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Multiscale Modeling in Engineering and Mechanics: From Microscopic to Macroscopic...
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Multiscale Modeling in Engineering and Mechanics: From Microscopic to Macroscopic Mechanical Modeling

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

Deadline for manuscript submissions: 31 May 2025 | Viewed by 11344

Special Issue Editors

Dr. Yuexiang Lin

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Guest Editor
1. School of Aeronautics and Astronautics, Sun Yat-sen University, Shenzhen 518107, China
2. Department of Civil and Environmental Engineering, The Hong Kong Polytechnic University, Hong Kong, China
Interests: multiscale modeling in geotechnique and underground space; heterogeneous geo-materials; finite-discrete element method
Special Issues, Collections and Topics in MDPI journals
Dr. Jianjun Ma

E-Mail Website
Guest Editor
School of Civil Engineering, Sun Yat-Sen University, Guangzhou 510275, China
Interests: multiscale modeling in geotechnique and underground space; rock mechanics; constitutive model; damage mechanics
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Mingfeng Lei

E-Mail Website
Guest Editor
School of Civil Engineering, Central South University, Changsha 410075, China
Interests: multiscale modeling in geotechnique and underground space; railway engineering; durability of tunnel strucutre
Special Issues, Collections and Topics in MDPI journals
Dr. Chengyong Cao

E-Mail Website
Guest Editor
Underground Polis Academy (UPA), Shenzhen University, Shenzhen 518060, China
Interests: data driving analysis; sensing technique; tunneling; deep excavation; numerical modeling
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Multiscale modeling has become an important tool in the field of engineering, allowing for a more accurate understanding and prediction of the behavior of materials and structures at different scales. This Special Issue aims to bring together the latest research on multiscale modeling in engineering, with a focus on both microscopic and macroscopic mechanical modeling. The Special Issue will cover a range of topics including, but not limited to, computational methods and techniques for multiscale modeling, mechanical modeling of materials and structures at different scales, morphology analysis of heterogeneous materials, and applications of multiscale modeling in engineering problems. In particular, we encourage submissions that highlight the deep mathematical connections between the microscopic and macroscopic scales. We hope this Special Issue will provide a platform for researchers to exchange ideas and knowledge, and to promote interdisciplinary collaborations to address complex engineering problems, which may contribute to the advancement of multiscale modeling in engineering and mechanics and inspire further research in this exciting field. We welcome original research papers, reviews, and case studies on topics including, but not limited to, the following:

  1. Computational methods and techniques for multiscale modeling;
  2. Microscopic and macroscopic mechanical modeling and mathematical connections;
  3. Multi-physics and multi-scale modeling of engineering;
  4. Computational morphology analysis and characterization of multi-scale materials and structures including defects and cracks;
  5. Data-driven approaches for multiscale modeling and mechanical property predictions;
  6. Uncertainty and randomness quantification and sensitivity analysis in multiscale modeling;
  7. Development of novel experimental techniques to probe the behavior of multiscale systems, such as in situ imaging and sensing methods;
  8. Future directions in the development and application of experimental and multiscale modeling methods, including the use of advanced imaging and sensing techniques, machine learning, and artificial intelligence.

We welcome contributions from researchers in various fields, such as mechanical engineering, materials science, physics, and applied mathematics. We hope this Special Issue will provide a valuable platform for researchers to share their findings and insights, and contribute to the advancement of multiscale modeling in engineering and mechanics.

Dr. Yuexiang Lin
Dr. Jianjun Ma
Prof. Dr. Mingfeng Lei
Dr. Chengyong Cao
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

  • multiscale modeling
  • mathematical connections
  • computational methods
  • data-driven approaches
  • artificial intelligence

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Published Papers (10 papers)

