On improving the conditioning of the method of fundamental solutions for biharmonic BVPs in 2D domains

Abstract High voltage terminations are commonly used in high-voltage (HV) connections such as cable -to-transformer connection to limit local electric field intensification that may result in electrical breakdown. The termination contains a conductor and multiple dielectric layers with an embedded semiconductive layer, which is essential for reliable insulation design and breakdown prevention. The traditional charge simulation method (CSM) for electric field calculation is limited to two pure dielectric media. The present paper is aimed at proposing a new formulation of the traditional CSM for calculation of electric field in HV terminations with more than two dielectric layers including a semiconductive one. The conductor, dielectric interfaces, and surrounding air are modelled using sets of fictitious ring charges. With consideration of the governing pertinent boundary conditions, namely the Dirichlet condition on the conductor surface, continuity of both the electric potential, and the normal electric flux density at dielectric interfaces, the simulation charges are evaluated and hence the electric field. To account for the presence of the semi-conductive layer with relatively high conductivity, a hybrid numerical approach combining CSM and the Finite Difference Method (FDM) is developed, where the semi-conductive region is solved using FDM and coupled to the CSM domain through continuity conditions of potential and normal flux density. The proposed formulation is validated through comparison with COMSOL Multiphysics simulations. The results demonstrate excellent agreement in potential and electric field distributions across all dielectric layers, including regions near to the interfaces and sheds. The new-formulated CSM with hybrid CSM–FDM provides an efficient and accurate tool for electric field calculation in complex plasma reactors and HV cable terminations accommodating multi-dielectric layers among them semiconductive ones.

Advanced methods of boundary integral equations for Laplace’s equation: Relations to method of fundamental solutions

This paper presents a focused pedagogical survey of fundamental solution methods for ordinary differential equations (ODEs), demonstrating the unifying and explanatory role of mathematical symmetry. The core of this study is that classical solution techniques are not mere algebraic manipulations but are inherently motivated by the equations' underlying invariant structure. We provide a targeted analysis showing that two critical classes of ODEs—the first-order homogeneous equation and the Cauchy-Euler equation are direct mathematical expressions of scale invariance. The substitution methods used to solve both types of equations are practical applications of exploiting this invariance property to transform the original non-separable equation into a simpler, solvable form. By explicitly reframing these core methods within a symmetry-based context, this work offers advanced practitioners and students a deeper conceptual foundation. This approach unifies seemingly disparate solution techniques under a single, powerful mathematical principle, thereby highlighting the significance of ODEs as mathematical models for analyzing physical systems that exhibit powerful geometrical and variational symmetries

We derive the closed-form fundamental solutions (Green's functions) of a linearized two-temperature model that captures heat transfer in rarefied polyatomic gases by treating translational and internal energy modes separately. These solutions exhibit both diffusive and wave-like behavior and include exponential terms that give rise to a nonequilibrium thermal layer near heat sources—analogous to the Knudsen layer in momentum transport. Unlike classical heat conduction, this layer reflects the localized mismatch between translational and internal temperatures arising from finite-rate energy exchange. To demonstrate their utility, we apply the derived solutions within the method of fundamental solutions, a meshless boundary method, to solve benchmark two- and three-dimensional problems. Numerical results show excellent agreement with analytical solutions and kinetic theory, validating both the model and the fundamental solutions. The work offers a physically consistent and computationally efficient framework for modeling heat transfer in rarefied polyatomic gases, with direct relevance to microscale thermal systems where nonequilibrium effects are prominent.

Density results using the method of fundamental solutions for a fluid-structure interaction problem

As global aquaculture continues to expand, there is increasing interest in sustainable and non-invasive tools for monitoring fish growth. Nile tilapia (Oreochromis niloticus) is one of the most farmed species worldwide. Its biomass estimation often relies on manual sampling or stereo-camera systems limited by water turbidity. This study establishes a robust relationship between lateral target strength (TS) and the total length (TL) and weight (W) of Nile tilapia using a cost-effective 201 kHz single-beam echosounder. Measurements were conducted with free-swimming fish in a controlled pond environment (TL range, 13–44 cm). The results show a strong linear correlation between acoustic and biometric data. Specifically, the relationship for mean TS was defined as TSmean = 20.4log(TL) − 68.8 (R2 = 0.93) and TSmean = 6.3log(W) − 55.4 (R2 = 0.96), proving the system’s accuracy for biomass estimation. Furthermore, the Method of Fundamental Solutions (MFS) was employed for numerical validation based on X-ray morphometry of the swim bladder. Very good agreement was observed between experimental data and numerical simulations, reinforcing the validity of the acoustic models despite the inherent complexity of biological targets. These findings demonstrate that calibrated single-beam acoustic systems provide a viable, non-intrusive tool for real-time monitoring in aquaculture ponds.

