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A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "E: Applied Mathematics".

Deadline for manuscript submissions: closed (30 April 2025) | Viewed by 13592

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Dr. Fajie Wang

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Guest Editor

College of Mechanical and Electrical Engineering, Qingdao University, Qingdao 266071, China
Interests: computational mechanics; numerical analysis; boundary element method; meshless method; acoustic propagation; heat and mass transfer
Special Issues, Collections and Topics in MDPI journals

Prof. Dr. Ji Lin

E-Mail Website
Guest Editor

College of Mechanics and Materials, Hohai University, Nanjing 211100, China
Interests: solid mechanics; computational mechanics; meshless method; wave propagation
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

We are delighted to announce the second volume of our Special Issue on "Computational Methods and Applications for Numerical Analysis". Building upon the success of the first volume, we aim to continue exploring the innovative computational methods and their diverse applications in numerical analysis

The scope encompasses a wide range of areas, including theory, algorithms, programming, coding, numerical simulation, and novel applications of computational techniques in engineering, science, and related disciplines. Contributions may explore various computational methods within applied mathematics and mechanics, including (but not limited to) finite element methods, finite difference methods, finite volume methods, meshless and particle methods, peridynamics, molecular dynamics, interpolation, approximation, optimization, quadrature methods, numerical linear algebra, and numerical methods for ordinary and partial differential equations. The goal is to showcase research that addresses real-world challenges and demonstrates the effectiveness of computational approaches for solving problems across scientific and engineering domains.

Researchers are encouraged to submit papers that include cutting-edge computational methods, novel algorithms, and successful applications in various fields, which will help advance the field of numerical analysis and its impact on scientific research and technological advancements.

Potential topics include (but are not limited to):

Computational methods.
Numerical analysis.
Finite element methods.
Finite difference methods.
Finite volume methods.
Meshless and particle methods.
Neural network algorithm.
High-performance computing techniques.
Optimization.
Interpolation.
Approximation.
Adaptive analysis.
Error estimation.
Convergence analysis.

We look forward to receiving your contributions.

Dr. Fajie Wang
Prof. Dr. Ji Lin
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.

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Related Special Issue

Computational Methods and Applications for Numerical Analysis in Mathematics (20 articles)

Published Papers (13 papers)

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Research

36 pages, 30116 KiB

Open AccessArticle

A 2.5D Generalized Finite Difference Method for Elastic Wave Propagation Problems

by Hao Chang, Fajie Wang, Xingxing Yue, Lin Qiu and Linlin Sun

Mathematics 2025, 13(8), 1249; https://doi.org/10.3390/math13081249 - 10 Apr 2025

Viewed by 239

Abstract

The analysis of elastic wave propagation is a critical problem in both science and engineering, with applications in structural health monitoring and seismic wave analysis. However, the efficient and accurate numerical simulation of large-scale three-dimensional structures has posed significant challenges to traditional methods, which often struggle with high computational costs and limitations. This paper presents a novel two-and-a-half-dimensional generalized finite difference method (2.5D GFDM) for efficient simulation of elastic wave propagation in longitudinally invariant structures. The proposed scheme integrates GFDM with 2.5D technology, reducing 3D problems to a series of 2D problems in the wavenumber domain via Fourier transforms. Subsequently, the solutions to the original 3D problems can be recovered by performing inverse Fourier transforms on the solutions obtained from the 2D problems. The 2.5D GFDM avoids the inherent challenge of mesh generation in traditional methods like FEM and FVM, offering a meshless solution for complex 3D problems. By employing sparse coefficient matrices, it offers significantly improved computational efficiency. The new approach achieves significant computational advantages while maintaining high accuracy, as validated through three representative examples, making it a promising tool for solving large-scale elastic wave propagation problems in longitudinally invariant structures. Full article

(This article belongs to the Special Issue Computational Methods and Applications for Numerical Analysis, 2nd Edition)

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21 pages, 8407 KiB

Open AccessFeature PaperArticle

An Artificial Neural Network Method for Simulating Soliton Propagation Based on the Rosenau-KdV-RLW Equation on Unbounded Domains

by Laurence Finch, Weizhong Dai and Aniruddha Bora

Mathematics 2025, 13(7), 1036; https://doi.org/10.3390/math13071036 - 22 Mar 2025

