Radial Point Interpolation Research Articles

Stabilized node‐based smoothed radial point interpolation method for micromechanical analysis of the magneto‐electro‐elastic structures in thermal environment

In this paper, the stabilized node‐based smoothed radial point interpolation method (SNS‐RPIM) is combined with asymptotic homogenization method (AHM) to investigate the static and transient responses of magneto‐electro‐elastic (MEE) structures in thermal environment. Stabilization terms are applied to construct a “close‐to‐exact” stiffness for the model; the temporal instability is also cured. To study the microscopic multi‐physics coupling problems more accurately, the effective material parameters related to the volume fraction of fiber are obtained based on AHM. Through numerical calculation, it is confirmed that SNS‐RPIM could simulate the responses of MEE structures well when combined with AHM. Therefore, the combination of SNS‐RPIM and AHM has great potential in studying nanostructures including nanobeams, nanoplates, and nanotubes.

Wave propagation in meshless numerical wave tank by using Hermite-type RPIM

In this paper, the Hermite-type radial point interpolation method (RPIM) is proposed to develop the meshless numerical wave tank (NWT) for simulating the propagation of fully nonlinear water waves. The Hermite-type interpolation scheme, the radial basis functions (RBFs), and the polynomial functions are used to construct shape functions. These shape functions have better stability and can use any distribution nodes in the computational domain. The strong form of governing equation is transformed to a linear system of algebraic equations by the collocation approach. The material node approach is used to describe the fully nonlinear free surface boundary conditions, and the fourth-order Runge-Kutta (RK4) scheme is applied to update the free surface particles and the velocity potential. The numerical results are compared with other available solutions to confirm the accuracy and efficiency of the proposed numerical procedure. In addition, the accuracy and stability of Hermite-type RPIM compared with RPIM in this study. The numerical results demonstrate that good improvement in the accuracy of the numerical results is obtained after using the Hermite interpolation scheme.

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.

The radial point interpolation mixed collocation method for the solution of transient diffusion problems

The Radial Point Interpolation Mixed Collocation (RPIMC) method is proposed in this paper for transient analysis of diffusion problems. RPIMC is an efficient purely meshfree method where the solution of the field variable is obtained through collocation. The field function and its gradient are both interpolated (mixed collocation approach) leading to reduced C-continuity requirement compared to strong-form collocation schemes. The accuracy of RPIMC was evaluated in heat conduction benchmark problems and compared against the Meshless Local Petrov–Galerkin Mixed Collocation (MLPG-MC) method and the Finite Element Method (FEM). These three methods were subsequently applied to solve electrical propagation in a cardiac tissue slab and a biventricular geometry. In benchmark problems, RPIMC achieved high accuracy, of the same order as FEM, and always higher than MLPG-MC due to the delta Kronecker property of RPIMC. In solving cardiac electrical propagation, RPIMC rendered activation time maps similar to those of FEM, with an improved performance compared to MLPG-MC. As a conclusion, RPIMC is shown to be a promising meshfree alternative to FEM for transient diffusion problems.

Numerical Simulation of Acoustic Equation Using Radial Point Interpolation Method with Discontinuous Galerkin Time Integration

The Numerical methods are research and industrial strategies commonly used by the finite difference method (FDM), finite element method (FEM) and finite volume method (FVM). The technique is a mesh based or formation of the domain. Owing to the complicated and time-consuming nature of the mesh method in the complex domain, it encounters numerous inconsistencies. One way of eluding this is by the use of a meshless method. This technique eliminates the use of but rather makes use of nodes in the distribution of its domain. This paper introduces the use of the radial point interpolation method (RPIM) to approximate the acoustic equations using the discontinuous Galerkin method (DGM) time integration. In order to determine the numerical behaviour, its results were simulated with the exact solution. The DGM time integration and order of accuracy is also compared with some commonly used procedures, such as the backward Euler and trapezoid methods. The size of support domain responsible for the numerical accuracy is also examined. Finally a comparison of numerical simulations of the exact results obtained during a specific time snapshot is displayed.

An Improved 2D Meshfree Radial Point Interpolation Method for Stress Concentration Evaluation of Welded Component

The study of characterizing the stress concentration effects at welds is one of the most important research directions for predicting the fatigue life of welded components. Stress solutions at the weld toe obtained from conventional meshfree methods are strongly influenced by parameters used in the methods as a result of stress singularity. In this study, an improved 2D meshfree radial point interpolation method (RPIM) is proposed for stress concentration evaluation of a welded component. The stress solutions are insensitive to parameters used in the improved RPIM. The improved RPIM-based scheme for consistently calculating stress concentration factor (SCF) and stress intensity factor at weld toe are presented. Our studies provide a novel approach to apply global weak-form meshfree methods in consistently computing SCFs and stress intensity factors at welds.

