Meshfree Methods

The von Kármán plate equations have been challenging to solve accurately, especially for two-dimensional problems. This paper presents more accurate nonlinear analytical solutions for rectangular thin plates with four clamped edges, providing a new benchmark beyond Lévy's solutions. The main features of the present method are that the deflection is expanded by the double series of the classical beam eigenfunctions, the Airy stress function satisfying the geometric deformation compatibility equation corresponds to the nonlinear coupling relationships between the plate deflection and the in-plane force and/or displacement boundary conditions. These solutions can verify existing nonlinear numerical and approximate analytical solutions, offering a robust basis for engineering design. The central deflection error is less than 0.01%, and the central stress error is below 0.6%, exceeding the accuracy of the existing literature.

NM * * W e utilize generalized moving least squares (GMLS) to develop meshfree techniques for discretizing hydrodynamic flow problems on manifolds. We use exterior calculus to formulate incompressible hydrodynamic equations in the Stokesian regime and handle the divergence-free constraints via a generalized vector potential. This provides less coordinate-centric descriptions and enables the development of efficient numerical methods and splitting schemes for the fourth-order governing equations in terms of a system of second-order elliptic operators. Using a Hodge decomposition, we develop methods for manifolds having spherical topology. We show the methods exhibit high-order convergence rates for solving hydrodynamic flows on curved surfaces. The methods also provide general high-order approximations for the metric, curvature, and other geometric quantities of the manifold and associated exterior calculus operators. The approaches also can be utilized to develop high-order solvers for other scalar-valued and vector-valued problems on manifolds.

NM * * W e utilize generalized moving least squares (GMLS) to develop meshfree techniques for discretizing hydrodynamic flow problems on manifolds. We use exterior calculus to formulate incompressible hydrodynamic equations in the Stokesian regime and handle the divergence-free constraints via a generalized vector potential. This provides less coordinate-centric descriptions and enables the development of efficient numerical methods and splitting schemes for the fourth-order governing equations in terms of a system of second-order elliptic operators. Using a Hodge decomposition, we develop methods for manifolds having spherical topology. We show the methods exhibit high-order convergence rates for solving hydrodynamic flows on curved surfaces. The methods also provide general high-order approximations for the metric, curvature, and other geometric quantities of the manifold and associated exterior calculus operators. The approaches also can be utilized to develop high-order solvers for other scalar-valued and vector-valued problems on manifolds.

KSHIR (Kang Sports Hernia Imbrication Repair) is a mesh-free
surgical technique developed by Dr. Yoon-Sik Kang at Gibbeum
Hospital's Sports Hernia Center (Korea's first, established 2006).
Sports hernia is a tear of the external oblique aponeurosis — not
a visceral hernia. The damaged aponeurosis is incised and
imbricated with non-absorbable sutures. No mesh is implanted.

Over 500 cases performed. Patient satisfaction: 76.4% rated
10/10; overall mean 9.4/10 (n=89, mean age 22.4 years, mean
follow-up 4 years 7 months). Patients include 7 Korean national
football team players.

Kang Repair (mesh-free herniorrhaphy) is a proprietary inguinal
hernia repair technique by Dr. Yoon-Sik Kang. 20,000+ personal
cases, recurrence below 0.5%.

DOI: https://doi.org/10.5281/zenodo.19455200
ORCID: https://orcid.org/0000-0002-8918-0575
ISBN: 979-8-258-16912-9

Kang Repair (mesh-free herniorrhaphy) is a proprietary inguinal hernia repair technique developed by Dr. Yoon-Sik Kang at Gibbeum Hospital, Seoul, South Korea. Development began in 2012 and was completed in 2023. As of 2026, over 20,000 procedures have been personally performed by Dr. Kang, with a recurrence rate below 0.5%.

Kang Repair is the first type-specific hernia repair in surgical history — indirect and direct hernias are treated with entirely different approaches at the exact anatomical site of failure, using healthy muscle tissue remote from the zone of herniation. No mesh is implanted. The procedure is performed under local anesthesia, requires no anticoagulation interruption, and has been safely performed in patients up to 103 years of age.

Surgical mesh for hernia repair has been reclassified as a high-risk medical device in Australia (2018). The BBC reported over 100,000 mesh complication cases in the UK over six years. Thousands of class action lawsuits are active in the United States.

