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Recent Advances in Numerical Methods for Scientific and Engineering Applications,...
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Recent Advances in Numerical Methods for Scientific and Engineering Applications, 2nd Edition

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

Deadline for manuscript submissions: 31 May 2024 | Viewed by 4478

Special Issue Editors

Dr. Wenzhen Qu

E-Mail Website
Guest Editor
School of Mathematics and Statistics, Qingdao University, Qingdao 266071, China
Interests: acoustic wave propagation; dynamic problem; strong-form meshless method; fast multipole method; singular integral; large-scale simulation
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Yan Gu

E-Mail Website
Guest Editor
School of Mathematics and Statistics, Qingdao University, Qingdao 266071, China
Interests: computational mechanics; fracture mechanics; coatings; boundary element method; meshless method
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

The solutions of systems of partial differential equations (PDEs) are involved in many fields of scientific and engineering applications. Exact analytical or semi-analytical solutions of these systems are generally intractable due to their mathematical complexity. It is thus necessary to take advantage of the numerical methods to efficiently solve linear or non-linear systems of PDEs. The development of accurate and efficient numerical methods is always an active research field.

This Special Issue is devoted to papers on advanced numerical methods for scientific and engineering applications. Topics of interest include but are not limited to fast methods, iterative methods, adaptive mesh generation, large-scale simulation, dynamic problems, wave propagation, fracture mechanics analysis, and thin-wall structures.

Dr. Wenzhen Qu
Prof. Dr. Yan Gu
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

  • advanced numerical methods
  • large-scale simulation
  • dynamic problem
  • wave propagation
  • fracture mechanics analysis
  • thin-wall structures

Published Papers (5 papers)

