Method Of Fundamental Solutions Research Articles

Novel numerical analysis of electric fields in the near field of very‐low‐frequency antennas based on the charge simulation method

AbstractVery‐low‐frequency (VLF) transmitting antennas have a crucial influence on the nearby electromagnetic environment. Commercial software packages, such as CST Studio Suite (CST) or Altair FEKO (FEKO), are always utilised to calculate the electric fields of VLF antennas, which consume considerable computing resources and time. The classical method of electric field calculation, the charge simulation method (CSM), is first applied to the calculation of electric field near the VLF transmitting antenna. The distribution characteristics of three‐dimensional electric fields were analysed. The electric field surrounding the antenna exhibited axial symmetry. The highest electric field strength was at the top of the antenna, and as the distance from the antenna increased, the electric field strength exponentially decreased. The comparisons are performed between the results of the CSM and those of commercial software, the average relative errors of CSM on two observation paths are very small, 2.20% and 2.21%, respectively. Meanwhile, the CSM shows considerable advantages in saving calculation resources and improving efficiency. Additionally, the comparison between CSM calculation and the measured data is carried out. The results show that this method can accurately calculate the electric field near VLF transmitting antenna and offer valuable insights and a new solution.

Stochastic Simulation Algorithm for Solving the System of Lame Equations for Two- and Three-Dimensional Domains by Combining the Slobodianskii Representation, the Method of Fundamental Solutions and a Stochastic Projection Method

Stochastic Simulation Algorithm for Solving the System of Lame Equations for Two- and Three-Dimensional Domains by Combining the Slobodianskii Representation, the Method of Fundamental Solutions and a Stochastic Projection Method

Numerical Analysis of the Plateau Problem by the Method of Fundamental Solutions

Toward identifying the number of minimal surfaces sharing the same boundary from the geometry of the boundary, we propose a numerical scheme with high speed and high accuracy. Our numerical scheme is based on the method of fundamental solutions. We establish the convergence analysis for Dirichlet energy and L∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^\\infty $$\\end{document}-error analysis for mean curvature. Each of the approximate solutions in our scheme is a smooth surface, a significant difference from previous studies that required mesh generation.

An inverse method of fundamental solutions for the identification of 2D elastic properties of anisotropic solids

An inverse method of fundamental solutions for the identification of 2D elastic properties of anisotropic solids

A novel hybrid SBM-MFS methodology for acoustic wave propagation problems

In this paper, a novel hybrid meshless approach that combines the singular boundary method (SBM) and the method of fundamental solutions (MFS) to deal with two-dimensional (2D) exterior acoustic wave propagation problems is proposed and studied. The methodology is particularly devised to solve problems with complex boundary geometries containing geometric singularities such as corners and sharp edges. It employs the SBM to model intricate segments of these geometries and the MFS for the smooth ones. The proposed hybrid SBM-MFS method is studied in a 2D context in the framework of three benchmark examples involving acoustic radiation problems of circular-, square- and L-shaped objects in a full-space acoustic medium. In addition, the applicability of the proposed hybrid SBM-MFS methodology to predict the acoustic performance of a T-shaped thin barrier is also investigated. These examples are specifically designed to assess the feasibility, validity and accuracy of the hybrid SBM-MFS approach in comparison with the available analytical solutions and alternative numerical strategies such as the MFS, the SBM and the boundary element method (BEM). Numerical simulations demonstrate that the proposed method can match the level of accuracy of the MFS while keeps the robustness of the SBM when dealing with complex geometries, overcoming one the most important drawbacks of the traditional MFS. Moreover, the proposed hybrid SBM-MFS naturally avoids the non-uniqueness problem arising at the fictitious eigenfrequencies associated with the corresponding interior problems, a feature that neither the SBM nor the BEM inherently possesses.

