Meshless Method Of Fundamental Solutions Research Articles

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.

Numerical Solution of Hydrodynamic Efficiency Equations for an Oscillating Water Column Wave Energy Converter Using the Method of Fundamental Solutions

The oscillating water column (OWC) converter, a prominent and widely adopted device for capturing wave energy, has yet to attain commercialization due to its intricate hydrodynamic behavior. Consequently, numerous laboratory and numerical studies have been carried out on OWC converters. One of the significant parameters in the study and comparison of different converters is their hydrodynamic efficiency parameter. Hence, the primary objective of this study is to examine the efficiency of an oscillating water column converter using the meshless method of fundamental solutions (MFS) to solve the governing equations. To address the challenges posed by the low wave steepness and to optimize computational efficiency, a complete linearization of two nonlinear sources at the boundary conditions inside and outside the OWC chamber was undertaken. Next, the boundary conditions were temporally discretized using three distinct methods: the Newmark, the forward differentiation formula, and the Runge-Kutta method of the fourth order. Subsequently, a comparison was made among these methods, revealing that the Newmark method exhibited a higher convergence rate to the solution. Using the developed numerical model, a more precise examination of the converter's efficiency, under various sloped bottom geometries, was conducted, surpassing the level of accuracy achieved in prior studies. Two key parameters, namely the opening, and slope, were identified as critical factors influencing the efficiency of converters considering sloped bottoms. The results indicated that the efficiency of the converter decreases with an increase in the slope and a decrease in the opening. Furthermore, to effectively interpret the efficiency of the converter, a suitable parameter was introduced the disparity in the free surface elevation inside and outside the chamber. This parameter proved valuable in providing insights into the converter's performance.

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.

Derivation and numerical validation of the fundamental solutions for constant and variable-order structural derivative advection–dispersion models

Various non-local structural derivative diffusion models have been proposed based on different kernel functions to describe the anomalous time dependence of the mean-squared displacements. In the present study, the fundamental solutions for constant and variable-order structural derivative advection–dispersion models are achieved via scaling transformation and the generalized non-Euclidean Hausdorff fractal distance. Comparative numerical investigations of the structural derivative models have been conducted to reveal the influences of various kernels via the meshless method of fundamental solutions. Numerical results verify the validity of the derived fundamental solutions and the rationality of the employed numerical method for structural derivative advection–dispersion models.

Meshless simulation of anti-plane crack problems by the method of fundamental solutions using the crack Green’s function

In this paper, we present a meshless method of fundamental solutions using the analytical crack Green’s function to solve anti-plane crack problems. The proposed scheme is a simple, powerful and effective collocation method for crack problems since it only requires the boundary discretization without special treatments of the crack. Three typical numerical examples, namely, a central crack, an off-central crack and a central slant crack are presented and discussed to illustrate the accuracy, convergence and stability of the proposed method.

Method of Fundamental Solutions Applied to 3-D Velocity Induced Eddy Current Problems

A meshless method of fundamental solutions (MFS) is applied for the determination of eddy currents inside a translatory moving conductor in the field of a permanent magnet. Three cases are considered: 1) a defect-free specimen; 2) a specimen with a spherical sub-surface defect; and 3) the difference between 1) and 2) as a defect response case. The latter is directly computed with a novel approach using two fictitious surfaces, inside and outside the defect, where the point sources of the MFS are randomly placed. We investigate the optimal distance of the fictitious surfaces containing the randomly placed point sources of the MFS. The eddy current field distributions are compared to a reference finite-element solution. We obtain mean normalized root-mean-square errors below 1% for all three cases. Our proposed meshless MFS for the defect response case allows for direct inclusion into inverse solutions for non-destructive evaluation applications.

How Accurate Is Inverse Electrocardiographic Mapping? A Systematic In Vivo Evaluation.

Inverse electrocardiographic mapping reconstructs cardiac electrical activity from recorded body surface potentials. This noninvasive technique has been used to identify potential ablation targets. Despite this, there has been little systematic evaluation of its reliability. Torso and ventricular epicardial potentials were recorded simultaneously in anesthetized, closed-chest pigs (n=5), during sinus rhythm, epicardial, and endocardial ventricular pacing (70 records in total). Body surface and cardiac electrode positions were determined and registered using magnetic resonance imaging. Epicardial potentials were reconstructed during ventricular activation using experiment-specific magnetic resonance imaging-based thorax models, with homogeneous or inhomogeneous (lungs, skeletal muscle, fat) electrical properties. Coupled finite/boundary element methods and a meshless approach based on the method of fundamental solutions were compared. Inverse mapping underestimated epicardial potentials >2-fold (P<0.0001). Mean correlation coefficients for reconstructed epicardial potential distributions ranged from 0.60±0.08 to 0.64±0.07 across all methods. Epicardial electrograms were recovered with reasonable fidelity at ≈50% of sites (median correlation coefficient, 0.69-0.72), but variation was substantial. General activation spread was reproduced (median correlation coefficient, 0.72-0.78 for activation time maps after spatio-temporal smoothing). Epicardial foci were identified with a median location error ≈16 mm (interquartile range, 9-29 mm). Inverse mapping with meshless method of fundamental solutions was better than with finite/boundary element methods, and the latter were not improved by inclusion of inhomogeneous torso electrical properties. Inverse potential mapping provides useful information on the origin and spread of epicardial activation. However the spatio-temporal variability of recovered electrograms limit resolution and must constrain the accuracy with which arrhythmia circuits can be identified independently using this approach.

