Mathematics

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15 pages, 5742 KiB

Open AccessArticle

A Novel RBF Collocation Method Using Fictitious Centre Nodes for Elasticity Problems

by Hui Zheng, Xiaoling Lai, Anyu Hong and Xing Wei

Mathematics 2022, 10(19), 3711; https://doi.org/10.3390/math10193711 - 10 Oct 2022

Cited by 1 | Viewed by 1389

The traditional radial basis function collocation method (RBFCM) has poor stability when solving two-dimensional elastic problems, and the numerical results are very sensitive to shape parameters, especially in solving elastic problems. In this paper, a novel radial basis function collocation method (RBFCM) using [...] Read more.

The traditional radial basis function collocation method (RBFCM) has poor stability when solving two-dimensional elastic problems, and the numerical results are very sensitive to shape parameters, especially in solving elastic problems. In this paper, a novel radial basis function collocation method (RBFCM) using fictitious centre nodes is applied to the elastic problem. The proposed RBFCM employs fictitious centre nodes to interpolate the unknown coefficients, and is much less sensitive to the shape parameter compared with the traditional RBFCM. The details of the shape parameters are discussed for the novel RBFCM in elastic problems. Elastic problems with and without analytical solutions are given to show the effectiveness of the improved RBFCM. Full article

(This article belongs to the Special Issue Recent Advances in Numerical Methods for Scientific and Engineering Applications)

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14 pages, 3898 KiB

Open AccessArticle

Analysis of Jet Wall Flow and Heat Transfer Conveying ZnO-SAE50 Nano Lubricants Saturated in Darcy-Brinkman Porous Medium

by Umair Khan, Aurang Zaib, Anuar Ishak, Iskandar Waini, El-Sayed M. Sherif and Ioan Pop

Mathematics 2022, 10(17), 3201; https://doi.org/10.3390/math10173201 - 5 Sep 2022

Cited by 6 | Viewed by 1566

Abstract

The problem of 2D (two-dimensional) wall jet flow, along with heat transfer incorporated by nanofluid in a Darcy-Brinkman medium, while recognizing the requirement for efficient heating and cooling systems. Following the use of similarity variables, the resultant system of ODEs (ordinary differential equations) [...] Read more.

The problem of 2D (two-dimensional) wall jet flow, along with heat transfer incorporated by nanofluid in a Darcy-Brinkman medium, while recognizing the requirement for efficient heating and cooling systems. Following the use of similarity variables, the resultant system of ODEs (ordinary differential equations) is solved using the well-known and efficient bvp4c (boundary-value problem of the 4th order) technique. The significance of physical quantities for the under-consideration parameters is illustrated and explained. The findings show that the nanoparticle volume fraction and porosity parameters decrease the velocity, but increase the temperature. In addition, the temperature uplifts in the presence of radiation effect. The suction parameter initially decreases and then increases the velocity near the surface, while the temperature declines. Full article

(This article belongs to the Special Issue Recent Advances in Numerical Methods for Scientific and Engineering Applications)

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22 pages, 5209 KiB

Open AccessArticle

Localized Method of Fundamental Solutions for Two-Dimensional Inhomogeneous Inverse Cauchy Problems

by Junli Zhang, Hui Zheng, Chia-Ming Fan and Ming-Fu Fu

Mathematics 2022, 10(9), 1464; https://doi.org/10.3390/math10091464 - 27 Apr 2022

Cited by 1 | Viewed by 1447

Abstract

Due to the fundamental solutions are employed as basis functions, the localized method of fundamental solution can obtain more accurate numerical results than other localized methods in the homogeneous problems. Since the inverse Cauchy problem is ill posed, a small disturbance will lead [...] Read more.

