Journal Description
Modelling
Modelling
is an international, peer-reviewed, open access journal on theory and applications of modelling and simulation in engineering science, published quarterly online by MDPI.
- Open Access— free for readers, with article processing charges (APC) paid by authors or their institutions.
- High Visibility: indexed within ESCI (Web of Science), Scopus, Ei Compendex, EBSCO and other databases.
- Rapid Publication: manuscripts are peer-reviewed and a first decision is provided to authors approximately 19.5 days after submission; acceptance to publication is undertaken in 2.9 days (median values for papers published in this journal in the first half of 2025).
- Journal Rank: JCR - Q2 (Engineering, Multidisciplinary) / CiteScore - Q2 (Mathematics (miscellaneous))
- Recognition of Reviewers: APC discount vouchers, optional signed peer review and reviewer names are published annually in the journal.
Impact Factor:
1.5 (2024);
5-Year Impact Factor:
1.5 (2024)
Latest Articles
An Anisotropic Failure Characteristic- and Damage-Coupled Constitutive Model
Modelling 2025, 6(3), 75; https://doi.org/10.3390/modelling6030075 (registering DOI) - 1 Aug 2025
Abstract
►
Show Figures
This study proposes a coupled constitutive model that captures the anisotropic failure characteristics and damage evolution of nickel-based single-crystal (SX) superalloys under various temperature conditions. The model accounts for both creep rate and material damage evolution, enabling accurate prediction of the typical three-stage
[...] Read more.
This study proposes a coupled constitutive model that captures the anisotropic failure characteristics and damage evolution of nickel-based single-crystal (SX) superalloys under various temperature conditions. The model accounts for both creep rate and material damage evolution, enabling accurate prediction of the typical three-stage creep curves, macroscopic fracture morphologies, and microstructural features under uniaxial tensile creep for specimens with different crystallographic orientations. Creep behavior of SX superalloys was simulated under multiple orientations and various temperature-stress conditions using the proposed model. The resulting creep curves aligned well with experimental observations, thereby validating the model’s feasibility and accuracy. Furthermore, a finite element model of cylindrical specimens was established, and simulations of the macroscopic fracture morphology were performed using a user-defined material subroutine. By integrating the rafting theory governed by interfacial energy density, the model successfully predicts the rafting morphology of the microstructure at the fracture surface for different crystallographic orientations. The proposed model maintains low programming complexity and computational cost while effectively predicting the creep life and deformation behavior of anisotropic materials. The model accurately captures the three-stage creep deformation behavior of SX specimens and provides reliable predictions of stress fields and microstructural changes at critical cross-sections. The model demonstrates high accuracy in life prediction, with all predicted results falling within a ±1.5× error band and an average error of 14.6%.
Full article
Open AccessArticle
A Conservative Four-Dimensional Hyperchaotic Model with a Center Manifold and Infinitely Many Equilibria
by
Surma H. Ibrahim, Ali A. Shukur and Rizgar H. Salih
Modelling 2025, 6(3), 74; https://doi.org/10.3390/modelling6030074 (registering DOI) - 29 Jul 2025
Abstract
►▼
Show Figures
This paper presents a novel four-dimensional autonomous conservative model characterized by an infinite set of equilibrium points and an unusual algebraic structure in which all eigenvalues of the Jacobian matrix are zero. The linearization of the proposed model implies that classical stability analysis
[...] Read more.
This paper presents a novel four-dimensional autonomous conservative model characterized by an infinite set of equilibrium points and an unusual algebraic structure in which all eigenvalues of the Jacobian matrix are zero. The linearization of the proposed model implies that classical stability analysis is inadequate, as only the center manifolds are obtained. Consequently, the stability of the system is investigated through both analytical and numerical methods using Lyapunov functions and numerical simulations. The proposed model exhibits rich dynamics, including hyperchaotic behavior, which is characterized using the Lyapunov exponents, bifurcation diagrams, sensitivity analysis, attractor projections, and Poincaré map. Moreover, in this paper, we explore the model with fractional-order derivatives, demonstrating that the fractional dynamics fundamentally change the geometrical structure of the attractors and significantly change the system stability. The Grünwald–Letnikov formulation is used for modeling, while numerical integration is performed using the Caputo operator to capture the memory effects inherent in fractional models. Finally, an analog electronic circuit realization is provided to experimentally validate the theoretical and numerical findings.
Full article