Simultaneous numerical determination of a corroded boundary and its admittance

In this paper, an inverse geometric problem for Laplace’s equation arising in boundary corrosion detection is considered. This problem, which consists of determining an unknown corroded portion of the boundary of a bounded domain and its admittance Robin coefficient from two pairs of boundary Cauchy data (boundary temperature and heat flux), is solved numerically using the meshless method of fundamental solutions. A non-linear minimization of the objective function is regularized, and the stability of the numerical results is investigated with respect to noise in the input data and various values of the regularization parameters involved.

Topological Derivative for the Inverse Conductivity Problem: A Bayesian Approach

The employment of topological derivative concept is considered to propose a new optimization algorithm for the inverse conductivity problem. Since this inverse problem is nonlinear and ill-posed it is necessary to incorporate a prior knowledge about the unknown conductivity. In particular, we apply the Bayes theorem to add the assumption that we have just one small ball-shaped inclusion, which must be at a certain distance from the boundary of the domain. As the main emphasis of this paper is to investigate numerically the proposed approach, we shall use the meshless method of fundamental solutions to present some numerical results.

A meshless method for an inverse two-phase one-dimensional nonlinear Stefan problem

We extend a meshless method of fundamental solutions recently proposed by the authors for the one-dimensional two-phase inverse linear Stefan problem, to the nonlinear case. In this latter situation the free surface is also considered unknown which is more realistic from the practical point of view. Building on the earlier work, the solution is approximated in each phase by a linear combination of fundamental solutions to the heat equation. The implementation and analysis are more complicated in the present situation since one needs to deal with a nonlinear minimization problem to identify the free surface. Furthermore, the inverse problem is ill-posed since small errors in the input measured data can cause large deviations in the desired solution. Therefore, regularization needs to be incorporated in the objective function which is minimized in order to obtain a stable solution. Numerical results are presented and discussed.

Numerical reconstruction of an inhomogeneity in an elliptic equation

In this paper, the inverse problem which consists of reconstructing an unknown inner boundary of a domain from a single pair of boundary Cauchy data associated to an elliptic equation is solved numerically using the meshless method of fundamental solutions. A nonlinear minimization of the objective function is regularized. The stability of the numerical results is investigated for several test examples with respect to noise in the input data and various values of the regularization parameters.

A meshless method for solving a two‐dimensional transient inverse geometric problem

PurposeThe purpose of this paper is to apply the meshless method of fundamental solutions (MFS) to the two‐dimensional time‐dependent heat equation in order to locate an unknown internal inclusion.Design/methodology/approachThe problem is formulated as an inverse geometric problem, using non‐invasive Dirichlet and Neumann exterior boundary data to find the internal boundary using a non‐linear least‐squares minimisation approach. The solver will be tested when locating a variety of internal formations.FindingsThe method implemented was proven to be both stable and reasonably accurate when data were contaminated with random noise.Research limitations/implicationsOwing to limited computational time, spatial resolution of internal boundaries may be lower than some similar case investigations.Practical implicationsThis research will have practical implications to the modelling and monitoring of crystalline deposit formations within the nuclear industry, allowing development of future designs.Originality/valueSimilar work has been completed in regards to the steady state heat equation, however to the best of the authors' knowledge no previous work has been completed on a time‐dependent inverse inclusion problem relating to the heat equation, using the MFS. Preliminary results presented here will have value for possible future design and monitoring within the nuclear industry

The method of fundamental solutions for the identification of a sound-soft obstacle in inverse acoustic scattering

In this paper we propose a simple meshless method of fundamental solutions (MFS) for detecting a sound-soft scatterer embedded in a host acoustic homogeneous medium from the measurement of the far-field pattern of the scattered wave for only one incoming direction. Further, when this measurement is contaminated with noise, we augment the MFS with a nonlinear constrained regularized minimization for obtaining a stable numerical solution of the inverse problem. Numerical results are presented and discussed.