Due to the fundamental solutions are employed as basis functions, the localized method of fundamental solution can obtain more accurate numerical results than other localized methods in the homogeneous problems. Since the inverse Cauchy problem is ill posed, a small disturbance will lead to great errors in the numerical simulations. More accurate numerical methods are needed in the inverse Cauchy problem. In this work, the LMFS is firstly proposed to analyze the inhomogeneous inverse Cauchy problem. The recursive composite multiple reciprocity method (RC-MRM) is adopted to change original inhomogeneous problem into a higher-order homogeneous problem. Then, the high-order homogeneous problem can be solved directly by the LMFS. Several numerical experiments are carried out to demonstrate the efficiency of the LMFS for the inhomogeneous inverse Cauchy problems. Full article

(This article belongs to the Special Issue Recent Advances in Numerical Methods for Scientific and Engineering Applications)

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12 pages, 1962 KiB

Open AccessArticle

An Enriched Finite Element Method with Appropriate Interpolation Cover Functions for Transient Wave Propagation Dynamic Problems

by Jue Qu, Hongjun Xue, Yancheng Li and Yingbin Chai

Mathematics 2022, 10(9), 1380; https://doi.org/10.3390/math10091380 - 20 Apr 2022

Cited by 1 | Viewed by 1785

Abstract

A novel enriched finite element method (EFEM) was employed to analyze the transient wave propagation problems. In the present method, the traditional finite element approximation was enriched by employing the appropriate interpolation covers. We mathematically and numerically showed that the present EFEM possessed [...] Read more.

A novel enriched finite element method (EFEM) was employed to analyze the transient wave propagation problems. In the present method, the traditional finite element approximation was enriched by employing the appropriate interpolation covers. We mathematically and numerically showed that the present EFEM possessed the important monotonic convergence property with the decrease of the used time steps for transient wave propagation problems when the unconditional stable Newmark time integration scheme was used for time integration. This attractive property markedly distinguishes the present EFEM from the traditional FEM for transient wave propagation problems. Two typical numerical examples were given to demonstrate the capabilities of the present method. Full article

(This article belongs to the Special Issue Recent Advances in Numerical Methods for Scientific and Engineering Applications)

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23 pages, 7464 KiB

Open AccessArticle

The Instability and Response Studies of a Top-Tensioned Riser under Parametric Excitations Using the Differential Quadrature Method

by Yang Zhang, Qiang Gui, Yuzheng Yang and Wei Li

Mathematics 2022, 10(8), 1331; https://doi.org/10.3390/math10081331 - 17 Apr 2022

Cited by 2 | Viewed by 1776

Abstract

The differential quadrature method (DQM) is a numerical technique widely applied in structure mechanics problems. In this work, a top-tensioned riser conveying fluid is considered. The governing equation of this riser under parametric excitations is deduced. Through Galerkin’s method, the partial differential governing [...] Read more.

The differential quadrature method (DQM) is a numerical technique widely applied in structure mechanics problems. In this work, a top-tensioned riser conveying fluid is considered. The governing equation of this riser under parametric excitations is deduced. Through Galerkin’s method, the partial differential governing equation with respect to time t and vertical coordinate z is reduced into a 1D differential equation with respect only to time. Moreover, the DQM is applied to discretize the governing equation to give solution schemes for the risers’ parametric vibration problem. Furthermore, the instability region of Mathieu equation is studied by both the DQM and the Floquet theory to verify the effectiveness of the DQM, and the solutions of both methods show good consistency. After that, the influences of some factors such as damping coefficient, internal flow velocity, and wet-weight coefficient on the parametric instability of a top-tensioned riser are discussed through investigating the instability regions solved by the DQM solution scheme. Hence, conclusions are obtained that the increase of damping coefficient will save the riser from parametric resonance while increasing internal flow velocity, or the wet-weight coefficient will deteriorate the parametric instability of the riser. Finally, the time-domain responses of several specific cases in both stable region and unstable region are presented. Full article

(This article belongs to the Special Issue Recent Advances in Numerical Methods for Scientific and Engineering Applications)

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18 pages, 8777 KiB

Open AccessArticle

Multi-Material Topology Optimization of Thermo-Elastic Structures with Stress Constraint

by Jianliang Chen, Qinghai Zhao and Liang Zhang

Mathematics 2022, 10(8), 1216; https://doi.org/10.3390/math10081216 - 8 Apr 2022

Cited by 16 | Viewed by 3049

Abstract

This paper proposes a multi-material topology optimization formulation for thermo-elastic structures considering coupled mechanical and uniform thermal loads. The ordered-SIMP multiple materials interpolation model is introduced, combined with examples considering structural volume minimization under stress constraints. The p-norm function with the adjusted [...] Read more.

