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Advanced Numerical and Computational Methods for Engineering and Applied Mathematical...
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Advanced Numerical and Computational Methods for Engineering and Applied Mathematical Problems, 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: 31 July 2025 | Viewed by 2671

Special Issue Editors

Prof. Dr. Lihua Wang

E-Mail Website
Guest Editor
School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai 200092, China
Interests: computational mechanics; numerical methods; meshfree methods; structural dynamics; fluid-structure interaction
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Benny Yiu-Chung Hon

E-Mail Website
Guest Editor
Department of Mathematics, City University of Hong Kong, Hong Kong, China
Interests: meshfree methods; inverse problems; computational mechanics
Special Issues, Collections and Topics in MDPI journals
Dr. Sheng-Wei Chi

E-Mail Website
Guest Editor
Department of Civil, Materials, and Environmental Engineering, University of Illinois at Chicago, Chicago, IL 60607, USA
Interests: meshfree methods; collocation methods; generalized finite element methods; image-based computational methods; fragment and impact simulations
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

The development of meshfree/particle methods and advanced numerical methods has created new avenues for further research and spearheaded pioneering efforts in the fields of engineering and scientific computation. The ability of these methods to provide high-fidelity solutions for sophisticated engineering problems has also been recognized by many researchers and engineers as a driving force to push forward the boundary of modern industry.

This Special Issue, as a follow-up to the successful first edition “Advanced Numerical and Computational Methods for Engineering and Applied Mathematical Problems” aims to publish state-of-the-art fundamental development for these advanced numerical methods, prospective research directions, and applications for engineering and applied mathematical problems.

Contributions are solicited in all subjects related to meshfree and other advanced numerical methods and their numerical applications, which include, but are not limited to, the following topics:

  1. Meshfree methods;
  2. Particle methods;
  3. Machine learning methods;
  4. Smoothed particle hydrodynamics;
  5. Peridynamics;
  6. Material point method;
  7. Strong-form collocation meshfree methods;
  8. Stabilized collocation methods;
  9. Generalized finite difference methods;
  10. Method of fundamental solutions (MFS);
  11. Boundary element methods;
  12. Integration-based meshfree methods;
  13. Localized radial basis functions methods;
  14. Characterization and stabilization of numerical instabilities;
  15. Other advanced numerical methods;
  16. Applications of meshfree methods, machine learning methods and other numerical methods for advanced materials and structures, soft materials, inverse problems, fluid dynamics and fluid-structure interaction, geomechanics, large deformation and non-linear problems, multi-phase interactions, contact and impact, static and dynamic structural responses, manufacturing processes, nano-mechanics, etc.

Prof. Dr. Lihua Wang
Prof. Dr. Benny Yiu-Chung Hon
Dr. Sheng-Wei Chi
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

  • numerical methods
  • computational methods
  • machine learning
  • differential equations
  • inverse problems
  • engineering problems
  • computational mechanics
  • mathematical problems
  • coupled problems
  • numerical simulations
  • meshfree methods
  • finite element methods
  • radial basis functions methods
  • multiscale material modeling
  • machine learning methods
  • structural dynamics
  • material modelling
  • computational mechanics
  • fluid-structure interaction

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Further information on MDPI's Special Issue policies can be found here.

Related Special Issue

Published Papers (4 papers)