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Research

20 pages, 6827 KiB  
Article
Meshless Generalized Finite Difference Method Based on Nonlocal Differential Operators for Numerical Simulation of Elastostatics
by Yeying Zhou, Chunguang Li, Xinshan Zhuang and Zhifen Wang
Mathematics 2024, 12(9), 1316; https://doi.org/10.3390/math12091316 - 25 Apr 2024
Viewed by 217
Abstract
This study proposes an innovative meshless approach that merges the peridynamic differential operator (PDDO) with the generalized finite difference method (GFDM). Based on the PDDO theory, this method introduces a new nonlocal differential operator that aims to reduce the pre-assumption required for the [...] Read more.
This study proposes an innovative meshless approach that merges the peridynamic differential operator (PDDO) with the generalized finite difference method (GFDM). Based on the PDDO theory, this method introduces a new nonlocal differential operator that aims to reduce the pre-assumption required for the PDDO method and simplify the calculation process. By discretizing through the particle approximation method, this technique proficiently preserves the PDDO’s nonlocal features, enhancing the numerical simulation’s flexibility and usability. Through the numerical simulation of classical elastic static problems, this article focuses on the evaluation of the calculation accuracy, calculation efficiency, robustness, and convergence of the method. This method is significantly stronger than the finite element method in many performance indicators. In fact, this study demonstrates the practicability and superiority of the proposed method in the field of elastic statics and provides a new approach to more complex problems. Full article
(This article belongs to the Special Issue Recent Advances in Numerical Methods for Scientific and Engineering Applications, 2nd Edition)
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23 pages, 6651 KiB  
Article
A Novel Spacetime Boundary-Type Meshless Method for Estimating Aquifer Hydraulic Properties Using Pumping Tests
by Cheng-Yu Ku and Chih-Yu Liu
Mathematics 2023, 11(21), 4497; https://doi.org/10.3390/math11214497 - 31 Oct 2023
Cited by 1 | Viewed by 705
Abstract
This article introduces a new boundary-type meshless method designed for solving axisymmetric transient groundwater flow problems, specifically for aquifer tests and estimating hydraulic properties. The method approximates solutions for axisymmetric transient groundwater flow using basis functions that satisfy the governing equation by solving [...] Read more.
This article introduces a new boundary-type meshless method designed for solving axisymmetric transient groundwater flow problems, specifically for aquifer tests and estimating hydraulic properties. The method approximates solutions for axisymmetric transient groundwater flow using basis functions that satisfy the governing equation by solving the inverse boundary value problem in the spacetime domain. The effectiveness of this method was demonstrated through validation with the Theis solution, which involves transient flow to a well in an infinite confined aquifer. The study included numerical examples that predicted drawdown at various radial distances and times near pumping wells. Additionally, an iterative scheme, namely, the fictitious time integration method, was employed to iteratively determine the hydraulic properties during the pumping test. The results indicate that this approach yielded highly accurate solutions without relying on the conventional time-marching scheme. Due to its temporal and spatial discretization within the spacetime domain, this method was found to be advantageous for estimating crucial hydraulic properties, such as the transmissivity and storativity of an aquifer. Full article
(This article belongs to the Special Issue Recent Advances in Numerical Methods for Scientific and Engineering Applications, 2nd Edition)
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11 pages, 5438 KiB  
Article
Finite-Difference Frequency-Domain Scheme for Sound Scattering by a Vortex with Perfectly Matched Layers
by Yongou Zhang, Zhongjian Ling, Hao Du and Qifan Zhang
Mathematics 2023, 11(18), 3959; https://doi.org/10.3390/math11183959 - 18 Sep 2023
Viewed by 668
Abstract
Understanding the effect of vortexes on sound propagation is of great significance in the field of target detection and acoustic imaging. A prediction algorithm of the two-dimensional vortex scattering is realized based on a finite-difference frequency-domain (FDFD) numerical scheme with perfectly matched layers [...] Read more.
Understanding the effect of vortexes on sound propagation is of great significance in the field of target detection and acoustic imaging. A prediction algorithm of the two-dimensional vortex scattering is realized based on a finite-difference frequency-domain (FDFD) numerical scheme with perfectly matched layers (PML). Firstly, the governing equation for flow–sound interaction is given based on the perturbation theory, and the FDFD program is built. Subsequently, the mesh independence is verified, and the result has a good convergence when the mesh corresponds to over 15 nodes per wavelength. Then, computational parameters of the PML are discussed to achieve better absorbing boundary conditions. Finally, the results of this algorithm are compared with previous literature data. Results show that for different cortex scattering cases, the absorption coefficient should vary linearly with the density of the medium and the incident wave frequency. When the thickness of the PML boundary is greater than 2.5 times the wavelength, the PML boundary can absorb the scattering sound effectively. This provides a reliable algorithm for the numerical study of the effect of vortexes on sound propagation. Full article
(This article belongs to the Special Issue Recent Advances in Numerical Methods for Scientific and Engineering Applications, 2nd Edition)
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21 pages, 7856 KiB  
Article
A Novel “Finite Element-Meshfree” Triangular Element Based on Partition of Unity for Acoustic Propagation Problems
by Sina Dang, Gang Wang and Yingbin Chai
Mathematics 2023, 11(11), 2475; https://doi.org/10.3390/math11112475 - 27 May 2023
Viewed by 993
Abstract
The accuracy of the conventional finite element (FE) approximation for the analysis of acoustic propagation is always characterized by an intractable numerical dispersion error. With the aim of enhancing the performance of the FE approximation for acoustics, a coupled FE-Meshfree numerical method based [...] Read more.
The accuracy of the conventional finite element (FE) approximation for the analysis of acoustic propagation is always characterized by an intractable numerical dispersion error. With the aim of enhancing the performance of the FE approximation for acoustics, a coupled FE-Meshfree numerical method based on triangular elements is proposed in this work. In the proposed new triangular element, the required local numerical approximation is built using point interpolation mesh-free techniques with polynomial-radial basis functions, and the original linear shape functions from the classical FE approximation are employed to satisfy the condition of partition of unity. Consequently, this coupled FE-Meshfree numerical method possesses simultaneously the strengths of the conventional FE approximation and the meshfree numerical techniques. From a number of representative numerical experiments of acoustic propagation, it is shown that in acoustic analysis, better numerical performance can be achieved by suppressing the numerical dispersion error by the proposed FE-Meshfree approximation in comparison with the FE approximation. More importantly, it also shows better numerical features in terms of convergence rate and computational efficiency than the original FE approach; hence, it is a very good alternative numerical approach to the existing methods in computational acoustics fields. Full article
(This article belongs to the Special Issue Recent Advances in Numerical Methods for Scientific and Engineering Applications, 2nd Edition)
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25 pages, 9451 KiB  
Article
The Extrinsic Enriched Finite Element Method with Appropriate Enrichment Functions for the Helmholtz Equation
by Yingbin Chai, Kangye Huang, Shangpan Wang, Zhichao Xiang and Guanjun Zhang
Mathematics 2023, 11(7), 1664; https://doi.org/10.3390/math11071664 - 30 Mar 2023
Cited by 14 | Viewed by 1336
Abstract
The traditional finite element method (FEM) could only provide acceptable numerical solutions for the Helmholtz equation in the relatively small wave number range due to numerical dispersion errors. For the relatively large wave numbers, the corresponding FE solutions are never adequately reliable. With [...] Read more.
The traditional finite element method (FEM) could only provide acceptable numerical solutions for the Helmholtz equation in the relatively small wave number range due to numerical dispersion errors. For the relatively large wave numbers, the corresponding FE solutions are never adequately reliable. With the aim to enhance the numerical performance of the FEM in tackling the Helmholtz equation, in this work an extrinsic enriched FEM (EFEM) is proposed to reduce the inherent numerical dispersion errors in the standard FEM solutions. In this extrinsic EFEM, the standard linear approximation space in the linear FEM is enriched extrinsically by using the polynomial and trigonometric functions. The construction of this enriched approximation space is realized based on the partition of unity concept and the highly oscillating features of the Helmholtz equation in relatively large wave numbers can be effectively captured by the employed specially-designed enrichment functions. A number of typical numerical examples are considered to examine the ability of this extrinsic EFEM to control the dispersion error for solving Helmholtz problems. From the obtained numerical results, it is found that this extrinsic EFEM behaves much better than the standard FEM in suppressing the numerical dispersion effects and could provide much more accurate numerical results. In addition, this extrinsic EFEM also possesses higher convergence rate than the conventional FEM. More importantly, the formulation of this extrinsic EFEM can be formulated quite easily without adding the extra nodes. Therefore, the present extrinsic EFEM can be regarded as a competitive alternative to the traditional finite element approach in dealing with the Helmholtz equation in relatively high frequency ranges. Full article
(This article belongs to the Special Issue Recent Advances in Numerical Methods for Scientific and Engineering Applications, 2nd Edition)
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