Mathematics

Research

25 pages, 9449 KiB

Open AccessArticle

Global–Local Non Intrusive Analysis with 1D to 3D Coupling: Application to Crack Propagation and Extension to Commercial Software

by Matías Jaque-Zurita, Jorge Hinojosa and Ignacio Fuenzalida-Henríquez

Mathematics 2023, 11(11), 2540; https://doi.org/10.3390/math11112540 - 1 Jun 2023

Cited by 2 | Viewed by 1536

Abstract

Computational simulation is a highly reliable tool used to solve structural analysis problems. In recent times, several techniques have been developed in the field of computational mechanics in order to analyze non-linearities in less time, helping decision-making when structures suffer damage. The global–local [...] Read more.

Computational simulation is a highly reliable tool used to solve structural analysis problems. In recent times, several techniques have been developed in the field of computational mechanics in order to analyze non-linearities in less time, helping decision-making when structures suffer damage. The global–local analysis is a technique to increase the efficiency of computational simulations by using a global model to obtain boundary conditions in a coupling zone imposed on a local model. Coupling can be performed through the primal–dual method, which is used for crack propagation using 2D and 3D models with fine meshes, thus saving computational time. However, it has not been implemented at a commercial level to analyze large structures such as multi-story buildings with focused non-linearities. In this work, a global–local analysis with non-intrusive methodology and simplified models was implemented in a cracked framed structure, using a 1D (global) and 3D (local) coupling considering crack propagation with primal–dual interface conditions. Different lengths of the local model were analyzed, studying their influence on the convergence of the problem, and compared with a 3D monolithic model to check the reliability of the results. The results show that the proposed methodology solves the problem with an error less than 10%. Furthermore, it was determined that the dimensions of the local model affect the convergence of the problem. This work also provides an implementation of the method for large structures containing focused non-linearities and using commercial software, reducing computational time for the cracked structural analysis. Full article

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

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19 pages, 10382 KiB

Open AccessArticle

Lattice Boltzmann Numerical Study on Mesoscopic Seepage Characteristics of Soil–Rock Mixture Considering Size Effect

by Peichen Cai, Xuesong Mao, Ke Lou and Zhihui Yun

Mathematics 2023, 11(8), 1968; https://doi.org/10.3390/math11081968 - 21 Apr 2023

Cited by 2 | Viewed by 1712

Abstract

One of the hot topics in the study of rock and soil hydraulics is the size effect of a soil–rock mixture’s (SRM) seepage characteristics. The seepage process of the SRM was simulated from the pore scale through the lattice Boltzmann method (LBM) in [...] Read more.

One of the hot topics in the study of rock and soil hydraulics is the size effect of a soil–rock mixture’s (SRM) seepage characteristics. The seepage process of the SRM was simulated from the pore scale through the lattice Boltzmann method (LBM) in this paper to explore the internal influence mechanism of sample size effect on the SRM seepage characteristics. SRM samples were generated using the improved Monte Carlo method (IMCM), and through 342 simulation test conditions the influence of size feature parameters such as resolution (R), segmentation type, model feature size (S), feature length ratio (F), and soil/rock particle size feature ratio (P) was examined. The study demonstrated that as R increases, the permeability of the SRM gradually rises and tends to stabilize when R reaches 60 ppi. At the same S, the dispersion degree of model permeability obtained by the four segmentation types is in the order of center < random < equal < top. With an increase in S, the permeability (k) of the SRM gradually decreases, conforming to the dimensionless mathematical model,

k = a_{0} \cdot S^{- b_{0}}

, and tends to stabilize at S = 80 mm. With an increase in F and an increase in S, the permeability of the SRM exhibits a linear “zonal” distribution that declines in order. When F is greater than 12, the dispersion of the permeability value distribution is especially small. With an increase in P, the permeability of the SRM decreases gradually before rising abruptly. P is crucial for the grading and structural makeup of the SRM. Overall, this paper concludes that the conditions of R = 60 ppi, center segmentation type, S = 80 mm, F ≥ 12, and P set by demand can be used to select and generate the size of the SRM optimal representative elementary volume (REV) numerical calculation model. The SRM can serve as a general reference for test and engineering construction as a common geotechnical engineering material. Full article

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

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Graphical abstract

15 pages, 9306 KiB

Open AccessArticle

A Novel Coupled Meshless Model for Simulation of Acoustic Wave Propagation in Infinite Domain Containing Multiple Heterogeneous Media

by Cheng Chi, Fajie Wang and Lin Qiu

Mathematics 2023, 11(8), 1841; https://doi.org/10.3390/math11081841 - 12 Apr 2023

Viewed by 2024

Abstract

This study presents a novel coupled meshless model for simulating acoustic wave propagation in heterogeneous media, based on the singular boundary method (SBM) and Kansa’s method (KS). In the proposed approach, the SBM was used to model the homogeneous part of the propagation [...] Read more.