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Research

15 pages, 3471 KiB  
Article
Failure Mode of Tunnel Face Under Transient Unsaturated Seepage with Temperature Influence
by Yi Xie, De Zhou, Hong Liao and Jianqun Zhu
Mathematics 2025, 13(8), 1311; https://doi.org/10.3390/math13081311 - 16 Apr 2025
Viewed by 147
Abstract
The seepage caused by heavy rainfall and storm runoff is not a static phenomenon. On the contrary, it is a dynamic process known as unsaturated transient seepage. Under the condition, the spatiotemporal variations in suction stress cannot be overlooked. With the development of [...] Read more.
The seepage caused by heavy rainfall and storm runoff is not a static phenomenon. On the contrary, it is a dynamic process known as unsaturated transient seepage. Under the condition, the spatiotemporal variations in suction stress cannot be overlooked. With the development of tunnel mechanics, there has been an emergence of tunnels affected by high ground temperatures or temperature influences, highlighting the necessity of incorporating temperature effects into the analysis. This article proposes a new framework for the spatiotemporal response of tunnel face safety to temperature-affected and unsaturated transient seepage conditions. A one-dimensional transient seepage assumption is used to develop an analytical model describing unsaturated transient seepage, which is then integrated centered on suction stress strength theory for unsaturated soils to acquire suction stress variations with depth and time. The temperature impact on the unsaturated soil shear strength is incorporated, applying a temperature-sensitive effective stress model in conjunction with the soil–water characteristic curve to quantitatively analyze temperature-induced apparent cohesion changes. The 3D logarithmic spiral failure model is used to analyze the tunnel face stability. The validity of the proposed failure model is demonstrated through an engineering calculation. The rates of internal dissipation and external work are calculated, and a kinematic approach related to strength reduction is used to determine the safety factor of the tunnel face with zero support pressure. The results show that considering transient unsaturated seepage and temperature effects can increase the safety factor. The influence of these effects mainly depends on the soil type, tunnel geometric parameters, and seepage conditions. This work explores the influence of variations in a series of parameters on the failure mode of tunnel faces under temperature effects, taking into account unsaturated transient seepage, thereby providing valuable references for the design and construction of tunnels in the future. Full article
(This article belongs to the Special Issue Multiscale Modeling in Engineering and Mechanics: From Microscopic to Macroscopic Mechanical Modeling)
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20 pages, 6970 KiB  
Article
Analysis and Prediction of Grouting Reinforcement Performance of Broken Rock Considering Joint Morphology Characteristics
by Guanglin Liang, Linchong Huang and Chengyong Cao
Mathematics 2025, 13(2), 264; https://doi.org/10.3390/math13020264 - 15 Jan 2025
Cited by 1 | Viewed by 666
Abstract
In tunnel engineering, joint shear slip caused by external disturbances is a key factor contributing to landslides, instability of surrounding rock masses, and related hazards. Therefore, accurately characterizing the macromechanical properties of joints is essential for ensuring engineering safety. Given the significant influence [...] Read more.
In tunnel engineering, joint shear slip caused by external disturbances is a key factor contributing to landslides, instability of surrounding rock masses, and related hazards. Therefore, accurately characterizing the macromechanical properties of joints is essential for ensuring engineering safety. Given the significant influence of rock joint morphology on mechanical behavior, this study employs the frequency spectrum fractal dimension (D) and the frequency domain amplitude integral (Rq) as quantitative descriptors of joint morphology. Using Fourier transform techniques, a reconstruction method is developed to model joints with arbitrary shape characteristics. The numerical model is calibrated through 3D printing and direct shear tests. Systematic parameter analysis validates the selected quantitative indices as effective descriptors of joint morphology. Furthermore, multiple machine learning algorithms are employed to construct a robust predictive model. Machine learning, recognized as a rapidly advancing field, plays a pivotal role in data-driven engineering applications due to its powerful analytical capabilities. In this study, six algorithms—Random Forest (RF), Support Vector Regression (SVR), BP Neural Network, GA-BP Neural Network, Genetic Programming (GP), and ANN-based MCD—are evaluated using 300 samples. The performance of each algorithm is assessed through comparative analysis of their predictive accuracy based on correlation coefficients. The results demonstrate that all six algorithms achieve satisfactory predictive performance. Notably, the Random Forest (RF) algorithm excels in rapid and accurate predictions when handling similar training data, while the ANN-based MCD algorithm consistently delivers stable and precise results across diverse datasets. Full article
(This article belongs to the Special Issue Multiscale Modeling in Engineering and Mechanics: From Microscopic to Macroscopic Mechanical Modeling)
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14 pages, 3354 KiB  