Viewed by 246

Abstract

The simulation of wave propagation, such as soliton propagation, based on the Rosenau-KdV-RLW equation on unbounded domains requires a bounded computational domain. Therefore, a special boundary treatment, such as an absorbing boundary condition (ABC) or a perfectly matched layer (PML), is necessary to minimize the reflections of outgoing waves at the boundary, preventing interference with the simulation’s accuracy. However, the presence of higher-order partial derivatives, such as

u_{x x t}

and

u_{x x x x t}

in the Rosenau-KdV-RLW equation, raises challenges in deriving accurate artificial boundary conditions. To address this issue, we propose an artificial neural network (ANN) method that enables soliton propagation through the computational domain without imposing artificial boundary conditions. This method randomly selects training points from the bounded computational space-time domain, and the loss function is designed based solely on the initial conditions and the Rosenau-KdV-RLW equation itself, without any boundary conditions. We analyze the convergence of the ANN solution theoretically. This new ANN method is tested in three examples. The results indicate that the present ANN method effectively simulates soliton propagation based on the Rosenau-KdV-RLW equation in unbounded domains or over extended periods. Full article

(This article belongs to the Special Issue Computational Methods and Applications for Numerical Analysis, 2nd Edition)

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24 pages, 7357 KiB

Open AccessFeature PaperArticle

Simulation of Antiplane Piezoelectricity Problems with Multiple Inclusions by the Meshless Method of Fundamental Solution with the LOOCV Algorithm for Determining Sources

by Jingyi Zhang, Ji Lin, Fajie Wang and Yan Gu

Mathematics 2025, 13(6), 920; https://doi.org/10.3390/math13060920 - 10 Mar 2025

Viewed by 486

Abstract

This paper provides a high-accuracy and efficient method for addressing antiplane piezoelectricity problems with multiple inclusions. The method of fundamental solutions is a boundary-type meshless method that applies the linear combination of fundamental solutions as approximate solutions with the collocation method for determining the unknowns. To avoid the singularity of fundamental solutions, sources are placed away from the physical boundary. The leave-one-out cross-validation algorithm is employed to identify the optimal source placements to mitigate the influence of this singularity on numerical results. Numerical results of the stress concentration and electric field concentration at the interface between circular and elliptic inclusions and matrix are studied and compared well with references. Furthermore, the stability of the method is verified. Perturbations are added to the boundary conditions. Accuracy on the order of 10⁻¹¹ is obtained without noise. After adding the disturbance, the calculation accuracy is the same order of magnitude as the disturbance. Full article

(This article belongs to the Special Issue Computational Methods and Applications for Numerical Analysis, 2nd Edition)

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19 pages, 602 KiB

Open AccessArticle

The Numerical Solution of an Inverse Pseudoparabolic Problem with a Boundary Integral Observation

by Miglena N. Koleva and Lubin G. Vulkov

Mathematics 2025, 13(6), 908; https://doi.org/10.3390/math13060908 - 8 Mar 2025

Viewed by 539

Abstract

Direct and inverse problems for a pseudoparabolic equation are considered. The direct (forward) problem is to find the solution of the corresponding initial–boundary-value problem for known model parameters, as well as the initial and boundary conditions. The well-posedness of the direct problem is shown and a priori estimates of the solution are obtained. We study the inverse problem for identifying the flux on a part of the boundary of a rectangle, using integral measurement on the same part of the boundary. We first reduce the inverse problem to a direct one. The initial–boundary-value direct problem is with nonclassical (integrodifferential) boundary conditions. We develop a finite-difference scheme for numerically solving this problem. Numerical test examples demonstrate the effectiveness of the proposed method. It successfully handles the nonclassical integrodifferential boundary conditions and provides accurate numerical solutions. Full article

(This article belongs to the Special Issue Computational Methods and Applications for Numerical Analysis, 2nd Edition)

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23 pages, 14344 KiB

Open AccessArticle

Finite Element Analysis of Functionally Graded Mindlin–Reissner Plates for Aircraft Tapered and Interpolated Wing Defluxion and Modal Analysis

by Ali Hajjia, Mohammed Berrada Gouzi, Bilal Harras, Ahmed El Khalfi, Sorin Vlase and Maria Luminita

