Radial Point Interpolation Research Articles

Stochastic static analysis of functionally graded sandwich nanoplates based on a novel stochastic meshfree computational framework

In this study, the spatial variability of materials is incorporated into the static analysis of functionally graded sandwich nanoplates to achieve higher accuracy. Utilising a modified point estimation method and the radial point interpolation method, we develop a novel stochastic meshfree computational framework to deal with the material uncertainty. Higher-order shear deformation theory is employed to establish the displacement field of the plates. The elastic modulus of ceramic and metal (Ec and Em) are treated as separate random fields and discretized through the Karhunen-Loève expansion (KLE) method. To improve the performance of procedure, the Wavelet-Galerkin method is introduced to solve the second type of Fredholm integral equation. Subsequently, substituting the random variables obtained by KLE into the stochastic computational framework, a high accuracy stochastic response of structures can be acquired. By comparing computed findings with those of Monte Carlo simulation, the accuracy and efficiency of developed framework are verified. Moreover, the results indicate that the plate's deflection exhibits varying sensitivities to the random fields Ec and Em. Also, the sandwich configuration as well as power-law exponents affect the stochastic response of structures. These findings offer valuable insights for the optimized design of functionally graded sandwich nanoplates.

Nonlinear analysis of three-dimensional hyperelastic problems using radial point interpolation method

Hyperelastic materials are primarily common in real life as well as in industry applications, and studying this kind of material is still an active research area. Naturally, the characteristic of hyperelastic material will be expressed when it undergoes large deformation, so the geometrical nonlinear effect should be considered. To analyze the behavior of hyperelastic material, the Neo-Hookean model is imposed in this study because of its simplicity. The model shows the nonlinear behavior when the deformation becomes large due to the nonlinear displacement-strain relation. This constitutive relation also gives a good correlation with experimental data. This study performs a meshless method, namely the radial point interpolation method (RPIM), to analyze the nonlinear behavior of Neo-Hookean hyperelastic material under a finite deformation state in three-dimensional space. The standard Newton-Raphson technique is applied to obtain the nonlinear solutions. Unlike mesh-based approaches, the meshless method shows its advantages in large deformation problems due to its mesh independence. In this paper, the numerical results of three-dimensional problems that undergo large deformation will be calculated and validated with solutions derived from previous studies. By investigating the obtained results, the superior ability of RPIM in hyperelastic problems can be proved.

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.

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

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

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

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

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.

An in-depth analysis of the Radial Point Interpolation Method’s parameters in relation to the accuracy of fracture problem

The fracture problem has been modeled using Radial Point Interpolation Method (RPIM) to examine the effects of support domain size and shape parameter value on the calculation of Stress Intensity Factor (SIF). The study focuses on an edge crack problem subjected to normal load and shear loads, determining that the support domain size of ‘ α ≥ 3.5 and shape parameter value between ‘ n = 1.9 -2.1’ yield results of accuracy with error % ≤ 3 % . Additionally, research extended by varying the problem domain size and crack location to verify consistency. The findings indicate that selecting the optimal shape parameter value will ensure better accuracy over the enrichment techniques with lesser computational cost.

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

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

Buckling analysis of functionally graded sandwich thin plates using a meshfree Hermite Radial Point Interpolation Method

Buckling analysis of functionally graded sandwich thin plates using a meshfree Hermite Radial Point Interpolation Method

Three-dimensional elastoplastic post-buckling analysis of functionally graded plates using a novel meshfree Tchebychev-radial point interpolation approach