Article
Failure Mode of Deep-Buried Rectangular Chamber and Upper Bound Solution of Surrounding Rock Pressure
by Daobing Zhang, Linhai Zeng, Zhilin Lv, Xiaochuan Yu, Chang Liu, Anming Jiang, Xianyong Jiang, Qi Li and Yongxiang Yang
Mathematics 2025, 13(1), 69; https://doi.org/10.3390/math13010069 - 28 Dec 2024
Cited by 1 | Viewed by 592
Abstract
In order to obtain more reasonable failure modes and more precise surrounding rock pressures of deep-buried underground rectangular chambers, the failure mode of a “wedge-shaped collapse body + arc rotator + logarithmic spiral rotator” deep-buried chamber is constructed based on the analysis of [...] Read more.
In order to obtain more reasonable failure modes and more precise surrounding rock pressures of deep-buried underground rectangular chambers, the failure mode of a “wedge-shaped collapse body + arc rotator + logarithmic spiral rotator” deep-buried chamber is constructed based on the analysis of existing failure mode. The upper bound solution of the surrounding rock pressure is derived based on limit analysis. The validity of the proposed failure model and the reliability of the limit analysis approach are demonstrated through numerical simulations, theoretical verification, and comparisons with engineering practices. The influence of various parameters on surrounding rock pressure is analyzed. The results show that each parameter has different influence on the surrounding rock pressure. The surrounding rock pressure q and e increases linearly with the increase in rock gravity γ and chamber size, and decreases with the increase in cohesion c and internal friction angle φ. The surrounding rock pressure q decreases with the increase in lateral pressure coefficient K, while surrounding rock pressure e increases with the increase in lateral pressure coefficient K. Full article
(This article belongs to the Special Issue Multiscale Modeling in Engineering and Mechanics: From Microscopic to Macroscopic Mechanical Modeling)
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25 pages, 5988 KiB  
Article
Reliability Analysis of Deep Foundation Pit Using the Gaussian Copula-Based Bayesian Network
by Bin Tan and Qiyuan Peng
Mathematics 2024, 12(24), 3961; https://doi.org/10.3390/math12243961 - 17 Dec 2024
Cited by 1 | Viewed by 820
Abstract
Urban underground space development has heightened concerns over the safety of deep foundation pit construction. This study conducted time-series monitoring of critical safety-influencing factors and applied the Gaussian copula-based Bayesian network (GCBN) model for comprehensive reliability analysis of deep foundation pit support structures. [...] Read more.
Urban underground space development has heightened concerns over the safety of deep foundation pit construction. This study conducted time-series monitoring of critical safety-influencing factors and applied the Gaussian copula-based Bayesian network (GCBN) model for comprehensive reliability analysis of deep foundation pit support structures. The GCBN model, integrating the multivariate data management of pair copula with Bayesian network’s uncertainty handling, found that building settlement has the greatest impact on the safety of deep foundation pit and revealed a reliability index (β) of 0.44 in an actual case, suggesting a hazardous condition. Based on the reliability index β, emergency measures were promptly taken. Compared to traditional reliability methods, the approach presented in this paper takes into account the dependence among monitoring indicators, which is more aligned with actual engineering conditions and holds significant reference value for the safety assessment of underground engineering structures. Full article
(This article belongs to the Special Issue Multiscale Modeling in Engineering and Mechanics: From Microscopic to Macroscopic Mechanical Modeling)
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14 pages, 868 KiB  
Article
Generalized Multiscale Finite Element Method and Balanced Truncation for Parameter-Dependent Parabolic Problems
by Shan Jiang, Yue Cheng, Yao Cheng and Yunqing Huang
Mathematics 2023, 11(24), 4965; https://doi.org/10.3390/math11244965 - 15 Dec 2023
Cited by 2 | Viewed by 1188
Abstract
We propose a generalized multiscale finite element method combined with a balanced truncation to solve a parameter-dependent parabolic problem. As an updated version of the standard multiscale method, the generalized multiscale method contains the necessary eigenvalue computation, in which the enriched multiscale basis [...] Read more.
We propose a generalized multiscale finite element method combined with a balanced truncation to solve a parameter-dependent parabolic problem. As an updated version of the standard multiscale method, the generalized multiscale method contains the necessary eigenvalue computation, in which the enriched multiscale basis functions are picked up from a snapshot space on users’ demand. Based upon the generalized multiscale simulation on the coarse scale, the balanced truncation is applied to solve its Lyapunov equations on the reduced scale for further savings while ensuring high accuracy. A θ-implicit scheme is utilized for the fully discretization process. Finally, numerical results validate the uniform stability and robustness of our proposed method. Full article
(This article belongs to the Special Issue Multiscale Modeling in Engineering and Mechanics: From Microscopic to Macroscopic Mechanical Modeling)
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22 pages, 7158 KiB  
Article