ABSTRACT In this paper, we propose a decompositional (phase‐wise split) approach to solve a two‐dimensional, two‐phase (solid and liquid, say), nonlinear inverse Stefan problem. The first step is to approximate the unknown moving boundary between the two phases and the Stefan condition on that boundary using the overspecified boundary and initial data on the solid. The second and final step is then to reconstruct the temperature and heat flux on the fixed liquid boundary using the approximated Stefan conditions and given initial data. In each phase, we obtain the Tikhonov‐regularized approximations using the method of fundamental solutions (MFS) and formulate heuristic residual a posteriori estimators to quantify the errors in the approximations. The MFS parameters for controlling the error are detected automatically in a systematic way, by virtue of a mean‐filtering algorithm and a deterministic optimization strategy; this is in stark contrast to the less systematic way employed in existing nonlinear optimization algorithms. Numerical results demonstrate the effectiveness of the proposed approach.

In this work, we present a numerical solution to the linear Richards' equation in a two-dimensional bounded soil domain, modelling unsaturated flow through a homogeneous rectangular medium under various infiltration scenarios. The proposed method employs a two-step approach: time discretization is performed using a finite difference scheme, followed by spatial discretization using the method of fundamental solutions (MFS), a meshless technique known for its simplicity and accuracy. The resulting scheme is easy to implement and yields reliable estimates of soil moisture dynamics. The effectiveness and accuracy of the method are demonstrated through several test cases, including comparisons with available analytical solutions.

In this paper, the method of fundamental solutions (MFS) is first developed for solving direct problems in bi-layer materials in the biomedical field of optical fluorescence. The governing system of second-order linear partial differential equations (PDEs) for the emission and excitation fluences is transformed into a single fourth-order PDE with appropriate boundary and interface matching conditions. The MFS is subsequently further developed, in conjunction with a constrained minimization regularization procedure, to solve nonlinear inverse optical fluorescence tomography problems. Numerical results confirm the accuracy, stability and versatility of the proposed meshless technique.

The paper considers numerical modeling of the approximation of a potential field with a non-smooth boundary having corners in the dirichlet problem. The methods considered are based on the method of fundamental solutions with some modifications. A method of increasing the accuracy of approximation by supplementing a set of singularity points around corners and a method of supplementing the classical basis with harmonic functions that consider the size and orientation of the corner are considered. The supplement of the classical basis with harmonic functions that, in the domain, are the potential of a double layer with a density in the form of a b-spline function are also considered.

A modification of the fundamental solution method is considered, which is applied to the approximation of a potential field within the domain of definition with known boundary conditions in the presence of discontinuities of the boundary function. For this purpose, it is proposed to enrich the approximation basis with functions that are double layer potential within the domain, in which the density function is a B-spline function. The supports of these B-spline functions must cover the boundary section where we want to reduce the approximation error and where the condition of continuity of the boundary function is satis-fied. The proposed method with an enriched basis, as well as its basic version, leads to the solution of an ill-conditioned system of linear equations, which is an ill-posed problem. The TSVD and Tikhonov regularization methods are used for the solution.

A numerical solution of the interior Dirichlet problem for the homogeneous conductivity equation is considered. After introducing certain assumptions and discretization of the domain, the boundary value problem for a second-order elliptic equation with variable coefficients is reduced to a set of coupled problems with the Helmholtz-type equation defined on multiple disjoint subdomains. The coefficients in the transmission coupled problems are numerically approximated using the Gauss-Legendre and the trapezoid quadrature rules. The coupled problems are simultaneously solved using the method of fundamental solutions, in which the unknown functions are approximated by linear combinations of fundamental solutions, and the coefficients are determined using the collocation method. The applicability and efficiency of the proposed approach are confirmed by the results of numerical experiments.

