Point Interpolation Meshless Method Research Articles

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.

Radial basis point interpolation for strain field calculation in digital image correlation

In order to extract smooth and accurate strain fields from the noisy displacement fields obtained by digital image correlation (DIC), a point interpolation meshless (PIM) method with a radial basis function (RBF) is introduced for full-field strain calculation, which overcomes the problems of slow calculation speed and unstable matrix inverse calculation of the element-free Galerkin method (EFG). The radial basis point interpolation method (RPIM) with three different radial basis functions and the moving least squares (MLS) and pointwise least squares (PLS) methods are compared by analyzing and validating the strain fields with high-strain gradients in simulation experiments. The results indicate that the RPIM is nearly 80% more computationally efficient than the MLS method when a larger support domain is used, and the efficiency of the RPIM is nearly 26% higher than that of the MLS method when a smaller support domain is used; the strain calculation accuracy is slightly lower than that of the MLS method by 0.3–0.5%, but the stability of the calculation is significantly improved. In contrast with the PLS method, which is easily affected by the noise and the size of the strain calculation window, the RPIM is insensitive to the displacement noise and the size of the support domain and can obtain a similar calculation accuracy. The RPIM with multiquadric (MQ) radial basis functions performs well in balancing the computational accuracy and efficiency and is insensitive to shape parameters. The application cases show that the method can effectively compute the strain field at the crack tip, validating its applicability to the study of the plastic region at the crack tip. In conclusion, the proposed RPIM-based method provides an accurate, practical, and robust approach for full-field strain measurements.

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

Implementation of point interpolation meshless method for failure analysis of laminated composite beams

The present study develops computationally efficient meshless model for the first-ply failure analysis and predicts the weakest ply, based on various failure criteria via one-dimensional, higher-order beam theory. The effective property of lamina is obtained with the help of micromechanics based Eshelby’s-Mori-Tanaka model. The governing equation is obtained with the help of point interpolation method based on polynomial basis function. The effects of different essential and natural boundary conditions, aspect ratio, lamination angle and volume fraction on the first-ply failure load of composite beam is studied.

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.

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.

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 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.

Geometrically Nonlinear Analysis for Elastic Beam Using Point Interpolation Meshless Method

The intrinsic beam theory, as one of the exact beam formulas, is quite suitable to describe large deformation of the flexible curved beam and has been widely used in many engineering applications. Owing to the advantages of the intrinsic beam theory, the resulted equations are expressed in first‐order partial differential form with second‐order nonlinear terms. In order to solve the intrinsic beam equations in a relative simple way, in this paper, the point interpolation meshless method was employed to obtain the discretization equations of motion. Different from those equations by using the finite element method, only the differential of the shape functions are needed to form the final discrete equations. Thus, the present method does not need integration process for all elements during each time step. The proposed method has been demonstrated by a numerical example, and results show that this method is highly efficient in treating this type of problem with good accuracy.

Combining radial point interpolation meshless method with a new homogenization technique for trabecular bone multiscale structural analyses.

Bone tissue is a dynamic tissue, possessing different functional requirements at different scales. This layered organization indicates the existence of a hierarchical structure, which can be characterized to distinguish macro-scale from micro-scale levels. Structurally, both scales can be linked by the use of classic multiscale homogenization techniques. Since in bone tissue each micro-scale domain is distinct form its neighbour, applying a classic multiscale homogenization technique to a complete bone structure could represent an inadmissible computational cost. Thus, this work proposes a homogenization methodology that is computationally efficient, presenting a reduced computational cost, and is capable to define the homogenized microscale mechanical properties of the trabecular bone highly heterogeneous medium. The methodology uses the fabric tensor concept in order to define the material principal directions. Then, using an anisotropic phenomenological law for bone tissue correlating the local apparent density with directional elasticity moduli, the anisotropic homogenized material properties of the micro-scale patch are fully defined. To validate the developed methodology, several numerical tests were performed, measuring the sensitivity of the technique to changes in the micro-patch size and preferential orientation. The results show that the developed technique is robust and capable to provide a consistent material homogenization. Additionally, the technique was combined with two discrete numerical techniques: the finite element method and radial point interpolation meshless method. Structural analyses were performed using real trabecular patches, showing that the proposed methodology is capable to accurately predict the micro-scale patch mechanical behavior in a fraction of the time required by classic homogenization techniques.