Figure 1
Open AccessArticle
Enhanced Cooling Performance in Cutting Tools Using TPMS-Integrated Toolholders: A CFD-Based Thermal-Fluidic Study
by
Haiyang Ji, Zhanqiang Liu, Jinfu Zhao and Bing Wang
Modelling 2025, 6(3), 73; https://doi.org/10.3390/modelling6030073 - 28 Jul 2025
Abstract
►▼
Show Figures
The efficient thermal management of cutting tools is critical for ensuring dimensional accuracy, surface integrity, and tool longevity, especially in the high-speed dry machining process. However, conventional cooling methods often fall short in reaching the heat-intensive zones near the cutting inserts. This study
[...] Read more.
The efficient thermal management of cutting tools is critical for ensuring dimensional accuracy, surface integrity, and tool longevity, especially in the high-speed dry machining process. However, conventional cooling methods often fall short in reaching the heat-intensive zones near the cutting inserts. This study proposes a novel internal cooling strategy that integrates triply periodic minimal surface (TPMS) structures into the toolholder, aiming to enhance localized heat removal from the cutting region. The thermal-fluidic behaviors of four TPMS topologies (Gyroid, Diamond, I-WP, and Fischer–Koch S) were systematically analyzed under varying coolant velocities using computational fluid dynamics (CFD). Several key performance indicators, including the convective heat transfer coefficient, Nusselt number, friction factor, and thermal resistance, were evaluated. The Diamond and Gyroid structures exhibited the most favorable balance between heat transfer enhancement and pressure loss. The experimental validation confirmed the CFD prediction accuracy. The results establish a new design paradigm for integrating TPMS structures into toolholders, offering a promising solution for efficient, compact, and sustainable cooling in advanced cutting applications.
Full article

Figure 1
Open AccessArticle
EMB System Design and Clamping Force Tracking Control Research
by
Junyi Zou, Haojun Yan, Yunbing Yan and Xianping Huang
Modelling 2025, 6(3), 72; https://doi.org/10.3390/modelling6030072 - 25 Jul 2025
Abstract
►▼
Show Figures
The electromechanical braking (EMB) system is an important component of intelligent vehicles and is also the core actuator for longitudinal dynamic control in autonomous driving motion control. Therefore, we propose a new mechanism layout form for EMB and a feedforward second-order linear active
[...] Read more.
The electromechanical braking (EMB) system is an important component of intelligent vehicles and is also the core actuator for longitudinal dynamic control in autonomous driving motion control. Therefore, we propose a new mechanism layout form for EMB and a feedforward second-order linear active disturbance rejection controller based on clamping force. This solves the problem of excessive axial distance in traditional EMB and reduces the axial distance by 30%, while concentrating the PCB control board for the wheels on the EMB housing. This enables the ABS and ESP functions to be integrated into the EMB system, further enhancing the integration of line control and active safety functions. A feedforward second-order linear active disturbance rejection controller (LADRC) based on the clamping force of the brake caliper is proposed. Compared with the traditional clamping force control methods three-loop PID and adaptive fuzzy PID, it improves the response speed, steady-state error, and anti-interference ability. Moreover, the LADRC has more advantages in parameter adjustment. Simulation results show that the response speed is increased by 130 ms, the overshoot is reduced by 9.85%, and the anti-interference ability is increased by 41.2%. Finally, the feasibility of this control algorithm was verified through the EMB hardware-in-the-loop test bench.
Full article

Figure 1
Open AccessArticle
SpatioConvGRU-Net for Short-Term Traffic Crash Frequency Prediction in Bogotá: A Macroscopic Spatiotemporal Deep Learning Approach with Urban Factors
by
Alejandro Sandoval-Pineda and Cesar Pedraza
Modelling 2025, 6(3), 71; https://doi.org/10.3390/modelling6030071 - 25 Jul 2025
Abstract
Traffic crashes represent a major challenge for road safety, public health, and mobility management in complex urban environments, particularly in metropolitan areas characterized by intense traffic flows, high population density, and strong commuter dynamics. The development of short-term traffic crash prediction models represents
[...] Read more.
Traffic crashes represent a major challenge for road safety, public health, and mobility management in complex urban environments, particularly in metropolitan areas characterized by intense traffic flows, high population density, and strong commuter dynamics. The development of short-term traffic crash prediction models represents a fundamental line of analysis in road safety research within the scientific community. Among these efforts, macro-level modeling plays a key role by enabling the analysis of the spatiotemporal relationships between diverse factors at an aggregated zonal scale. However, in cities like Bogotá, predicting short-term traffic crashes remains challenging due to the complexity of these spatiotemporal dynamics, underscoring the need for models that more effectively integrate spatial and temporal data. This paper presents a strategy based on deep learning techniques to predict short-term spatiotemporal traffic crashes in Bogotá using 2019 data on socioeconomic, land use, mobility, weather, lighting, and crash records across TMAU and TAZ zones. The results showed that the strategy performed with a model called SpatioConvGru-Net with top performance at the TMAU level, achieving = 0.983, MSE = 0.017, and MAPE = 5.5%. Its hybrid design captured spatiotemporal patterns better than CNN, LSTM, and others. Performance improved at the TAZ level using transfer learning.
Full article
(This article belongs to the Special Issue Advanced Modelling Techniques in Transportation Engineering)
►▼
Show Figures