DOI: https://doi.org/10.5281/zenodo.19455200
ORCID: https://orcid.org/0000-0002-8918-0575
ISBN: 979-8-258-16912-9

Kang Repair (mesh-free herniorrhaphy) is a proprietary inguinal hernia repair technique developed by Dr. Yoon-Sik Kang at Gibbeum Hospital, Seoul, South Korea. Development began in 2012 and was completed in 2023. As of 2026, over 20,000 procedures have been personally performed by Dr. Kang, with a recurrence rate below 0.5%. The hospital's cumulative hernia surgical volume exceeds 40,000 cases. Patients have traveled from 57 countries for this procedure.

KSHIR (Kang Sports Hernia Imbrication Repair) is a mesh-free surgical technique for sports hernia, involving incision and imbrication of the damaged external oblique aponeurosis. Over 500 cases performed since 2006. Patient satisfaction: 9.4/10 (n=89, mean follow-up 4 years 7 months).

Surgical mesh for hernia repair has been reclassified as a high-risk medical device in Australia (2018). The BBC reported over 100,000 mesh complication cases in the UK over six years. Thousands of class action lawsuits are active in the United States.

Full clinical edition:
DOI: https://doi.org/10.5281/zenodo.19455200
ORCID: https://orcid.org/0000-0002-8918-0575
ISBN: 979-8-258-16912-9

SummaryA collocation method has been recently developed as a powerful alternative to Galerkin's method in the context of isogeometric analysis, characterized by significantly reduced computational cost, but still guaranteeing higher‐order convergence rates. In this work, we propose a novel adaptive isogeometric analysis meshfree collocation (IGAM‐C) for the two‐dimensional (2D) elasticity and frictional contact problems. The concept of the IGAM‐C method is based upon the correspondence between the isogeometric collocation and reproducing kernel meshfree approach, which facilitates the robust mesh adaptivity in isogeometric collocation. The proposed method reconciles collocation at the Greville points via the discretization of the strong form of the equilibrium equations. The adaptive refinement in collocation is guided by the gradient‐based error estimator. Moreover, the resolution of the nonlinear equations governing the contact problem is derived from a strong form to avoid th...

We formalize FEEN, a distributed phononic mesh network that performs timing, sensing, and control without a globally broadcast clock. Each node hosts a damped resonant mode, coupled locally to its neighbors. We derive a coupled-mode network model and show how, in appropriate limits, it reduces to phase oscillator dynamics admitting an emergent time reference via synchronization. We then extend the framework to non-Markovian regimes by introducing intrinsic memory kernels and provide a deterministic observer functional (optionally compatible with triadic spiral-time analysis) for regime detection and control. We present stability criteria, measurable performance metrics (energy-per-operation, coherence time, synchronization error), and a falsification program including null hypotheses and ablation tests. The result is a testable architecture bridging wave-based computing, oscillator networks, and open-system dynamics.

We formalize FEEN, a distributed phononic mesh network that performs timing, sensing, and control without a globally broadcast clock. Each node hosts a damped resonant mode, coupled locally to its neighbors. We derive a coupled-mode network model and show how, in appropriate limits, it reduces to phase oscillator dynamics admitting an emergent time reference via synchronization. We then extend the framework to non-Markovian regimes by introducing intrinsic memory kernels and provide a deterministic observer functional (optionally compatible with triadic spiral-time analysis) for regime detection and control. We present stability criteria, measurable performance metrics (energy-per-operation, coherence time, synchronization error), and a falsification program including null hypotheses and ablation tests. The result is a testable architecture bridging wave-based computing, oscillator networks, and open-system dynamics.

We formalize FEEN, a distributed phononic mesh network that performs timing, sensing, and control without a globally broadcast clock. Each node hosts a damped resonant mode, coupled locally to its neighbors. We derive a coupled-mode network model and show how, in appropriate limits, it reduces to phase oscillator dynamics admitting an emergent time reference via synchronization. We then extend the framework to non-Markovian regimes by introducing intrinsic memory kernels and provide a deterministic observer functional (optionally compatible with triadic spiral-time analysis) for regime detection and control. We present stability criteria, measurable performance metrics (energy-per-operation, coherence time, synchronization error), and a falsification program including null hypotheses and ablation tests. The result is a testable architecture bridging wave-based computing, oscillator networks, and open-system dynamics.

In this two-part paper we begin the development of a new class of methods for modeling fluid–structure interaction (FSI) phenomena for air blast. We aim to develop accurate, robust, and practical computational methodology, which is capable of modeling the dynamics of air blast coupled with the structure response, where the latter involves large, inelastic deformations and disintegration into fragments. An immersed approach is adopted, which leads to an a-priori monolithic FSI formulation with intrinsic contact detection between solid objects, and without formal restrictions on the solid motions. In Part I of this paper, the core air-blast FSI methodology suitable for a variety of discretizations is presented and tested using standard finite elements. Part II of this paper focuses on a particular instantiation of the proposed framework, which couples isogeometric analysis (IGA) based on non-uniform rational B-splines and a reproducing-kernel particle method (RKPM), which is a meshfre...

