Meshless Approach Research Articles

Meshless method and shifted Chebyshev polynomials for solving a class of VIEs of the third kind

The main goal of this study is to solve a class of linear and nonlinear Volterra integral equations (VIEs) of the third kind. The proposed technique estimates the solution of these equations using a technique based on MLS approximation and shifted Chebyshev polynomials. The approach is independent of the geometry of the domain and does not require any background meshes; therefore, it can be considered a meshless approach. The new method is effective and more flexible for many types of VIEs of the third kind, and its algorithm can be simply implemented on computers. The construction of a new technique for the proposed equations has been introduced. The convergence analysis is also studied. The convergence precision of the new approach is checked on classes of VIEs of the third kind, and the results validate the theoretical error estimates. Finally, numerical tests are provided to illustrate the accuracy and reliability of this approach.

A meshless model for three-dimensional direct numerical simulation of local scour around cylinder bridge foundation

A meshless local scour model for cylinder bridge foundation based on the smoothed particle hydrodynamics (SPH) method is proposed in this paper. Differing from the traditional Eulerian mesh model used for monopile local scour, it overcomes the high requirements of mesh quality and density for capturing large deformations at the fluid interface. Unlike the transient sediment erosion SPH model used for dam-break, sediment flushing, and water jet, the meshless approach is innovatively applied to the continuous simulation of bridge local scour under steady flow conditions. The present SPH model is established based on the SPH multi-phase fluid model, integrating a classical fluid model combined with numerical stabilization algorithms to govern water particle motion and a sediment model from Zhang et al. (2024) to govern sediment evolution, and tests a new scheme for boundary treatment in the model based on DualSPHysics. Furthermore, a novel scour visualization method that aims at extracting both visible and actual bed surfaces from discrete particles is proposed. The model is validated through comparison with rigid bed and live bed experiments, demonstrating favorable agreement in terms of flow and scour simulation. Importantly, the model results reveal a discrepancy between the elevation of the post-scour visible bed surface derived from the conventional Eulerian mesh model and experimental data, compared to the actual bed surface unaffected by water flow erosion. This indicates the necessity for incorporating a safety margin in foundation design when utilizing experimental results to determine the maximum scour depth.

Investigation of micropolar dusty fluid flow in a rotational frame with magnetic field: A meshless radial basis function pseudospectral approach

AbstractRotational flow has numerous applications across a wide range of fields, including industrial processes for mixing, pumping, and processing materials, geophysical fluid dynamics for studying the Earth's atmosphere and oceans, aerospace engineering for designing and testing aircraft and spacecraft, medical imaging for creating detailed images of the human body, and environmental engineering for studying the transport of pollutants in water and air. In rotational flow, pumping power is an important parameter that measures the amount of energy required to maintain the flow of the fluid in a rotating frame. It is significant because the presence of the Coriolis force and other rotational effects can substantially alter the fluid dynamics, making it difficult to maintain a steady flow. The present study deals with the flow of micropolar dusty fluid in a rotating frame under the influence of the magnetic field, applied transversely to the flow in a horizontal channel. The fluid is induced to flow between two parallel plates by the movement of the upper plate and a constant pressure gradient. A meshless radial basis function pseudospectral approach is employed to derive a set of partial differential equations that determine the primary and secondary velocity profiles of the fluid, as well as the particle and microrotation velocities and temperature distribution. The pumping power required to sustain the flow in the absence of a pressure gradient is also calculated based on these velocity profiles. The findings provide valuable insights into the behavior of micropolar dusty fluids in a rotational channel and can potentially inform the design of industrial processes involving such fluids.

A meshless and binless approach to compute statistics in 3D ensemble PTV

A meshless and binless approach to compute statistics in 3D ensemble PTV

A novel meshless radial basis reproducing kernel particle approach for 3D thermo-mechanical analysis of mushy zone phase-change problems

This paper presents a three-dimensional thermo-elastoplastic computational model for the analysis of phase change problems involving the mushy zone. The model utilizes a novel meshless radial basis reproducing kernel particle approach, incorporating a high-order kernel that integrates Gaussian and cosine functions. An energy balance equation for the entire domain is formulated using the effective heat capacity method, based on the enthalpy formulation. The governing equations are discretized using the Galerkin weak form to compute thermal residual stresses, with essential boundary conditions enforced through the penalty method. Plastic deformation is characterized using the von Mises criterion with isotropic hardening, and the strain term resulting from temperature-dependent material properties is incorporated into the incremental solution process. Determination of the plastic strain increment is based on the Prandtl–Reuss flow rule. The accuracy and efficiency of the proposed meshless approach were rigorously evaluated through numerical examples encompassing two- and three-dimensional problems. The computational results were compared against available analytical and validated numerical solutions. This comprehensive assessment substantiated the robustness and precision of the developed technique. The case studies presented elucidate the capacity of the novel meshless model to effectively predict temperature distributions and residual stress fields within mushy zone phase change phenomena, accommodating both regular and irregular geometries under various boundary conditions. Notably, the model demonstrated consistent stability and high accuracy across the spectrum of simulated scenarios, thus underscoring its potential as a valuable tool for investigating complex phase change processes.