Study on the Stiffness and Dynamic Characteristics of a Bridge Approach Zone: Tests and Numerical Analyses
by Ping Hu, Wei Liu, Huo Liu, Leixue Wu, Yang Wang and Wei Guo
Mathematics 2023, 11(19), 4202; https://doi.org/10.3390/math11194202 - 8 Oct 2023
Viewed by 1220
Abstract
This study focuses on the stiffness and dynamic characteristic rules of a bridge approach zone in a high-speed railway (HSR). Indoor and in situ tests were performed to explore the stiffness and dynamic characteristics of the roadbed filling. Based on the test results, [...] Read more.
This study focuses on the stiffness and dynamic characteristic rules of a bridge approach zone in a high-speed railway (HSR). Indoor and in situ tests were performed to explore the stiffness and dynamic characteristics of the roadbed filling. Based on the test results, an effective track-subgrade finite element model (FEM) of a high-speed train (HST) was established. The FEM simulated the train load and model boundaries based on the obtained loads and viscoelastic artificial boundaries. Suitable elements were then selected to simulate the various components of the system and the constraint equations were established and solved using multi-point constraints. The model was verified by comparing the time–history curve characteristics, the frequency-domain characteristics and the results obtained from different modeling methods with the measured results. The influence of stiffness on the dynamic characteristics of the bridge approach zone were subsequently analyzed based on the aforementioned tests and simulations. The results indicate that (i) the model produced reliable results using the proposed approach; (ii) the influence of train load on the embankment was generally reflected in the upper part of the structure, and thus, bed structures are recommended to be strengthened; and (iii) under stationarity, the stiffness ratio between the bridge and normal subgrade is recommended as 1:6, with a transition length of 25 m. Full article
(This article belongs to the Special Issue Multiscale Modeling in Engineering and Mechanics: From Microscopic to Macroscopic Mechanical Modeling)
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18 pages, 4883 KiB  
Article
Simulation Analysis of Dynamic Damage Probability Modelling for Laser Systems
by Jiaowei Shi, Shiyan Sun, Jun Xie and Chaobing Zheng
Mathematics 2023, 11(19), 4097; https://doi.org/10.3390/math11194097 - 27 Sep 2023
Viewed by 1645
Abstract
The paper proposes a method that analyses the dynamic damage probability of laser systems to address the shortcomings of the quantitative model for the damage probability of laser systems. Firstly, the far-field energy density distribution model is constructed according to the power spectrum [...] Read more.
The paper proposes a method that analyses the dynamic damage probability of laser systems to address the shortcomings of the quantitative model for the damage probability of laser systems. Firstly, the far-field energy density distribution model is constructed according to the power spectrum inversion method. Then, the instantaneous on-target spot power density distribution is equivalently portrayed based on the combination of the far-field power density and the missile target characteristics. Next, the instantaneous on-target spot is combined with the tracking and aiming error to obtain the probability distribution of the energy density of the long-period on-target spot. Finally, the temperature probability distribution is obtained by analyzing the relation between the target energy density and the temperature of the inner wall of the warhead. Consequently, the damage probability was calculated. The simulation shows that there is a unique maximum damage probability when the target is flying in a straight line and the laser system strikes the missile sideways. The method can provide support for the shooting timing of high-energy laser systems. Full article
(This article belongs to the Special Issue Multiscale Modeling in Engineering and Mechanics: From Microscopic to Macroscopic Mechanical Modeling)
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19 pages, 6985 KiB  
Article
Study on the Influence of an Under-Crossing Parallel Double-Line Shield Tunnel on the Existing Tunnel Structure
by Linhai Zeng, Daobing Zhang, Changjiang Lian, Jiahua Zhang and Huadong Yin
Mathematics 2023, 11(14), 3125; https://doi.org/10.3390/math11143125 - 14 Jul 2023
Cited by 8 | Viewed by 1628
Abstract
Extra care should be taken when new tunnels pass through an existing tunnel. If it is not handled properly, this will affect the operation safety of the existing line, and bring security risks to the train’s operation. In order to study the impact [...] Read more.
Extra care should be taken when new tunnels pass through an existing tunnel. If it is not handled properly, this will affect the operation safety of the existing line, and bring security risks to the train’s operation. In order to study the impact of an under-crossing parallel double-line shield tunnel on the existing shield tunnel structure, the influence of tunnel construction on the deformation of overlying strata was analyzed, and the formula for estimating the formation settlement at depth Z below the surface, caused by the excavation of a double-tunnel parallel tunnel, was deduced. Then, a series of three-dimensional finite element numerical simulations were carried out. We analyzed and systematically studied the adverse effects of the tunnel structure of Guangzhou subway Line 5, caused by the tunneling of subway Line 18, evaluated