Meshless Radial Point Interpolation Method Research Articles

Meshfree method for bending and free vibration analysis of laminated plates using the Reissner’s mixed variational theorem

Free vibration and bending behavior of laminated plates are investigated using a combination of Reissner’s mixed variational theorem and the meshless radial point interpolation method. By introducing Reissner’s mixed variational theorem, a refined transverse shear stress field is derived based on three-dimensional (3D) elasticity equations. The efficiency and accuracy of this model are demonstrated through numerical examples. It is found that accurate prediction of shear stresses significantly improves the prediction of natural frequency for composite laminated plates. The difference between shear stresses derived from 3D equilibrium equations and that from constitutive equations is significant.

Extending the Natural Neighbour Radial Point Interpolation Meshless Method to the Multiscale Analysis of Sandwich Beams with Polyurethane Foam Core

This work investigates the mechanical behaviour of sandwich beams with cellular cores using a multiscale approach combined with a meshless method, the Natural Neighbour Radial Point Interpolation Method (NNRPIM). The analysis is divided into two steps, aiming to analyse the efficiency of NNRPIM formulation when combined with homogenisation techniques for a multiscale computational framework of large-scale sandwich beam problems. In the first step, the cellular core material undergoes a controlled modification process in which circular holes are introduced into bulk polyurethane foam (PUF) to create materials with varying volume fractions. Subsequently, a homogenisation technique is combined with NNRPIM to determine the homogenised mechanical properties of these PUF materials with different porosities. In this step, NNRPIM solutions are compared with high-order FEM simulations. While the results demonstrate that RPIM can approximate high-order FEM solutions, it is observed that the computational cost increases significantly when aiming for comparable smoothness in the approximations. The second step applies the homogenised mechanical properties obtained in the first step to analyse large-scale sandwich beam problems with both homogeneous and functionally graded cores. The results reveal the capability of NNRPIM to closely replicate the solutions obtained from FEM analyses. Furthermore, an analysis of stress distributions along the beam thickness highlights a tendency for some NNRPIM formulations to yield slightly lower stress values near the domain boundaries. However, convergence towards agreement among different formulations is observed with mesh refinement. The findings of this study show that NNRPIM can be used as an alternative numerical method to FEM for analysing sandwich structures.

A Meshless Radial Point Interpolation Method for Solving Fractional Navier–Stokes Equations

This paper aims to develop a meshless radial point interpolation (RPI) method for obtaining the numerical solution of fractional Navier–Stokes equations. The proposed RPI method discretizes differential equations into highly nonlinear algebraic equations, which are subsequently solved using a fixed-point method. Furthermore, a comprehensive analysis regarding the effects of spatial and temporal discretization, polynomial order, and fractional order is conducted. These factors’ impacts on the accuracy and efficiency of the solutions are discussed in detail. It can be shown that the meshless RPI method works quite well for solving some benchmark problems accurately.

A computationally efficient Element Edge point numerical integration scheme in the meshless method framework for solving fracture problems

The fracture problem is modeled using the Radial Point Interpolation Meshless Method (RPIM) to solve for displacements. Further, the stresses and Stress Intensity Factor (SIF) are calculated using obtained displacements. An effective numerical integral quadrature called the Element Edge point(EE) scheme is used as an alternative to conventional Gauss Quadrature to improve computational efficiency. A comparative study based on two numerical integration schemes, the Element Edge point scheme and Gauss Quadrature, is conducted on four benchmark problems of thick, cracked plates owing to plane strain conditions. The study reveals that the proposed Element Edge point (EE) scheme is computationally efficient and works well in the meshless method framework.