This study presents a novel coupled meshless model for simulating acoustic wave propagation in heterogeneous media, based on the singular boundary method (SBM) and Kansa’s method (KS). In the proposed approach, the SBM was used to model the homogeneous part of the propagation domain, while KS was employed to model a heterogeneity. The interface compatibility conditions associated with velocities and pressures were imposed to couple the two methods. The proposed SBM–KS coupled approach combines the respective advantages of the SBM and KS. The SBM is especially suitable for solving external sound field problems, while KS is attractive for nonlinear problems in bounded non-homogeneous media. Moreover, the new methodology completely avoids grid generation and numerical integration compared with the finite element method and boundary element method. Numerical experiments verified the accuracy and effectiveness of the proposed scheme. Full article

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

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20 pages, 584 KiB

Open AccessArticle

Defect Analysis of a Non-Iterative Co-Simulation

by Slaven Glumac and Zdenko Kovačić

Mathematics 2023, 11(6), 1342; https://doi.org/10.3390/math11061342 - 9 Mar 2023

Viewed by 1337

Abstract

This article presents an analysis of co-simulation defects for a system of coupled ordinary differential equations. The research builds on the theorem that the co-simulation error is bounded if the co-simulation defect is bounded. The co-simulation defect can be divided into integration, output, [...] Read more.

This article presents an analysis of co-simulation defects for a system of coupled ordinary differential equations. The research builds on the theorem that the co-simulation error is bounded if the co-simulation defect is bounded. The co-simulation defect can be divided into integration, output, and connection defects, all of which can be controlled. This article proves that the output and connection defect can be controlled by the co-simulation master by varying the communication step size. A non-iterative co-simulation method with variable communication step size is presented to demonstrate the applicability of the presented research. The orders of the interpolation polynomials used in the co-simulation method are varied in the experimental analysis. The experimental analysis shows how each component of a co-simulation defect affects the co-simulation error. The analysis presented is used to verify the applicability of the proposed approach and to provide guidelines for the configuration of the co-simulation. Full article

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

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17 pages, 3616 KiB

Open AccessArticle

Numerical Analysis and Structure Optimization of Concentric GST Ring Resonator Mounted over SiO₂ Substrate and Cr Ground Layer

by Khaled Aliqab, Bo Bo Han, Ammar Armghan, Meshari Alsharari, Jaymit Surve and Shobhit K. Patel

Mathematics 2023, 11(5), 1257; https://doi.org/10.3390/math11051257 - 5 Mar 2023

Cited by 9 | Viewed by 1927

Abstract

Since the introduction of Metal-Insulator-Metal (MIM) absorbers, most of the structures demonstrated a narrowband absorption response which is not suitable for potential applications in photovoltaic systems, as it requires higher energy to enhance its performance. Very little research is being conducted in this [...] Read more.

Since the introduction of Metal-Insulator-Metal (MIM) absorbers, most of the structures demonstrated a narrowband absorption response which is not suitable for potential applications in photovoltaic systems, as it requires higher energy to enhance its performance. Very little research is being conducted in this direction; to address this issue, we exhibit a broadband solar absorber designed using a concentric GST ring resonator placed upon a silicon dioxide substrate layer with chromium used as a ground plane. It was analyzed using the finite element method. The design is also optimized by using a nonlinear parametric optimization algorithm. Comparatively less work has been focused on solar absorbers designed with the help of GST material, and here we have compared the effect of two different phases of GST, i.e., amorphous (aGST) and crystalline (cGST); the results indicate the higher performance of aGST phase. Parametric optimization has been adapted to identify the optimal design to attain high performance at minimal resources. The absorption response is angle insensitive for 0 to 60 degrees, and at the same time for both TE and TM modes, the design provides identical results, indicating the polarization-insensitive properties. The electric field intensity changes at the six peak wavelengths are also demonstrated for the authentication of the high performance. Thus, the proposed concentric GST ring resonator solar absorber can present a higher solar energy absorption rate than other solar structure designs. This design can be applied for improving the performance of photovoltaic systems. Full article

