not cover an explicit design method. Experimental bending tests were performed on rectangular and circular hollow sections having the end plate fixed inside the profile and bolted by four and one high-strength bolts, respectively. The joint separation represents a serviceability criterion which was monitored using digital image correlation technique. Based on experimental results, a numerical model was validated using the finite element method. After the validation of the numerical model based on the experimental results, a parametric investigation was conducted to study the influence of the access hole, the preload level, the end plate thickness, and the axial force. The results show the small influence of the bolt preload, but the end plate thickness was of major importance. A reduction of the assembly rigidity was also caused by the manhole. The study shows the feasibility of the connection configuration with the end plate positioned inside the hollow profile. Full article

(This article belongs to the Special Issue Numerical and Computational Methods in Structural Engineering and Mechanics)

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15 pages, 4685 KiB

Open AccessArticle

A Self-Evolving Neural Network-Based Finite-Time Control Technique for Tracking and Vibration Suppression of a Carbon Nanotube

by Fawaz W. Alsaade, Mohammed S. Al-zahrani, Qijia Yao and Hadi Jahanshahi

Mathematics 2023, 11(7), 1581; https://doi.org/10.3390/math11071581 - 24 Mar 2023

Cited by 2 | Viewed by 1448

Abstract

The control of micro- and nanoscale systems is a vital yet challenging endeavor because of their small size and high sensitivity, which make them susceptible to environmental factors such as temperature and humidity. Despite promising methods proposed for these systems in literature, the chattering in the controller, convergence time, and robustness against a wide range of disturbances still require further attention. To tackle this issue, we present an intelligent observer, which accounts for uncertainties and disturbances, along with a chatter-free controller. First, the dynamics of a carbon nanotube (CNT) are examined, and its governing equations are outlined. Then, the design of the proposed controller is described. The proposed approach incorporates a self-evolving neural network-based methodology and the super-twisting sliding mode technique to eliminate the uncertainties’ destructive effects. Also, the proposed technique ensures finite-time convergence of the system. The controller is then implemented on the CNT and its effectiveness in different conditions is investigated. The numerical simulations demonstrate the proposed method’s outstanding performance in both stabilization and tracking control, even in the presence of uncertain parameters of the system and complicated disturbances. Full article

(This article belongs to the Special Issue Numerical and Computational Methods in Structural Engineering and Mechanics)

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Review

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38 pages, 14063 KiB

Open AccessReview

Review of Experimental, Theoretical and Numerical Advances in Multi-Crack Fracture Mechanics

by Zhao-Wei Wang, D. M. Li, Yi-Fan Zhong, Yi-Kung Liu and Yu-Nong Shao

Mathematics 2024, 12(24), 3881; https://doi.org/10.3390/math12243881 - 10 Dec 2024

Viewed by 1435

Abstract

In engineering fracture problems, cracks tend to interact with each other rather than exist singly. In recent years, the phenomenon of multi-cracking has received attention from both academia and industry. This article firstly emphasizes the importance and research trends of crack interactions. The article then discusses the experimental observation and theoretical modeling of the multi-crack problem and compares the different numerical methods in detail. Finally, this paper offers a comprehensive summary and in-depth analysis of the advancements in multi-crack fracture mechanics, aiming to provide reliable support for solving the multi-crack problem in engineering. Full article

(This article belongs to the Special Issue Numerical and Computational Methods in Structural Engineering and Mechanics)

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