Numerical Solution of Natural Convection Problems Using Radial Point Interpolation Meshless (RPIM) Method Combined with Artificial-Compressibility Model

A numerical method is used to solve the thermal analysis of natural convection in enclosures. This paper proposes the use of an implicit artificial-compressibility model in conjunction with the Radial Point Interpolation Meshless (RPIM) method to mimic laminar natural convective heat transport. The technique couples the pressure with the velocity components using an artificial compressibility model. The RPIM is used to discretize the spatial terms of the governing equation. We solve the semi-algebraic system implicitly in backward Euler pseudo-time. The proposed method solves two test problems—natural convection in the annulus of concentric circular cylinders and trapezoidal cavity. Additionally, the results are validated using experimental and numerical data available in the literature. Excellent agreement was seen between the numerical results acquired with the suggested method and those obtained through the standard techniques found in the literature.

Elastoplastic Analysis of Plates with Radial Point Interpolation Meshless Methods

For both linear and nonlinear analysis, finite element method (FEM) software packages, whether commercial or in-house, have contributed significantly to ease the analysis of simple and complex structures with various working conditions. However, the literature offers other discretization techniques equally accurate, which show a higher meshing flexibility, such as meshless methods. Thus, in this work, the radial point interpolation meshless method (RPIM) is used to obtain the required variable fields for a nonlinear elastostatic analysis. This work focuses its attention on the nonlinear analysis of two benchmark plate-bending problems. The plate is analysed as a 3D solid and, in order to obtain the nonlinear solution, modified versions of the Newton–Raphson method are revisited and applied. The material elastoplastic behaviour is predicted assuming the von Mises yield surface and isotropic hardening. The nonlinear algorithm is discussed in detail. The analysis of the two benchmark plate examples allows us to understand that the RPIM version explored is accurate and allows to achieve smooth variable fields, being a solid alternative to FEM.

Simulation of coupled flow and contaminant transport in an unconfined aquifer using the local radial point interpolation meshless method

Simulation of coupled flow and contaminant transport in an unconfined aquifer using the local radial point interpolation meshless method

Dispersion error reduction for interior acoustic problems using the radial point interpolation meshless method with plane wave enrichment functions

In acoustic analyses the standard radial point interpolation meshless method (RPIM) is able to outperform the classic finite element method (FEM) because the numerical dispersion can be significantly mitigated by increasing the size of the interpolation domain. However, this operation usually leads to a surge in computational cost. In this work an enriched RPIM (e-RPIM) is developed by using the general multiple wave patterns but no extra degrees of freedom are needed. In this enriched scheme the traditional radial basis function interpolation is augmented by a set of plane wave enrichment functions which helps to reflect the oscillation characteristics of the wave propagation. This e-RPIM can obviously alleviate the numerical dispersion and yield satisfactory solution accuracy by using a relatively small size interpolation domain. Although extra computational cost is required in constructing the enriched interpolation shape functions, the computational efficiency of the original RPIM is evidently improved by the proposed e-RPIM if the computation accuracy is taken into consideration. Several numerical experiments validate that the proposed e-RPIM is a promising competitor to the standard RPIM and FEM in acoustic analyses.

Optimization of a Radial Point Interpolation Meshless strategy for strain gradient nanoplates

A parameter optimization of the Radial Point Interpolation Meshless Method (RPIM) is presented in this work for solving the static bending analysis of Kirchhoff nanoplates which include the first order strain gradient theory. Optimization in meshless strategies is often required since shape parameters, of the Radial Basis Functions (RBFs) used, strongly influence the numerical results when different geometries, boundary conditions or various mechanical aspects are considered. Due to the introduction of a first order strain gradient theory within Kirchhoff plate framework, to approximate the bending degrees of freedom, a Hermite RPIM is used. The static deflection of thin plates with different aspect ratios and boundary conditions is analysed. The results of the static analysis are compared with the solutions available in the literature and good agreement among all presented results as well as convergence behaviour is shown.