The numerical analysis of symmetric cross-ply laminates using the natural neighbour radial point interpolation method and high-order shear deformation theories

Composite structures are commonly analysed using the Finite Element Method (FEM). However, new accurate and efficient discrete numerical techniques have appeared recently – the meshless methods. Thus, this work uses a meshless method – the Natural Neighbour Radial Point Interpolation Method (NNRPIM) – to perform an elasto-static analysis of composite laminated plates. Meshless methods only require an unstructured nodal distribution to discretize the problem domain. In order to numerically integrate the integro-differential equation from the Galerkin weak formulation, a background integration mesh is constructed using the Voronoï diagram. Then, the nodal connectivity is enforced using the ‘influence-cell’ concept and the shape functions are obtained. In this work, laminated composite plates are analysed using distinct equivalent single layer theories, considering different transverse high-order shear deformation laws. Thus, several third-order, exponential and trigonometric transverse shear deformation theories are combined with the NNRPIM to analyse the structural response of composite laminated plates. In the end, composite laminated plates are numerically analysed and the meshless solutions are compared with the analytical solution available in the literature. Therefore, this works contributes with new solutions for classic composite symmetric cross-ply laminated plates and provides a comparative study on the accuracy of some high-order shear deformation theories (HSDTs).

A modified incremental creep integral approach for meshfree analysis of viscoelastic problems

In the present work, an improved meshfree radial point interpolation method (RPIM) is developed for the analysis of homogeneous, as well as non-homogeneous viscoelastic problems. A novel strategy for implementing integral-based viscoelastic constitutive theories using the meshfree methods is presented. This strategy makes use of the integral formulation of Schapery-type constitutive theories and higher-order finite difference (FD) schemes. The Cartesian transformation method (CTM) is adopted for accurate and efficient calculation of the domain integrals in the formulation of the meshfree RPIM. The superior performance of this modified algorithm is assessed through some benchmark examples and by comparing the obtained results with the existing analytical or numerical solutions. Through these comparisons, an improvement over the conventional integral-based recursive strategies of viscoelastic materials is observed.

A preliminary study of endothelial cell migration during angiogenesis using a meshless method approach

Angiogenesis, the development of new blood capillaries, is crucial for the wound healing process. This biological process allows the proper blood supply to the tissue, essential for cell proliferation and viability. Several biological factors modulate angiogenesis, however the vascular endothelial growth factor (VEGF) is the main one. Given the complexity of angiogenesis, in the last years, computational modelling aroused the interest of scientists since it allows to model this process with different, more economic and faster methodologies, comparatively to experimental approaches. In this work, a mathematical model motivated by the analysis of the effect of VEGF diffusion gradient in endothelial cell migration is presented. This is the process that allows capillary formation and it is essential for angiogenesis. The proposed mathematical model is combined with the Radial Point Interpolation Method, being the area discretized considering an unorganized nodal cloud and a background mesh of integration points, without predefined relations. The nodal connectivity was achieved using the "influence-domain" approach. The interpolation functions were constructed using the Radial Point Interpolators techniques. This method combines a radial basis functions with a polynomial functions to obtain the approximation. This preliminary work does not account for the whole complexity of cell and tissue biology, and numerical results are presented for an idealised two-dimensional setting. Nevertheless, the developed RPIM software is a valid numerical tool that can be adjusted to biological problems and may also be able to complement the biological and medical subjects.

Two versions of the extended Hill’s lemma for non-Cauchy continua based on the couple stress theory

Two versions of the extended Hill’s lemma for non-Cauchy continua satisfying the couple stress theory are proposed. Each version can be used to determine two effective elasticity (stiffness) tensors: one classical and the other higher order. The classical elasticity tensor relates the symmetric part of the force stress to the symmetric strain, whereas the higher-order elasticity tensor links the deviatoric part of the couple stress to the non-symmetric curvature. Four sets of boundary conditions (BCs) are identified using Version I, and three sets of BCs are obtained from Version II of the extended Hill’s lemma, which can all satisfy the Hill–Mandel condition. For each BC set selected, admissibility and average field requirements are checked. Furthermore, the equilibrium is examined for the cases with the kinetic BCs, and the compatibility is checked for the cases with the kinematic BCs. To illustrate the two newly proposed versions of the extended Hill’s lemma, a homogenization analysis is conducted for a two-phase composite using a meshfree radial point interpolation method. The effective elastic constants obtained in this analysis are compared with those predicted by the Voigt and Reuss bounds and computed through a finite element model constructed using COMSOL, which verifies and supports the current method.