A 2.5D hybrid SBM-MFS methodology for elastic wave propagation problems

This paper proposes a novel hybrid methodology that combines the singular boundary method (SBM) and the method of fundamental solutions (MFS) for the computational simulation of elastic wave propagation. Particularly, the methodology aims to address radiation or scattering problems of longitudinally invariant systems in the wavenumber–frequency domain involving boundaries with intricate geometries. The approach uses the SBM to deal with the complex parts of these geometries and the MFS for the smooth ones. The method is studied in the framework of three case studies involving longitudinally infinite cavities in a homogeneous full-space with circular, square-shaped and five-cusped hypocycloid cross-sections. These three examples are selected to assess the accuracy and robustness of the hybrid SBM-MFS approach in comparison with alternative modelling strategies. The comparisons show that the proposed method inherits the accuracy of the MFS while keeping the robustness of the SBM when dealing with complex geometries. The method is found to be computationally more efficient than the SBM or the boundary element method (BEM). Moreover, the hybrid approach naturally mitigates the effect of fictitious eigenfrequencies, a feature that neither conventional versions of the SBM nor the BEM have.

An Eulerian–Lagrangian method of fundamental solutions for the advection–diffusion equation with time dependent coefficients

This paper investigates the application of the Eulerian–Lagrangian method of fundamental solutions (ELMFS), a truly meshless method that couples the Euler–Lagrange Method (ELM) with the time-dependent method of fundamental solution (MFS). It is applied to solve the transient advection–diffusion equation with transport coefficients exhibiting constant and temporally exponential behaviors. The stability and accuracy of the proposed method were analyzed in terms of condition numbers, location of source points, and error metrics. The stability results for constant coefficients of partial differential equations are presented in a map and compared with well-established stability criteria for explicit in-time mesh methods, such as the Courant–Friedrichs–Lewy and Von Neumann criteria. These comparisons demonstrate that ELMFS has larger stable regions.

Application of the localized method of fundamental solutions to heat transfer problems

A localized version of the Method of Fundamental Solutions is applied to the 2D steady heat transfer equation with spatially varying thermal conductivity. Though the corresponding fundamental solution cannot be computed in general, the localization splits the original problem into several subproblems defined on small subdomains; in these subdomains, the subproblems are approximately converted to certain convection-diffusion equations with constant coefficients, so that the Method of Fundamental Solutions is applicable. In each subdomain, the corresponding subproblem is solved separately. This results in an iterative method, which mimics the overlapping (alternating) Schwarz method. Due to its advantageous numerical properties, the technique seems a useful generalization of the Method of Fundamental Solutions. The method is illustrated through numerical examples.

Improved linear notch mechanics for lightweight design using MFS

This paper is a study of a new prediction method for accurately evaluating the brittle fracture strength of notched materials with multiple holes using the methods of fundamental solutions(MFS). Conventional linear notch mechanics (LNM) was proposed in 1980’s to clarify the distinction between the fracture and non-fracture zones using a master-curve. It is effective for evaluating the fracture strength of notched materials with a single hole. When multiple holes are adjacent to each other, the conventional LNM is less accurate in predicting the fracture strength. To eliminate the shortcomings, I proposed the improved linear notch mechanics (ILNM), which uses the stress gradient χ at the bottom of the notch instead of a notch radius ρ. For lightweight design, the applicability of ILNM and its effectiveness are verified for notched materials with multiple holes by some experiments and some calculations.

Solution of the Cauchy problem for the Brinkman equations using an alternating method of fundamental solutions

Solution of the Cauchy problem for the Brinkman equations using an alternating method of fundamental solutions

Online estimation of inlet contaminant concentration using Eulerian-Lagrangian method of fundamental solutions and Bayesian inference

The advection-diffusion equation is fundamental to modeling various transport phenomena, including the distribution of chemical species in surface or groundwater flow. In cases where the concentration at the source is unknown, inverse problem formulations are required to estimate the desired states by assimilating concentration monitoring data from specific points along the watercourse using a Bayesian approach. This paper proposes a combination of the Eulerian-Lagrangian method and the meshless method of fundamental solutions to solve the advection-diffusion equation. Moreover, the method was used as an evolution model for the sequential importance resampling particle filter algorithm to reconstruct the time-dependent inlet pollutant concentration in inverse problems. Numerical smooth and discontinuous inlet function results show that the particle filter - Eulerian-Lagrangian method of fundamental solutions combination can reconstruct inlet concentration time series.