Figure 1
Open AccessArticle
Failure Mode Discrimination and Stochastic Behavior Study of RC Beams Under Impact Loads
by
Taochun Yang, Yating Jiang, Xiaoyan Zhang, Qinghai Liu and Yin Wang
Modelling 2025, 6(3), 70; https://doi.org/10.3390/modelling6030070 - 22 Jul 2025
Abstract
To clarify the potential failure modes of reinforced concrete (RC) beams under impact and understand their impact resistance safety, a comprehensive study was conducted by focusing on the failure mode discrimination and failure probability of RC beams under impact loads. This research utilized
[...] Read more.
To clarify the potential failure modes of reinforced concrete (RC) beams under impact and understand their impact resistance safety, a comprehensive study was conducted by focusing on the failure mode discrimination and failure probability of RC beams under impact loads. This research utilized drop hammer impact tests, ABAQUS2022 software, and theoretical methods. The study examined three typical failure modes of RC beams under impact loads: flexural failure, flexural-shear failure, and shear failure. A discrimination criterion based on the flexural-shear capacity–effect curve was developed. Utilizing this criterion, along with the basic principles of structural reliability theory, the failure probability of RC beams under impact loads was calculated and analyzed using the Monte Carlo method. The results indicate that the criterion based on the flexural-shear capacity–effect curve can be used for discriminating failure modes of RC beams under impact loads. The impact velocity and stirrup ratio were identified as crucial factors that influenced the failure modes of RC beams under impact. Specifically, an increase in the stirrup spacing reduced the reliability of the RC beams under impact, while an increase in the stirrup ratio could significantly enhance their impact resistance. Furthermore, with a constant impact energy, an increase in beam span correlated with the improved reliability of RC beams under impact, where larger spans yielded a better impact resistance.
Full article
(This article belongs to the Special Issue Finite Element Simulation and Analysis)
►▼
Show Figures

Graphical abstract
Open AccessArticle
Quantum-Enhanced Attention Neural Networks for PM2.5 Concentration Prediction
by
Tichen Huang, Yuyan Jiang, Rumeijiang Gan and Fuyu Wang
Modelling 2025, 6(3), 69; https://doi.org/10.3390/modelling6030069 - 21 Jul 2025
Abstract
►▼
Show Figures
As industrialization and economic growth accelerate, PM2.5 pollution has become a critical environmental concern. Predicting PM2.5 concentration is challenging due to its nonlinear and complex temporal dynamics, limiting the accuracy and robustness of traditional machine learning models. To enhance prediction accuracy,
[...] Read more.
As industrialization and economic growth accelerate, PM2.5 pollution has become a critical environmental concern. Predicting PM2.5 concentration is challenging due to its nonlinear and complex temporal dynamics, limiting the accuracy and robustness of traditional machine learning models. To enhance prediction accuracy, this study focuses on Ma’anshan City, China and proposes a novel hybrid model (QMEWOA-QCAM-BiTCN-BiLSTM) based on an “optimization first, prediction later” approach. Feature selection using Pearson correlation and RFECV reduces model complexity, while the Whale Optimization Algorithm (WOA) optimizes model parameters. To address the local optima and premature convergence issues of WOA, we introduce a quantum-enhanced multi-strategy improved WOA (QMEWOA) for global optimization. A Quantum Causal Attention Mechanism (QCAM) is incorporated, leveraging Quantum State Mapping (QSM) for higher-order feature extraction. The experimental results show that our model achieves a MedAE of 1.997, MAE of 3.173, MAPE of 10.56%, and RMSE of 5.218, outperforming comparison models. Furthermore, generalization experiments confirm its superior performance across diverse datasets, demonstrating its robustness and effectiveness in PM2.5 concentration prediction.
Full article