Analysis of antisymmetric cross-ply laminates using high-order shear deformation theories: a meshless approach

For many years finite element method (FEM) was the chosen numerical method for the analysis of composite structures. However, in the last 20 years, the scientific community has witnessed the birth and development of several meshless methods, which are more flexible and equally accurate numerical methods. The meshless method used in this work is the natural neighbour radial point interpolation method (NNRPIM). In order to discretize the problem domain, the NNRPIM only requires an unstructured nodal distribution. Then, using the Voronoi mathematical concept, it enforces the nodal connectivity and constructs the background integration mesh. The NNRPIM shape functions are constructed using the radial point interpolation technique. In this work, the displacement field of composite laminated plates is defined by high-order shear deformation theories. In the end, several antisymmetric cross-ply laminates were analysed and the NNRPIM solutions were compared with the literature. The obtained results show the efficiency and accuracy of the NNRPIM formulation.

Material non-linearity in the numerical analysis of SLJ bonded with ductile adhesives: A meshless approach

Modern transportation, medical, and leisure equipment are often manufactured with non-metallic and composite materials bonded with adhesives. Numerical analyses of these joints are commonly performed using the Finite Element Method (FEM). In adhesive joints, the bond line is small compared to the substrates requiring highly refined meshes to perform the analyses. Also, adhesives can reach higher strains compared to the substrates, and so causing mesh distortion and compromising the solution. Alternatively, meshless methods (MM) can be employed. Single-lap joints (SLJ), during loading, present tensile and compressive stresses along the adhesive layer. Commonly, in elastic-plastic analyses, equal adhesive's strength in tension and compression is considered (using von Mises (vM) yield criterion), but it has been reported that adhesives are stronger in compression. Thus, a yielding criterion such as the Exponent Drucker-Prager (EDP) can be used; however, no MM implementations are available in the literature. This work seeks to implement the EDP into a MM, the Natural Neighbour Radial Point Interpolation Method (NNRPIM). Then, using the NNRPIM elastic-plastic parametric analyses of SLJ with aluminium substrates were performed considering four overlap lengths (LO) and two different ductile adhesives. All the cases were also evaluated using MM and the vM yield criterion, as a benchmark. Stress and strain distributions along the bond-line were obtained; afterwards, joint strength (Pmax) was determined by using continuum mechanics failure criteria, and then, evaluated and compared. Then, the results were compared against experimental data. The cases solved using the EDP predicted Pmax closer to the experimental data; the joints with intermediate overlap lengths LO=25.0 mm presented the closest values. In conclusion, the results' correlation with experimental data indicates the EDP is closer than vM to the actual adhesive behaviour in the joints. Also, the EDP was successfully implemented. Overall, the NNRPIM is a strong and accurate alternative for elastic-plastic analyses of adhesive joints.

On the analysis of a kind of nonlinear Sobolev equation through locally applied pseudo‐spectral meshfree radial point interpolation

AbstractIn this study, we develop an approximate formulation for two‐dimensional nonlinear Sobolev problems by focusing on pseudospectral meshless radial point interpolation (PSMRPI) which is a kind of locally applied radial basis function interpolation and truthfully a meshless approach. In the PSMRPI method, the nodal points do not need to be regularly distributed and can even be quite arbitrary. It is easy to have high order derivatives of unknowns in terms of the values at nodal points by constructing operational matrices. The convergence and stability of the technique in some sense are studied via some examples to show the validity and trustworthiness of the PSMRPI technique.