Using a radial point interpolation meshless method and the finite element method for application of a bio-inspired remodelling algorithm in the design of optimized bone scaffold

The design of bone scaffold involves the analysis of stress shielding, which can occur when the Young’s modulus of the implant is higher than the Young’s modulus of the bone it is replacing, leading to bone decay in the surrounding tissue. It is therefore very important that the material is adequately designed to match the properties of the surrounding tissue, allowing an appropriate load transfer. While some approaches exist in the literature exploring functional gradients of material density, there are much less solutions based on biological laws. A homogenized model of gyroid infill obtained with PLA (E = 3145 MPa) was obtained through mechanical tests of 3D printed specimens, namely tensile and compression, and the obtained model was implemented in a bone remodelling algorithm. The homogenized law was compared to the results obtained with a bone tissue law to assess the equivalence of density distribution and mechanical properties. Through a radial point interpolation method, it was found that similar density fields were obtained for the gyroid infill and for bone tissue when subject to the same boundary conditions. The finite element method was also used for comparison and validation. With the density field results, the gyroid mechanical behaviour was extrapolated to other materials, and similar stiffness values were obtained for bone tissue and titanium alloy (E = 110 GPa) scaffold, which justify this proposal of gyroid scaffolds for mimicking bone properties.

Groundwater flow simulation in a confined aquifer using Local Radial Point Interpolation Meshless method (LRPIM)

Groundwater flow problems are generally solved using analytical or numerical methods. Though analytical solutions are exact and preferable, they are not available for complex field problems. Hence numerical methods such as Finite Element and Finite Difference methods are used to solve complex groundwater problems. These conventional mesh/ grid-based numerical methods need construction of a detailed mesh/ grid. On the other hand, the meshless approach creates a system of algebraic equations on a collection of distributed nodes in the problem area and the boundary. As a result, it is easy to incorporate any modifications to the model at a later time by simply adding nodes to the domain. In this study a weak form meshless method known as local radial point interpolation method (LRPIM) which uses radial basis functions for approximation or interpolation is developed to solve the groundwater flow problems in a confined aquifer. The results obtained from the LRPIM model has been compared with other numerical methods for benchmark and real field problems, and are found to be satisfactory. Implementation of the essential boundary conditions was relatively easier in LRPIM and gave good accuracy for the problems considered. LRPIM can potentially be used as an alternative to the other conventional methods, especially where the domain boundary is irregular or varying with time.

Elasto-plastic adhesive joint design approach by a radial point interpolation meshless method

ABSTRACT For efficient use of adhesive joints, reliable prediction techniques should be made available to the designer. Simulation of these joints’ behaviour is usually performed using the Finite Element Method (FEM). However, it is known that, in adhesive joints, the adhesive thickness (t A) is much smaller than the adherend thickness (t P), thus requiring a highly refined mesh to produce good results. Linked to this, the adhesive has to withstand high strains, causing mesh distortion and hindering the resolution. In these cases, meshless methods can be a good alternative. This work aims to implement the von Mises (vM) and Exponent Drucker-Prager (EDP) criteria combined with a meshless formulation based on the Radial Point Interpolation Method (RPIM), for the strength prediction of adhesively-bonded single-lap joints (SLJ). Validation with experiments is undertaken for joints with brittle to ductile adhesives, with varying overlap lengths (L O). Stress and strain distributions were plotted in the adhesive layer, and the failure load (P m) was assessed by strength of materials failure criteria. Significant adhesive and L O effects were found on P m. The RPIM proved to be a promising tool to predict the behaviour of bonded joints, although some limitations were found by using strength of materials criteria.

Bending and free vibration and analysis of laminated plates on Winkler foundations based on meshless layerwise theory

Free vibration and bending of laminated plates on Winkler foundations are investigated based on a combination of generalized layerwise theory (LW) for higher-order shear deformation theory (HSDT) and the meshless radial point interpolation method (RPIM). The shape function of the meshless radial interpolation method is used to discretize the weak form control equations derived from the principle of virtual work. The shape functions have the properties of Kronecker functions. The slope components associated with the first-order derivatives of the transverse displacements are defined as approximate variables. The transverse shear stresses at each layer interface are continuous. A comparative study with literature solutions verified the numerical stability and applicability of the combined generalized LW-HSDT and meshless RPIM methods. The effect of Winkler modulus parameters on the vibration performance and bending performance of the laminated plate was investigated. The results of bending, vibration frequencies and modes of the laminated plates on different Winkler foundations are presented. It is found that initially, the Winker foundation had a significant effect on the frequency magnitude and bending deformation, but little influence on vibration mode. After the influence of the foundation on the frequency enters the stable period, the influence of foundation on the mode shape will be significant. With the relaxation of the boundary conditions, the vibration mode will still be affected when the foundation parameters are small. Highlights It is for the first time that the laminated plates on Winker foundations were investigated using a combination of higher-order layerwise theory and meshless radial interpolation method. In the proposed model, the slope component related to the first-order derivative of the transverse displacement is defined as an approximate variable and does not require the boundary to be treated by the isogeometric method. The transverse shear stresses at the interface of each layer are continuous for the bending analysis of laminated panels. The meshless results reveal the influence of the Winker foundation parameters on the vibration characteristics of the laminated plate