Graphical abstract
Open AccessArticle
Non-Fourier Thermoelastic Peridynamic Modeling of Cracked Thin Films Under Short-Pulse Laser Irradiation
by
Tao Wu, Tao Xue, Yazhou Wang and Kumar Tamma
Modelling 2025, 6(3), 68; https://doi.org/10.3390/modelling6030068 - 15 Jul 2025
Abstract
In this paper, we develop a peridynamic computational framework to analyze thermomechanical interactions in fractured thin films subjected to ultrashort-pulsed laser excitation, employing nonlocal discrete material point discretization to eliminate mesh dependency artifacts. The generalized Cattaneo–Fourier thermal flux formulation uncovers contrasting dynamic responses:
[...] Read more.
In this paper, we develop a peridynamic computational framework to analyze thermomechanical interactions in fractured thin films subjected to ultrashort-pulsed laser excitation, employing nonlocal discrete material point discretization to eliminate mesh dependency artifacts. The generalized Cattaneo–Fourier thermal flux formulation uncovers contrasting dynamic responses: hyperbolic heat propagation ( ) generates intensified temperature localization and elevates transient crack-tip stress concentrations relative to classical Fourier diffusion ( ). A GSSSS (Generalized Single Step Single Solve) i-Integration temporal scheme achieves oscillation-free numerical solutions across picosecond-level laser–matter interactions, effectively resolving steep thermal fronts through adaptive stabilization. These findings underscore hyperbolic conduction’s essential influence on stress-mediated fracture evolution during ultrafast laser processing, providing critical guidelines for thermal management in micro-/nano-electromechanical systems.
Full article
(This article belongs to the Special Issue The 5th Anniversary of Modelling)
►▼
Show Figures

Figure 1
Open AccessArticle
Digital Twin-Driven SimLean-TRIZ Framework in Cold Room Door Production
by
Thenarasu M, Sumesh Arangot, Narassima M S, Olivia McDermott and Arjun Panicker
Modelling 2025, 6(3), 67; https://doi.org/10.3390/modelling6030067 - 14 Jul 2025
Abstract
►▼
Show Figures
The study aims to increase productivity in the cold room door manufacturing industry by addressing non-value-adding operations, identifying bottlenecks, and reducing processing time through digital twin (DT)-based simulation. The goal is to eliminate the need for supply chain outsourcing and increase overall efficiency.
[...] Read more.
The study aims to increase productivity in the cold room door manufacturing industry by addressing non-value-adding operations, identifying bottlenecks, and reducing processing time through digital twin (DT)-based simulation. The goal is to eliminate the need for supply chain outsourcing and increase overall efficiency. The research involves developing a DT of the existing production process for five distinct categories of cold room doors: flush door, single door, double door, face-mounted door, and sliding door. Simulation was used to uncover problems at multiple stations, encompassing curing, welding, and packing. Lean principles were used to identify the causes of inefficiency, and the process was improved using TRIZ principles. These changes produced a 42.90% improvement in productivity, a 20% dependence reduction on outsourcing and an increase of 10.5% added inventory to the shortage demand level. The approach presented is provided for a particular manufacturer of cold room doors, but the methods and techniques used are generally applicable to other manufacturing companies to support systematic innovation. Combining DT simulation, lean techniques and TRIZ principles, this study presents a strong approach to addressing the productivity challenges in manufacturing. The incorporation of these methods has brought considerable operational efficiency and has minimised dependency on external outsourcing.
Full article

Figure 1
Open AccessArticle
Direct Numerical Simulation of the Differentially Heated Cavity and Comparison with the κ-ε Model for High Rayleigh Numbers
by
Fernando Iván Molina-Herrera and Hugo Jiménez-Islas
Modelling 2025, 6(3), 66; https://doi.org/10.3390/modelling6030066 - 11 Jul 2025
Abstract
►▼
Show Figures
This study presents a numerical comparison between Direct numerical simulation (DNS) and the standard κ-ε turbulence model to evaluate natural convection in a two-dimensional, differentially heated, air-filled cavity over the Rayleigh number range 103 to 1010. The objective is to
[...] Read more.
This study presents a numerical comparison between Direct numerical simulation (DNS) and the standard κ-ε turbulence model to evaluate natural convection in a two-dimensional, differentially heated, air-filled cavity over the Rayleigh number range 103 to 1010. The objective is to assess the predictive capabilities of both methods across laminar and turbulent regimes, with a particular emphasis on the quantitative comparison of thermal characteristics under high Rayleigh number conditions. The Navier–Stokes and energy equations were solved using the finite element method with Boussinesq approximation, employing refined meshes near the hot and cold walls to resolve thermal and velocity boundary layers. The results indicate that for Ra ≤ 106, the κ-ε model significantly underestimates temperature gradients, maximum velocities, and average Nusselt numbers, with errors up to 19.39%, due to isotropic assumptions and empirical formulation. DNS, in contrast, achieves global energy balance errors of less than 0.0018% across the entire range. As Ra increases, the κ-ε model predictions converge to DNS, with Nusselt number deviations dropping below 1.2% at Ra = 1010. Streamlines, temperature profiles, and velocity distributions confirm that DNS captures flow dynamics more accurately, particularly near the wall vortices. These findings validate DNS as a reference solution for high-Ra natural convection and establish benchmark data for assessing turbulence models in confined geometries
Full article

