Meshless Method Research Articles

Deformation monitoring for fixed-wing UAS through Inverse Mesh-free Method

This study proposes the Inverse Local Radial Point Interpolation Method (iLRPIM) for reconstructing the deformation of irregular plate structures. The proposed methodology eliminates the drawbacks of the existing inverse finite-element method (iFEM), where the Jacobian matrix results in reduced computational accuracy for irregular plate-shape sensing. To this end, this paper released from the element and used the Mesh-Free method to describe the calculated strains. Subsequently, the system equation was formulated by minimizing the error between the measured and calculated strains. The deformation parameters were then computed by solving this system equation. Additionally, compared to iFEM, iLRPIM allows for easier adjustment of the interpolation function, making it more suitable for real-world engineering applications. The effectiveness of iLRPIM has been validated through simulations and practical experiments using four typical models. In simulations of the UAS wing box equivalent model, peak errors were below 0.8 mm for a deformation range of 25 mm. In practical scenarios, peak errors were below 3.1 mm for a deformation range of 85 mm. These results underscore the system’s capability to accurately sense the shapes of irregular plate structures using iLRPIM.

Enhanced meshfree method with nodal integration for analysis of functionally graded material sandwich curved shells

This paper presents a nodal integration technique, the sub-domain stabilized conforming integration (SSCI), for the meshfree radial point interpolation method (RPIM) applied to the static and modal analysis of functionally graded material (FGM) sandwich curved shells. FGM sandwich shells with different kinds of core and face sheets are considered in this work while the interested curved shell is formulated by the first-order shear deformation theory. The numerical integration technique to compute the stiffness and mass matrices in the equilibrium equation is the SSCI, which is a stabilized nodal integration with strain smoothing to preserve the accuracy and stability of the numerical results. The RPIM shape functions are utilized in this study for interpolating both the field variables and the geometry of the curved shell due to their ability to satisfy the Kronecker delta property, a rare advantage among meshfree methods. The static and modal analysis of different geometry curved shells with various sandwich FGMs are conducted. Through several numerical examples, the accuracy and efficiency of the SSCI technique in the meshfree RPIM are demonstrated and discussed.

Progressive Failure of Water-Resistant Stratum in Karst Tunnel Construction Using an Improved Meshfree Method Considering Fluid–Solid Interaction

An improved meshfree method that considers cracking, contact behaviour and fluid–solid interaction (FSI) was developed and employed to shed light on the progressive failure of the water-resistant stratum and inrush process in a karst tunnel construction. Hydraulic fracturing tests considering different scenarios and inrush events of the field-scale Jigongling karst tunnel in three scenarios verify the feasibility of the improved meshfree method. The results indicate that the brittle fracture characteristics of the rock mass are captured accurately without grid re-meshing by improving the kernel function of the meshfree method. The complex contact behaviour of rock along the fracture surface during inrush is correctly captured through the introduction of Newton’s law-based block contact algorithms. FSI processing during inrush is accurately modelled by an improved two-phase adaptive adjacent method considering the discontinuous particles without coupling other solvers and additional artificial boundaries, which improves computational efficiency. Furthermore, the improved meshfree method simultaneously captures the fast inrush and rock failure in the Jigongling karst tunnel under varying thicknesses and strengths of water-resistant rocks and sizes of karst caves. As the thickness and strength of water-resistant rock increase, the possibility of an inrush disaster in the tunnel decreases, and a drop in the water level and an increase in the maximum flow velocity have significant delayed effects during the local inrush stage.

Three-dimensional buckling analysis of stiffened plates with complex geometries using energy element method

A novel numerical method, energy element method (EEM), is proposed for the three-dimensional (3D) buckling analysis of stiffened plates with complex geometries. The problem is formulated in a cuboidal domain, and any complex geometric stiffened plate is modeled by assigning cutouts within the cuboidal domain. The stiffened plate is considered as an energy body and is discretized using Gauss points with variable stiffness properties to simulate its energy distribution. Incorporating the extended interval integration, Gauss quadrature, variable stiffness properties, and Chebyshev polynomials, the strain energy of stiffened plates with complex geometries can be numerically simulated by putting the stiffness and thickness of Gauss points in the cutouts to zero in the cuboidal domain. Using the principle of minimum potential energy and Ritz solution procedure, the deformation and buckling behaviors of stiffened plates with complex geometries can be captured. As a result of the new formulations in EEM, new standard energy functionals and solving procedures have been developed. In addition, Gauss points are generated within the energy elements accounting for the geometric boundaries of the stiffened plate, which are characterized by level set functions. EEM is employed to investigate complex-shaped stiffened plates with straight or curvilinear stiffeners, and the results are compared to those obtained using FEM or mesh-free method. The precision, generalization, and stability of EEM are demonstrated.

