Radial Point Interpolation Method Research Articles

Capillary formation in tumor angiogenesis through meshless weak and strong local radial point interpolation

This article is devoted to showing the efficiency and accuracy of two numerical procedures for the numerical solution of the capillary formation in tumor angiogenesis. A meshless local radial point interpolation (MLRPI) scheme based on Galerkin weak form is analyzed. The reason for choosing MLRPI approach is that it does not require any background integration cells; instead, integrations are implemented over local quadrature domains, which are further simplified for reducing the complication of computation using regular and simple shape. The spectral meshless radial point interpolation (SMRPI) method does not need any integration. The radial point interpolation method is proposed to construct shape functions for MLRPI and basis functions for SMRPI. A weak formulation with a Heaviside step function transforms the set of governing equations into local integral equations on local subdomains in MLRPI, whereas the operational matrices convert easily the governing equations (even high order) into a linear system of equations in SMRPI. The finite difference technique is employed to approximate the time derivatives. A stability analysis technique based on the eigenvalue of the matrices is employed to check the stability condition for the presented meshless methods. Furthermore, the methods are successfully applied to solve the problem with high values of the cell diffusion constant which many of the available methods did not investigate for solving these cases. Because of non-availability of the exact solution, we consider two strategies for checking the stability of time difference scheme and for surveying the convergence of the fully discrete scheme. From numerical results, it is considerable that the present methods provide the similar results. Moreover, less CPU time and less computational complexity are two advantages of the SMRPI method with respect to the MLRPI method.

Continuous–discontinuous hybrid boundary node method for frictional contact problems

This paper presents a continuous–discontinuous hybrid boundary node method for frictional contact problems. In this method, the outer and internal boundaries are divided into several individual segments, for a continuous segment on outer boundary, the radial point interpolation method (RPIM) is employed for shape function construction, for discontinuous segments, the enriched discontinuous basis functions combined with RPIM are developed, in order to reflect the local field property of displacement and stress around crack tip, different basis functions for displacement and traction are developed for shape function construction on discontinuous segments individually. And the near tip asymptotic field functions and Heaviside function are employed for simulating the high gradient of stress field and discontinuous displacement field on contact surfaces. Besides a frictional contact theory and complementation detail for the present method is proposed, and some additional equations are developed for frictional contact iteration. Based on above technique and theory, a continuous–discontinuous hybrid boundary node method is proposed for frictional contact problems. Some numerical examples are shown that the present method is effective and can be widely used for some frictional contact engineering.

Simulating fracture propagation in brittle materials using a meshless approach

In the present work, a meshless method is combined with a crack growth algorithm, considering a linear elastic model to simulate the propagation of cracks in brittle materials. The proposed methodology is automatic, which means that it does not require the user intervention. In this study, the meshless method used is the Natural Neighbour Radial Point Interpolation Method (NNRPIM), an efficient discrete numeric method. Being a truly meshless method, the NNRPIM only requires an unstructured nodal distribution to fully discretize the problem domain. With the coordinates of the nodes, the NNRPIM formulation is autonomously capable: to define the nodal connectivity; to build the background integration mesh; and to construct the shape functions. The crack propagation algorithm here proposed simulates the crack growth by displacing iteratively the crack tip. Updating the crack tip location requires a local remeshing, which does not represent a numeric difficulty for the NNRPIM. The updated position of the crack tip depends on the estimated stress field in the vicinity of the crack tip. Thus, in each iteration, the stress field in the vicinity of the crack tip is estimated, and then, the direction of the crack propagation is calculated using the maximum circumferential stress criterion.

A novel hybrid stress meshless method for two-dimension elastic structure with a traction-free circular boundary

PurposeTo increase the use of the meshless method, a hybrid stress method is introduced into the meshless method.Design/methodology/approachThe method is based on the radial point interpolation method (RPIM). According to the Hellinger Reissner principle, stress functions are introduced into the solution procedure. Finite elements are used as background cells for integration. All cells are divided into two types – the H cells, which are around the traction-free circular boundary, and the G cells. For the H cells, stress functions in polar coordinates are created. For the G cells, 12-parameter stress functions in Cartesian coordinates are used. Stress functions are based on equilibrium equations and stress compatible equation.FindingsNumerical results show that this method is reliable.Originality/valueHybrid stress methods have been applied to finite element methods, but the finite element methods have not been applied into meshless methods.