Graphical abstract
Open AccessArticle
Numerical Simulation of Impermeability of Composite Geomembrane in Rigid Landfills
by
Ming Huang, Teng Tu, Yueling Jing and Fan Yang
Modelling 2025, 6(3), 65; https://doi.org/10.3390/modelling6030065 - 10 Jul 2025
Abstract
To investigate the impermeability characteristics of composite geomembranes in rigid landfills, a three-dimensional finite element seepage analysis model, which incorporates a composite geomembrane, was established based on a case study of a rigid landfill project in Tongling. Utilizing the seepage mechanism of the
[...] Read more.
To investigate the impermeability characteristics of composite geomembranes in rigid landfills, a three-dimensional finite element seepage analysis model, which incorporates a composite geomembrane, was established based on a case study of a rigid landfill project in Tongling. Utilizing the seepage mechanism of the composite geomembrane, the seepage distribution patterns of the hazardous waste leachate within the unit cell were computed under representative operating conditions. Different thickness amplification factor schemes for the equivalent treatment of the composite geomembrane were comparatively analyzed, considering both isotropic and anisotropic seepage conditions. The relationships between the seepage flow rate, velocity, and thickness amplification factor were determined. The results showed that the leachate experiences a rapid drop in the water head as it passes through the composite geomembrane, with a low seepage flow rate and velocity, highlighting the membrane’s significant impermeability effect. The finite element analysis indicated that thickness amplification of the composite geomembrane based on the flow equivalence is feasible to some degree, but treating the geomembrane as an anisotropic material during the equivalent process better approximates the actual conditions.
Full article
(This article belongs to the Special Issue Finite Element Simulation and Analysis)
►▼
Show Figures

Graphical abstract
Open AccessArticle
Modification of Airfoil Thickness and Maximum Camber by Inverse Design for Operation Under Icing Conditions
by
Ibrahim Kipngeno Rotich and László E. Kollár
Modelling 2025, 6(3), 64; https://doi.org/10.3390/modelling6030064 - 8 Jul 2025
Abstract
Wind turbine performance in cold regions is affected by icing which can lead to power reduction due to the aerodynamic degradation of the turbine blade. The development of airfoil shapes applied as blade sections contributes to improving the aerodynamic performance under a wide
[...] Read more.
Wind turbine performance in cold regions is affected by icing which can lead to power reduction due to the aerodynamic degradation of the turbine blade. The development of airfoil shapes applied as blade sections contributes to improving the aerodynamic performance under a wide range of weather conditions. The present study considers inverse design coupled with numerical modelling to simulate the effects of varying airfoil thickness and maximum camber. The inverse design process was implemented in MATLAB R2023a, whereas the numerical models were constructed using ANSYS Fluent and FENSAP ICE 2023 R1. The inverse design process applied the modified Garabedian–McFadden (MGM) iterative technique. Shear velocities were calculated from the flow over an airfoil with slip conditions, and then this velocity distribution was modified according to the prevailing icing conditions to obtain the target velocities. A parameter was proposed to consider the airfoil thickness as well when calculating the target velocities. The airfoil generated was then exposed to various atmospheric conditions to check the improvement in the aerodynamic performance. The ice mass and lift-to-drag ratio were determined considering cloud characteristics under varying liquid water content (LWC) from mild to severe (0.1 g/m3 to 1 g/m3), median volume diameter (MVD) of 50 µm, and two ambient temperatures (−4 °C and −20 °C) that characterize freezing drizzle and in-cloud icing conditions. The ice mass on the blade section was not significantly impacted by modifying the shape after applying the process developed (i.e., <5%). However, the lift-to-drag ratio that describes the aerodynamic performance may even be doubled in the icing scenarios considered.
Full article
(This article belongs to the Section Modelling in Engineering Structures)
►▼
Show Figures

Figure 1
Open AccessArticle
Coupled Study on the Building Load Dynamics and Thermal Response of Ground Sources in Shallow Geothermal Heat Pump Systems Under Severe Cold Climate Conditions
by
Jianlin Li, Xupeng Qi, Xiaoli Li, Huijie Huang and Jian Gao
Modelling 2025, 6(3), 63; https://doi.org/10.3390/modelling6030063 - 7 Jul 2025
Abstract
►▼
Show Figures
To address thermal imbalance and ground temperature degradation in shallow geothermal heat pump (GSHP) systems in severely cold climates, this study analyzes a typical logistics building using an hourly dynamic load model. Multiyear simulations were conducted to investigate the coupling between building load
[...] Read more.
To address thermal imbalance and ground temperature degradation in shallow geothermal heat pump (GSHP) systems in severely cold climates, this study analyzes a typical logistics building using an hourly dynamic load model. Multiyear simulations were conducted to investigate the coupling between building load variation and soil thermal response. The results indicate that with a cumulative heating load of 14.681 million kWh and cooling load of 6.3948 million kWh, annual heat extraction significantly exceeds heat rejection, causing ground temperature to decline by about 1 °C per year. Over five and ten years, the cumulative drops reached 2.65 °C and 4.71 °C, respectively, leading to a noticeable reduction in borehole heat exchanger performance and system COP. The study quantitatively evaluates ground temperature and heat exchange degradation, highlighting the key role of load imbalance. To mitigate long-term thermal deterioration, strategies such as load optimization, summer heat reinjection, and operational adjustments are proposed. The findings offer guidance for the design and sustainable operation of GSHP systems in cold regions.
Full article