its structure and operational safety, and provided suggestions for site construction. This research demonstrates the following conclusions. (1) The tunnel structures of subway Line 5 and Line 18 are mainly in the strongly weathered argillaceous siltstone stratum, and lightly weathered argillaceous siltstone stratum, respectively, and the stratum where the Line 18 tunnel is located is relatively safe. (2) According to three-dimensional finite element numerical simulation analysis, during the shield-tunneling process of subway Line 18, the maximum X horizontal displacement, the maximum Y horizontal displacement, and the maximum Z vertical displacement of the tunnel structure in subway Line 5 are 1.09, 3.50, and 4.55 mm, respectively. It is considered that the impact of the shield-tunnel penetration of subway Line 18 on the tunnel structure of subway Line 5 is relatively controllable, and does not affect the structure and operational safety. (3) It is suggested that settlement monitoring should be strengthened within the range of 12 m (about 1.5 tunnel diameter D). before and after the excavation axis of the underpass tunnel, and it is necessary to carry out local reinforcement treatment, to prevent adverse effects on the operation of the existing tunnel. Full article
(This article belongs to the Special Issue Multiscale Modeling in Engineering and Mechanics: From Microscopic to Macroscopic Mechanical Modeling)
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15 pages, 3573 KiB  
Article
Numerical Simulation of Mechanical Properties of Soil Considering the Effect of Internal Erosion
by Mao-Wen Li, Sheng-Liang Hu and Chen-Xi Tong
Mathematics 2023, 11(13), 2959; https://doi.org/10.3390/math11132959 - 3 Jul 2023
Viewed by 1311
Abstract
The loss of fine particles from the skeleton formed by coarse particles due to seepage action significantly affects the grading, void ratio, and mechanical properties of soil. This results in several issues of engineering hazards. In order to analyze the effect of internal [...] Read more.
The loss of fine particles from the skeleton formed by coarse particles due to seepage action significantly affects the grading, void ratio, and mechanical properties of soil. This results in several issues of engineering hazards. In order to analyze the effect of internal erosion on the mechanical properties of gap-graded soils from macro and micro perspectives, triaxial consolidation and drainage shear tests were simulated in this paper using the particle flow discrete element software PFC3D. A linear contact model was employed to simulate internal erosion by randomly removing fine particles. The results showed that the void ratio of the specimens increased with the erosion degree. The variation in void ratios of the specimens with the erosion degree before loading was greater than those after loading. The peak deviatoric stresses of the specimens decreased with the increase of the erosion degrees. The larger the erosion degree, the more the maximum volumetric strain and the resistance capacity to deformation was also reduced. The average particle coordination number (Z) of the specimens generally tended to decrease as the erosion degree increased. When the average effective stress was not large, the critical state line gradually increased with the erosion degree, while the void ratio was also found to correlate with the erosion degree under the critical state of the specimens with zero average effective stress. Full article
(This article belongs to the Special Issue Multiscale Modeling in Engineering and Mechanics: From Microscopic to Macroscopic Mechanical Modeling)
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14 pages, 15880 KiB  
Article
Hydro-Mechanical Coupling of Cement-Based Slurry Grouting in Saturated Geomaterials
by Haitao Wang, Lei Kou and Hongkang Zhu
Mathematics 2023, 11(13), 2877; https://doi.org/10.3390/math11132877 - 27 Jun 2023
Viewed by 1144
Abstract
A mathematical model is proposed to simulate the fully hydro-mechanical coupling of two-phase cement-based slurry migration in saturated deformable geomaterials from microscopic to macroscopic scale. The mass conservation equations and the momentum balance equations for cement-based slurry and geomaterials are derived based on [...] Read more.
A mathematical model is proposed to simulate the fully hydro-mechanical coupling of two-phase cement-based slurry migration in saturated deformable geomaterials from microscopic to macroscopic scale. The mass conservation equations and the momentum balance equations for cement-based slurry and geomaterials are derived based on the thermodynamically constrained averaging theory (TCAT). The Galerkin discretization of the governing equations of hydro-mechanical coupling are developed by the isogeometric analysis (IGA) integrated with the Bézier extraction operator, and the numerical calculation is implemented with the generalized backward Euler method. The presented modeling is verified by comparison of the numerical calculation with the experimental tests, and the pore fluid pressure of the stratum and the slurry concentration of cement-based slurry migration in saturated deformable geomaterials are discussed. The modeling presented provides an effective alternative method to simulate cement-based slurry migration and explore isothermal multiphase coupled problems. Full article
(This article belongs to the Special Issue Multiscale Modeling in Engineering and Mechanics: From Microscopic to Macroscopic Mechanical Modeling)
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