This paper proposes a multi-material topology optimization formulation for thermo-elastic structures considering coupled mechanical and uniform thermal loads. The ordered-SIMP multiple materials interpolation model is introduced, combined with examples considering structural volume minimization under stress constraints. The p-norm function with the adjusted coefficient was adopted to measure the global maximum stress. The adjoint variable method is presented to discuss the sensitivity of stress constraints, and the method of moving asymptotes (MMA) is utilized to update the design variables. The results demonstrate that clear topologies are obtained for complicated multiple material combinations with various temperature values. Meanwhile, the optimized configuration with stress constraints has clear sensitivity to uniform temperature variations. Therefore, the proposed model demonstrates the necessity of a thermo-elastic model influenced by temperature in optimization. Full article

(This article belongs to the Special Issue Recent Advances in Numerical Methods for Scientific and Engineering Applications)

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24 pages, 6254 KiB

Open AccessArticle

Solving the Eigenfrequencies Problem of Waveguides by Localized Method of Fundamental Solutions with External Source

by Ke Sun, Shuang Ding, Junli Zhang and Yan-Cheng Liu

Mathematics 2022, 10(7), 1128; https://doi.org/10.3390/math10071128 - 1 Apr 2022

Viewed by 1800

Abstract

The localized method of fundamental solutions (LMFS) is a domain-type, meshless numerical method. Compared with numerical methods that have a high grid dependence, it does not require grid generation and numerical integration, so it can effectively improve computational efficiency and avoid complex integration [...] Read more.

The localized method of fundamental solutions (LMFS) is a domain-type, meshless numerical method. Compared with numerical methods that have a high grid dependence, it does not require grid generation and numerical integration, so it can effectively improve computational efficiency and avoid complex integration processes. Moreover, it is formed using the traditional method of fundamental solutions (MFS) and the localization approach. Previous studies have shown that the MFS may produce a dense and ill-conditioned matrix. However, the proposed LMFS can yield a sparse system of linear algebraic equations, so it is more suitable and effective in solving complicated engineering problems. In this article, LMFS was used to solve eigenfrequency problems in electromagnetic waves, which were controlled using two-dimensional Helmholtz equations. Additionally, the resonant frequencies of the eigenproblem were determined by the response amplitudes. In order to determine the eigenfrequencies, LMFS was applied for solving a sequence of inhomogeneous problems by introducing an external source. Waveguides with different shapes were analyzed to prove the stability of the present LMFS in this paper. Full article

(This article belongs to the Special Issue Recent Advances in Numerical Methods for Scientific and Engineering Applications)

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15 pages, 6556 KiB

Open AccessArticle

A Localized Method of Fundamental Solution for Numerical Simulation of Nonlinear Heat Conduction

by Feng Wang, Yan-Cheng Liu and Hui Zheng

Mathematics 2022, 10(5), 773; https://doi.org/10.3390/math10050773 - 28 Feb 2022

Cited by 2 | Viewed by 2237

Abstract

In this study, an efficient localized method of fundamental solution (LMFS) is applied to nonlinear heat conduction with mixed boundary conditions. Since the thermal conductivity is temperature-dependent, the Kirchhoff transformation is used to transform the nonlinear partial differential equations (PDEs) into Laplace equations [...] Read more.