Figure 1
Open AccessArticle
Application of the Ant Colony Optimization Metaheuristic in Transport Engineering: A Case Study on Vehicle Routing and Highway Service Stations
by
Luiz Vicente Figueira de Mello Filho, Felipe Pastori Lopes de Sousa, Gustavo de Godoi, William Machado Emiliano, Felippe Benavente Canteras, Vitor Eduardo Molina Júnior, João Roberto Bertini Junior and Yuri Alexandre Meyer
Modelling 2025, 6(3), 62; https://doi.org/10.3390/modelling6030062 - 3 Jul 2025
Abstract
►▼
Show Figures
Efficient logistics and transport infrastructure are critical in contemporary urban and interurban scenarios due to their impact on economic development, environmental sustainability, and quality of life. This study explores the use of the Ant Colony Optimization (ACO) metaheuristic applied to the Vehicle Routing
[...] Read more.
Efficient logistics and transport infrastructure are critical in contemporary urban and interurban scenarios due to their impact on economic development, environmental sustainability, and quality of life. This study explores the use of the Ant Colony Optimization (ACO) metaheuristic applied to the Vehicle Routing Problem (VRP) and the strategic positioning of service stations along major highways. Through a systematic mapping of the literature and practical application to a real-world scenario—specifically, a case study on the Bandeirantes Highway (SP348), connecting Limeira to São Paulo, Brazil—the effectiveness of ACO is demonstrated in addressing complex logistical challenges, including capacity constraints, route optimization, and resource allocation. The proposed method integrates graph theory principles, entropy concepts from information theory, and economic analyses into a unified computational model implemented using Python (version 3.12), showcasing its accessibility for educational and practical business contexts. The results highlight significant improvements in operational efficiency, cost reductions, and optimized service station placement, emphasizing the algorithm’s robustness and versatility. Ultimately, this research provides valuable insights for policymakers, engineers, and logistics managers seeking sustainable and cost-effective solutions in transport infrastructure planning and management.
Full article

Figure 1
Open AccessArticle
Reliability Analysis of Interface Oxidation for Thermal Barrier Coating Based on Proxy Model
by
Juan Ma, Anyi Wang, Philipp Junker, Anas W. Alshawawreh, Qingya Li, Haoqi Xu and Runzhuo Xue
Modelling 2025, 6(3), 61; https://doi.org/10.3390/modelling6030061 - 3 Jul 2025
Abstract
Thermal barrier coatings have been widely used in industrial fields where thermal damage occurs, and they are crucial for insulation technology and for the safe service of high-temperature components. So, it is critical to accurately predict the reliability of thermal barrier coatings. In
[...] Read more.
Thermal barrier coatings have been widely used in industrial fields where thermal damage occurs, and they are crucial for insulation technology and for the safe service of high-temperature components. So, it is critical to accurately predict the reliability of thermal barrier coatings. In this work, an adaptive reliability analysis method based on radial basis functions is proposed, in which different shape parameters and subsets are used to initiate different radial basis function models for multiple predictions. An active learning function that comprehensively considers local uncertainty, limit state function information, and distance among samples is then used for sequential sampling, and the proposed method is validated via a four-branch series connection system. Finally, a reliability analysis is conducted on the failure of interface oxidation in thermal barrier coatings, which verifies the feasibility of the proposed method.
Full article
(This article belongs to the Special Issue The 5th Anniversary of Modelling)
►▼
Show Figures