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

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26 pages, 415 KiB

Open AccessArticle

The Novel Analytical–Numerical Method for Multi-Dimensional Multi-Term Time-Fractional Equations with General Boundary Conditions

by Ji Lin, Sergiy Reutskiy, Yuhui Zhang, Yu Sun and Jun Lu

Mathematics 2023, 11(4), 929; https://doi.org/10.3390/math11040929 - 12 Feb 2023

Cited by 9 | Viewed by 1776

Abstract

This article presents a simple but effective two-step analytical–numerical algorithm for solving multi-dimensional multi-term time-fractional equations. The first step is to derive an analytic representation that satisfies boundary requirements for 1D, 2D, and 3D problems, respectively. The second step is the meshless approximation [...] Read more.

This article presents a simple but effective two-step analytical–numerical algorithm for solving multi-dimensional multi-term time-fractional equations. The first step is to derive an analytic representation that satisfies boundary requirements for 1D, 2D, and 3D problems, respectively. The second step is the meshless approximation where the Müntz polynomials are used to form the approximate solution and the unknown parameters are obtained by imposing the approximation for the governing equations. We illustrate first the detailed derivation of the analytic approximation and then the numerical implementation of the solution procedure. Several numerical examples are provided to verify the accuracy, efficiency, and adaptability to problems with general boundary conditions. The numerical results are compared with exact solutions and numerical methods reported in the literature, showing that the algorithm has great potential for multi-dimensional multi-term time-fractional equations with various boundary conditions. Full article

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

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19 pages, 24047 KiB

Open AccessArticle

C¹-Cubic Quasi-Interpolation Splines over a CT Refinement of a Type-1 Triangulation

by Haithem Benharzallah, Abdelaziz Mennouni and Domingo Barrera

Mathematics 2023, 11(1), 59; https://doi.org/10.3390/math11010059 - 23 Dec 2022

Cited by 1 | Viewed by 1834

Abstract

C^{1}

continuous quasi-interpolating splines are constructed over Clough–Tocher refinement of a type-1 triangulation. Their Bernstein–Bézier coefficients are directly defined from the known values of the function to be approximated, so that a set of appropriate basis functions is not required. The resulting [...] Read more.

C^{1}

continuous quasi-interpolating splines are constructed over Clough–Tocher refinement of a type-1 triangulation. Their Bernstein–Bézier coefficients are directly defined from the known values of the function to be approximated, so that a set of appropriate basis functions is not required. The resulting quasi-interpolation operators reproduce cubic polynomials. Some numerical tests are given in order to show the performance of the approximation scheme. Full article

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

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27 pages, 3936 KiB

Open AccessArticle

The Finite Element Method with High-Order Enrichment Functions for Elastodynamic Analysis

by Xunbai Du, Sina Dang, Yuzheng Yang and Yingbin Chai

Mathematics 2022, 10(23), 4595; https://doi.org/10.3390/math10234595 - 4 Dec 2022

Cited by 3 | Viewed by 2122

Abstract

Elastodynamic problems are investigated in this work by employing the enriched finite element method (EFEM) with various enrichment functions. By performing the dispersion analysis, it is confirmed that for elastodynamic analysis, the amount of numerical dispersion, which is closely related to the numerical [...] Read more.

Elastodynamic problems are investigated in this work by employing the enriched finite element method (EFEM) with various enrichment functions. By performing the dispersion analysis, it is confirmed that for elastodynamic analysis, the amount of numerical dispersion, which is closely related to the numerical error from the space domain discretization, can be suppressed to a very low level when quadric polynomial bases are employed to construct the local enrichment functions, while the amount of numerical dispersion from the EFEM with other types of enrichment functions (linear polynomial bases or first order of trigonometric functions) is relatively large. Consequently, the present EFEM with a quadric polynomial enrichment function shows more powerful capacities in elastodynamic analysis than the other considered numerical techniques. More importantly, the attractive monotonic convergence property can be broadly realized by the present approach with the typical two-step Bathe temporal discretization technique. Three representative numerical experiments are conducted in this work to verify the abilities of the present approach in elastodynamic analysis. Full article

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

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13 pages, 636 KiB

Open AccessArticle

The Shortest-Edge Duplication of Triangles

by Miguel Ángel Padrón, Francisco Perdomo, Ángel Plaza and José Pablo Suárez

Mathematics 2022, 10(19), 3643; https://doi.org/10.3390/math10193643 - 5 Oct 2022

Viewed by 1346

Abstract

We introduce a new triangle transformation, the shortest-edge (SE) duplication, as a natural way of mesh derefinement suitable to those meshes obtained by iterative application of longest-edge bisection refinement. Metric properties of the SE duplication of a triangle in the region of normalised [...] Read more.