In this study, an efficient localized method of fundamental solution (LMFS) is applied to nonlinear heat conduction with mixed boundary conditions. Since the thermal conductivity is temperature-dependent, the Kirchhoff transformation is used to transform the nonlinear partial differential equations (PDEs) into Laplace equations with nonlinear boundary conditions. Then the LMFS is applied to the governing equation, and the nonlinear equations are treated by the fictitious time integration method (FTIM). Both 2D and 3D numerical examples are proposed to verify the effectiveness of the LMFS. Full article

(This article belongs to the Special Issue Recent Advances in Numerical Methods for Scientific and Engineering Applications)

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19 pages, 3326 KiB

Open AccessArticle

Thermal Conductivity Identification in Functionally Graded Materials via a Machine Learning Strategy Based on Singular Boundary Method

by Wenzhi Xu, Zhuojia Fu and Qiang Xi

Mathematics 2022, 10(3), 458; https://doi.org/10.3390/math10030458 - 30 Jan 2022

Cited by 8 | Viewed by 2461

Abstract

A machine learning strategy based on the semi-analytical singular boundary method (SBM) is presented for the thermal conductivity identification of functionally graded materials (FGMs). In this study, only the temperature or heat flux on the surface or interior of FGMs can be measured [...] Read more.

A machine learning strategy based on the semi-analytical singular boundary method (SBM) is presented for the thermal conductivity identification of functionally graded materials (FGMs). In this study, only the temperature or heat flux on the surface or interior of FGMs can be measured by the thermal sensors, and the SBM is used to construct the database of the relationship between the thermal conductivity and the temperature distribution of the functionally graded structure. Based on the aforementioned constructed database, the artificial neural network-based machine learning strategy was implemented to identify the thermal conductivity of FGMs. Finally, several benchmark examples are presented to verify the feasibility, robustness, and applicability of the proposed machine learning strategy. Full article

(This article belongs to the Special Issue Recent Advances in Numerical Methods for Scientific and Engineering Applications)

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21 pages, 4361 KiB

Open AccessArticle

Free and Forced Vibration Analysis of Two-Dimensional Linear Elastic Solids Using the Finite Element Methods Enriched by Interpolation Cover Functions

by Yancheng Li, Sina Dang, Wei Li and Yingbin Chai

Mathematics 2022, 10(3), 456; https://doi.org/10.3390/math10030456 - 30 Jan 2022

Cited by 35 | Viewed by 3316

Abstract

In this paper, a novel enriched three-node triangular element with the augmented interpolation cover functions is proposed based on the original linear triangular element for two-dimensional solids. In this enriched triangular element, the augmented interpolation cover functions are employed to enrich the original [...] Read more.

In this paper, a novel enriched three-node triangular element with the augmented interpolation cover functions is proposed based on the original linear triangular element for two-dimensional solids. In this enriched triangular element, the augmented interpolation cover functions are employed to enrich the original standard linear shape functions over element patches. As a result, the original linear approximation space can be effectively enriched without adding extra nodes. To eliminate the linear dependence issue of the present method, an effective scheme is used to make the system matrices of the numerical model completely positive-definite. Through several typical numerical examples, the abilities of the present enriched three node triangular element in forced and free vibration analysis of two-dimensional solids are studied. The results show that, compared with the original linear triangular element, the present element can not only provide more accurate numerical results, but also have higher computational efficiency and convergence rate. Full article

(This article belongs to the Special Issue Recent Advances in Numerical Methods for Scientific and Engineering Applications)

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14 pages, 1495 KiB

Open AccessArticle

Directional Difference Convolution and Its Application on Face Anti-Spoofing

by Mingye Yang, Xian Li, Dongjie Zhao and Yan Li

Mathematics 2022, 10(3), 365; https://doi.org/10.3390/math10030365 - 25 Jan 2022

Cited by 1 | Viewed by 2647

Abstract

In practical application, facial image recognition is vulnerable to be attacked by photos, videos, etc., while some currently used artificial feature extractors in machine learning, such as activity detection, texture descriptors, and distortion detection, are insufficient due to their weak detection ability in [...] Read more.