Graphical abstract
Open AccessArticle
Quasi-LPV Approach for the Stabilization of an Innovative Quadrotor
by
Said Chaabani and Naoufel Azouz
Modelling 2025, 6(3), 60; https://doi.org/10.3390/modelling6030060 - 1 Jul 2025
Abstract
►▼
Show Figures
In recent decades, the deployment of quadcopters has significantly expanded, particularly in outdoor applications such as parcel delivery. These missions require highly stable aerial platforms capable of maintaining balance under diverse environmental conditions, ensuring the safe operation of both the drone and its
[...] Read more.
In recent decades, the deployment of quadcopters has significantly expanded, particularly in outdoor applications such as parcel delivery. These missions require highly stable aerial platforms capable of maintaining balance under diverse environmental conditions, ensuring the safe operation of both the drone and its payload. This paper focuses on the stabilization of a quadcopter designed for outdoor use. A detailed dynamic model of a compact vertical takeoff and landing (VTOL) drone forms the basis for a non-linear control strategy targeting stability during the critical takeoff phase. The control law is designed using a quasi-linear parameter-varying (quasi-LPV) model that captures the system’s non-linear dynamics. Lyapunov theory and linear matrix inequalities (LMIs) are employed to validate the stability and design the controller. Numerical simulations demonstrate the controller’s effectiveness, and a comparative study is conducted to benchmark its performance against a reference quadrotor model.
Full article

Figure 1
Open AccessArticle
Thermo-Hydro-Mechanical–Chemical Modeling for Pressure Solution of Underground sCO2 Storage
by
Selçuk Erol
Modelling 2025, 6(3), 59; https://doi.org/10.3390/modelling6030059 - 1 Jul 2025
Cited by 1
Abstract
►▼
Show Figures
Underground production and injection operations result in mechanical compaction and mineral chemical reactions that alter porosity and permeability. These changes impact the flow and, eventually, the long-term sustainability of reservoirs utilized for CO2 sequestration and geothermal energy. Even though mechanical and chemical
[...] Read more.
Underground production and injection operations result in mechanical compaction and mineral chemical reactions that alter porosity and permeability. These changes impact the flow and, eventually, the long-term sustainability of reservoirs utilized for CO2 sequestration and geothermal energy. Even though mechanical and chemical deformations in rocks take place at the pore scale, it is important to investigate their impact at the continuum scale. Rock deformation can be examined using intergranular pressure solution (IPS) models, primarily for uniaxial compaction. Because the reaction rate parameters are estimated using empirical methods and the assumption of constant mineral saturation indices, these models frequently overestimate the rates of compaction and strain by several orders of magnitude. This study presents a new THMC algorithm by combining thermo-mechanical computation with a fractal approach and hydrochemical computations using PHREEQC to evaluate the pressure solution. Thermal stress and strain under axisymmetric conditions are calculated analytically by combining a derived hollow circle mechanical structure with a thermal resistance model. Based on the pore scale, porosity and its impact on the overall excessive stress and strain rate in a domain are estimated by applying the fractal scaling law. Relevant datasets from CO2 core flooding experiments are used to validate the proposed approach. The comparison is consistent with experimental findings, and the novel analytical method allows for faster inspection compared to numerical simulations.
Full article

Figure 1
Open AccessArticle
A Modified Nonlocal Macro–Micro-Scale Damage Model for the Simulation of Hydraulic Fracturing
by
Changgen Liu and Xiaozhou Xia
Modelling 2025, 6(3), 58; https://doi.org/10.3390/modelling6030058 - 26 Jun 2025
Abstract
►▼
Show Figures
The nonlocal macro–meso-scale damage (NMMD) model, implemented in the framework of the finite element method, has been demonstrated to be a promising numerical approach in simulating crack initiation and propagation with reliable efficacy and high accuracy. In this study, the NMMD model was
[...] Read more.
The nonlocal macro–meso-scale damage (NMMD) model, implemented in the framework of the finite element method, has been demonstrated to be a promising numerical approach in simulating crack initiation and propagation with reliable efficacy and high accuracy. In this study, the NMMD model was further enhanced by employing an identical degradation mechanism for both the tensile and shear components of shear stiffness, thereby overcoming the limitation of equal degradation in shear and tensile stiffness inherent in the original model. Additionally, a more refined and physically sound seepage evolution function was introduced to characterize the variation in permeability in porous media with geometric damage, leading to the development of an improved NMMD model suitable for simulating coupled seepage–stress problems. The reliability of the enhanced NMMD model was verified by the semi-analytical solutions of the classical KGD problem. Finally, based on the modified NMMD model, the effects of preset fracture spacing and natural voids on hydraulic fracture propagation were investigated.
Full article