We introduce a new triangle transformation, the shortest-edge (SE) duplication, as a natural way of mesh derefinement suitable to those meshes obtained by iterative application of longest-edge bisection refinement. Metric properties of the SE duplication of a triangle in the region of normalised triangles endowed with the Poincare hyperbolic metric are studied. The self-improvement of this transformation is easily proven, as well as the minimum angle condition. We give a lower bound for the maximum of the smallest angles of the triangles produced by the iterative SE duplication

α = \frac{π}{6}

. This bound does not depend on the shape of the initial triangle. Full article

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

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29 pages, 911 KiB

Open AccessArticle

A Modified Inverse Iteration Method for Computing the Symmetric Tridiagonal Eigenvectors

by Wei Chu, Yao Zhao and Hua Yuan

Mathematics 2022, 10(19), 3636; https://doi.org/10.3390/math10193636 - 5 Oct 2022

Cited by 2 | Viewed by 1666

Abstract

This paper presents a novel method for computing the symmetric tridiagonal eigenvectors, which is the modification of the widely used Inverse Iteration method. We construct the corresponding algorithm by a new one-step iteration method, a new reorthogonalization method with the general Q iteration [...] Read more.

This paper presents a novel method for computing the symmetric tridiagonal eigenvectors, which is the modification of the widely used Inverse Iteration method. We construct the corresponding algorithm by a new one-step iteration method, a new reorthogonalization method with the general Q iteration and a significant modification when calculating severely clustered eigenvectors. The numerical results show that this method is competitive with other existing methods, especially when computing part eigenvectors or severely clustered ones. Full article

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

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20 pages, 2783 KiB

Open AccessArticle

A Modified Radial Point Interpolation Method (M-RPIM) for Free Vibration Analysis of Two-Dimensional Solids

by Tingting Sun, Peng Wang, Guanjun Zhang and Yingbin Chai

Mathematics 2022, 10(16), 2889; https://doi.org/10.3390/math10162889 - 12 Aug 2022

Cited by 2 | Viewed by 1638

Abstract

The classical radial point interpolation method (RPIM) is a powerful meshfree numerical technique for engineering computation. In the original RPIM, the moving support domain for the quadrature point is usually employed for the field function approximation, but the local supports of the nodal [...] Read more.

The classical radial point interpolation method (RPIM) is a powerful meshfree numerical technique for engineering computation. In the original RPIM, the moving support domain for the quadrature point is usually employed for the field function approximation, but the local supports of the nodal shape functions are always not in alignment with the integration cells constructed for numerical integration. This misalignment can result in additional numerical integration error and lead to a loss in computation accuracy. In this work, a modified RPIM (M-RPIM) is proposed to address this issue. In the present M-RPIM, the misalignment between the constructed integration cells and the nodal shape function supports is successfully overcome by using a fixed support domain that can be easily constructed by the geometrical center of the integration cell. Several numerical examples of free vibration analysis are conducted to evaluate the abilities of the present M-RPIM and it is found that the computation accuracy of the original RPIM can be markedly improved by the present M-RPIM. Full article

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

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14 pages, 1761 KiB

Open AccessArticle

An Extrinsic Approach Based on Physics-Informed Neural Networks for PDEs on Surfaces

by Zhuochao Tang, Zhuojia Fu and Sergiy Reutskiy

Mathematics 2022, 10(16), 2861; https://doi.org/10.3390/math10162861 - 11 Aug 2022

Cited by 8 | Viewed by 2307

Abstract

In this paper, we propose an extrinsic approach based on physics-informed neural networks (PINNs) for solving the partial differential equations (PDEs) on surfaces embedded in high dimensional space. PINNs are one of the deep learning-based techniques. Based on the training data and physical [...] Read more.