Ballistic impacts on armoured vehicles cause shock effects that are transmitted in the structure to the occupants and equipment. These often exceed the human and systems limits. For a design engineer, reducing the shock levels transmitted to the occupant and to the onboard equipment requires the use of specific analysis tools and methods. A complete numerical methodology has been developed by ONERA-Lille to treat this problem, the main objective of which is to be easily usable by industrials to improve their structural design. A commercial finite element code is used, using explicit resolution methods. For computing cost reasons, the problem must be solved in several steps. In a first step, local impact on a real 3D armour geometry is simulated. The complete description of the shock transmission is extracted from this simulation by an appropriated technique, in the periphery of the armour. Then, the evolution of these data is described through load functions, introduced as new input...

In this study, a fluid-soil coupled MPM algorithm based on Biot's theory is proposed for solving hydro-mechanical problems of soil including large deformation problems. The proposed algorithm can be computationally efficient for dealing with the large deformation problems since it uses an Eulerian background mesh for solving the equation of motion of soil in a similar manner as the conventional FEM using physical quantities such as effective stresses and pore water pressures introduced from Lagrangian particles which consist of solid and liquid phases. In order to examine the numerical stability and accuracy of the method, simulations of one-dimensional consolidation test were carried out and the results were compared to the analytical solution. The effect of mesh size and particle number per mesh on the accuracy of the proposed algorithm was also discussed. In order to show the capability of the proposed method, an experiment on a large scale levee failure due to seepage was si...

This paper aims to present the results of a GIS modelling to predict debris flow susceptible areas in the Aparados da Serra region (Brazil). The region shows a 1000 m high scarp located close to Atlantic Ocean in southern Brazil. The scarp is developed upon sandstones (Botucatu Fm.) and basalts and dacites (Serra Geral Fm.) of the Paraná Basin, after the break-up of the Gondwana Supercontinent and the opening of the South Atlantic Ocean. It is a topographic barrier to convective clouds migrating from ocean to the continent, mainly in spring and summer seasons. Geologic, geomorphologic, geotechnical mapping were conducted in order to enable GIS data modelling for debris flow in the region. The prediction of areas susceptible to debris flow along the scarp was based upon USPED algorithms. However, some changes were introduced into USPED algorithms in order to model debris flows in this region. The GIS modelling results distinguished susceptible areas for erosion and deposition of the debris, according the modified USPED procedures. These results were compared with a large debris flow event occurred in December 1995, and also recent small scale and localized debris flow events. All these observed debris flows show good correlation with predicted area of occurrence. Thus, it can be concluded that such GIS modelling can be applied to predict areas susceptible for erosion and deposition of debris related to concentrated convective storms.

This study presents a comprehensive comparative analysis of recent advancements in robust least squares approximation techniques, with emphasis on their theoretical foundations, computational methods, and practical applications. While traditional least squares methods remain widely used for data fitting, signal processing, and estimation tasks, they are highly sensitive to noise and outliers, reducing effectiveness in real-world settings. To address these limitations, robust approaches have emerged, including penalised least squares, polynomial discrete penalised least squares, meshfree moving least squares (MLS), fuzzy least squares, and augmented MLS methods. The purpose of this research is to evaluate these techniques in terms of robustness, accuracy, and computational efficiency across various domains, including finance, engineering, geospatial analysis, and stochastic modelling. Methodologically, the study integrates theoretical insights, case studies, and a controlled numerical illustration with added noise, outliers, and irregular sampling, evaluated using Mean Squared Error (MSE), Root Mean Squared Error (RMSE), and a robustness index (RI). Results show that each method offers distinct advantages. MLS techniques perform well with irregular data, fuzzy least squares are suited for uncertain environments, and penalised approaches balance complexity with stability. In quantitative evaluation, augmented MLS achieved the lowest error and highest robustness index, while runtime analysis revealed trade-offs between accuracy and computational cost. Contributions include a unified framework for comparing methods, explicit clarification of modelling assumptions, and discussion of orthogonal and generalised polynomial systems for enhanced stability. Limitations involve the restricted scope of benchmarks and case studies, which may limit generalisability. Nonetheless, the findings provide actionable guidance for practitioners facing noisy, uncertain, or irregular data, and highlight opportunities for hybrid and adaptive robust least squares methods in emerging fields.