With the aim of investigating numerical solutions for groundwater flow in confined aquifers, this study compares the performance of the Finite Difference Method (FDM) and the Method of Fundamental Solutions (MFS). The governing partial differential equation models subsurface flow involving water extraction or recharge through a well, combining mass conservation with Darcy’s Law. The main objective is to perform simulations and numerical comparisons using the two aforementioned numerical methods. Both approaches are applied to a two-dimensional steady-state formulation. A key difference between the methods lies in their spatial discretization: FDM requires the construction of an interconnected mesh of points (or nodes), where approximate values of the solution function are computed. In contrast, MFS does not require mesh generation; instead, it estimates the solution at freely distributed nodes in the domain, based on the fundamental solution and the problem's boundary conditions. However, for the purpose of direct comparison between the methods, the MFS nodes were selected to coincide with those used in the FDM, ensuring spatial equivalence between the numerical solutions. This strategy aims to highlight the strengths and limitations of each approach. Both methods were compared with each other and with analytical solutions, whenever available, yielding satisfactory results for the proposed applications. The analysis showed that each method has specific advantages and disadvantages, and that the choice depends on the particular characteristics of the problem under consideration.

Abstract The charge simulation method (CSM) was first introduced for field calculation in high-voltage (HV) arrangements involving electrodes and two dielectrics at most. Each electrode is simulated by a set of charges inside it. The interface between the two dielectrics is simulated by two sets of charges, one set in each dielectric. The proposed method aims to extend the CSM for the first time to apply to arrangements with many electrodes and multi-dielectric layers. This represents the novelty of the method. Its intelligence lies in the proper selection of the simulation charges to be used for calculating the electric potential and field anywhere within the HV arrangement, following a systematic procedure. The method predicts potential and field values that coincide with their respective exact values in a single-core cable with multi coaxial-dielectric layers. For a dielectric-barrier discharge (DBD) arrangement having multi parallel-flat-dielectric layers with and without embedded electrodes, the method also predicts potential and field values that agree reasonably with those obtained using COMSOL software. The effectiveness of the embedded electrode in decreasing the field at the edge of the stressed electrode is verified by the proposed method in agreement with the experimental observations recorded for the investigated DBD arrangement.

Global rapid thermomechanical decoupling method based on adaptive localized method of fundamental solutions and sparse embedded FBG in thermal protection materials for aerospace vehicles

The method of fundamental solutions for plate bending under arbitrary distributed lateral forces on partial areas

A special localization technique is presented for solving steady heat transfer problems, in which the thermal conductivity may depend on space variables. The original problem is split into several subproblems defined on much smaller subdomains. The subproblems are solved using the Method of Fundamental Solutions, which is a truly meshless method. This leads to a Seidel-like iterative technique, which mimics the classical Schwarz overlapping method. The problems associated with large, dense and ill-conditioned matrices are avoided. The method is embedded into a multi-level context, which significantly reduces the computational complexity.

Abstract This work suggests different Monte Carlo algorithms for solving large systems of linear algebraic equations arising from the numerical solution of the Dirichlet problem for the Helmholtz equation. Approach based on boundary integral representations, vector randomization algorithm, method of fundamental solutions, stochastic projection algorithm, and randomized singular value decomposition are applied. It is shown that the use of stochastic iterative refinement and preconditioning can significantly improve the accuracy and stability of the computations. Simulation results are presented, demonstrating the effectiveness of the proposed methods.

This paper introduces a singular distance function rs in terms of a symmetric non-negative metric tensor S. If S satisfies a quadratic matrix equation involving a parameter β, then for the Laplace equation rsβ is a non-singular generalized radial basis function solution if 2 > β > 0, and a weaker singularity fundamental solution if −1 < β < 0. With a unit vector as a medium to express S, we can derive the metric tensor in closed form and prove that S is a singular projection operator. For the anisotropic Laplace equation, the corresponding closed-form representation of S is also derived. The concept of non-singular generalized radial basis function solution for the Laplace-type equations is novel and useful, which has not yet appeared in the literature. In addition, a logarithmic type method of fundamental solutions is developed for the anisotropic Laplace equation. Owing to non-singularity and weaker singularity of the bases of solutions, numerical experiments verify the accuracy and efficiency of the proposed methods.