Random thermal-vibration mechanisms of sandwich ventral fin-type plate-shell systems with porous functionally graded core

The stochastic thermal-vibration mechanisms within a sandwich ventral fin-type plate-shell system, featuring a porous functionally graded (FG) core, are exhaustively analyzed under various random loading conditions employing an innovative node-based, meshless computational approach. The studied structure is decoupled into several plates and open cylindrical panel according to the geometric characteristics, and the mechanical relationships at the structural boundaries or connection interfaces are equivalently simulated by using penalty parameters. Following the general Hamilton's principle, the meshless approach combined with the first-order shear deformation theory (FSDT) incorporating thermal effects is employed to derive the vibration equations of the sandwich ventral fin-type plate-shell systems. Also, the pseudo excitation method (PEM) is introduced to calculate stationary and nonstationary random responses. In order to verify the accuracy of the meshless algorithm in this study, the convergence and correctness are studied comprehensively. And then, the effects of some parameters such as temperature variation, porosity parameters, power-law index and random excitations on the thermal vibration characteristics of the sandwich ventral fin-type plate-shell systems with porous FG core are presented. The results show that the power-law index and temperature can increase the frequency parameter of the structure. The smooth power spectral density (PSD) excitation only affects the amplitude of the response curve, and does not affect the frequency corresponding to the peak. In the analysis of non-stationary random vibration, the influence of modulation parameters on response is very significant.

Numerical solution of the boundary value problem of elliptic equation by Levi function scheme

AbstractFor boundary value problem of an elliptic equation with variable coefficients describing the physical field distribution in inhomogeneous media, the parametrix can represent the solution in terms of volume and surface potentials, with the drawback that the volume potential involving in the solution expression requires heavy computational costs as well as the solvability of the integral equations with respect to the density pair. We introduce an modified integral expression for the solution to an elliptic equation in divergence form under the parametrix framework. The well‐posedness of the linear integral system with respect to the density functions to be determined is rigorously proved. Based on the singularity decomposition for the parametrix, we propose two schemes to deal with the volume integrals so that the density functions can be solved efficiently. One method is an adaptive discretization scheme for computing the integrals with continuous integrands, leading to the uniform accuracy of the integrals in the whole domain, and consequently the efficient computations for the density functions. The other method is the dual reciprocity method which is a meshless approach converting the volume integrals into boundary integrals equivalently by expressing the volume density as the combination of the radial basis functions determined by the interior grids. The proposed schemes are justified numerically to be of satisfactory computation costs. Numerical examples in 2‐dimensional and 3‐dimensional cases are presented to show the validity of the proposed schemes.

A meshless approach based on fractional interpolation theory and improved neural network bases for solving non-smooth solution of 2D fractional reaction–diffusion equation with distributed order

The primary objective of this research paper is to present a novel and effective meshless numerical approach for solving the 2D time fractional reaction diffusion system with distributed order on an arbitrary domain. Gauss–Legendre quadrature formula is applied to discretize distributed-order derivative integral. We establish the piecewise parabolic fractional interpolation theory and with its assistance, the proposed approach can proficiently solve the non-smooth solutions of the equations and more accurately approximate the Caputo fractional derivative. This meshless method based on improved neural network bases combines the high accuracy approximation advantage and strong express ability of neural network to construct the basis functions set on arbitrary domains, which significantly reduces a computational consumption. The bases constructed based on the neural network allow the selection of 12 bases numbers to achieve an approximation equivalent to that of 1000 ordinary bases. The theoretical analyses of error and convergence order for the meshless approach are carried out. Numerical examples are implemented to validate the high precision and capability of the meshless numerical approach.

A Full Meshless Algorithm For Super-Resolution In Image Velocimetry Based On KNN-PTV And Constrained RBFs

"A novel full meshless super-resolution method for image velocimetry is proposed. This method builds upon the combination, presented in Tirelli et al. (2023a), of K-nearest neighbour Particle Tracking Velocimetry (Tirelli et al., 2023b, KNN-PTV) and constrained Radial Basis Functions regression (Sperotto et al., 2022, c-RBFs). The main novelty is that the algorithm is implemented here in a fully meshless version, i.e. it does not require the definition of a common Eulerian grid at any step of the process. KNN-PTV enhances the spatial resolution of vector fields by exploiting data coherence in space and time, even for non-time-resolved measurements. On the other and, c-RBFs provide an analytical representation of the field from scattered data while allowing the introduction of physical constraints. In this new version, the dictionary to blend data from different snapshots is computed with a meshless POD approach developed by the authors and presented in another contribution to this conference. This approach removes the last constraint of Eulerian grids and paves the way for a fully meshless algorithm. The algorithm is validated on a challenging 2D synthetic case, such as turbulent channel flow, and an experimental case involving the separation bubble around the frontal portion of a Ground Transportation System (GTS)."