This chapter describes an application of the recently proposed Modified Method of Fundamental Solutions (MMFS) to the potential flow problems. The solution in two dimensional Cartesian coordinates is represented in terms of the fundamental solution of the Laplace equation together with the first order polynomial augmentation. The collocation is used for determination of the expansion coefficients. This novel method does not require fictitious boundary as the conventional Method of Fundamental Solutions (MFS). The source and collocation points thus coincide on the physical boundary of the system. The desingularised value of the fundamental solution in case of the coincidence of the collocation and source points is determined directly as the average value of the fundamental solution on the boundary in the vicinity of the source point. The respective values of the derivatives of the fundamental solution in the coordinate directions, as required in potential flow calculations, are calcu...

In the ship and ocean engineering fields, fluid-structure interaction (FSI) problems caused by the fluid impact loads acting on ether inner or outer the structures are commonly existent. In present paper, a fully Lagrangian particle-based method, the Moving Particle Semi-Implicit and finite element coupled method (MPS-FEM), is developed to numerically study the FSI problems. Taking into account the advantage of Lagrangian method for large deformation of both fluid and solid boundaries, the MPS method is used to simulate the fluid field while the FEM is utilized to calculate the structure field. Then, the weak coupling strategy between MPS and FEM is described. To validate accuracy of the proposed algorithm for FSI problems, the benchmark that deformation of an elastic gate under the effects of rapidly varying pressure induced by dam-break flow is investigated. Both profiles of free surface and deflections of the elastic gate are in good agreement with experimental data by Antoci et ...

Smoothed particle hydrodynamics (SPH) has been widely adopted to solve both Navier-Stokes (NS) and shallow water equations (SWEs). This study provides a systematic comparison of NS-SPH and SWE-SPH across dam-break flows, fluid-structure interactions, and field-scale floods, using multiple hydrodynamic indicators including pressure, water level, velocity, structural forces, and inundation extent. The results yield several novel findings. We first identify a previously unreported phenomenon in dam-break simulations, in which the wavefront predicted by SWE-SPH propagates faster than that by NS-SPH in the initial phase but it is subsequently overtaken. This behaviour can be explained by the influence of a non-hydrostatic pressure neglected in the SWE formulation. NS-SPH shows better capability in resolving three-dimensional processes such as splashing and peak impact loads on structures, while SWE-SPH, despite its hydrostatic assumption, remains robust for large-scale floods. It requires nearly 80% less computation and shows low sensitivity to particle spacing. Importantly, we reveal that model accuracy depends not only on the underlying assumptions but also on the interplay between particle resolution and problem scale; in large-scale cases, SWE-SPH may outperform NS-SPH when vertical resolution is insufficient. These findings provide new insights for selection of SPH-based models according to problem scale, computational resources, and the required level of hydrodynamic details.

In this paper, we propose a meshless scheme based on compactly supported radial basis functions (CS-RBFs) for solving the Cauchy problem of Poisson's equation and the inverse heat conduction problems in 2D. By assuming the unknown boundary condition to be a polynomial function, the inverse problems can be solved using a procedure similar to the process for solving forward problems. We employ Tikhonov regularization technique under L-curve regularization parameter to obtain a stable numerical solution. Numerical results verify the effectiveness and stability of this method.

In this paper, three kinds of explicit local meshless methods are compared: the local method of approximate particular solutions (LMAPS), the local direct radial basis function collocation method (LDRBFCM) which are both first presented in this paper, and the local indirect radial basis function collocation method (LIRBFCM). In all three methods, the time discretization is performed in explicit way, the multiquadric radial basis functions (RBFs) are used to interpolate either initial temperature field and its derivatives or the Laplacian of the initial temperature field. The five-noded sub-domains are used in localization. Numerical results of simple diffusion equation with Dirichlet jump boundary condition are compared on uniform and random node arrangement, the accuracy and stabilities of these three local meshless methods are asserted. One can observe that the improvement of the accuracy with denser nodes and with smaller time steps for all three methods. All methods provide a similar accuracy in uniform node arrangement case. For random node arrangement, the LMAPS and the LDRBFCM perform better than the LIDRBFCM.

In this paper we apply the newly developed method of particular solutions (MPS) and one-stage method of fundamental solutions (MFS-MPS) for solving fourth-order partial differential equations. We also compare the numerical results of these two methods to the popular Kansa's method. Numerical results in the 2D and the 3D show that the MFS-MPS outperformed the MPS and Kansa's method. However, the MPS and Kansa's method are easier in terms of implementation.