Numerical computational technique for solving Volterra integro-differential equations of the third kind using meshless collocation method

The primary goal of this study is to give an approximate algorithm for solving Volterra integro-differential equations (VIDEs) of the third kind using meshless collocation techniques. The basic framework of the novel approach is based on a collocation scheme and radial basis functions (RBFs) created on scattered points. This technique requires no background approximation cells, and the algorithm is powerful, has greater stability, and does not require much computer memory. This approach represents the solution of VIDEs of the third kind by interpolating the RBFs based on the Gauss–Legendre quadrature formula. The problem is reduced to a system of algebraic equations that can be easily solved. A description of the technique for the proposed equations is provided. Furthermore, the error analysis of this scheme is examined. A few numerical experiments are presented to prove the reliability and precision of the suggested approach for solving VIDEs of the third kind. Certain problems were compared with analytical solutions, the moving least squares method, and other methods to prove the effectiveness and applicability of the approach described.

A novel meshless method for numerical simulation of fractured-vuggy reservoirs

This paper proposes a novel meshless numerical simulation method for fractured-vuggy reservoirs. Initially, model discretization involves extracting fracture and vug feature nodes along the contours of the reservoir bodies and in alignment with fracture orientations, tailored to the characteristics of the fractured-vuggy oil deposit. Subsequently, a connection element model is established within the influence domain. Based on the seepage equations of the connection system, a computational method for the parameters of the connection elements (connection transmissibility and connection volume) is defined. In addition, a sequential method is employed to solve for the pressure and saturation, achieving rapid prediction of the reservoir's production dynamics. On this basis, an automatic history matching algorithm is utilized for real-time fitting of oil–water dynamic indicators, inverting the characteristic parameters of the connection element model, and quantitatively characterizing the injection–production connection elements. The research findings indicate that the method is capable of rapidly fitting and predicting the dynamics of oil reservoir production. Furthermore, conceptual model examples have substantiated that it achieves comparable computational efficiency and superior computational accuracy when compared to the traditional finite difference (volume) method under the same nodal conditions. Additionally, the node arrangement for fractures and vugs is comparatively flexible, and it can ensure the integrity of the flow paths with a limited number of nodes to enhance predictive accuracy. Therefore, this method can effectively meet the rapid prediction requirements for fractured-vuggy reservoirs and also provides a novel meshless computational approach for the numerical simulation of such reservoirs.

A continuous-discontinuous coupling computational method for multi-material mixtures

Considering the important role of particle-reinforced composites (PRCs), and to accurately assess the mechanical properties of PRC with arbitrarily complex internal structures, this paper proposes a new continuous-discontinuous coupled computational method, i.e., the multi-material integrated analysis (MIA) method, by combining the advantages of the discontinuous deformation analysis (DDA) method, the numerical manifold method (NMM), and the meshless method, for the continuous-discontinuous dual state of this kind of materials when they are subjected to forces. The core idea of this method is by establishing a single physical cover (PC) on each particle and defining a unified and piecewise approximation function over it to achieve displacement coordination between the particles and the matrix. In this model, the particles can maintain the contact and separation characteristics like the blocks, while the matrix is able to achieve the bonding function by overlapping different covers. Two numerical integration schemes, namely single-point Gaussian integration and Monte Carlo integration, are proposed to address the integration issues in stiffness matrix calculations. Meanwhile by considering the similarity between NMM and meshless method interpolation function construction, this paper combines circular coverage and 0th-order moving least squares to construct the interpolation function to form the displacement approximation function for the whole solution space. Moreover, the method of lattice retrieval (also Nearest Neighbor Search, NNS) is invoked for particle contact determination to improve the efficiency of contact and coverage relationship determination. The validity of the model is verified by taking asphalt mixture, a typical PRC, as a demonstrative case study. The results show that the MIA method solves the one-sided problem that the existing algorithms usually only consider the continuous or discontinuous properties of PRC, and provides a new method for comprehensively analyzing the micromechanical response of PRC and solving its large deformation problem.