Electric Charge Gaussian Gradation Method (ECGGM): A technique for improving material interface representations for the radial point interpolation method

AbstractAn improved meshless discretization methodology, based on the Coulomb's law discretization method, is introduced. With the presented improvement, it is possible to naturally and controllably increase the density of nodes around edges and corners of scatterers immersed in analysis space. This is achieved by gradually modifying node's charges across space according to Gaussian functions. It is observed that higher concentration of nodes in the neighborhood of media interfaces substantially improves the precision of numerical solutions of Maxwell's equations obtained with the radial point interpolation method. This improvement is justified by refined calculation of intense spatial field variations near the boundaries. Several other relevant benefits resulting from the new technique are observed and highlighted.

Analysis of transient dynamic fracture parameters of cracked functionally graded composites by improved meshfree methods

In this paper, computation and analysis for transient dynamic stress intensity factors (DSIFs) of two-dimensional fracture problems of functionally graded materials (FGMs) by extended meshfree methods are described. The extended moving Kriging based meshfree method (X-MK) is introduced and compared with recently developed extended meshfree radial point interpolation method (X-RPIM) in the dynamical fracture context of FGM models. Moreover, in the present study, the X-MK is improved by introducing three new different types of correlation function that fully eliminate the user parameter for MK shape functions. The standard basis functions are used as the enrichment for asymptotic near crack tip fields to accurately capture the singularities at the crack tip and the Heaviside function is used to enrich the displacement discontinuity along the crack faces. We apply the dynamic form of non-homogenous interaction integral in conjunction with the asymptotic near crack tip field to extract the DSIFs. The DSIFs obtained from the proposed methods are further compared with reference solutions derived from other existing methods, including analytical method, (extended-) finite element method, boundary element method, and scaled boundary finite element method.

Meshfree Method in Geophysical Electromagnetic Prospecting: The 2D Magnetotelluric Example

As an important supplement and development of traditional methods, the meshfree method has received a great deal of attention in the field of engineering calculation, and has been successfully used to solve many problems which traditional methods have difficulty in solving. However, the application of meshfree method is relatively less in the area of geophysics. In this paper, we apply the meshfree method to the numerical simulation of geophysical electromagnetic prospecting, taking the 2D magnetotelluric as an example and deduce the corresponding meshfree radial point interpolation method (RPIM) equivalent linear equations in detail. The high-efficiency and accurate solutions of large-scale sparse linear equations are solved by the quasi-minimal residual method based on Krylov subspace. The optimal values of the shape parameters are given by numerical experiments. The correctness of the meshfree method is verified by a layered model. The root mean square error of the calculation results is no more than 0.35%, its accuracy is superior to the finite element method. We also compare the meshfree solution with FEM solution by calculating an inclined vein body model, and the calculation results are in good agreement. A continuously changing fault model and undulating terrain model which traditional methods have difficulty in simulating are respectively calculated, the sectional profiles of the apparent resistivity accurately reflect the trend of the anomalies. The meshfree method does not require the complicated mesh generation, and the physical parameters are loaded at a series of points, thus it is especially suitable for the calculation of the complex geological models. With the rapid development of computational science, the meshfree techniques will certainly become a new robust numerical simulation method in geophysical electromagnetic prospecting.

The analysis of composite laminated beams using a 2D interpolating meshless technique

Laminated composite materials are widely implemented in several engineering constructions. For its relative light weight, these materials are suitable for aerospace, military, marine, and automotive structural applications. To obtain safe and economical structures, the modelling analysis accuracy is highly relevant. Since meshless methods in the recent years achieved a remarkable progress in computational mechanics, the present work uses one of the most flexible and stable interpolation meshless technique available in the literature—the Radial Point Interpolation Method (RPIM). Here, a 2D approach is considered to numerically analyse composite laminated beams. Both the meshless formulation and the equilibrium equations ruling the studied physical phenomenon are presented with detail. Several benchmark beam examples are studied and the results are compared with exact solutions available in the literature and the results obtained from a commercial finite element software. The results show the efficiency and accuracy of the proposed numeric technique.

A Radial Point Interpolation Method for Pricing Options on a Dividend Paying Asset

A Radial Point Interpolation Method for Pricing Options on a Dividend Paying Asset

Novel adaptive surrogate model based on LRPIM for probabilistic analysis of turbine disc

Probabilistic lifetime assessment for aero-engine turbine disc is required to ensure structural safety and reliability. For probabilistic analysis of aero-engine turbine disc, a large amount of random variables involving load, geometry, and material properties result in a high dimensional nonlinear state function for the fatigue lifetime, which can become prohibitively expensive. This paper presents a novel adaptive surrogate model for the probabilistic analysis of an aero-engine turbine disc by integrating the local radial point interpolation method (LRPIM) and directional sampling technique. The directional sampling technique includes initial sampling, limit state recognition and subsequent sampling. In order to implement the high-dimension-probabilistic analysis for the turbine disc, an adaptive scheme is proposed involving three parts, i.e. scale adjustment of local support domain, convergence test and repeated procedure of subsequent sampling. Applied to an aero-engine turbine disc probabilistic analysis problem with 11 dimensional random variables, it is demonstrated that the novel approach proposed improves the accuracy and computational efficiency with reduced sampling amount as compared to other models such as response surface method (RSM), Kriging model (KM) and artificial neural network model (ANNM). A leave-one-out (LOO) validation test is performed to verify the robustness of the prediction of the adaptive surrogate model in the probabilistic analysis process of aero-engine turbine discs.