Mathematics 2025, 13(4), 620; https://doi.org/10.3390/math13040620 - 13 Feb 2025

Viewed by 720

Abstract

This paper explores and discusses how wing structures vibrate by using the Mindlin–Reissner plate theory, which takes into consideration the effects of transverse shear deformation and rotary inertia. This theory works well for thicker structures, like aircraft wings, where it gives accuracy by detecting shear and rotation effects. FGMs, or functionally graded materials, are used in aviation to enhance structural patterns and reduce stress points by gradually changing material properties along the wing thickness based on the volume fraction index. Finite element method (FEM) simulations were conducted to compare the natural frequencies and mode shapes of tapered and interpolated wing geometries. The results indicate that interpolated meshes exhibit higher natural frequencies due to increased stiffness, whereas tapered meshes show lower frequencies due to their flexibility. Validation through ANSYS simulations confirms the accuracy of the FEM results, highlighting the influence of geometry and material gradation on vibrational behavior. The findings offer valuable insights for aerospace applications, supporting the development of lightweight and efficient wing designs. Full article

(This article belongs to the Special Issue Computational Methods and Applications for Numerical Analysis, 2nd Edition)

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23 pages, 1026 KiB

Open AccessArticle

Construction of Uniform Designs over a Domain with Linear Constraints

by Luojing Yang, Xiaoping Yang and Yongdao Zhou

Mathematics 2025, 13(3), 438; https://doi.org/10.3390/math13030438 - 28 Jan 2025

Viewed by 625

Abstract

Uniform design is a powerful and robust experimental methodology that is particularly advantageous for multidimensional numerical integration and high-level experiments. As its applications expand across diverse disciplines, the theoretical foundation of uniform design continues to evolve. In real-world scenarios, experimental factors are often subject to one or more linear constraints, which pose challenges in constructing efficient designs within constrained high-dimensional experimental spaces. These challenges typically require sophisticated algorithms, which may compromise uniformity and robustness. Addressing these constraints is critical for reducing costs, improving model accuracy, and identifying global optima in optimization problems. However, existing research primarily focuses on unconstrained or minimally constrained hypercubes, leaving a gap in constructing designs tailored to arbitrary linear constraints. This study bridges this gap by extending the inverse Rosenblatt transformation framework to develop innovative methods for constructing uniform designs over arbitrary hyperplanes and hyperspheres within unit hypercubes. Explicit construction formulas for these constrained domains are derived, offering simplified calculations for practitioners and providing a practical solution applicable to a wide range of experimental scenarios. Numerical simulations demonstrate the feasibility and effectiveness of these methods, setting a new benchmark for uniform design in constrained experimental regions. Full article

(This article belongs to the Special Issue Computational Methods and Applications for Numerical Analysis, 2nd Edition)

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19 pages, 13916 KiB

Open AccessArticle

Hole Appearance Constraint Method in 2D Structural Topology Optimization

by Lei Zhu, Tongxing Zuo, Chong Wang, Qianglong Wang, Zhengdong Yu and Zhenyu Liu

Mathematics 2024, 12(17), 2645; https://doi.org/10.3390/math12172645 - 26 Aug 2024

Viewed by 1046

Abstract

A 2D topology optimization algorithm is proposed, which integrates the control of hole shape, hole number, and the minimum scale between holes through the utilization of an appearance target image. The distance between the structure and the appearance target image is defined as the hole appearance constraint. The appearance constraint is organized as inequality constraints to control the performance of the structure in an iterative optimization. Specifically, hole shapes are controlled by matching adaptable equivalent shape templates, the minimum scales between holes are controlled by a hole shrinkage strategy, and the hole number is controlled by a hole number calculation and filling method. Based on the SIMP interpolation topology optimization model, the effectiveness of the proposed method is verified through numerical examples. Full article

(This article belongs to the Special Issue Computational Methods and Applications for Numerical Analysis, 2nd Edition)

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21 pages, 793 KiB

Open AccessArticle

A Path-Conservative ADER Discontinuous Galerkin Method for Non-Conservative Hyperbolic Systems: Applications to Shallow Water Equations

by Xiaoxu Zhao, Baining Wang, Gang Li and Shouguo Qian

Mathematics 2024, 12(16), 2601; https://doi.org/10.3390/math12162601 - 22 Aug 2024