A meshless study of antisymmetric angle-ply laminates using high-order shear deformation theories

Composite laminated plates have found several applications in engineering field. For instance, aerospace and automotive industry use these structures in several components of their products due to their exceptional specific properties along the direction of the fibres. There are several two-dimensional plate models that aim to more accurately and more efficiently predict the kinematics of those structures. Unlike the Classical Plate Theory (CLPT), which can only be applied to thin laminates, the First-Order Shear Deformation Theory (FSDT), developed by Reissner and Mindlin, already accounts shear effects and, due to that, it can describe in a satisfactory way the kinematic of a generic thick laminated plate. Nevertheless, the FSDT considers linear shape functions to describe the in-plane displacements through the plate thickness, which results in constant shear strains. Thus, the traction boundary conditions on the top and bottom surface of plate are violated. Additionally, the FSDT needs shear correction factors, which are not easily obtained for a generic problem. Equivalent single layer (ESL) theories following high-order shear deformation theories (HSDTs) have been proposed and more efficiently applied to composite laminated plates. These theories allow to represent the nonlinear variation of transverse shear stresses through thickness of plate because they possess nonlinear shape functions interpolating the in-plane displacements. Thus, the kinematics of these structural elements can be better described and more accurate stress fields can be obtained for problems that cannot be handled with the FSDT.These structures have been analysed for several years using the Finite Element Method (FEM), which is the most popular numerical tool in structural analysis. However, several other advanced numerical techniques, such as meshless methods, can also handle this kind of problems and, in some cases, in a more efficient way. This work makes use of a recently developed meshless method – the Natural Neighbour Radial Point Interpolation Method (NNRPIM) (Belinha, 2014) – to study the bending of antisymmetric angle-ply composite laminates using distinct HSDTs.

Dynamic analysis of magneto-electro-elastic nanostructures using node-based smoothed radial point interpolation method combined with micromechanics-based asymptotic homogenization technique

The node-based smoothed radial point interpolation method combined with the asymptotic homogenization method was proposed, as an addition to the finite element method, to address the static and dynamic reactions of magneto-electro-elastic coupling micromechanical problems. First, several basic equations for relevant problems were derived. Second, asymptotic homogenization method was utilized to determine the material properties of magneto-electro-elastic nanomaterials. Third, node-based smoothed radial point interpolation method was applied to obtain the discrete equations of magneto-electro-elastic nanostructures. Then, the Newmark method was introduced to solve the response of microcosmic problems. Finally, several numerical examples were calculated to prove the accuracy, convergence, and reliability of node-based smoothed radial point interpolation method by comparing the results of node-based smoothed radial point interpolation method with those of finite element method.

Analyses of thermal buckling and secondary instability of post-buckled S-FGM plates with porosities based on a meshfree method

The present research deals with thermal buckling and postbuckling of sigmoid functionally graded material (S-FGM) plates with porosities resting on elastic foundation based on the popular third-order shear deformation theory and the well-known von Kármán large deflection assumption. Special attention is paid to scrutinize the secondary instability that occurs during postbuckling regime. In addition, the transitions of deflection shapes of the plates under various boundary conditions between primary and secondary postbuckling paths are investigated. A displacement control strategy and a meshfree radial point interpolation method (RPIM) which employs a radial basis function without any adaptive parameters are utilized synthetically to carry out thermal buckling loads and postbuckling paths of the plates numerically. Numerical comparisons are performed to verify the convergence of the modified RPIM and the accuracy of iteration method in postbuckling path analysis. The influences of functionally graded index, porosity coefficient and parameters of elastic foundation to behaviors of the thermal buckling and especially the secondary instability occurs during postbuckling regime are investigated. It is found that the secondary instability is sensitive to the boundary conditions.

Abstract 224: A Deep Learning Approach to Left-Ventricular Chamber Quantification for Fully Automated Three Dimensional Strain Analysis in Cardiotoxicity

This study investigated MRI and semantic segmentation-based deep-learning (SSDL) automation for left-ventricular chamber quantifications (LVCQ) and low longitudinal strain (LLS) determination, thus eliminating user-bias by providing an automated tool to detect cardiotoxicity (CT) in breast cancer patients treated with antineoplastic agents. Displacement Encoding with Stimulated Echoes-based (DENSE) myocardial images from 26 patients were analyzed with the tool’s Convolution Neural Network with underlying Resnet-50 architecture. Quantifications based on the SSDL tool’s output were for LV end-diastolic diameter (LVEDD), ejection fraction (LVEF), and mass (LVM) (see figure for phase sequence). LLS was analyzed with Radial Point Interpolation Method (RPIM) with DENSE phase-based displacements. LVCQs were validated by comparison to measurements obtained with an existing semi-automated vendor tool (VT) and strains by 2 independent users employing Bland-Altman analysis (BAA) and interclass correlation coefficients estimated with Cronbach’s Alpha (C-Alpha) index. F1 score for classification accuracy was 0.92. LVCQs determined by SSDL and VT were 4.6 ± 0.5 vs 4.6 ± 0.7 cm (C-Alpha = 0.93 and BAA = 0.5 ± 0.5 cm) for LVEDD, 58 ± 5 vs 58 ± 6 % (0.90, 1 ± 5%) for LVEF, 119 ± 17 vs 121 ± 14 g (0.93, 5 ± 8 g) for LV mass, while LLS was 14 ± 4 vs 14 ± 3 % (0.86, 0.2 ± 6%). Hence, equivalent LV dimensions, mass and strains measured by VT and DENSE imaging validate our unique automated analytic tool. Longitudinal strains in patients can then be analyzed without user bias to detect abnormalities for the indication of cardiotoxicity and the need for therapeutic intervention even if LVEF is not affected.