The method of fundamental solutions for the detection of rigid inclusions and cavities in plane linear elastic bodies

We investigate the numerical reconstruction of smooth star-shaped voids (rigid inclusions and cavities) which are compactly contained in a two-dimensional isotropic linear elastic body from a single non-destructive measurement of both the displacement and traction vectors (Cauchy data) on the external boundary. The displacement vector satisfying the Lamé system in linear elasticity is approximated using the meshless method of fundamental solutions (MFS). The fictitious source points are located both outside the (known) outer boundary of the body and inside the (unknown) void. The inverse geometric problem is then reduced to finding the minimum of a nonlinear least-squares functional that measures the gap between the given and computed data, penalized with respect to both the MFS constants and the derivative of the radial polar coordinates describing the position of the star-shaped void. The interior source points are anchored and move with the void during the iterative reconstruction procedure. The stability of the numerical method is investigated by inverting measurements contaminated with noise.

A meshless method for an inverse two-phase one-dimensional linear Stefan problem

We extend the meshless method of fundamental solutions proposed in [B.T. Johansson, D. Lesnic, and T. Reeve, A method of fundamental solutions for the one-dimensional inverse Stefan problem, Appl. Math. Model. 35 (2011), pp. 4367–4378] for the one-dimensional one-phase inverse Stefan problem to the two-phase change case. The implementation and analysis are more complicated and meaningful since one needs to handle composite layered materials. Furthermore, the inverse problem is ill-posed since small errors in the input data lead to large deviations in the solution. Therefore, the inverse problem is intractable to classical methods of linear inversion, and regularization is employed in our study. Numerical results obtained when the input data is either exact or contaminated with random noise are compared with the analytical solution, where available, or with the numerical results obtained by other methods, [D.D. Ang, A. Pham Ngoc Dinh, and D.N. Tranh, Regularization of an inverse two-phase Stefan problem, Nonlinear Anal. 34 (1998), pp. 719–731], otherwise.

Inverse shape and surface heat transfer coefficient identification

In this paper, an inverse geometric problem for the modified Helmholtz equation arising in heat conduction in a fin is considered. This problem which consists of determining an unknown inner boundary of an annular domain and possibly its surface heat transfer coefficient from one or two pairs of boundary Cauchy data (boundary temperature and heat flux) is solved numerically using the meshless method of fundamental solutions (MFS). A nonlinear unconstrained minimisation of the objective function is regularised when noise is added to the input boundary data. The stability of the numerical results is investigated for several test examples with respect to noise in the input data and various values of the regularisation parameters.

Method of Fundamental Solutions for Stokes Problems by the Pressure-Stream Function Formulation

ABSTRACTThe main purpose of this paper is to investigate the pressure-stream function formulation to solve 2D and 3D Stokes flows by the meshless numerical scheme of the method of fundamental solutions (MFS). The MFS can be regarded as a truly scattered, grid-free (or meshless) and non-singular numerical scheme. By the proposed algorithm, the stream function is governed by the bi-harmonic equation while the pressure is governed by the Laplace equation. The velocity field is then obtained by the curl of the stream function for 2D flows and curl of the vector stream function for 3D flows. We can simultaneously solve the pressure, velocity, vorticity, stream function and traction forces fields. Furthermore during the present numerical procedure no pressure boundary condition is needed which is a tedious and forbidden task. The developed algorithm is used to test several numerical experiments for the benchmark examples, including (1) the driven circular cavity, (2) the circular cavity with eccentric rotating cylinder, (3) the square cavity with traction boundary conditions and (4) the uniform flow past a sphere. The results compare very well with the solutions obtained by analytical or other numerical methods such as finite element method (FEM). It is found that the meshless MFS will give a simpler and more efficient and accurate solutions to the Stokes flows investigated in this study.

Method of fundamental solutions for multidimensional Stokes equations by the dual-potential formulation

A simple and accurate boundary-type meshless method of fundamental solutions (MFS) is applied to solve both 2D and 3D Stokes flows based on the dual-potential formulation of velocity potential and stream function vector. Using the dual-potential concept, the solutions of both 2D and 3D Stokes flows are obtained by combining the much simpler fundamental solutions of Laplace (potential) and bi-harmonic equations without using the complicated singular fundamental solutions such as Stokeslets and their derivatives as well as source doublet hypersingularity. The developed algorithm is used to test five numerical experiments for 2D flows: (1) circular cavity, (2) wave-shaped bottom cavity and (3) circular cavity with eccentric rotating cylinder; and for 3D flows: (4) a uniform flow passing a sphere and (5) a uniform flow passing a pair of spheres. Good results are obtained as comparing with solutions of analytical and numerical methods such as FEM, BEM and other meshfree schemes.