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Mathematics
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Multiscale Modeling in Engineering and Mechanics, 2nd Edition
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Multiscale Modeling in Engineering and Mechanics, 2nd Edition

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

Deadline for manuscript submissions: 20 April 2026 | Viewed by 847

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 geotechniques and underground space; heterogeneous geomaterials; 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 geotechniques and underground space; rock mechanics; constitutive model; damage mechanics
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Prof. Dr. Mingfeng Lei

E-Mail Website
Guest Editor
School of Civil Engineering, Central South University, Changsha 410075, China
Interests: multiscale modeling in geotechniques and underground space; railway engineering; durability of tunnel structure
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Dr. Chengyong Cao

E-Mail Website
Guest Editor
Underground Polis Academy, Shenzhen University, Shenzhen 518060, China
Interests: data-driven 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. Multiphysics and multiscale modeling of engineering;
  4. Computational morphology analysis and characterization of multiscale 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, biomedicine and applied mathematics. We hope that 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 250 words) can be sent to the Editorial Office for assessment.

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 modeling
  • computational methods
  • data-driven approaches
  • artificial intelligence
  • micro–macro mechanics
  • mechanical modeling
  • biomechanics
  • bioengineering

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Related Special Issue

Published Papers (2 papers)

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Research

29 pages, 5537 KB  
Article
A Multi-Scale Approach for the Piezoelectric Modal Analysis in Periodically Perforated Structures
by Mengyu Zhang, Shuyu Ye and Qiang Ma
Mathematics 2025, 13(24), 3967; https://doi.org/10.3390/math13243967 - 12 Dec 2025
Viewed by 229
Abstract
Piezoelectric composites have found a wide range of applications in smart structures and devices and effective numerical methods should be developed to simulate not only the macroscopic coupled piezoelectric performances, but also the details of the local distributions of the stress and electric [...] Read more.
Piezoelectric composites have found a wide range of applications in smart structures and devices and effective numerical methods should be developed to simulate not only the macroscopic coupled piezoelectric performances, but also the details of the local distributions of the stress and electric field. In this paper, we proposed a multi-scale asymptotic algorithm based on the Second-Order Two-Scale (SOTS) analysis method for the piezoelectric eigenvalue problem in perforated domain with periodic micro-configurations. The eigenfunctions and eigenvalues are expanded to the second-order terms and the homogenized eigensolutions; the expressions of the first- and second-order correctors are derived successively. The first- and second-order correctors of the eigenvalues are determined according to the integration forms of the correctors of the corresponding eigenfunctions. Explicit expressions of the homogenized material coefficients are derived for the laminated structures and the finite element procedures are proposed to compute the homogenized solutions and the correctors numerically. The error estimations for the approximations of eigenvalues are proved under some regularity assumptions and a typical numerical experiment is carried out for the two-dimensional perforated domain. The computed results show that the SOTS analysis method is efficient in identifying the piezoelectric eigenvalues accurately and reproducing the original eigenfunctions effectively. This approach also provides an efficient computational tool for piezoelectric eigenvalue analysis and can extend to other multi-physics problems with complex microstructures. Full article
(This article belongs to the Special Issue Multiscale Modeling in Engineering and Mechanics, 2nd Edition)
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19 pages, 2515 KB  
Article
Failure Mode of Rectangular Shallow Tunnel in Unsaturated Fine-Grained Soil Under Temperature Influence
by Wenjun Shao, Hong Liao and De Zhou
Mathematics 2025, 13(22), 3671; https://doi.org/10.3390/math13223671 - 16 Nov 2025
Viewed by 278
Abstract
Failure analysis has always been among the key research focuses in underground tunneling, particularly in forecasting the collapse risk of tunnel crowns, which bears great engineering and practical significance for tunnel safety assessment. In practical engineering, the soil surrounding shallow tunnels and other [...] Read more.
Failure analysis has always been among the key research focuses in underground tunneling, particularly in forecasting the collapse risk of tunnel crowns, which bears great engineering and practical significance for tunnel safety assessment. In practical engineering, the soil surrounding shallow tunnels and other underground chambers is typically unsaturated. With the advancement of tunneling technology, shallow tunnels affected by ground temperatures are increasingly common, making it essential to incorporate temperature effects into the stability analysis of unsaturated shallow tunnels. This paper proposes a novel framework for analyzing the stability of shallow rectangular tunnel crowns under temperature influence. By adopting a temperature-dependent effective stress model for unsaturated soils combined with the soil–water characteristic curve, temperature influence is integrated into the calculation of apparent cohesion in unsaturated soils. The upper bound theorem and a multi-rigid-block failure mechanism are adopted to assess crown stability, with the geometry of the failure mechanism determined through a compatible velocity field. New analytical expressions are derived. Through calculating the internal energy dissipation rate, considering temperature effects and external work rate, the critical support pressure at the tunnel crown is obtained using the Sequential Quadratic Programming (SQP). Discussions of temperature and other unsaturated soil parameters are carried out to explore their effects on the stability of shallow tunnels. Results demonstrate that temperature significantly influences the tunnel’s critical support pressure, with the extent of this impact primarily dependent on the unsaturated soil type and seepage conditions. Furthermore, the theoretical framework developed in this study provides a more accurate description for unsaturated fine-grained soils. This study introduces a novel integration of thermal influences into the upper bound theorem, applying this enhanced methodology to the stability assessment of shallow rectangular tunnel crowns. The resulting failure model and analytical framework establish a rigorous upper bound solution for crown stability, thereby furnishing a more accurate theoretical foundation for subsequent tunnel face support strategies. Full article
(This article belongs to the Special Issue Multiscale Modeling in Engineering and Mechanics, 2nd Edition)
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