On applicability of MQ-RPIM and MLPG meshless methods with 3D extended-enriched base functions for estimation of mode I stress intensity factor and fatigue crack growth in cyclic tensile and bending load of an un-notched and notched shaft

In this study, a numerical meshless method is used to solve the weak form of the linear elastic equations in solid mechanics. Evaluation and comparison of the numerical meshless methods have been carried out via the radial point interpolation meshless method with multi-quadrics base functions (MQ-RPIM) and meshless local Petrov-Galerkin method (MLPG). Using these two methods, stress intensity factors in an elastic medium containing geometric discontinuities and cracks are estimated based on tensile and bending cyclic loading. The analysis domain has been identified via three-dimensional modeling of the notched and un-notched shafts with an initial surface semi-elliptical crack subjected to tensile or bending cyclic loadings. To enhance the accuracy of calculations, the RPIM meshless method is applied using polynomial and extended-enriched 3D base functions. Shape functions have been developed using standard and optimal parameters and values with Mono-Objective Function in PSO algorithm. In the MLPG meshless method with the extended-enriched functions, discretization is performed via direct and penalty factor methods, to reach more efficient results and meet the boundary conditions. Efficiency comparison of the selected numerical methods with the experimental findings and the numerical analysis of finite elements method indicates that in comparison with the MLPG method, MQ-RPIM enriched meshless method can be utilized with fewer nodes in the analysis domain while reaching the accuracy and convergence with lower stress intensity factors and gentler slope. However, the processing time of the MLPG meshless method is lower than that of the other methods.

An improved meshless method for finite deformation problem in compressible hyperelastic media

Hyperelastic materials are considered as special category of elastic solid materials because of their nonlinear complicated constitutive laws. Due to large strain state, the behaviour of such materials is often considered in finite deformation analysis. The nonlinear large deformation behavior of such materials is important. In this study, a novel meshless radial point interpolation method (RPIM) enhanced by Cartesian transformation method (CTM), an effective numerical integration, is presented for nonlinear behavior of hyperelastic media under finite deformation state with total Lagrange formulation. Unlike the mesh-based approaches, the meshless methods have shown their advantages in analysis of large deformation problems. The developed CTM-based RPIM is thus free from the need for background cells, which are often used for numerical integration in many conventional meshfree approaches. The developed meshfree method owns some desirable features of an effective technique in solving large deformation, which will be illustrated through the numerical experiments in which our computed results are validated against reference solutions derived from other approaches.

Simulation of the viscoplastic extrusion process using the radial point interpolation meshless method

Additive manufacturing is an emergent technology, which witnessed a large growth demanded by the consumer market. Despite this growth, the technology needs scientific regulation and guidelines to be reliable and consistent to the point that is feasible to be used as a source of manufactured end-products. One of the processes that has seen the most significant development is the fused deposition modeling, more commonly known as 3D printing. The motivation to better understand this process makes the study of extrusion of materials important. In this work, the radial point interpolation method, a meshless method, is applied to the study of extrusion of viscoplastic materials, using the formulation originally intended for the finite element method, the flow formulation. This formulation is based on the reasoning that solid materials under those conditions behave like non-Newtonian fluids. The time stepped analysis follows the Lagrangian approach taking advantage of the easy remeshing inherent to meshless methods. To validate the newly developed numerical tool, tests are conducted with numerical examples obtained from the literature for the extrusion of aluminum, which is a more common problem. Thus, after the performed validation, the algorithm can easily be adapted to simulate the extrusion of polymers in fused deposition modeling processes.