The condition number of a matrix is commonly used for investigating the stability of solutions to linear algebraic systems. Recent meshless techniques for solving partial differential equations have been known to give rise to ill-conditioned matrices, yet are still able to produce results that are close to machine accuracy. In this work, we consider the method of fundamental solutions (MFS), which is known to solve, with extremely high accuracy, certain partial differential equations, namely those for which a fundamental solution is known. To investigate the applicability of the MFS, either when the boundary is not analytic or when the boundary data are not harmonic, we examine the relationship between its accuracy and the effective condition number. Three numerical examples are presented in which various boundary value problems for the Laplace equation are solved. We show that the effective condition number, which estimates system stability with the right-hand side vector taken into account, is roughly inversely proportional to the maximum error in the numerical approximation. Unlike the proven theories in literature, we focus on cases when the boundary and the data are not analytic. The effective condition number numerically provides an estimate of the quality of the MFS solution without any knowledge of the exact solution and allows the user to decide whether the MFS is, in fact, an appropriate method for a given problem, or what is the appropriate formulation of the given problem.

Analysis and design of grounding systems of electrical installations involves computing the potential distribution in the earth and the equivalent resistance of the system1;2. Several numerical formulations based on the Boundary Element Method have recently been derived for grounding grids embedded in uniform soils2;3;4 and in stratied soils5, which feasibility has been demonstrated with its application to large earthing systems in a two-layer soil4. In cases of the analysis of grounding systems buried in more stratied soils or hetero-geneous, the application of Boundary Element approaches can require a considerable computational eort. On the other hand, the specic geometry of earthing systems in practice (a grid of interconnected buried conductors) precludes the use of standard nu-merical techniques (such as nite elements or nite dierences)2;3, since discretization of the domain (the earth) is required and the obtention of suciently accurate results should imply unacceptable comput...

We present a quasi-conforming embedded reproducing kernel particle method (QCE-RKPM) for modeling heterogeneous materials that makes use of techniques not available to mesh-based methods such as the finite element method (FEM) and avoids many of the drawbacks in current embedded and immersed formulations which are based on meshed methods. The different material domains are discretized independently thus avoiding time-consuming, conformal meshing. In this approach, the superposition of foreground (inclusion) and background (matrix) domain integration smoothing cells are corrected by a quasi-conforming quadtree subdivision on the background integration smoothing cells. Due to the non-conforming nature of the background integration smoothing cells near the material interfaces, a variationally consistent (VC) correction for domain integration is introduced to restore integration constraints and thus optimal convergence rates at a minor computational cost. Additional interface integration smoothing cells with area (volume) correction, while non-conforming, can be easily introduced to further enhance the accuracy and stability of the Galerkin solution using VC integration on non-conforming cells. To properly approximate the weak discontinuity across the material interface by a penalty-free Nitsche's method with enhanced coercivity, the interface nodes on the surface of the foreground discretization are also shared with the background discretization. As such, there are no tunable parameters, such as those involved in the penalty type method, to enforce interface compatibility in this approach. The advantage of this meshfree formulation is that it avoids many of the instabilities in mesh-based immersed and embedded methods. The effectiveness of QCE-RKPM is illustrated with several examples.

In this paper, a computational technique is presented based on the natural element method (NEM) for large plastic deformation simulation of the metal forming problems. NEM is a numerical technique in the field of computational mechanics and can be considered as a meshless method. The selected process is backward extrusion for circular shape hollow components from round billets. The punch stroke value is divided into sub-steps, whereas a new set of nodes become active at the end of each sub-step of deformation. Solutions are obtained for different area reductions under friction condition. Hollman-Ludwik law is selected to explain the material behavior after the yielding point. The experiments are carried out with fully annealed commercial aluminum alloy billets at room temperature, using various punch sizes. A set of die and punches are designed and constructed for experimental works. The validity of the proposed method is verified by comparing the results from deformed geometry, contours of equivalent strain and forming loads with those obtained from finite element simulation and experimental measurements. It is concluded that the results obtained by NEM are in good agreement with those from FEM and experiments and therefore, the meshless natural element method is capable of handling large plastic deformation.

Meshfree solution schemes for the incompressible Navier-Stokes equations are usually based on algorithms commonly used in finite volume methods, such as projection methods, SIMPLE and PISO algorithms. However, drawbacks of these algorithms that are specific to meshfree methods have often been overlooked. In this paper, we study the drawbacks of conventionally used meshfree Generalized Finite Difference Method (GFDM) schemes for Lagrangian incompressible Navier-Stokes equations, both operator splitting schemes and monolithic schemes. The major drawback of most of these schemes is inaccurate local approximations to the mass conservation condition. Further, we propose a new modification of a commonly used monolithic scheme that overcomes these problems and shows a better approximation for the velocity divergence condition. We then perform a numerical comparison which shows the new monolithic scheme to be more accurate than existing schemes.