A truly meshless approach to structural topology optimization based on the Direct Meshless Local Petrov–Galerkin (DMLPG) method

A truly meshless approach to structural topology optimization based on the Direct Meshless Local Petrov–Galerkin (DMLPG) method

A mesh-free framework for high-order simulations of viscoelastic flows in complex geometries

The accurate and stable simulation of viscoelastic flows remains a significant computational challenge, exacerbated for flows in non-trivial and practical geometries. Here we present a new high-order meshless approach with variable resolution for the solution of viscoelastic flows across a range of Weissenberg numbers. Based on the Local Anisotropic Basis Function Method (LABFM) of King et al. (2020), highly accurate viscoelastic flow solutions are found using Oldroyd B and PPT models for a range of two dimensional problems — including Kolmogorov flow, planar Poiseulle flow, and flow in a representative porous media geometry. Convergence rates up to 9th order are shown. Three treatments for the conformation tensor evolution are investigated for use in this new high-order meshless context (direct integration, Cholesky decomposition, and log-conformation), with log-conformation providing consistently stable solutions across test cases, and direct integration yielding better accuracy for simpler unidirectional flows. The final test considers symmetry breaking in the porous media flow at moderate Weissenberg number, as a precursor to a future study of fully 3D high-fidelity simulations of elastic flow instabilities in complex geometries. The results herein demonstrate the potential of a viscoelastic flow solver that is both high-order (for accuracy) and meshless (for straightforward discretisation of non-trivial geometries including variable resolution). In the near-term, extension of this approach to three dimensional solutions promises to yield important insights into a range of viscoelastic flow problems, and especially the fundamental challenge of understanding elastic instabilities in practical settings.

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.

Meshless Generalized Finite Difference Method Based on Nonlocal Differential Operators for Numerical Simulation of Elastostatics

This study proposes an innovative meshless approach that merges the peridynamic differential operator (PDDO) with the generalized finite difference method (GFDM). Based on the PDDO theory, this method introduces a new nonlocal differential operator that aims to reduce the pre-assumption required for the PDDO method and simplify the calculation process. By discretizing through the particle approximation method, this technique proficiently preserves the PDDO’s nonlocal features, enhancing the numerical simulation’s flexibility and usability. Through the numerical simulation of classical elastic static problems, this article focuses on the evaluation of the calculation accuracy, calculation efficiency, robustness, and convergence of the method. This method is significantly stronger than the finite element method in many performance indicators. In fact, this study demonstrates the practicability and superiority of the proposed method in the field of elastic statics and provides a new approach to more complex problems.

Three-dimensional elastoplastic post-buckling analysis of porous FG plates resting on Winkler/Pasternak foundation using meshless RRKPM

This paper presents a three-dimensional (3D) analysis of the elastoplastic post-buckling behavior of porous functionally graded (FG) plates resting on Winkler/Pasternak foundations under uniaxial and biaxial in-plane loadings using an enhanced meshless approach. The principle of virtual work is used to derive the governing equations, and 3D nonlinear Green–Lagrange strains are considered. Plastic deformation is modeled employing the Prandtl–Reuss flow law and the isotropic hardening von Mises criterion. A novel meshless radial basis reproducing kernel particle approach is used to obtain the discretized system of equations. The Newton–Raphson method, coupled with the arc-length technique, is used to compute post-buckling paths of porous FG plates. Numerical assessments show that post-buckling paths are significantly influenced by porosity parameter, porosity distribution, foundation parameters, material gradient, plate thickness, loading ratio, and boundary conditions (BCs).

PARALLEL MESHLESS RADIAL BASIS FUNCTION COLLOCATION METHOD FOR NEUTRON DIFFUSION PROBLEMS

The meshless global radial basis function (RBF) collocation method is widely used to model physical phenomena in science and engineering. The method produces highly accurate solutions with an exponential convergence rate. However, due to the global approximation structure of the method, dense node distributions lead to long computation times and hinder the applicability of the technique. In order to overcome this issue, this study proposes a parallel meshless global RBF collocation algorithm. The algorithm is applied to 2-D neutron diffusion problems. The multiquadric is used as the RBF. The algorithm is developed with Mathematica and eight virtual processors are used in calculations on a multicore computer with four physical cores. The method provides accurate numerical results in a stable manner. Parallel speedup increases with the number of processors up to five and seven processors for external and fission source problems, respectively. The speedup values are limited by the constrained resource sharing of the multicore computer’s memory. On the other hand, significant time savings are achieved with parallel computation. For the four-group fission source problem, when 4316 interpolation nodes are employed, the utilization of seven processors instead of sequential computation decreases the computation time of the meshless approach by 716 s.