In practical application, facial image recognition is vulnerable to be attacked by photos, videos, etc., while some currently used artificial feature extractors in machine learning, such as activity detection, texture descriptors, and distortion detection, are insufficient due to their weak detection ability in feature extraction from unknown attack. In order to deal with the aforementioned deficiency and improve the network security, this paper proposes directional difference convolution for the deep learning in gradient image information extraction, which analyzes pixel correlation within the convolution domain and calculates pixel gradients through difference calculation. Its combination with traditional convolution can be optimized by a parameter

θ

. Its stronger ability in gradient extraction improves the learning and predicting ability of the network, whose performance testing on CASIA-MFSD, Replay-Attack, and MSU-MFSD for face anti-spoofing task shows that our method outperforms the current related methods. Full article

(This article belongs to the Special Issue Recent Advances in Numerical Methods for Scientific and Engineering Applications)

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18 pages, 5246 KiB

Open AccessArticle

A Pseudo-Spectral Fourier Collocation Method for Inhomogeneous Elliptical Inclusions with Partial Differential Equations

by Xiao Wang, Juan Wang, Xin Wang and Chujun Yu

Mathematics 2022, 10(3), 296; https://doi.org/10.3390/math10030296 - 19 Jan 2022

Cited by 20 | Viewed by 3009

Abstract

Inhomogeneous elliptical inclusions with partial differential equations have aroused appreciable concern in many disciplines. In this paper, a pseudo-spectral collocation method, based on Fourier basis functions, is proposed for the numerical solutions of two- (2D) and three-dimensional (3D) inhomogeneous elliptic boundary value problems. [...] Read more.

Inhomogeneous elliptical inclusions with partial differential equations have aroused appreciable concern in many disciplines. In this paper, a pseudo-spectral collocation method, based on Fourier basis functions, is proposed for the numerical solutions of two- (2D) and three-dimensional (3D) inhomogeneous elliptic boundary value problems. We describe how one can improve the numerical accuracy by making some extra “reconstruction techniques” before applying the traditional Fourier series approximation. After the particular solutions have been obtained, the resulting homogeneous equation can then be calculated using various boundary-type methods, such as the method of fundamental solutions (MFS). Using Fourier basis functions, one does not need to use large matrices, making accrual computations relatively fast. Three benchmark numerical examples involving Poisson, Helmholtz, and modified-Helmholtz equations are presented to illustrate the applicability and accuracy of the proposed method. Full article

(This article belongs to the Special Issue Recent Advances in Numerical Methods for Scientific and Engineering Applications)

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9 pages, 548 KiB

Open AccessArticle

Precorrected-FFT Accelerated Singular Boundary Method for High-Frequency Acoustic Radiation and Scattering

by Weiwei Li and Fajie Wang

Mathematics 2022, 10(2), 238; https://doi.org/10.3390/math10020238 - 13 Jan 2022

Cited by 13 | Viewed by 1629

Abstract

This paper presents a precorrected-FFT (pFFT) accelerated singular boundary method (SBM) for acoustic radiation and scattering in the high-frequency regime. The SBM is a boundary-type collocation method, which is truly free of mesh and integration and easy to program. However, due to the [...] Read more.

This paper presents a precorrected-FFT (pFFT) accelerated singular boundary method (SBM) for acoustic radiation and scattering in the high-frequency regime. The SBM is a boundary-type collocation method, which is truly free of mesh and integration and easy to program. However, due to the expensive CPU time and memory requirement in solving a fully-populated interpolation matrix equation, this method is usually limited to low-frequency acoustic problems. A new pFFT scheme is introduced to overcome this drawback. Since the models with lots of collocation points can be calculated by the new pFFT accelerated SBM (pFFT-SBM), high-frequency acoustic problems can be simulated. The results of numerical examples show that the new pFFT-SBM possesses an obvious advantage for high-frequency acoustic problems. Full article

(This article belongs to the Special Issue Recent Advances in Numerical Methods for Scientific and Engineering Applications)

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