Figure 1
Open AccessArticle
Numerical Simulation and Performance Analysis of DesanderDuring Tight Gas Provisional Process
by
Gang Sun, Hua Li, Hongcheng Liu, Fuchun Li, Huanhuan Wang, Jun Zhou and Guangchuan Liang
Modelling 2025, 6(3), 57; https://doi.org/10.3390/modelling6030057 - 26 Jun 2025
Abstract
►▼
Show Figures
Tight gas wells in Southwest oil and gas fields have significant production and high sand output intensity. The sand out of the wellhead has a certain erosion effect on the downstream pipeline, the equipment, and affects the normal production. This paper models and
[...] Read more.
Tight gas wells in Southwest oil and gas fields have significant production and high sand output intensity. The sand out of the wellhead has a certain erosion effect on the downstream pipeline, the equipment, and affects the normal production. This paper models and simulates the desander used at the wellhead according to the real parameters of the tight gas wellhead, and explores the effects of gas production, pressure, temperature, sand particle size, water content, and other factors on the desander’s sand removal efficiency. This paper combines the principle of fluid dynamics to analyze the internal mechanism of the effect trend and according to the simulation results uses the Pearson correlation coefficient quantification of the effect of each operating parameter to explore the optimal boundary condition parameters applicable to the desander. From the simulation results, it can be seen that the separation efficiency of the desander is the highest when the gas production rate is 4 × 104 m3/d, the pressure is 7 MPa, and the lower the working temperature is, the larger is the gravel particle size. Combined with the sand management problems occurring in the field of tight gas wells, suggestions are made for the optimization of the operating parameters and structure of the desander, which will provide a basis for supporting the rapid production and large-scale beneficial development of tight gas fields.
Full article

Figure 1
Open AccessArticle
Minimizing Waste and Costs in Multi-Level Manufacturing: A Novel Integrated Lot Sizing and Cutting Stock Model Using Multiple Machines
by
Nesma Khamis, Nermine Harraz and Hadi Fors
Modelling 2025, 6(3), 56; https://doi.org/10.3390/modelling6030056 - 26 Jun 2025
Abstract
►▼
Show Figures
Lot sizing and cutting stock problems are critical for manufacturing companies seeking to optimize resource utilization and minimize waste. This paper addresses the interconnected nature of these problems, often occurring sequentially in industries involving cut items or packaging. We propose a novel mixed
[...] Read more.
Lot sizing and cutting stock problems are critical for manufacturing companies seeking to optimize resource utilization and minimize waste. This paper addresses the interconnected nature of these problems, often occurring sequentially in industries involving cut items or packaging. We propose a novel mixed integer linear programming (MILP) model that integrates the capacitated lot sizing problem with the one-dimensional cutting stock problem within a multi-level manufacturing framework. The cutting stock problem is addressed using an arc flow formulation. Our model aims to minimize setup, production, holding, and waste material costs while incorporating capacity constraints, setup requirements, inventory balance, and the use of various cutting machines. The effectiveness of our model is demonstrated through numerical experiments using a commercial optimization package. While the model efficiently generates optimal solutions for most scenarios, larger instances pose challenges within the specified time limits. Sensitivity analysis is conducted to evaluate the effect of changing essential parameters of the integrated problem on model performance and to provide managerial insights for real-life applications.
Full article

Graphical abstract
Highly Accessed Articles
Latest Books
E-Mail Alert
News
Topics
Topic in
Energies, Sensors, Electronics, Modelling, Electricity
EMC and Reliability of Power Networks
Topic Editors: Antonella Ragusa, Alistair DuffyDeadline: 31 October 2025
Topic in
Applied Sciences, Electronics, Energies, Modelling, Systems
Agents and Multi-Agent Systems
Topic Editors: He Cai, Maobin LvDeadline: 30 November 2025
Topic in
Applied Sciences, Energies, Fluids, Modelling, Processes
Oil and Gas Pipeline Network for Industrial Applications
Topic Editors: Enbin Liu, Hongfang Lu, Weibiao Qiao, Shanbi PengDeadline: 31 December 2025
Topic in
AI, Algorithms, Modelling, Applied Sciences
Advancements in Coupling Simulation and Artificial Intelligence: Methods, Applications, and Future Directions
Topic Editors: Greg Zacharewicz, Nicolas Daclin, Vittorio Solina, Diana Maria Torres RicaurteDeadline: 30 June 2026

Conferences
Special Issues
Special Issue in
Modelling
Feature Papers of Computational Modelling and Simulation for Fatigue and Fracture of Engineering Materials and Structures
Guest Editors: Abílio M. P. De Jesus, Reza AbediDeadline: 30 September 2025
Special Issue in
Modelling
Advanced Modelling Techniques in Transportation Engineering
Guest Editors: Konstantinos Gkyrtis, Andreas Nikiforiadis, George N. Botzoris, Alexandros KokkalisDeadline: 31 October 2025
Special Issue in
Modelling
The 5th Anniversary of Modelling
Guest Editors: Alfredo Cuzzocrea, Wei GaoDeadline: 31 December 2025
Special Issue in
Modelling
Advances in Discrete Event Simulation
Guest Editors: Sadok Rezig, Nadia Ndhaief, Greg ZacharewiczDeadline: 31 December 2025