This paper presents a three-dimensional (3D) analysis of the post-buckling behavior of functionally graded (FG) plates for the first time using an elastoplasticity-based meshfree formulation. A novel 3D Tchebychev-radial point interpolation approach, which combines radial basis functions with Tchebychev polynomials, is developed to solve the nonlinear post-buckling problem. The incremental plastic deformation is modeled by the Prandtl–Reuss flow rule along the isotropic hardening von Mises criterion. The governing equations are derived using the principle of virtual work by considering the 3D full Green–Lagrange strain components. The Newton–Raphson technique along with the arc-length method is used to determine post-buckling equilibrium paths of FG plates. The effective elastoplastic properties of the functionally graded material are assessed by exploiting the homogenization method, named Tamura–Tomota–Ozawa (TTO) model. It has been demonstrated that the accuracy of the solution is improved by incorporating Tchebychev polynomials into the radial point interpolation method (RPIM) shape function, and the stability and robustness of the results are independent of variations in the shape parameter. Furthermore, it is confirmed that TRPIM, with a slightly higher CPU time compared to RPIM, exhibits a higher convergence rate. The excellent agreement of the results with those existing in the literature shows that the proposed meshfree method can be used as a robust and accurate numerical tool to predict the elastoplastic post-buckling behavior of FG plates. Further numerical assessments indicate that post-buckling paths are significantly affected by factors such as geometric parameters, material gradient, loading ratio, and boundary conditions (BCs).

A robust radial point interpolation method empowered with neural network solvers (RPIM-NNS) for nonlinear solid mechanics

In this work, we proposed a robust radial point interpolation method empowered with neural network solvers (RPIM-NNS) for solving highly nonlinear solid mechanics problems. It is enabled by neural network solvers via minimizing an energy-based functional loss. The RPIM-NNS has the following key ingredients: (1) It uses radial basis functions (RBFs) for displacement interpolation at arbitrary points in the problem domain, permitting irregular node distributions. (2) Nodes are placed also beyond the domain boundary, allowing the convenient implementation of boundary conditions of both Dirichlet and Neumann types. (3) It uses strain energy in an integral form as a part of the loss function, ensuring solution stability. (4) A well-developed gradient descendant algorithm in machine learning is employed to find the optimal solution, enabling robustness and ease in handling material and geometrical nonlinearities. (5) The proposed RPIM-NNS is compatible with parallel computing schemes. The performance of this method is tested using nonlinear problems including Cook's membrane and 3D twisting rubber problems, demonstrating its remarkable stability and robustness. This work, which seamlessly integrates the neural network solvers with mechanics governing equations and computational mechanics techniques, offers an excellent alternative for nonlinear solid mechanics problems. MATLAB codes are made available at https://github.com/JinshuaiBai/RPIM_NNS for free downloading.

Predicting the mixed-mode fracture propagation in single-leg bending using a radial point interpolation meshless method

Adhesive bonding is a widely used joining technique with increasing applications in the automotive and aircraft industries. However, the numerical simulation of adhesively bonded joints is challenging due to the intricate mechanical behavior of the bonding layer, especially when structures are subjected to mixed-mode loading. This work proposes a meshless fracture propagation algorithm to simulate mixed-mode fracture propagation in single-leg bending (SLB) adhesive joints. The proposed algorithm combines the radial point interpolation method (RPIM) with a mixed-mode strain-based criterion to predict crack growth. The RPIM permits a flexible discretization of the fracture region and eases the geometric remeshing of the integration cells to account for the crack tip propagation. Additionally, the RPIM provides accurate and smooth strain/stress fields, allowing implementation of the mixed-mode fracture criterion. The numerical model was validated against experimental data for SLB adhesive joints with a brittle adhesive. The results showed that the proposed algorithm can accurately predict the resistance curves and critical energy release rates of the adhesive joints.