A novel meshless numerical simulation of oil-water two-phase flow with gravity and capillary forces in three-dimensional porous media

This paper presents a novel fully implicit scheme for simulating three-dimensional (3D) oil-water two-phase flow with gravity and capillary forces using the meshless generalized finite difference method (GFDM). The approach combines an implicit Eulerian scheme in time with a GFDM discretization method in space to compute implicit solutions for the pressure and saturation in the flow control equations. The research introduces an L2 norm error formula and conducts a sensitivity analysis on the impact of varying influence domain radii on computational accuracy within the Cartesian node collocation scheme. Findings suggest that larger influence domain radii correspond to reduced computational accuracy, providing a preliminary guideline for selecting the domain radius in 3D GFDM applications. Overall, this paper presents an effective and precise meshless method for addressing two-phase flow challenges in 3D porous media, highlighting the promising prospects of GFDM in numerical simulations.

A radial basis function-finite difference method for solving Landau–Lifshitz–Gilbert equation including Dzyaloshinskii-Moriya interaction

This paper investigates a numerical method for solving the two-dimensional Landau–Lifshitz–Gilbert (LLG) equation, governing the dynamics of the magnetization in ferromagnetic materials. Specifically, we incorporate the Dzyaloshinskii–Moriya interaction into the LLG equation—a crucial factor for the creation and stabilization of magnetic skyrmions. We propose a local meshless method that utilizes radial basis function-finite difference (RBF-FD) for spatial discretization and the Crank–Nicolson scheme for temporal discretization, along with an extrapolation technique to handle the nonlinear terms. We demonstrate the method’s accuracy, efficiency, and adaptability through numerical tests on domains of various shapes, showcasing its practical utility in simulating real-world magnetic phenomena and advanced materials.

Stepwise Alternating Direction Implicit Method of the Three Dimensional Convective-Diffusion Equation

A stepwise alternating direction implicit method of the three dimensional convective-diffusion equation is considered in this paper. We constructed an implicit difference scheme and analyzes it's truncation error, convergence and stabilities. The theoretical and numerical analysis shows that the implicit difference scheme is unconditional stable. Then the Greedy Algorithm is proposed to solve the numerical solution on x,y and z axis separately by using implicit difference scheme and the numerical solution is convergent theoretically, however with no physical meaning. The Stepwise Alternating Direction Implicit Method (SADIM) is proposed, which uses the implicit difference scheme in this paper. Using Sauls scheme to pretreat the initial-boundary condition before iterating, thus eliminate the numerical oscillation caused by discontinuous initial boundary conditions. This SADIM is at least six ordered convergent, and is one of high ordered numerical methods for three dimensional problem. Our implicit difference scheme is more ideal than the standard Galerkin centered on finite difference scheme, quicker than SOR iteration method. The convergence of our implicit scheme is better than finite element method, characteristic line method, and mesh-less method. Our method eliminates the numerical oscillation caused by the convection dominant, resists the dispersion effectively and addresses dissipation caused by diffusion dominant.The implicit difference scheme has good theoretical and practical value.

A new space–time localized meshless method based on coupling radial and polynomial basis functions for solving singularly perturbed nonlinear Burgers’ equation

In this paper, the singularly perturbed nonlinear Burgers’ problem (SPBP) with small kinematic viscosity 0<ϵ≪1 is solved using a new Space–Time Localized collocation method based on coupling Polynomial and Radial Basis Functions (STLPRBF). To our best knowledge, it is the first time that the solution of SPBP is accurately approximated using the space–time meshless method without applying any adaptive refinement technique. The method is based on solving the problem without distinguishing between space and time variables, which eliminates the need for time discretization schemes. To address the inherent non-linearity of the problem, the method employs an iterative algorithm based on quasilinearization technique. The efficiency and accuracy of the proposed method are demonstrated by solving different examples of one- and two-dimensional SPBP with very small ϵ up to 10−10. Additionally, the numerical convergence of the method with respect to ϵ and also to the number of collocation points has been investigated. The comparison of the STLPRBF results with other published ones is presented.

Numerical analysis of nonlinear Klein–Gordon equations by a meshless superconvergent finite point method

In this paper, we introduce a meshless method for numerical simulation of nonlinear Klein–Gordon equations. The method begins with a temporal discretization to address time derivatives. The stability and error of the temporal discretization scheme are theoretically analyzed. Subsequently, meshless algebraic systems of Klein–Gordon solitons are established by using the superconvergent finite point method (SFPM) for spatial discretization. The moving least squares approximation and its smoothed derivatives are adopted in the SFPM to ensure the high accuracy and remarkable superconvergence. Accuracy and convergence of the meshless numerical simulation for nonlinear Klein–Gordon equations are analyzed in theory. Numerical results validate the superconvergence and effectiveness of the method and confirm the theoretical analysis.

Multiscale Analysis of Sandwich Beams with Polyurethane Foam Core: A Comparative Study of Finite Element Methods and Radial Point Interpolation Method.