In this paper, we propose an extrinsic approach based on physics-informed neural networks (PINNs) for solving the partial differential equations (PDEs) on surfaces embedded in high dimensional space. PINNs are one of the deep learning-based techniques. Based on the training data and physical models, PINNs introduce the standard feedforward neural networks (NNs) to approximate the solutions to the PDE systems. Using automatic differentiation, the PDEs information could be encoded into NNs and a loss function. To deal with the surface differential operators in the loss function, we combine the extrinsic approach with PINNs and then express that loss function in extrinsic form. Subsequently, the loss function could be minimized extrinsically with respect to the NN parameters. Numerical results demonstrate that the extrinsic approach based on PINNs for surface problems has good accuracy and higher efficiency compared with the embedding approach based on PINNs. In addition, the strong nonlinear mapping ability of NNs makes this approach robust in solving time-dependent nonlinear problems on more complex surfaces. Full article

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

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22 pages, 492 KiB

Open AccessArticle

A Novel Divisional Bisection Method for the Symmetric Tridiagonal Eigenvalue Problem

by Wei Chu, Yao Zhao and Hua Yuan

Mathematics 2022, 10(15), 2782; https://doi.org/10.3390/math10152782 - 5 Aug 2022

Cited by 1 | Viewed by 1879

Abstract

The embarrassingly parallel nature of the Bisection Algorithm makes it easy and efficient to program on a parallel computer, but with an expensive time cost when all symmetric tridiagonal eigenvalues are wanted. In addition, few methods can calculate a single eigenvalue in parallel [...] Read more.

The embarrassingly parallel nature of the Bisection Algorithm makes it easy and efficient to program on a parallel computer, but with an expensive time cost when all symmetric tridiagonal eigenvalues are wanted. In addition, few methods can calculate a single eigenvalue in parallel for now, especially in a specific order. This paper solves the issue with a new approach that can parallelize the Bisection iteration. Some pseudocodes and numerical results are presented. It shows our algorithm reduces the time cost by more than 35–70% compared to the Bisection algorithm while maintaining its accuracy and flexibility. Full article

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

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12 pages, 1210 KiB

Open AccessArticle

Simultaneous Design of the Host Structure and the Polarisation Profile of Piezoelectric Sensors Applied to Cylindrical Shell Structures

by David Ruiz, Sergio Horta Muñoz and Reyes García-Contreras

Mathematics 2022, 10(15), 2753; https://doi.org/10.3390/math10152753 - 3 Aug 2022

Cited by 2 | Viewed by 1687

Abstract

Piezoelectric actuators and sensors are applied in many fields in order to produce forces or displacements with the aim of sensing, manipulating or measurement, among other functions. This study presents the numerical methodology to optimize the static response of a thick-shell structure consisting [...] Read more.

Piezoelectric actuators and sensors are applied in many fields in order to produce forces or displacements with the aim of sensing, manipulating or measurement, among other functions. This study presents the numerical methodology to optimize the static response of a thick-shell structure consisting of piezoelectric sensors, based on the maximisation of the electric charge while controlling the amount of piezoelectric and material required. Two characteristic functions are involved, determining the topology of the sensor and the polarisation profile. Constraints over the reaction force are included in the optimisation problem in order to avoid singularities. The topology optimisation method is used to obtain the optimal results, where regularisation techniques (density filtering and projection) are used to avoid hinges. The minimum length scale can be controlled by the use of three different projections. As the main novelty, a displacement-controlled scheme is proposed in order to generate a robust algorithm for future studies including non-linearities. Full article

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

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19 pages, 6380 KiB

Open AccessArticle

Failure Transition and Validity of Brazilian Disc Test under Different Loading Configurations: A Numerical Study

by Peng Xiao, Guoyan Zhao and Huanxin Liu

Mathematics 2022, 10(15), 2681; https://doi.org/10.3390/math10152681 - 29 Jul 2022

Cited by 10 | Viewed by 2451

Abstract

The Brazilian disc test is a popular tensile strength test method for engineering materials. The fracture behavior of specimens in the Brazilian disc test is closely related to the validity of the test results. In this paper, the fracture process of granite discs [...] Read more.

