Method Of Fundamental Solutions Research Articles

The meshless method of fundamental solutions applied to couple stress problems

In this paper, the meshless method of fundamental solutions is developed to solve couple stress problems even with the existence of body forces. Suitable new particular solutions are derived to deal with the non-homogeneous partial differential equations keeping the meshless essence of the developed technique. The derivation of the new particular solutions for generalizing displacements and tractions are discussed in detail. Several examples including stress concentration and problems with body forces are tested to demonstrate the efficiency and strength of the derived formulation.

The method of fundamental solutions for the high frequency acoustic-elastic problem and its relationship to a pure acoustic problem

This paper firstly documents the method of fundamental solutions for the two dimensional high frequency acoustic-elastic interaction problem, and investigates its relationship to a pure acoustic problem with a sound-hard boundary. We approximate the displacements by the linear combination of the fundamental matrix tensors of the Navier equation. The acoustic scattered field outside the elastic solid is approximated by the linear combination of the fundamental solutions of the Helmholtz equation. At last, we give some examples using the real data in four cases, including the copper alloy in the water, in the oil, in the cork wood and in the air. Especially in the first example, an exact solution is available for checking the accuracy of the presented method. We also find that the acoustic-elastic interaction problem is close to a pure acoustic problem with a sound-hard boundary as the ratio of two medium density increasing.

Approximation of three‐dimensional nonlinear wave equations by fundamental solutions and weighted residuals process

In this paper, the localized method of fundamental solutions (LMFS) with coupling the dual reciprocity method (DRM) is applied to simulate three‐space dimensional nonlinear wave equations. First, DRM, which is a popular meshless method based on polyharmonic splines (PhS), is applied to obtain the particular solution. After evaluating the particular solution, 3D‐LMFS method can be employed to evaluate the homogeneous solution. The proposed 3D‐LMFS algorithms construct 3D artificial surfaces where source points and then the collocation procedure propose by using the fundamental solutions in each 3D surface. Straightforwardly, a solution to the 3D wave equation is approximated by a sum of the combination of PhS and fundamental solutions using LMFS. Eventually, the scheme has been prosperously tested with selected examples.

Fundamental solutions of an extended hydrodynamic model in two dimensions: Derivation,theory,andapplications.

The inability of the Navier-Stokes-Fourier equationsto capture rarefaction effects motivates us to adopt the extended hydrodynamic equations. In the present work, a hydrodynamic model, which consists of the conservation laws closed with the recently propounded coupled constitutive relations (CCR), is utilized. This model is referred to as the CCR model and is adequate for describing moderately rarefied gas flows. A numerical framework based on the method of fundamental solutions is developed to solve the CCR model for rarefied gas flow problems in quasi two dimensions. To this end, the fundamental solutions of the linearized CCR model are derived in two dimensions. The significance of deriving the two-dimensional fundamental solutions is that they cannot be deduced from their three-dimensional counterparts that do exist in literature. As applications, the developed numerical framework based on the derived fundamental solutions is used to simulate (i) a rarefied gas flow between two coaxial cylinders with evaporating walls and (ii) a temperature-driven rarefied gas flow between two noncoaxial cylinders. The results for both problems have been validated against those obtained with the other classical approaches. Through this, it is shown that the method of fundamental solutions is an efficient tool for addressing quasi-two-dimensional multiphase microscale gas flow problems at a low computational cost. Moreover, the findings also show that the CCR model solved with the method of fundamental solutions is able to describe rarefaction effects, like transpiration flows and thermal stress, generally well.

Transient Source Terms Estimate in Advection-Diffusion Problems Using the Method of Fundamental Solutions and an Eulerian-Lagrangian Transformation

In this work, transient source terms in advection-diffusion problems are recovered. The solution technique proposed relies on the implicit finite difference method coupled with the method of fundamental solutions, a technique for solving partial differential equations when their fundamental solution is known. The Truncated Singular Value Decomposition was used to overcome the ill-conditioned resulting linear system. The advection-diffusion inverse problem was solved using a Eulerian–Lagrangian method. Several numerical examples were considered for testing purposes by using simulated measurements perturbed with a Gaussian random noise for different Reynolds numbers. Results show that the proposed method provides stable and accurate estimates for the source term.

Insertion loss prediction of sonic crystal noise barriers covered by porous concrete using the Method of Fundamental Solutions

Noise attenuation by sonic crystal noise barriers occurs mainly in specific frequency bands (bandgaps), due to a mechanism called Bragg scattering, which is the result of destructive interference between multiple reflections. To improve their performance, sound-absorbing materials may be used to coat the rigid scatterers, increasing attenuation in other frequency bands. In this work, the Method of Fundamental Solutions (MFS) is used to evaluate the performance of sonic crystal noise barriers with cylinders covered by porous and granular materials simulating the volume of the absorbent material using equivalent fluid models for porous or granular materials. The proposed numerical models were validated by experimental tests and, afterwards, these numerical models were verified through the comparison with results obtained using the Finite Element Method (FEM). Finally, several numerical tests were performed in order to understand/predict the acoustic behavior of the sonic crystal noise barriers covered with porous concrete.