A Novel Meshfree Method for Nonlinear Equations in Flow through Porous Media and Electrohydrodynamic Flows

In this study, an efficient meshfree numerical method is introduced for solving the nonlinear boundary value problems. The method of fundamental solutions (MFS) is one of the most popular among meshfree methods. While traditionally limited to linear and homogeneous problems, this study extends the applicability of the MFS to include nonhomogeneous and nonlinear equations. To achieve this, an extended MFS is combined with a fixed-point iteration scheme. This developed framework is benchmarked to address two different flow problems. The first involves fluid flow through porous media in a channel governed by the nonlinear Brinkman-Forchheimer equation. The second problem pertains to electrohydrodynamic (EHD) flow in a circular conduit. The obtained solutions are compared with the finite element method and the solutions available in the existing literature.

A decomposition method for two and three dimensional fluid-solid interaction scattering problem

In this paper, we propose a decomposition formula for the fluid-solid interaction scattering problem. By the decomposition of the displacement, the corresponding problem is transformed into a coupled boundary value problem. For the two dimensional case, the coupled boundary value problem is about two scalar Helmholtz equations, and the other one is a scalar Helmholtz equation and a vector Helmholtz equation for the three dimensional. Then the coupled boundary value problem is solved by the method of fundamental solution. At last we give some examples to show the effectiveness of the present method in two and three dimensional case. A benchmark example is available for checking the feasibility and accuracy of the presented method. In the numerical examples, a part of exact solution for the boundary value problem is including for checking the accuracy.

Method of fundamental solutions: New approximation results and applications

The approximation properties of shifted fundamental solutions are stated for a general class of second order linear elliptic differential operators. Motivated by the method of the fundamental solutions, the singularities of the fundamental solutions are located in the vicinity of a fixed surface, which is the boundary of a 3-dimensional Lipschitz domain. The approximation is estimated first in Ck-norm. The corresponding convergence rate is determined depending on the density of the singularities. This delivers a solid theoretical basis for the method of fundamental solutions. The results are extended to exterior domains and as a new application, the numerical solution of Dirichlet-to-Neumann problems are constructed and analyzed on unbounded domains. The efficiency of this new method is demonstrated in series of numerical experiments.

Mitigating train-induced building vibrations with rubber bearings designed for horizontal earthquake isolation

As rail transportation, including high-speed railways and urban metros, rapidly expands, its proximity to residential areas decreases, raising concerns about ground-borne vibrations. This study investigates the main characteristics of ground-borne vibrations induced by high-speed railways and metros through substructure method based on 2.5D FEM-MFS (finite element method-the method of fundamental solutions). Besides, the study also delves into the vertical vibration behaviors of buildings caused by train operations, evaluating the effect of laminated rubber bearings designed for horizontal seismic isolation on the train-induced building vibration mitigation. Our findings highlight that the broadband vibrations from train operations might coincide with the natural vibration frequencies of buildings, potentially leading to resonance. Implementing isolators allows the natural frequencies of the vibration modes with larger effective masses to be significantly lower than the dominant frequencies of train-induced ground-borne vibration, reducing the high-frequency responses. Additionally, the vibration mitigation enhances with the increase in the thickness of rubber layer. This research not only aims to reduce vibration levels in buildings adjacent to railways, enhancing the living conditions and health of residents but also underscores the engineering innovations necessary for sustainable urban development.