Fractional Boundary Element Solution for Nonlinear Nonlocal Thermoelastic Problems of Anisotropic Fibrous Polymer Nanomaterials

This paper provides a new fractional boundary element method (BEM) solution for nonlinear nonlocal thermoelastic problems with anisotropic fibrous polymer nanoparticles. This comprehensive BEM solution comprises two solutions: the anisotropic fibrous polymer nanoparticles problem solution and the nonlinear nonlocal thermoelasticity problem. The nonlinear nonlocal thermoelasticity problem solution separates the displacement field into complimentary and specific components. The overall displacement is obtained using the boundary element methodology, which solves a Navier-type problem, and the specific displacement is derived using the local radial point interpolation method (LRPIM). The new modified shift-splitting (NMSS) technique, which minimizes memory and processing time requirements, was utilized to solve BEM-created linear systems. The performance of NMSS was evaluated. The numerical results show how fractional and graded parameters influence the thermal stresses of nonlinear nonlocal thermoelastic issues involving anisotropic fibrous polymer nanoparticles. The numerical findings further reveal that the BEM results correlate very well with the finite element method (FEM) and analytical results, demonstrating the validity and correctness of the proposed methodology.

A meshless computational framework for a modified dynamic system of vehicle coupled with plate structure

In previous simulations of train–bridge interaction systems (TBIS), the supporting system for the train are commonly treated as beam structures, leading to less accurate results, particularly for small-span cases. To address this limitation, a modified vertical TBIS is proposed. In the presented TBIS, the supporting system is considered as a Reissner–Mindlin plate, and the displacement field is described by first-order shear deformation theory (FSDT). To establish the model, radial point interpolation method (RPIM), a meshless method, is employed. Finally, a coupled dynamic equation is established to calculate various responses of the system. Several numerical examples are presented to illustrate the disparities between the system based on plate model and traditional beam model. The results indicate that the beam model yields higher estimates of the mid-span vertical displacement of the bridge, while the peak of the mid-span vertical acceleration is smaller compared to the plate model; additionally, it is observed that the carbody is primarily influenced by rail irregularities. Consequently, the proposed plate model offers distinct advantages over the beam model in providing comprehensive structural response information, thereby offering novel insights into bridge design and analysis. Additionally, this marks the inaugural application of the meshless method in the field of TBIS, which further extends the application scope of meshless methods.

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.

An efficient meshless numerical method with the error estimate for two-dimensional Schrödinger equation

In this study, we present a capable meshfree numerical approach to numerical solving of the two-dimensional Schrödinger equation (SE). Firstly, the radial point interpolation is obtained to build the shape functions for discretization in the mentioned equation. These meshless shape functions have Kronecker delta attributes. Secondly, the local weak form is constructed over the local sub-domain for the purpose of diminishing the solution of the time-dependent SE in a two-dimensional case and transforming it into a system of algebraic linear equations. Also, semi and fully discrete schemes with their error estimates for this meshless method are obtained. We indicate the proposed method square of error has the order of O(h2m+τ2) in the L2 norm. Moreover, some numerical tests are provided, and their exact and finite difference solutions are compared to verify the efficiency of our algorithm.

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.

Dynamic Analysis of an FGM Beam Undergoing Large Overall Motion in Temperature Field

The dynamic analysis of a functionally gradient material (FGM) beam undergoing large overall motion based on meshless methods is studied in the variable temperature field environment. Considering the high-order terms of nonlinear coupling deformation which were often ignored in previous studies, the high-order rigid–flexible coupled (HOC) dynamic model of a rotating FGM beam with a lumped mass in a temperature field is established by employing Lagrange’s equations of the second kind. Compared with the first-order approximate coupled (FOAC) dynamic model, the HOC dynamic model is more accurate and has a wider scope of application. The validity of the point interpolation (PIM) method and radial point interpolation method (RPIM) in this paper is verified by comparison with simulation results of the assumed mode method (AMM) and the finite element method (FEM). On this basis, the factors affecting the dynamic characteristics of the FGM beam such as the form of temperature field, the functionally gradient index, and the location of added lumped mass are discussed. In this paper, the meshless method with higher computational efficiency and the HOC dynamic model with more accurate results are adopted. This study will provide guidance and reference for dealing with problems in engineering, machinery, aerospace, and other fields in complex environment.