The Brazilian disc test is a popular tensile strength test method for engineering materials. The fracture behavior of specimens in the Brazilian disc test is closely related to the validity of the test results. In this paper, the fracture process of granite discs under different loading configurations is simulated by using a coupled finite–discrete element method. The results show that the maximum tensile stress value is located within 18 mm (0.7 times the disc radius) of the vertical range of the disc center under different loading configurations. In small diameter rods loading, the invalid tensile strength is obtained because the crack initiation and plastic strain is at the end of the disc. The crack initiation points of flat platen loading and curved jaws loading are all within the center of the disc, and the valid tensile strength can be obtained. The tensile strength test results under different loading configurations show that the error of small diameter rods loading is 13%, while the errors of flat platen loading and curved jaws loading are both 1%. The curved jaws loading is the most suitable for measuring the tensile strength of brittle materials such as rock, followed by flat platen loading. The small diameter rods loading is not recommended for the Brazilian test. Full article

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

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17 pages, 774 KiB

Open AccessArticle

Solving Inverse Conductivity Problems in Doubly Connected Domains by the Homogenization Functions of Two Parameters

by Jun Lu, Lianpeng Shi, Chein-Shan Liu and C. S. Chen

Mathematics 2022, 10(13), 2256; https://doi.org/10.3390/math10132256 - 27 Jun 2022

Cited by 2 | Viewed by 1552

Abstract

In the paper, we make the first attempt to derive a family of two-parameter homogenization functions in the doubly connected domain, which is then applied as the bases of trial solutions for the inverse conductivity problems. The expansion coefficients are obtained by imposing [...] Read more.

In the paper, we make the first attempt to derive a family of two-parameter homogenization functions in the doubly connected domain, which is then applied as the bases of trial solutions for the inverse conductivity problems. The expansion coefficients are obtained by imposing an extra boundary condition on the inner boundary, which results in a linear system for the interpolation of the solution in a weighted Sobolev space. Then, we retrieve the spatial- or temperature-dependent conductivity function by solving a linear system, which is obtained from the collocation method applied to the nonlinear elliptic equation after inserting the solution. Although the required data are quite economical, very accurate solutions of the space-dependent and temperature-dependent conductivity functions, the Robin coefficient function and also the source function are available. It is significant that the nonlinear inverse problems can be solved directly without iterations and solving nonlinear equations. The proposed method can achieve accurate results with high efficiency even for large noise being imposed on the input data. Full article

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

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25 pages, 2257 KiB

Open AccessArticle

epSFEM: A Julia-Based Software Package of Parallel Incremental Smoothed Finite Element Method (S-FEM) for Elastic-Plastic Problems

by Meijun Zhou, Jiayu Qin, Zenan Huo, Fabio Giampaolo and Gang Mei

Mathematics 2022, 10(12), 2024; https://doi.org/10.3390/math10122024 - 11 Jun 2022

Cited by 2 | Viewed by 2527

Abstract

In this paper, a parallel Smoothed Finite Element Method (S-FEM) package epSFEM using incremental theory to solve elastoplastic problems is developed by employing the Julia language on a multicore CPU. The S-FEM, a new numerical method combining the Finite Element Method (FEM) and [...] Read more.

In this paper, a parallel Smoothed Finite Element Method (S-FEM) package epSFEM using incremental theory to solve elastoplastic problems is developed by employing the Julia language on a multicore CPU. The S-FEM, a new numerical method combining the Finite Element Method (FEM) and strain smoothing technique, was proposed by Liu G.R. in recent years. The S-FEM model is softer than the FEM model for identical grid structures, has lower sensitivity to mesh distortion, and usually produces more accurate solutions and a higher convergence speed. Julia, as an efficient, user-friendly and open-source programming language, balances computational performance, programming difficulty and code readability. We validate the performance of the epSFEM with two sets of benchmark tests. The benchmark results indicate that (1) the calculation accuracy of epSFEM is higher than that of the FEM when employing the same mesh model; (2) the commercial FEM software requires 10,619 s to calculate an elastoplastic model consisting of approximately 2.45 million triangular elements, while in comparison, epSFEM requires only 5876.3 s for the same computational model; and (3) epSFEM executed in parallel on a 24-core CPU is approximately 10.6 times faster than the corresponding serial version. Full article

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

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23 pages, 4763 KiB

Open AccessArticle

A Simplified Radial Basis Function Method with Exterior Fictitious Sources for Elliptic Boundary Value Problems

by Chih-Yu Liu and Cheng-Yu Ku

Mathematics 2022, 10(10), 1622; https://doi.org/10.3390/math10101622 - 10 May 2022

Cited by 6 | Viewed by 1949

Abstract

In this article, we propose a simplified radial basis function (RBF) method with exterior fictitious sources for solving elliptic boundary value problems (BVPs). Three simplified RBFs, including Gaussian, multiquadric (MQ), and inverse multiquadric (IMQ) without the shape parameter, are adopted in this study. [...] Read more.