The Split-Field PML Absorbing Boundary Condition for the Unconditionally Stable Node-Based LOD-RPIM Method

To numerically model open-domain structures and radiation problems, an absorbing boundary condition (ABC) has to be applied to emulate infinite open space. In this letter, the split-field perfectly matched layer (PML) ABC is developed and presented for the unconditionally stable node-based locally one-dimensional radial point interpolation meshless method. Numerical experiments are conducted to demonstrate the performance of the proposed PML ABC; they show that the proposed PML method has a good absorbing capability.

An Unconditionally Stable Radial Point Interpolation Meshless Method Based on the Crank–Nicolson Scheme Solution of Wave Equation

A 2-D unconditionally stable radial point interpolation meshless method (RPIM) based on the Crank–Nicolson (CN) scheme is presented. The CN algorithm in the proposed method is applied to only one of the Maxwell equations. It leads to solving the second-order vector wave equation in the time domain. Therefore, only one electromagnetic field is implicitly updated at each iteration. Furthermore, the computational cost of the CN-RPIM scheme is lesser than the conventional RPIM. The Courant–Friedrichs–Lewy condition in the proposed vector wave equation method does not constrain the time step due to its implicit formulation. Moreover, the stability condition of the proposed method is investigated in this paper through its amplification matrix. A 2-D waveguide and a filter are used as examples to validate and to demonstrate the effectiveness and the accuracy of the proposed method. Numerical results are compared with those obtained by the explicit RPIM method and the conventional finite-difference time-domain method.

Extending radial point interpolating meshless methods to the elasto-plastic analysis of aluminium alloys

This study performs an elasto-plastic numerical analysis of an aluminium alloy using two efficient meshless methods. The obtained numerical solutions are validated using experimental results due to the lack of a suitable elasto-plastic analytical solution. Thus, an experimental test was performed in order to obtain the force/displacement curve and the corresponding mechanical properties by means of Digital Image Correlation (DIC). In this work, two advanced discretization techniques, the Radial Point Interpolation Meshless Method (RPIM) and Natural Neighbour Radial Point Interpolation Meshless Method (NNRPIM), were extended to elasto-plasticity. In order to assess the performance of the numerical methods, the problem was additionally solved with the Finite Element Method (FEM) formulation in ABAQUS©. The success of the obtained numerical results validates the experimental solution contributing to accomplish the study objectives. Additionally, meshless methods produce accurate results compared to the experimental solution and FEM, proving that the presented methodology is feasible and robust.

SOLVE ABEL’S INTEGRAL EQUATION USING POINT INTERPOLATION MESHLESS METHOD

In this paper the numerical method for solving Abel,s integral equation are introduced ,this method is based on point interpolation meshless method. Also Radial basis function, zeros of the shifted Legendre polynomial as the collocation points utilized to apply for solving Abel,s integral equation of the first and second kind .The result of numerical experiment show that the numerical scheme is very effective and convenient of this method.

A radial point interpolation meshless method extended with an elastic rate-independent continuum damage model for concrete materials

ABSTRACTAn advanced discretization meshless technique, the radial point interpolation method (RPIM), is applied to analyze concrete structures using an elastic continuum damage constitutive model. Here, the theoretical basis of the material model and the computational procedure are fully presented. The plane stress meshless formulation is extended to a rate-independent damage criterion, where both compressive and tensile damage evolutions are established based on a Helmholtz free energy function. Within the return-mapping damage algorithm, the required variable fields, such as the damage variables and the displacement field, are obtained. This study uses the Newton–Raphson nonlinear solution algorithm to achieve the nonlinear damage solution. The verification, where the performance is assessed, of the proposed model is demonstrated by relevant numerical examples available in the literature.