Numerical Simulations of Tank Sloshing Problems Based on Moving Pseudo-Boundary Method of Fundamental Solution

The moving pseudo-boundary method of fundamental solutions (MFS) was employed to solve the Laplace equation, which describes the potential flow in a two-dimensional (2D) numerical wave tank. The MFS is known for its ease of programming and the advantage of its high precision. The solution of the boundary value can be expressed by a linear combination of the fundamental solutions. The major issue with such an implementation is the optimal distribution of source nodes in the pseudo-boundary. Traditionally, the positions of the source nodes are assumed to be fixed to keep the set of equations closed. However, in the moving boundary problem, the distribution of source nodes may influence the stability of numerical calculations. Moreover, MFS is unstable in time iterations. Hence, it is necessary to constantly revise the weighting coefficients of fundamental solutions. In this study, the source nodes were free, and their locations were determined by solving a nonlinear least-squares problem using the Levenberg–Marquardt algorithm. To solve the above least-squares problem, the MATLAB© routine lsqnonlin was adopted. Additionally, the weighting coefficients of fundamental solutions were solved as a nonlinear least-squares problem using the aforementioned method. The numerical results indicated that the numerical simulation method adopted in this paper is accurate and reliable in solving the problem of 2D tank sloshing. The main contribution of this study is to expand the application of the MFS in engineering by integrating it with the optimal configuration problem of pseudo-boundaries to solve practical engineering problems.

Special finite elements with arbitrarily-shaped holes for heat conduction analysis of porous materials

Special finite elements are developed which adequately represent the state of heat conduction in the region around an arbitrarily-shaped hole. This is achieved with two assumed temperature fields. The frame field is defined along the element boundary while the intra-element field is independently approximated by using truncated T-complete solutions as trial functions for capturing hole effects. The geometrical irregularity of holes contribute to the difficulty in seeking T-complete solutions. By means of the charge simulation method and numerical conformal mapping technique, the systematic construction of T-complete solutions is shown which satisfy a priori the governing differential equation. Based on the modified variational principle, the two independent temperature fields are linked together at the element level, ensuring conformity across inter-element boundaries. The resultant element stiffness equation involves boundary integrals only, which provides a great flexibility in mesh generation with polygons. Numerical examples for the problems with a single, dual, and multiple holes are given to illustrate the efficiency and accuracy of the proposed method.

Numerical Solution of the Electric Field and Dielectrophoresis Force of Electrostatic Traveling Wave System.

Electrostatic traveling wave (ETW) methods have shown promising performance in dust mitigation of solar panels, particle transport and separation in in situ space resource utilization, cell manipulation, and separation in biology. The ETW field distribution is required to analyze the forces applied to particles and to evaluate ETW design parameters. This study presents the numerical results of the ETW field distribution generated by a parallel electrode array using both the charge simulation method (CSM) and the boundary element method (BEM). A low accumulated error of the CSM is achieved by properly arranging the positions and numbers of contour points and fictitious charges. The BEM can avoid the inconvenience of the charge position required in the CSM. The numerical results show extremely close agreement between the CSM and BEM. For simplification, the method of images is introduced in the implementation of the CSM and BEM. Moreover, analytical formulas are obtained for the integral of Green's function along boundary elements. For further validation, the results are cross-checked using the finite element method (FEM). It is found that discrepancies occur at the ends of the electrode array. Finally, analyses are provided of the electric field and dielectrophoretic (DEP) components. Emphasis is given to the regions close to the electrode surfaces. These results provide guidance for the fabrication of ETW systems for various applications.

A meshless regularized method of fundamental solution for electromagnetic scattering problems of three-dimensional perfect electric conductor targets

In this study, a meshless regularized method of fundamental solution (RMFS) is proposed to solve the electromagnetic scattering problem for three-dimensional perfect electric conductor (PEC) targets. The method is based on integral equation and constructed solutions using de-singularized subtraction and add-back techniques to regularize the singularity and hyper-singularity of the fundamental solution. The performance of the RMFS in terms of convergence and error is evaluated and compared with the Mie series solution, MFS, and the EM simulation software FEKO on the PEC sphere. Finally, several test examples are given. The results show the potential of RMFS in solving three-dimensional electromagnetic scattering problems.