In this article, we propose a simplified radial basis function (RBF) method with exterior fictitious sources for solving elliptic boundary value problems (BVPs). Three simplified RBFs, including Gaussian, multiquadric (MQ), and inverse multiquadric (IMQ) without the shape parameter, are adopted in this study. With the consideration of many exterior fictitious sources outside the domain, the radial distance of the RBF is always greater than zero, such that we can remove the shape parameter from RBFs. Additionally, simplified Gaussian, MQ, and IMQ RBFs and their derivatives in the governing equation are always smooth and nonsingular. Comparative analysis is conducted for three different collocation types, including conventional uniform centers, randomly fictitious centers, and exterior fictitious sources. Numerical examples of elliptic BVPs in two and three dimensions are carried out. The results demonstrate that the proposed simplified RBFs with exterior fictitious sources can significantly improve the accuracy, especially for the Laplace equation. Furthermore, the proposed simplified RBFs exhibit the simplicity of solving elliptic BVPs without finding the optimum shape parameter. Full article

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

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22 pages, 7566 KiB

Open AccessArticle

Reliability-Based Topology Optimization of Thermo-Elastic Structures with Stress Constraint

by Liang Zhang, Qinghai Zhao and Jianliang Chen

Mathematics 2022, 10(7), 1091; https://doi.org/10.3390/math10071091 - 28 Mar 2022

Cited by 12 | Viewed by 2357

Abstract

Traditional topology optimization of thermo-elastic structures is based on deterministic conditions, without considering the influence of uncertainty factors. To address the impact uncertainty on structural strength, a reliability-based topology optimization of thermo-elastic structure with stress constraint is proposed. The probabilistic uncertainty quantities are [...] Read more.

Traditional topology optimization of thermo-elastic structures is based on deterministic conditions, without considering the influence of uncertainty factors. To address the impact uncertainty on structural strength, a reliability-based topology optimization of thermo-elastic structure with stress constraint is proposed. The probabilistic uncertainty quantities are associated with the structural material property, mechanical loads and the thermal stress coefficient with the topology optimization formulation considering volume minimization and stress constraint. The relaxation stress method combined with normalized p-norm function is adopted to condense whole element stresses into the global stress measurement that approximates the maximum stress. The adjoint variable method is utilized to derive the sensitivity of the stress constraint and the optimization problem is solved by the method of moving asymptote (MMA). Finally, several numerical examples are presented to demonstrate the effectiveness and validity of the proposed approach. Compared with the deterministic design, the reliability design has distinct topological configurations and the optimized structures maintain a higher reliability level. Full article

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

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9 pages, 2197 KiB

Open AccessArticle

A Simple, Accurate and Semi-Analytical Meshless Method for Solving Laplace and Helmholtz Equations in Complex Two-Dimensional Geometries

by Xingxing Yue, Buwen Jiang, Xiaoxuan Xue and Chao Yang

Mathematics 2022, 10(5), 833; https://doi.org/10.3390/math10050833 - 5 Mar 2022

Cited by 2 | Viewed by 2463

Abstract

A localized virtual boundary element–meshless collocation method (LVBE-MCM) is proposed to solve Laplace and Helmholtz equations in complex two-dimensional (2D) geometries. “Localized” refers to employing the moving least square method to locally approximate the physical quantities of the computational domain after introducing the [...] Read more.

A localized virtual boundary element–meshless collocation method (LVBE-MCM) is proposed to solve Laplace and Helmholtz equations in complex two-dimensional (2D) geometries. “Localized” refers to employing the moving least square method to locally approximate the physical quantities of the computational domain after introducing the traditional virtual boundary element method. The LVBE-MCM is a semi-analytical and domain-type meshless collocation method that is based on the fundamental solution of the governing equation, which is different from the traditional virtual boundary element method. When it comes to 2D problems, the LVBE-MCM only needs to calculate the numerical integration on the circular virtual boundary. It avoids the evaluation of singular/strong singular/hypersingular integrals seen in the boundary element method. Compared to the difficulty of selecting the virtual boundary and evaluating singular integrals, the LVBE-MCM is simple and straightforward. Numerical experiments, including irregular and doubly connected domains, demonstrate that the LVBE-MCM is accurate, stable, and convergent for solving both Laplace and Helmholtz equations. Full article

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

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