Improved dynamic analysis for plane elasticity problems with a boundary meshfree method and distributed sources

In this paper, we present an improved dynamic analysis method to address plane elasticity problems. Distributed sources are employed in a boundary-free mesh-based approach known as Boundary Distributed Source (BDS). This method, which builds upon a modified Method of Fundamental Solutions (MFS), aims to eliminate singularities without the reliance on fictitious boundaries. Distributed sources are strategically positioned directly on the real boundary, centered over disks. The method's efficiency and simplicity are highlighted by its ability to operate without the need for precise data concerning adjacent points. The dual reciprocity method (DRM) is utilized to approximate the non-homogeneous inertial term. Furthermore, the present formulation investigates several factors that influence the convergence of the proposed method, including the number of collocation points, internal points, the radius of the circular disk, and the time step size. Based on the numerical results, it is evident that the method exhibits remarkable accuracy and efficiency when applied to plane elasticity problems including forced, harmonic, and free analyses. As compared to conventional numerical methods for dynamic analysis, the method outperforms in both accuracy and convergence rate.

A Viewpoint on Thermally-Induced Transport in Rarefied Gases through the Method of Fundamental Solutions

Some phenomena pertaining to rarefied gases are beyond the reach of traditional fluid dynamics described, e.g., by the Euler or Navier–Stokes–Fourier equations. Therefore we adopt a recently developed model—referred to as the CCR model—to investigate thermally-induced transport in rarefied gases. To this end, the method of fundamental solutions is employed on the CCR model to investigate two problems: (i) a rarefied gas flow confined between two coaxial cylinders having different temperatures with the inner cylinder being circular while the outer being elliptical, and (ii) evaporation/condensation process in a rarefied vapor confined between two coaxial cylinders, again with the inner cylinder being circular and the outer being elliptical. Through a comprehensive analysis, the efficiency of the method of fundamental solutions is assessed. The work contributes toward a better understanding of thermally-induced confined rarefied gas flows.

Modeling of Chemical Vapor Infiltration for Fiber-Reinforced Silicon Carbide Composites Using Meshless Method of Fundamental Solutions

In this study, the Method of Fundamental Solutions (MFSs) is adopted to model Chemical Vapor Infiltration (CVI) in a fibrous preform. The preparation of dense fiber-reinforced silicon carbide composites is considered. The reaction flux at the solid surface is equal to the diffusion flux towards the surface. The Robin or third-type boundary condition is implemented into the MFS. From the fibers’ surface concentrations obtained by MFS, deposition rates are calculated, and the geometry is updated at each time step, modeling the pore filling over time. The MFS solution is verified by comparing the results to a known analytical solution for a simplified geometry of concentric cylinders with a concentration set at the outer cylinder and a reaction at the inner cylinder. MFS solutions are compared to published experimental data. Porosity transients are obtained by a combination of MFSs with surface deposition to show the relation between the initial and final porosities.

An efficient localized Trefftz method for the simulation of two-dimensional sloshing behaviors

This study proposes a new method for effectively and accurately simulating sloshing in a two-dimensional numerical tank. This technique uses the localized Trefftz method (LTM), a meshless numerical approach. Sloshing in a numerical tank is mathematically described as a space-time boundary-value problem rooted in potential flow theory. This occurrence is governed by a second-order partial differential equation and two nonlinear free-surface boundary conditions. In this study, the moving boundary problem undergoes discretization in time and space by using the LTM and the explicit Euler method, respectively. After discretization with the explicit Euler method, the elevation of the free surface is updated, and a boundary-value problem is generated at each time step. This boundary-value problem can be effectively examined using the recently developed LTM, which eliminates the need for complex meshing. Contrary to conventional methods, such as the method of fundamental solutions and the boundary element method, the LTM generates sparse matrices, allowing large-scale numerical simulations. Four numerical examples are presented to demonstrate the clarity and accuracy of the proposed meshless approach.

Wave scattering problems in exterior domains with the method of fundamental solutions

The method of fundamental solutions has been mainly applied to wave scattering problems in bounded domains and to our knowledge there have not been works addressing density results for general shapes, or addressing the calculation of the complex resonance frequencies that occur in exterior problems. We prove density and convergence of the fundamental solutions approximation in the context of wave scattering problems, with and without a priori knowledge of the frequency, which is of particular importance to detect resonance frequencies for trapping domains. We also present several numerical results that illustrate the good performance of the method in the calculation of complex resonance frequencies for trapping and non trapping domains in 2D and 3D.