Self-Consistent Positive Streamer-Leader Propagation Model Based on Finite Element Method (FEM) and Voltage Distortion Method (VDM)

Researchers have worked on positive leader propagation models and proposed different theoretical and numerical approaches. The charge simulation method (CSM) has traditionally been chosen to model the quasi-static electric field of each stage of leader propagation. The biggest drawback of the CSM is that the calculation is complicated and time-consuming when dealing with asymmetric electric field structures. On the contrary, the finite element method (FEM) is more suitable and reliable for solving electrostatic field problems with asymmetric and complex boundary conditions, avoiding the difficulties of virtual charge configuration and electric field calculation under complex boundary conditions. This paper modeled a self-consistent streamer-leader propagation model in an inverted rod-plane air gap based on FEM and the voltage distortion method (VDM). The voltage distortion coefficient was analyzed to calculate the streamer length and space charge. The physical dynamic process of the discharge was simulated with the help of COMSOL Multiphysics and MATLAB co-simulation technology. The results show that the initial voltage of the first corona is -1036 kV, close to the experiment value of -1052 kV. The breakdown voltage of -1369 kV is highly consistent with the experimental value of -1365 kV. The largest streamer length is 2.72 m, slightly higher than the experimental value of 2.3 m. The leader velocity is 2.43×104 m/s, close to the experiment value of 2.2×104 m/s. This model has simple calculations and can be used in complex electrode configurations and arbitrary boundary conditions without simplifying the model structure, making the model more flexible.

The method of fundamental solutions for electromagnetic scattering by a perfect conductor in chiral environment

The electromagnetic scattering problem for a time-harmonic plane wave by a perfectly conductive body embedded in a chiral medium is considered. Potentials in terms of the fundamental solution in dyadic form for chiral media are defined. The well-posedness of the scattering problem is studied using defined potentials in terms of the fundamental solution for chiral environment and the boundary integral operators associated with them. An extension of the method of fundamental solutions for electromagnetic scattering in chiral media is applied in order to approximate the solution. The electric scattered field is approximated by a finite linear combination of the dyadic fundamental solutions of the equation governing the problem for singularities placed on an auxiliary surface outside the domain of the problem and for unknown coefficients determined by the boundary conditions. Density properties on the surface of the scatterer are proven for a defined system of functions consisting of the tangential components of the dyadic fundamental solution. These density properties together with the well-posedness of the scattering problem are used to reduce the approximation of the solution to the approximation of the known boundary data. The linear system obtained from the boundary conditions in order to determine the appropriate unknown coefficients of the finite linear combination is presented. Using these results we approximate the far-field patterns and the scattering cross section. Numerical results for the special case of a spherical scatterer and an ellipsoidal scatterer are presented.

Electroelastic Analysis of Two-Dimensional Piezoelectric Structures by the Localized Method of Fundamental Solutions

Electroelastic Analysis of Two-Dimensional Piezoelectric Structures by the Localized Method of Fundamental Solutions

A local meshfree radial point interpolation method for Berger equation arising in modelling of thin plates

In this study, two-dimensional (2D) Berger equation arising in deflection of thin plates is solved numerically. A local meshfree collocation technique based on radial point interpolation is developed for the Berger equation. As radial basis function thin plate splines are used to avoid selecting optimal shape parameter. The method is used to solve both linear and non-linear Berger equation on regular and irregular geometries. Obtained results during numerical simulations, are compared with different methods exist in literatrue such as method of fundamental solutions, local Kansa’s method and Pascal polynomial based meshless method. The comparisons affirm the accuracy and reliability of the suggested method.

Simulation Study on Safe Distance between Crane Boom and High-Voltage Line Based on Electric Field Intensity

Aiming at the safety accident caused by the close distance between the crane arm and the high-voltage line when the crane is working near the high-voltage transmission line, this paper proposes a simulation model of the electric field intensity distribution of the high-voltage transmission line. First, through the simplification and equivalent treatment of the high-voltage line, the simulation modeling, and analysis of the electric field around the high-voltage line are carried out. Then, the theory of solving the electric field intensity of any point around the high-voltage line based on the charge simulation method is deduced, and the conclusion that the electric field intensity at the equal distance of the high-voltage line is approximately equal is obtained. Furthermore, the influence of the crane on the electric field strength of a high-voltage line is analyzed, and the linear correction function of electric field strength is obtained. Based on the research in this paper, the field strength sensor can be used to obtain the value of the electric field strength, calculate the distance between the crane and the high-voltage line, and make a safety warning response. The simulation results verify the feasibility and effectiveness of the proposed model and method.