This study presents a comprehensive multiscale analysis of sandwich beams with a polyurethane foam (PUF) core, delivering a numerical comparison between finite element methods (FEMs) and a meshless method: the radial point interpolation method (RPIM). This work aims to combine RPIM with homogenisation techniques for multiscale analysis, being divided in two phases. In the first phase, bulk PUF material was modified by incorporating circular holes to create PUFs with varying volume fractions. Then, using a homogenisation technique coupled with FEM and four versions of RPIM, the homogenised mechanical properties of distinct PUF with different volume fractions were determined. It was observed that RPIM formulations, with higher-order integration schemes, are capable of approximating the solution and field smoothness of high-order FEM formulations. However, seeking a comparable field smoothness represents prohibitive computational costs for RPIM formulations. In a second phase, the obtained homogenised mechanical properties were applied to large-scale sandwich beam problems with homogeneous and approximately functionally graded cores, showing RPIM's capability to closely approximate FEM results. The analysis of stress distributions along the thickness of the beam highlighted RPIM's tendency to yield lower stress values near domain edges, albeit with convergence towards agreement among different formulations. It was found that RPIM formulations with lower nodal connectivity are very efficient, balancing computational cost and accuracy. Overall, this study shows RPIM's viability as an alternative to FEM for addressing practical elasticity applications.

Stress-based topology optimization using maximum entropy basis functions-based meshless method

AbstractThis paper presents volume-constrained stress minimization-based, topology optimization. The maximum entropy (maxent) basis functions-based meshless method for two-dimensional linear elastic structures is explored. This work focuses to test the effectiveness of the meshless method in handling the stress singularities during the topology optimization process. The commonly used moving least square basis functions are replaced with maximum entropy basis functions, as the latter possess weak Kronecker delta property which leads to the finite element method (FEM) like displacement boundary conditions imposition. The maxent basis functions are calculated once at the beginning of the simulation and then used in optimization at every iteration. Young’s modulus for each background cell is interpolated using the modified solid isotropic material with penalization approach. An open source pre-processor CUBIT is used. A comparison of the proposed approach with the FEM is carried out using a diverse set of problems with simple and complex geometries of structured and unstructured discretization, to establish that maxent-based meshless methods perform better in tackling the stress singularities due to its smooth stress field.

Multiquadric Collocations for Nonlinear Vibration of Functionally Graded Plates of Ti–6Al–4V Metal

While Finite Element Method (FEM) has proven reliable for analyzing linear or nonlinear stresses in materials, structures, and fluid flows, its reliance on mesh generation can escalate project simulation costs. In contrast, mesh- free techniques offer an alternative by directly generating system algebraic equations for the entire issue domain, bypassing the need for a preset mesh. This study focuses on investigating the nonlinear free vibration response of shear-deformable Functionally Graded Material (FGM) plates. These plates are expected to exhibit varying material properties, following a power-law distribution linked to the volume fractions of constituent materials as plate thickness increases. Utilizing shear deformation theories and von Karman nonlinear kinematics, the study seeks a mathematical solution for the real physical issues encountered in these plates. To address these challenges, a meshless method employing the Multi Quadric Radial Basis Function (MQRBF) is employed for evaluation. The computed results reveal insights into how different material properties and amplitudes influence the nonlinear free vibration response of composite plates. Comparing these outcomes to prior research demonstrates promising progress in this area.

Numerical modeling of earthquake-induced landslides using updated Lagrangian nonlocal general particle dynamics method

Developing a robust numerical method to model earthquake-induced landslides has long been a persistent challenge in the field of computational geotechnical engineering. Recently, the meshless methods based on nonlocal theory have piqued the interest of researchers. However, the application of nonlocal theory in seismic analysis is currently limited. This paper proposed a numerical framework based on the updated Lagrangian nonlocal general particle dynamics (UL-NGPD) method to analyze earthquake-induced landslide problems. The UL-NGPD method benefiting from the update support domain can capture the whole process of slope run-out induced by earthquakes. To enhance the numerical stability of the proposed method, several optimizing strategies are proposed. In the current framework, the seismic waves are input through the proposed boundary treatments of nonlocal form. Besides, a nonlocal friction model with velocity-weakening is proposed to accurately simulate the movement process of soils on a rocky sliding bed. The proposed framework is validated by the simulations of several classic problems, including the collapse of sand and the shake table test. The performance of the proposed approach is further demonstrated through simulating the landslides induced by earthquakes. The numerical results consistent with recorded data indicate that the UL-NGPD method has an excellent capacity to deal with earthquake-induced landslide problems.