Shock-induced stochastic dynamic analysis of cylinders made of saturated porous materials using MLPG method: considering uncertainty in mechanical properties

The stochastic meshless local Petrov–Galerkin method is employed for dynamic analysis of cylinders made of fully saturated porous materials with considering uncertainties in the constitutive mechanical properties. The porous cylinder is assumed to be under shock loading. To approximate the trial functions in the radial point interpolation method, the radial basis functions are utilized. The Monte Carlo simulation is used to generate the random fields for mechanical properties. The results are obtained for various random variables, which are simulated by uniform, normal and lognormal probability density functions with various coefficients of variation (COV), changing from 0 to 20%. The obtained results from the presented stochastic analysis are compared to those obtained from analysis with considering deterministic mechanical properties. The results show that the uncertainty in mechanical properties has a significant effect on the structural responses, especially for big values of COVs.

Meshfree radial point interpolation method for analysis of viscoplastic problems

A truly meshfree radial point interpolation method (RPIM) for analysis of viscoplastic problems is presented. Without recourse to any background cell, a fast and accurate meshfree integration method is employed for computation of the domain integrals in the RPIM. The effects of the total number of nodes and the parameters of the meshfree method on the accuracy of the solutions are studied too. Moreover, the effects of the viscoplastic material parameters on the results are investigated. To demonstrate the accuracy of the proposed method and provide a comprehensive investigation of the viscoplastic and meshfree parameters, several numerical examples are presented. Excellent agreements are observed through comparing the obtained results with those retrieved by the finite element method.

Thermal buckling analyses of FGM sandwich plates using the improved radial point interpolation mesh-free method

This study reports thermal buckling analyses of functionally graded material (FGM) sandwich plates using an improved mesh-free radial point interpolation method (RPIM). The buckling formulation of FGM sandwich plates is derived from the improved RPIM which employs a new radial basis function enabling the shape functions to be built without any supporting fixing parameters based on the higher-order shear deformation plate theory. Two types of FGM sandwich plates with different composition scheme, i.e. one with FGM skins and homogenous core and the other composed of homogenous skins and FGM core are considered in the analyses. The simulated results by the improved RPIM are compared with the analytical solutions found in the literature for the verification purpose. Detailed parametric studies are then carried out to scrutinize the effects of the volume fraction, plate length-to-thickness ratio, aspect ratio, boundary condition and FGM constituents on the critical buckling temperature changes of the FGM sandwich plates under various types of temperature variation through the thickness. Results demonstrate that the improved mesh-free RPIM can effectively predict the thermal buckling behavior of the FGM sandwich plates.

A continuous–discontinuous hybrid boundary node method for solving stress intensity factor

A novel boundary type meshless method called continuous–discontinuous hybrid boundary node method is proposed in this paper, in which the enriched discontinuous shape function is developed to solve linear elastic crack problems. Firstly, the whole boundary is divided into several individual segments, and variables on each one of those segments are interpolated, respectively. For continuous segments, radial point interpolation method is employed. In regard to discontinuous segments, the enriched discontinuous basis functions combining with radial point interpolation method are developed for simulating the discontinuity of displacement and stress field on surfaces of crack, and the near tip asymptotic field functions are employed for simulating the high gradient of stress field around crack tip, so that high accuracy and discontinuity property of a crack can be easily described. Stress intensity factors are calculated directly using displacement extrapolation by displacement field near crack tip. Some numerical examples are shown that the present method is effective and can be widely applied in some practical engineering.

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.