Viewed by 735

Abstract

In this article, we propose a new path-conservative discontinuous Galerkin (DG) method to solve non-conservative hyperbolic partial differential equations (PDEs). In particular, the method here applies the one-stage ADER (Arbitrary DERivatives in space and time) approach to fulfill the temporal discretization. In addition, this method uses the differential transformation (DT) procedure rather than the traditional Cauchy–Kowalewski (CK) procedure to achieve the local temporal evolution. Compared with the classical ADER methods, the current method is free of solving generalized Riemann problems at inter-cells. In comparison with the Runge–Kutta DG (RKDG) methods, the proposed method needs less computer storage, thanks to the absence of intermediate stages. In brief, this current method is one-step, one-stage, and fully-discrete. Moreover, this method can easily obtain arbitrary high-order accuracy both in space and in time. Numerical results for one- and two-dimensional shallow water equations (SWEs) show that the method enjoys high-order accuracy and keeps good resolution for discontinuous solutions. Full article

(This article belongs to the Special Issue Computational Methods and Applications for Numerical Analysis, 2nd Edition)

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18 pages, 2073 KiB

Open AccessArticle

Numerical Reconstruction of Time-Dependent Boundary Conditions to 2D Heat Equation on Disjoint Rectangles at Integral Observations

by Miglena N. Koleva and Lubin G. Vulkov

Mathematics 2024, 12(10), 1499; https://doi.org/10.3390/math12101499 - 11 May 2024

Cited by 5 | Viewed by 1159

Abstract

In this paper, two-dimensional (2D) heat equations on disjoint rectangles are considered. The solutions are connected by interface Robin’s-type internal conditions. The problem has external Dirichlet boundary conditions that, in the forward (direct) formulation, are given functions. In the inverse problem formulation, the Dirichlet conditions are unknown functions, and the aim is to be reconstructed upon integral observations. Well-posedness both for direct and inverse problems is established. Using the given 2D integrals of the unknown solution on each of the domains and the specific interface boundary conditions, we reduce the 2D inverse problem to a forward heat 1D one. The resulting 1D problem is solved using the explicit Saul’yev finite difference method. Numerical test examples are discussed to illustrate the efficiency of the approach. Full article

(This article belongs to the Special Issue Computational Methods and Applications for Numerical Analysis, 2nd Edition)

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20 pages, 5568 KiB

Open AccessArticle

Intelligent Low-Consumption Optimization Strategies: Economic Operation of Hydropower Stations Based on Improved LSTM and Random Forest Machine Learning Algorithm

by Hong Pan, Jie Yang, Yang Yu, Yuan Zheng, Xiaonan Zheng and Chenyang Hang

Mathematics 2024, 12(9), 1292; https://doi.org/10.3390/math12091292 - 24 Apr 2024

Cited by 4 | Viewed by 1526

Abstract

The economic operation of hydropower stations has the potential to increase water use efficiency. However, there are some challenges, such as the fixed and unchangeable flow characteristic curve of the hydraulic turbines, and the large number of variables in optimal load distribution, which limit the progress of research. In this paper, we propose a new optimal method of the economic operation of hydropower stations based on improved Long Short-Term Memory neural network (I-LSTM) and Random Forest (RF) algorithm. Firstly, in order to accurately estimate the water consumption, the LSTM model’s hyperparameters are optimized using improved particle swarm optimization, and the I-LSTM method is proposed to fit the flow characteristic curve of the hydraulic turbines. Secondly, the Random Forest machine learning algorithm is introduced to establish a load-distribution model with its powerful feature extraction and learning ability. To improve the accuracy of the load-distribution model, we use the K-means algorithm to cluster the historical data and optimize the parameters of the Random Forest model. A Hydropower Station in China is selected for a case study. It is shown that (1) the I-LSTM method fits the operating characteristics under various working conditions and actual operating characteristics of hydraulic turbines, ensuring that they are closest to the actual operating state; (2) the I-LSTM method is compared with Support Vector Machine (SVM), Extreme Learning Machine (ELM) and Long Short-Term Memory neural network (LSTM). The prediction results of SVM have a large error, but compared with ELM and LSTM, MSE is reduced by about 46% and 38% respectively. MAE is reduced by about 25% and 21%, respectively. RMSE is reduced by about 27% and 24%, respectively; (3) the RF algorithm performs better than the traditional dynamic programming algorithm in load distribution. With the passage of time and the increase in training samples, the prediction accuracy of the Random Forest model has steadily improved, which helps to achieve optimal operation of the units, reducing their average total water consumption by 1.24%. This study provides strong support for the application of intelligent low-consumption optimization strategies in hydropower fields, which can bring higher economic benefits and resource savings to renewable energy production. Full article

(This article belongs to the Special Issue Computational Methods and Applications for Numerical Analysis, 2nd Edition)