A new modified Bessel‐type radial basis function for meshless methods: Utilized in the analysis of free vibration in 2D functionally graded Euler–Bernoulli beams

AbstractRadial basis functions (RBFs) are extensively employed in mesh‐free methods owing to their distinct properties. This study presents a novel RBF formulation based on a modified first‐kind Bessel function, introduced for the first time. The efficacy and precision of the proposed function are assessed through an examination of the free vibrations of Euler–Bernoulli beams composed of two‐directional functionally graded materials. Longitudinal and thickness property variations are modeled in polynomial and exponential forms, respectively. The performance of the novel RBF is scrutinized under various boundary conditions (clamped, simply supported, and free), and comparative analyses are conducted against similar investigations and an RBF based on the first‐kind Bessel function. Convergence analysis of the proposed modified first‐kind Bessel function‐based RBF reveals superior convergence rates compared to the first‐kind Bessel function‐based RBF. Moreover, a comparison between results obtained from modeling using the proposed RBF and exact solutions underscores the adequacy of this approach, with a maximum discrepancy of 4.933% observed under clamped‐free boundary conditions. In essence, the findings suggest that the proposed modified first‐kind Bessel function‐based RBF holds promise for analyzing the free vibrations of functionally graded Euler–Bernoulli beams. The primary aim of this research is to introduce and validate a new RBF based on a modified first‐kind Bessel function for the analysis of free vibrations in Euler–Bernoulli beams made of two‐directional functionally graded materials. The study focuses on evaluating the performance and accuracy of this novel RBF in comparison with existing RBFs and exact solutions. By addressing the limitations of conventional RBFs and proposing an innovative approach, this research aims to enhance the accuracy and efficiency of meshless methods in structural vibration analysis.

PERSPECTIVE: Cryosurgery process applications: a mathematical review

The present study reviews some of the prominent mathematical models that are used to simulate the cryosurgery treatment of tumor tissues, i.e., destruction of tumor tissues via controlled freezing with cryoprobes with minimizing the impact on surrounding healthy tissues. Numerical simulation of the appropriate mathematical models that reflect practical situations may help the physicians to design a planning framework for the treatment, which includes total number of cryoprobes to be used, their placement design and the duration of optimal freezing, etc. Finite element method, meshfree method, and finite volume method are some of the suitable numerical techniques for simulating bio-heat transfer process within complex tissues during treatment.

Three-dimensional continuum point cloud method for large deformation and its verification

This study presents a strong form based meshfree collocation method, which is named Continuum Point Cloud Method, to solve nonlinear field equations derived from classical mechanics for deformed bodies in three-dimensional Euclidean space. The method and its implementation are benchmarked against a nonlinear vector field using manufactured solutions. The analysis of mechanical fields firstly focuses on the study of St. Venant Kirchhoff and compressible neo-Hookean materials. Results for various initial boundary value problems are presented, including benchmark cases involving unidirectional tension and simple shear. Subsequently, the study concludes with an analysis of a displacement-controlled simulation of a compressible neo-Hookean material, specifically a bar that is pulled to 50% of its original length and rotated 90°. The pure tension case yields a 1.5% error in displacement between computed and expected values and a combined tension and torsion loading case provides further insight into material behavior under complex loading conditions. The resulting normal axial and transverse stress-strain curves are also presented. Finally, the consistency and robustness of the proposed nonlinear numerical schemes are successfully demonstrated through various numerical experiments.

Gaussian smoothed particle hydrodynamics: A high-order meshfree particle method

Smoothed particle hydrodynamics (SPH) has attracted significant attention in recent decades, and exhibits special advantages in modeling complex flows with multiphysics processes and complex phenomena. Its accuracy depends heavily on the distribution of particles, and will generally be lower if the particles are distributed non-uniformly. A high-order SPH scheme is proposed in the present work for simulating both compressible and incompressible flows. It uses a Gaussian quadrature rule to perform the particle approximation of SPH by introducing Gaussian nodes. Unfortunately, the Gaussian nodes hardly overlap with SPH particles due to the Lagrangian feature, and thus we use a high-order interpolation method to obtain the corresponding physical quantities at the Gaussian nodes. The accuracy and robustness of the proposed Gaussian SPH are demonstrated by several numerical tests, including the Sod problem, Poiseuille flow, Couette flow, cavity flow, Taylor–Green vortex and dam break flow, and a convergence analysis is also conducted to evaluate the effects of particle resolution and distribution for reconstructing a given function. The simulation results for each test case are in good agreements with the available analytical, experimental or numerical results, showing that the proposed Gaussian SPH method is accurate and reliable but expensive for simulating compressible and incompressible flow problems.