A modified Kirchhoff plate theory for analyzing thermo-mechanical static and buckling responses of functionally graded material plates

In this paper, we present a modified Kirchhoff theory for analyzing thermo-mechanical static and buckling responses of functionally graded material (FGM) isotropic and sandwich plates. In comparison to the classical Kirchhoff theory, the proposed method is taken account into the shear deformation effects, then it can be applied for both moderately thick and thin plates. On the other hand, comparing to the first order shear deformation theory (FSDT) or Reissner-Mindlin plate theory, the number of independent unknowns reduces one variable and retains four degrees of freedom (Dofs) per node. The shear locking phenomenon is also negligible due to extending from the classical Kirchhoff theory. Furthermore, a new normalization of a quartic form is used to build the shape functions in the radial point interpolation method (RPIM). The novel imposing essential boundary conditions related to slopes of deflections has been proposed, then the enforcement of these gradient deflections becomes easily and directly as the same in finite element analysis. The obtained numerical results for the thermo-mechanical static and buckling problems from the proposed method are stable and well accurate prediction as comparing to the exact solutions and other published analyses.

The numerical analysis of piezoelectric ceramics based on the Hermite-type RPIM

In this paper, the Hermite-type radial point interpolation method (RPIM) is applied to analyze the property of piezoelectric ceramics in order to overcome the defects of finite element method. In this method, the inside and boundary of the problem domain are discreted by a distribution of nodes, and then the interpolation function of nodes are constructed to solve the displacement of the evaluation nodes. Compared with the finite element method, it is easier and faster for the Hermite-type RPIM to accurately achieve solution of the local regions. In contrast with the existing meshless methods, this method would not cause singularity in the process of evaluating the shape function. Furthermore, the shape function of the Hermite-type RPIM has a better stability and it can adapt to any distribution of nodes. In addition, the accuracy and stability of the method are proved by the numerical simulation.

The Natural Neighbor Radial Point Interpolation Method in Computational Fracture Mechanics: A 2D Preliminary Study

In this work, the natural neighbor radial point interpolation method (NNRPIM) is extended to the numeric analysis of crack propagation problems. Here, the advanced discretization meshless technique is combined with a linear elastic crack growth algorithm. The algorithm simulates the crack propagation by displacing iteratively the crack tip, which consequently requires a local remeshing. In each iteration, it is estimated the stress state in the crack tip and afterwards the direction of the crack propagation is obtained considering the maximum circumferential stress criterion.The required local remeshing does not represent a numeric difficulty for the NNRPIM. The main advantage of the NNRPIM is its capability to fully discretize the problem domain using only an unstructured nodal distribution. Being a truly meshless method, the NNRPIM is able to define autonomously the nodal connectivity and the background integration mesh.The classic NNRPIM formulation permits to enforce the nodal connectivity by means of two kind of influence-cells: first degree influence-cells or second degree influence-cells. This work investigates the influence of the nodal connectivity on the simulated crack propagation path. Thus, demanding benchmark crack propagation examples are studied and the obtained results are compared with reference solutions available in the literature.

Meshless local radial point interpolation (MLRPI) for generalized telegraph and heat diffusion equation with non-local boundary conditions

In this paper, the meshless local radial point interpolation (MLRPI) method is formulated to the generalized one-dimensional linear telegraph and heat diffusion equation with non-local boundary conditions. The MLRPI method is categorized under meshless methods in which any background integration cells are not required, so that all integrations are carried out locally over small quadrature domains of regular shapes, such as lines in one dimensions, circles or squares in two dimensions and spheres or cubes in three dimensions. A technique based on the radial point interpolation is adopted to construct shape functions, also called basis functions, using the radial basis functions. These shape functions have delta function property in the frame work of interpolation, therefore they convince us to impose boundary conditions directly. The time derivatives are approximated by the finite difference time- -stepping method. We also apply Simpson’s integration rule to treat the non-local boundary conditions. Convergency and stability of the MLRPI method are clarified by surveying some numerical experiments.

A scaled boundary radial point interpolation method for 2‐D elasticity problems

SummaryThe scaled boundary radial point interpolation method (SBRPIM), a new semi‐analytical technique, is introduced and applied to the analysis of the stress–strain problems. The proposed method combines the advantages of the scaled boundary finite element method and the boundary radial point interpolation method. In this method, no mesh is required, nodes are scattered only on the boundary of the domain, no fundamental solution is required, and as the shape functions constructed based on the radial point interpolation method possess the Kronecker delta function property, the boundary conditions of problems can be imposed accurately without additional efforts. The main ideas of the SBRPIM are introducing a new method based on boundary scattered nodes without the need to element connectivity information, satisfying Kronecker delta function property, and being capable to handle singular problems. The equations of the SBRPIM in stress–strain fields are outlined in this paper. Several benchmark examples of 2‐D elastostatic are analyzed to validate the accuracy and efficiency of the proposed method. It is found that the SBRPIM is very easy to implement and the obtained results of the proposed method show a very good agreement with the analytical solution. Copyright © 2017 John Wiley & Sons, Ltd.