New regularization techniques for ill-conditioning problems and their applications: Choices of regularization parameters

This is a continued study of Li et al. (2011) for the regularization of ill-conditioning problems, where the minimal singular value σmin of discrete matrices is infinitesimal. To remove the effects of high frequency caused by the singular vector of σmin better, we combine the Tikhonov regularization (TR) with the truncated singular value decomposition (TSVD) (simply denoted by T-TR). New computational formulas of the traditional condition number (Cond) and the effective condition number (Cond_eff) are derived, and a brief error analysis is made. The regularization parameter λ is involved in the TR, and its better (or optimal) choices are essential in both theory and computation. For ill-conditioning problems, stability is more imperative than accuracy. The most important criterion is to reduce the Cond. Consider the linear algebraic equations Ax = b. The optimal regularization parameter for the TR is derived as λ=σmaxσmin, where σmax and σmin(>0) are the maximal and the minimal singular values of matrix A, respectively. The L-curve techniques emphasize the ill-conditioning of ‖xλ‖ (see Hansen (1998)). Note that the norm ‖xλ‖ is only part of the Cond in stability. In fact, by using regularization, the Cond may be greatly reduced, while the errors do not increase much. The second important criterion is for both stability and solution accuracy. The sensitivity index proposed in Zhang et al. (2021) indicates the severity of ill-conditioning via accuracy for comparing different numerical methods and techniques. The sensitivity index is effective not only to select the source nodes in the method of fundamental solutions (MFS), but also to select the regularization parameter in the TR. The regularization choices of the T-TR are similar to those of the TR because differences of the Cond and the errors between the TR and the T-TR are insignificant. Numerical experiments are reported by the MFS for Laplace’s equation, to support the new regularization techniques proposed. For data fitting, imaging processing, pattern recognition, and machine learning, the ‖xλ‖ is, however, more important than Cond for ill-conditioning. The L-curve techniques are used in wide applications, and discussed in Hansen (1998). The parameter choices are rather complicated if the crossover region is not small. In this paper, we apply the sensitivity index to L-curves, the optimal parameter λL−curve can be found by the minimal sensitivity index. The λL−curve may also be found by the lowest line that passes the origin. The new algorithms are simple and effective.

Application of the error estimation technique in the method of fundamental solutions for solving incident wave problem with multiple cylinders

This article attempts to solve the problem of oblique incident wave with multiple cylinders using the method of fundamental solutions (MFS) combined with a proposed error estimation technique. In real engineering, many problems only have an empirical result without an exact solution. There are three advantages to the proposed error estimation technique. First, the error estimation technique can provide a reliable solution to approximate the analytical solution. Second, the optimal number of points for the source and collocation points can be determined. Third, a scheme refinement of adaptive distribution of source points can be developed employing the error estimation technique. The theory of single-layer and double-layer potential formulas in the MFS method is considered in this paper. The numerical examples include water wave passing through one, two, and four circular cylinders, respectively. The results show the resultant forces on the cylinder and free surface elevation around the cylinder, respectively. The results are then presented.

A Localized Version of the Method of Fundamental Solutions in a Multi-level Context

The Method of Fundamental Solutions is applied to the Laplace equation. Instead of using the traditional approach with external source points and boundary collocation points, the original domain decomposed into a lot of smaller, overlapping subdomains, and the Method of Fundamental Solutions is used to the individual local subdomains. After eliminating the local source points, local schemes are obtained. Instead of constructing a global scheme, the local subproblems are solved sequentially, in an iterative way. This mimics a multiplicative Schwarz method with overlapping subdomains, which assures the convergence of the method. Combining the iteration with a simple Seidel-type method, the resulting iteration is used as a smoothing procedure of a multi-level method. The points belonging to the coarse and fine levels are defined by a quadtree-generated cell system controlled by the boundary of the original domain. The multi-level character of the obtained method makes it possible to reduce the necessary number of iterations, that is, the overall computational cost can be significantly reduced. Moreover, the solution of large and ill-conditioned systems is completely avoided. The method is illustrated through several numerical test examples.

The enhanced boundary knot method with fictitious sources for solving Helmholtz-type equations

The boundary knot method (BKM) is a boundary meshless method for solving homogeneous boundary value problems. The fact that the BKM uses non-singular general solution without the need of source points outside the domain is at the expense of the accuracy when compared to the well-known method of fundamental solutions. In this paper, a new approach called the enhanced BKM (EBKM) is proposed to greatly improve the accuracy of the BKM by introducing fictitious sources which can be placed inside and/or outside the domain quite freely. We also adopt various techniques to place these fictitious sources in an optimal way.

Pricing Multidimensional American Options

A new explicit form is provided for the solution of optimal stopping problems involving a multidimensional geometric Brownian motion. A free-boundary value approach is adopted and the value function is obtained via fundamental solution methods. There are many applications for the valuation of perpetual options of American style, which are of interest for finance and managerial decisions.