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Research

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24 pages, 1392 KiB  
Article
Multi-Dimensional Analytic Functions for Laplace Equations and Generalized Cauchy–Riemann Equations
by Chein-Shan Liu, Zhuojia Fu and Chung-Lun Kuo
Mathematics 2025, 13(8), 1246; https://doi.org/10.3390/math13081246 - 10 Apr 2025
Viewed by 115
Abstract
A new concept of projective solution is introduced for the multi-dimensional Laplace equations. We project the field point onto a characteristic vector to obtain a projective variable, which can be used to reduce the Laplace equations to a second-order ordinary differential equation with [...] Read more.
A new concept of projective solution is introduced for the multi-dimensional Laplace equations. We project the field point onto a characteristic vector to obtain a projective variable, which can be used to reduce the Laplace equations to a second-order ordinary differential equation with only a leading term multiplied by the squared norm of the characteristic vector. The projective solutions involve characteristic vectors as parameters, which must be complex numbers to satisfy a null equation. Since the projective variable is a complex variable, we can construct the analytic function based on the conventional complex analytic function theory. Both the analytic function and the Cauchy–Riemann equations are generalized for the multi-dimensional Laplace equations. A powerful numerical technique to solve the 3D Laplace equation with high accuracy is available by further developing the Trefftz-type bases. Numerical experiments confirm the accuracy and efficiency of the projective solutions method (PSM). Full article
(This article belongs to the Special Issue Advanced Numerical and Computational Methods for Engineering and Applied Mathematical Problems, 2nd Edition)
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31 pages, 993 KiB  
Article
Integral Representation for Three-Dimensional Steady-State Couple-Stress Size-Dependent Thermoelasticity
by Ali R. Hadjesfandiari, Arezoo Hajesfandiari and Gary F. Dargush
Mathematics 2025, 13(4), 638; https://doi.org/10.3390/math13040638 - 15 Feb 2025
Viewed by 360
Abstract
Boundary element methods provide powerful techniques for the analysis of problems involving coupled multi-physical response. This paper presents the integral equation formulation for the size-dependent thermoelastic response of solids under steady-state conditions in three dimensions. The formulation is based upon consistent couple stress [...] Read more.
Boundary element methods provide powerful techniques for the analysis of problems involving coupled multi-physical response. This paper presents the integral equation formulation for the size-dependent thermoelastic response of solids under steady-state conditions in three dimensions. The formulation is based upon consistent couple stress theory, which features a skew-symmetric couple-stress pseudo-tensor. For general anisotropic thermoelastic material, there is not only thermal strain deformation, but also thermal mean curvature deformation. Interestingly, in this size-dependent multi-physics model, the thermal governing equation is independent of the deformation. However, the mechanical governing equations depend on the temperature field. First, thermal and mechanical weak forms and reciprocal theorems are developed for this theory. Then, an integral equation formulation for three-dimensional size-dependent thermoelastic isotropic materials is derived, along with the corresponding singular infinite-space fundamental solutions or kernel functions. For isotropic materials, there is no thermal mean curvature deformation, and the thermoelastic effect is solely the result of thermal strain deformation. As a result, the size-dependent behavior is specified entirely by a single characteristic length scale parameter l, while the thermal coupling is defined in terms of the thermal expansion coefficient α, as in the classical theory of steady-state isotropic thermoelasticity. Full article
(This article belongs to the Special Issue Advanced Numerical and Computational Methods for Engineering and Applied Mathematical Problems, 2nd Edition)
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38 pages, 9211 KiB  
Article
Transfinite Patches for Isogeometric Analysis
by Christopher Provatidis
Mathematics 2025, 13(3), 335; https://doi.org/10.3390/math13030335 - 21 Jan 2025
Viewed by 555
Abstract
This paper extends the well-known transfinite interpolation formula, which was developed in the late 1960s by the applied mathematician William Gordon at the premises of General Motors as an extension of the pre-existing Coons interpolation formula. Here, a conjecture is formulated, which claims [...] Read more.
This paper extends the well-known transfinite interpolation formula, which was developed in the late 1960s by the applied mathematician William Gordon at the premises of General Motors as an extension of the pre-existing Coons interpolation formula. Here, a conjecture is formulated, which claims that the meaning of the involved blending functions can be enhanced, such that it includes any linear independent and complete set of functions, including piecewise-linear, trigonometric functions, Bernstein polynomials, B-splines, and NURBS, among others. In this sense, NURBS-based isogeometric analysis and aspects of T-splines may be considered as special cases. Applications are provided to illustrate the accuracy in the interpolation through the L2 error norm of closed-formed functions prescribed at the nodal points of the transfinite patch, which represent the solution of partial differential equations under boundary conditions of the Dirichlet type. Full article
(This article belongs to the Special Issue Advanced Numerical and Computational Methods for Engineering and Applied Mathematical Problems, 2nd Edition)
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Review

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66 pages, 8492 KiB  
Review
An Overview of Underwater Optical Wireless Communication Channel Simulations with a Focus on the Monte Carlo Method
by Intesar Ramley, Hamdah M. Alzayed, Yas Al-Hadeethi, Mingguang Chen and Abeer Z. Barasheed
Mathematics 2024, 12(24), 3904; https://doi.org/10.3390/math12243904 - 11 Dec 2024
Cited by 1 | Viewed by 1192
Abstract
Building a reliable and optimum underwater optical wireless communication (UOWC) system requires identifying all potential factors that cause the attenuation and dispersion of the optical signal. The radiative transfer equation (RTE) solution can be utilised to conclude these essential design parameters to build [...] Read more.
Building a reliable and optimum underwater optical wireless communication (UOWC) system requires identifying all potential factors that cause the attenuation and dispersion of the optical signal. The radiative transfer equation (RTE) solution can be utilised to conclude these essential design parameters to build an optimum UOWC system. RTE has various numerical and simplified analytical solutions with varying reliability and capability scope. Many scientists consider the Monte Carlo simulation (MCS) method to be a consistent and widely accepted approach to formulating an RTE solution, which models the propagation of photons through various underwater channel environments. MCS recently attracted attention because we can build a reliable model for underwater environments. Based on such a model, this report demonstrates the resulting received optical power distribution as an output for an array of emulation inputs, including transmitted light’s spatial and temporal distribution, channel link regimes, and associated impairments. This study includes a survey component, which presents the required framework’s foundation to establish a valid RTE model, which leads to solutions with different scopes and depths that can be drawn for practical UOWC use cases. Hence, this work shows how underlying modelling elements can influence a solution technique, including inherent optical properties (IOPs), apparent optical properties (AOPs), and the potential limitations of various photon scattering function formats. The work introduces a novel derivation of mathematical equations for single- and multiple-light-pulse propagation in homogeneous and inhomogeneous channels, forming the basis for MCS-based UOWC studies. The reliability of MCS implementation is assessed using compliance with the Central Limit Theorem (CLT) and leveraging the Henyey–Greenstein phase function with full-scale random selection. As part of the tutorial component in this work, the MCS inner working is manifested using an object-oriented design method. Therefore, this work targets researchers interested in using MCS for UOWC research in general and UOWC photon propagation in seawater channel modelling in general. Full article
(This article belongs to the Special Issue Advanced Numerical and Computational Methods for Engineering and Applied Mathematical Problems, 2nd Edition)
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