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.

Elasto-Static Analysis of Composite Restorations in a Molar Tooth: A Meshless Approach.

Dental caries and dental restorations possess a long history and over the years, many materials and methods have been invented. In recent decades, modern techniques and materials have brought complexity to this issue, which has created the necessity to investigate more and more to achieve durability, consistency, proper mechanical properties, efficiency, beauty, good colour, and reduced costs and time. Combined with the recent advances in the medical field, mechanical engineering plays a significant role in this topic. This work aims at studying the elasto-static response of a human molar tooth as a case study, respecting the integral property of the tooth and different composite materials of the dental restoration. The structural integrity of the case study will be assessed through advanced numerical modelling resorting to meshless methods within the stress analysis on the molar tooth under different loading conditions. In this regard, bruxism is considered as being one of the most important cases that cause damage and fracture in a human tooth. The obtained meshless methods results are compared to the finite element method (FEM) solution. The advantages and disadvantages of the analysed materials are identified, which could be used by the producers of the studied materials to improve their quality. On the other hand, a computational framework, as the one presented here, would assist the clinical practice and treatment decision (in accordance with each patient's characteristics).

Generalized Thermoelasticity of Two-Dimensional Bounded Media Under Moving Heat Source: A Meshless Approach

In this work, the coupled generalized thermoelasticity of two-dimensional finite domain subjected to moving heat source is investigated. The generalized thermoelasticity is based on Lord–Shulman (LS) theory, and the moving heat source is considered to move along the [Formula: see text]-axis at the middle of the domain with constant velocity. The meshless method, because of continuous moving of approximations over the problem domain, is the best numerical method for problems with moving loads. So, for solving the governing equations, the meshless Galerkin weak form is employed. In the applied meshless method, the moving least square (MLS) shape functions are used for approximating the field variables in each influence domain. Also, for solving the final equations in the time domain, the Newmark time marching method is used. The results show that a moving source with a velocity equal to the temperature’s wave speed exhibit the maximum peak values of temperature, displacement and stress in the domain, and when the temperature wave speed and the elastic wave speed are equal, the moving heat source at this speed drastically increases the peak values of stress, displacement and temperature.

Aerodynamic stability analysis of the concrete ceiling reinforced with advanced functionally graded nano-materials: A sustainable approach for construction materials

Proper consideration should be given to the stability of new concrete in order to assure the quality of building engineering, while also demanding its excellent flowability. The inadequate initial state stability negatively impacts the long-term durability of reinforced concrete structures, although this issue has not been well addressed. This work focuses on assessing the aerodynamic response of concrete ceilings reinforced with advanced functionally graded nano-materials. As the reinforcement of the current concrete structure, graphene oxide powders (GOPs) are used with improved material properties than other types of reinforcement. For mathematical modeling of the current structure, Reddy’s Higher-order Shear Deformation Theory is used to model the current work’s displacement fields. Also, the perturbation aerodynamic force is mathematically described using the Bernoulli equation for potential flow. After that using a method that involves separating variables, we may get the answer for the aerodynamic pressure in its ultimate form. Haber-Schaim foundation made of auxetic material in the Cartesian coordinate system is used to model the foundation of the current work with high accuracy. After obtaining the governing equations and associated boundary conditions, a meshless approach with weighted orthogonal basis and Kronecker delta-based shape functions are used to solve the equations. Finally, some suggestions for improving the aerodynamics and flutter velocity of airflow of the presented concrete plate reinforced by GOPs are presented for related aerodynamics industries.

Efficient simulation of two-dimensional time-fractional Navier–Stokes equations using RBF-FD approach

This study is focused on the Navier–Stokes equations. The numerical method employed in this study for solving the Navier–Stokes equations is a meshless approach that combines the radial basis function (RBF) and finite difference (FD) schemes on two-dimensional arbitrary domains. This method, known as radial basis function-generated finite differences (RBF-FD), utilizes the finite difference technique to discretize the time domain and the RBF-FD approximation to discretize the space domain. The RBF-FD method offers several advantages, including high-order convergence rates and the ability to effectively handle sparse node layouts within arbitrary domains. The efficacy and accuracy of the suggested approach are evaluated through the analysis of four unique instances: comprising Taylor–Green vortices, lid-driven cavity flow, backward-facing step flow, and flow around a square cylinder. In the last example, we examined a simulation of air pollution around buildings.