Application of the smoothed point interpolation methods in computational geomechanics: A comparative study

In this study, computational efficiency and performance of the smoothed point interpolation methods (SPIMs) in coupled problems of geomechanics are examined and compared with those of the standard linear finite element method (FEM). In the SPIMs, the problem domain is first discretised using a triangular background mesh, and the smoothing domains are then constructed using the cells of the background mesh. Different approaches for construction of the smoothing domains result in different SPIMs including Edge-based SPIM (ESPIM), cell-based SPIM (CSPIM), and node-based SPIM (NSPIM), each featuring different computational characteristics. For each type of the SPIM, only the simplest and most widely used supporting node selection technique is adopted in this study. The field function is approximated using shape functions constructed adopting either the polynomial point interpolation method (PIM), or radial point interpolation method (RPIM). The governing equations are derived from the equilibrium equation of the solid phase, and momentum and mass balance equations for the flow phase. The equations are then discretised in space using the generalized smoothed Galerkin (GS-Galerkin) method, and in time using the three-point time discretisation technique. The performance of the SPIMs in coupled geomechanical problems are then thoroughly assessed and compared to those of the linear FEM through extensive numerical analyses of four benchmark problems in saturated porous media covering both linear and non-linear material behaviours, and the conclusions drawn by analysing the results.

Static responses of magneto-electro-elastic structures in moisture field using stabilized node-based smoothed radial point interpolation method

In the present work, the static magneto-electro-hygro-elastic (MEHE) multi-field coupling problems are investigated by a stabilized node-based smoothed radial point interpolation method (SNS-RPIM). The equilibrium equations of magneto-electro-elastic (MEE) material in moisture field are obtained using G space theory, weakened weak (W2) formulation and constitutive equations of MEE structures. In order to further improve the accuracy of the algorithm, the stable item is introduced into the system stiffness matrix. The influence of the moisture concentration on the elastic coefficient is considered, and the results show that moisture environment could affect the stiffness of the structure and cause a change in its natural frequency. The calculation results of SNS-RPIM and finite element method (FEM) are given to study the effect of moisture load on the structural responses. In addition, it can be seen that the developed method could obtain higher accuracy results than FEM with a small number of nodes. And SNS-RPIM is more robust against mesh distortion, so that problems with complex geometries could be solved well. This article can provide a reference for the design and analysis of MEE devices in moisture environment.

Meshfree radial point interpolation method for the vibration and buckling analysis of FG-GPLRC perforated plates under an in-plane loading

A meshfree radial point interpolation method (RPIM) based on a higher-order shear deformation theory (HSDT) is here developed to investigate the free vibration and buckling behavior of multilayer functionally graded graphene platelets reinforced composite (FG-GPLRC) perforated plates under an in-plane edge loading. The non-uniform initial stresses in the multilayer FG-GPLRC perforated plates are evaluated accurately by solving the equilibrium equations derived from the same approach. The mixed collocation-Lagrange multiplayer technique is employed to enforce the various boundary conditions. The multilayer FG-GPLRC plates are built up as perfectly-bonded composite layers reinforced with a uniform distribution of graphene platelets (GPLs), randomly oriented. Perforated plates with three different cut-out shapes are analyzed. Several numerical examples are provided to demonstrate the high performance, the fast convergence rate and the high accuracy of the proposed method. Then, we study the influence of different GPLs distribution patterns, GPLs weight fraction, shape and size of cut-outs on the free vibration and buckling behavior of the multilayer FG-GPLRC perforated plates under an in-plane edge loading.