A meshless radial point interpolation method for finite deformation analysis of hyperelasticity

Hyperelastic materials are special materials that possess the non-linear material property. In these materials, the stress-strain relation is derived from the strain energy density function. An interesting property of these materials like rubber is the ability of elastic response when it is subjected to large deformations. That means when the load is removed, the material can easily return to the initial configuration. In addition, they also have some excellent mechanical properties like good tear and abrasion resistance, flexibility at ambient temperature. So hyperelastic materials are widely used in the industry. Due to this reason, it is needed to take a careful look at these materials, especially in the field of mechanical behavior. Because hyperelastic materials usually work under large deformation for almost all cases, their behavior is often considered in a highly non-linear elastic state. This paper presents a meshless radial point interpolation method for hyperelastic bodies with compressible and nearly-incompressible states. The weak form is obtained from the principle of minimum potential energy, and the finite deformation analysis of non-linear behavior is performed under the total Lagrange formulation. Radial point interpolation method shape functions are employed to approximate field nodes and derivatives. Due to possessing the Kronecker delta function property, the boundary conditions are imposed directly in the proposed method. Moreover, this method also shows its advantage for non-linear analysis, especially when the large deformation is considered and the highly-distorted nodal mesh is inherent in the structure. Two numerical examples are conducted with some distributed loads for both compressible and nearly-incompressible states. The obtained results show good agreement with the reference solution. That clearly demonstrates the efficiency and reliability of the proposed method for complex problems.

An efficient meshless radial point collocation method for nonlinear p-Laplacian equation

This paper considered the spectral meshless radial point interpolation (SMRPI) method to unravel for the nonlinear p-Laplacian equation with mixed Dirichlet and Neumann boundary conditions. Through this assessment, which includes meshless methods and collocation techniques based on radial point interpolation, we construct the shape functions, with the Kronecker delta function property, as basis functions in the framework of spectral collocation methods. Studies in this regard require one to evaluate the high-order derivatives without any kind of integration locally over the small quadrature domains. Finally, some examples are given to illustrate the low computing costs and high enough accuracy and efficiency of this method to solve a p-Laplacian equation and it would be of great help to fulfill the implementation related to the element-free Galerkin (EFG) method. Both the SMRPI and the EFG methods have been compared by similar numerical examples to show their application in strongly nonlinear problems.

A Time-Domain Meshless Method Without Discretization in Time

In this letter, we propose a new time-domain meshless method without discretization in time based on eigenexpansions of numerical solutions. As a result, numerical time-stepping is not required, and its associated errors are eliminated. Analytical solutions of the radial point interpolation meshless (RPIM) method are derived, and the solutions become always stable. In other words, electric and magnetic field values are available at any time, making late-time electromagnetic field analysis convenient and efficient. Numerical experiments are conducted to verify the effectiveness and accuracy of the proposed method.

PERFORMANCE OF RADIAL POINT INTERPOLATION METHOD IN SOLVING KINEMATIC WAVE EQUATION FOR HYDROLOGIC MODELLING

This paper presents the solution of the kinematic wave equation using a meshless radial point interpolation method (RPIM). The partial differential equation is discretized using a Galerkin weighted residual method employing RPIM shape functions. A forward difference scheme is used for temporal discretization, while the direct substitution method is employed to solve the nonlinear system at each time step. The formulation is validated against solutions from conventional numerical techniques and physical observation. In all cases, excellent agreements are achieved and hence the validation of the proposed formulation. Optimum values of the multi-quadrics shape parameters were then determined before the assessment of the performance of the method. Based on the convergence rate, it has been shown that the proposed method performs better than the finite difference method and equivalent to the finite element method. This highlights the potential of RPIM as an alternative method for hydrologic modeling.