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19 pages, 8027 KiB

Open AccessArticle

Dynamic Behavior of a 10 MW Floating Wind Turbine Concrete Platform under Harsh Conditions

by Xiaocui Chen, Qirui Wang, Yuquan Zhang and Yuan Zheng

Mathematics 2024, 12(3), 412; https://doi.org/10.3390/math12030412 - 26 Jan 2024

Cited by 1 | Viewed by 1640

Abstract

To ensure the safe and stable operation of a 10 MW floating wind turbine concrete platform under harsh sea conditions, the fluid–structure coupling theory was used to apply wind, wave, and current loads to a concrete semi-submersible floating platform, and strength analysis was performed to calculate its stress and deformation under environmental loads. Moreover, the safety factor and fatigue life prediction of the platform were also conducted. The results indicated that the incident angles of the environmental loads had a significant impact on motion response in the surge, sway, pitch, and yaw directions. As the incident angles increased, the motion response in the surge and pitch directions gradually decreased, the motion response in the sway direction gradually increased, and the yaw motion response showed a trend of first increasing and then decreasing. In addition, the maximum stress of the floating platform under harsh sea conditions was 12.718 MPa, mainly concentrated at the connection of the middle column and pontoon and the connection of the heave plate and Y-shaped pontoon, which meets the use strength requirements. However, the stress concentration zone exhibited a significantly shorter fatigue life with a magnitude of 10⁶. This implies a higher susceptibility to fatigue damage and the potential occurrence of structural failure. This research holds paramount significance in ensuring the safe and stable operation of floating wind turbine platforms, particularly under harsh sea conditions. Full article

(This article belongs to the Special Issue Computational Methods and Applications for Numerical Analysis, 2nd Edition)

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18 pages, 8825 KiB

Open AccessArticle

Numerical Investigation on Suction Flow Control Technology for a Blunt Trailing Edge Hydrofoil

by Peng Yang, Chiye Zhang, Hongyeyu Yan, Yifan Ren, Changliang Ye, Yaguang Heng and Yuan Zheng

Mathematics 2023, 11(16), 3618; https://doi.org/10.3390/math11163618 - 21 Aug 2023

Viewed by 1406

Abstract

The generation of hydro-mechanical resonance is related to the transition of the boundary layer and the development of vortex shedding. The application effect of suction control in hydrodynamics is equally deserving of consideration as an active control technique in aerodynamics. This study examines how suction control affects the flow field of the NACA0009 blunt trailing edge hydrofoil using the γ transition model. Firstly, the accuracy of the numerical method is checked by performing a three-dimensional hydrofoil numerical simulation. Based on this, three-dimensional hydrofoil suction control research is conducted. According to the results, the suction control increases the velocity gradient in the boundary layer and delays the position of transition. The frequency of vortex shedding in the wake region lowers, and the peak value of velocity fluctuation declines. The hydrofoil hydrodynamic performance may be successfully improved with a proper selection of the suction coefficient via research of the suction coefficient and suction position on the flow field around the hydrofoil. The lift/drag ratio goes up as the suction coefficient goes up. The boundary layer displacement thickness and momentum thickness are at their lowest points, and the velocity fluctuation amplitude in the wake region is at its lowest point as the suction coefficient C_μ = 0.003. When the suction slots are at the leading edge, the momentum loss in the boundary layer is minimal and the velocity fluctuation in the wake zone is negligible. Full article

(This article belongs to the Special Issue Computational Methods and Applications for Numerical Analysis, 2nd Edition)

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9 pages, 270 KiB

Open AccessArticle

A New Efficient Method for Absolute Value Equations

by Peng Guo, Javed Iqbal, Syed Muhammad Ghufran, Muhammad Arif, Reem K. Alhefthi and Lei Shi

Mathematics 2023, 11(15), 3356; https://doi.org/10.3390/math11153356 - 31 Jul 2023

Cited by 1 | Viewed by 1577

Abstract

In this paper, the two-step method is considered with the generalized Newton method as a predictor step. The three-point Newton–Cotes formula is taken as a corrector step. The proposed method’s convergence is discussed in detail. This method is very simple and therefore very effective for solving large systems. In numerical analysis, we consider a beam equation, transform it into a system of absolute value equations and then use the proposed method to solve it. Numerical experiments show that our method is very accurate and faster than already existing methods. Full article

(This article belongs to the Special Issue Computational Methods and Applications for Numerical Analysis, 2nd Edition)

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