Smoothed Point Interpolation Method Research Articles

A novel approach for application of smoothed point interpolation methods to axisymmetric problems in poroelasticity

A smoothed point interpolation method (SPIM) for the numerical modelling of saturated porous media in axisymmetric conditions is proposed, aiming to overcome the singularity problem encountered when using SPIMs in axisymmetric settings. The singularity is circumvented in this study by decomposition of the property matrices of the system to sub-matrices with smoothed terms and non-smoothed terms. The salient feature of the proposed method is that it neither incurs additional computation nor compromises on the accuracy of the method. The proposed method is examined by numerical modelling of several benchmark axisymmetric problems, along with a set of convergence studies.

Elastic-plastic analysis of multi-material structures using edge-based smoothed point interpolation method

An edge-based smoothed point interpolation method is formulated to deal with elastic-plastic analysis of multi-material structures. The problem domain is discretized using triangular elements and field functions are approximated using point interpolation method shape functions. Edge-based smoothing domains are constructed based on the edge of triangular cells and smoothing operations are then performed in these integral domains. Numerical examples with different kinds of material models are investigated to fully verify the validity of the present method. It is observed that all edge-based smoothed point interpolation method models can achieve much better accuracy and higher convergence rate than the standard finite element method, when dealing with elastic-plastic analysis of multi-material structures.

A Novel Pick-Out Theory and Technique for Constructing the Smoothed Derivatives of Functions for Numerical Methods

In order to solve partial differential equations (PDEs) numerically, one first needs to approximate the field functions (such as the displacement functions), and then obtain the derivatives of the field functions (such as the strains), by directly differentiating the field functions. Using such direct-derivatives in formulating a numerical method is common and is used in the standard finite element method (FEM), but such models are often found to be “stiff”. In the weakened weak (W2) formulations, it is found that the use of properly re-constructed derivatives can be beneficial in ways because the model can become “softer”. This paper presents a novel “pick-out” theory and technique for re-constructing the derivatives (such as the strains) of functions defined in a local domain, using smoothing operations. The local domain can be a smoothing domain used in the smoothed finite element methods (S-FEMs), smoothed point interpolation methods (S-PIMs), and smoothed particle hydrodynamics (SPH). It is discovered that through the use of a set of linearly independent smoothing functions that are continuous in the local domain, one can simply pick out various orders of smoothed derivatives (at the center of a domain) from any given function that may discontinuous (strictly) inside the local domain. As long as the smoothing function is continuous in the smoothing domain, the picked out “smoothed derivatives” are equivalent (in a local integral sense) to the compatible direct-derivatives, which ensures the convergence of the smoothed model (such as the S-FEM) when the smoothing domains shrinking to zero. The pick-out technique can be used in strong, weak, local weak, weak-strong, or weakened weak formulations to create stable and convergent numerical models. It may offer a new window of opportunity to develop new effective numerical models using smoothed derivatives (strains) that are “softer” and can produce accurate solutions also in the derivatives (strains and stresses) of the field functions (displacements).

Coupling immersed method with node-based partly smoothed point interpolation method (NPS-PIM) for large-displacement fluid-structure interaction problems

The newly developed node-based partly smoothed point interpolation method (NPS-PIM) and immersed technique are coupled to simulate fluid-structure interaction (FSI) problems associated with largely deformable solid/structure. Under the framework of immersed method, the governing equations can be decomposed into three components based on fictitious fluid assumption, i.e. equations for incompressible viscous fluid, FSI conditions and nonlinear solid. The semi-implicit characteristic-based split algorithm is used to solve the Navier-Stokes equation of incompressible viscous fluid, and the fictitious fluid domain can be used to calculate the FSI force. Owing to the properly softened stiffness by the node-based partly strain smoothing operation, NPS-PIM can effectively solve the large deformation problem of solids discretized with the simplest linear triangular mesh. The immersed framework makes the present method avoid the troublesome re-meshing process and the triangular mesh further simplifies the preprocessing operation. In comparison with the models using FEM and node-based smoothed point interpolation method (NS-PIM) as solid solver, numerical results have shown that the present method using NPS-PIM can provide more accurate solutions for both steady and transient problems with regard to large-displacement FSI problems.

A node-based smoothed point interpolation method for dynamic analysis of rotating flexible beams

We proposed a mesh-free method, the called node-based smoothed point interpolation method (NS-PIM), for dynamic analysis of rotating beams. A gradient smoothing technique is used, and the requirements on the consistence of the displacement functions are further weakened. In static problems, the beams with three types of boundary conditions are analyzed, and the results are compared with the exact solution, which shows the effectiveness of this method and can provide an upper bound solution for the deflection. This means that the NS-PIM makes the system soften. The NS-PIM is then further extended for solving a rigid-flexible coupled system dynamics problem, considering a rotating flexible cantilever beam. In this case, the rotating flexible cantilever beam considers not only the transverse deformations, but also the longitudinal deformations. The rigid-flexible coupled dynamic equations of the system are derived via employing Lagrange’s equations of the second type. Simulation results of the NS-PIM are compared with those obtained using finite element method (FEM) and assumed mode method. It is found that compared with FEM, the NS-PIM has anti-ill solving ability under the same calculation conditions.

An immersed smoothed point interpolation method (IS‐PIM) for fluid‐structure interaction problems

SummaryAn immersed smoothed point interpolation method using 3‐node triangular background cells is proposed to solve 2D fluid‐structure interaction problems for solids with large deformation/displacement placed in incompressible viscous fluid. In the framework of immersed‐type method, the governing equations can be decomposed into 3 parts on the basis of the fictitious fluid assumption. The incompressible Navier‐Stokes equations are solved using the semi‐implicit characteristic‐based split scheme, and solids are simulated using the newly developed edge‐based smoothed point interpolation method. The fictitious fluid domain can be used to calculate the coupling force. The numerical results show that immersed smoothed point interpolation method can avoid remeshing for moving solid based on immersed operation and simulate the contact phenomenon without an additional treatment between the solid and the fluid boundary. The influence from information transfer between solid domain and fluid domain on fluid‐structure interaction problems has been investigated. The numerical results show that the proposed interpolation schemes will generally improve the accuracy for simulating both fluid flows and solid structures.

A cell-based smoothed point interpolation method (CS-PIM) for 2D thermoelastic problems

PurposeDue to the strong reliance on element quality, there exist some inherent shortcomings of the traditional finite element method (FEM). The model of FEM behaves overly stiff, and the solutions of automated generated linear elements are generally of poor accuracy about especially gradient results. The proposed cell-based smoothed point interpolation method (CS-PIM) aims to improve the results accuracy of the thermoelastic problems via properly softening the overly-stiff stiffness.Design/methodology/approachThis novel approach is based on the newly developed G space and weakened weak (w2) formulation, and of which shape functions are created using the point interpolation method and the cell-based gradient smoothing operation is conducted based on the linear triangular background cells.FindingsOwing to the property of softened stiffness, the present method can generally achieve better accuracy and higher convergence results (especially for the temperature gradient and thermal stress solutions) than the FEM does by using the simplest linear triangular background cells, which has been examined by extensive numerical studies.Practical implicationsThe CS-PIM is capable of producing more accurate results of temperature gradients as well as thermal stresses with the automated generated and unstructured background cells, which make it a better candidate for solving practical thermoelastic problems.Originality/valueIt is the first time that the novel CS-PIM was further developed for solving thermoelastic problems, which shows its tremendous potential for practical implications.

An edge-based smoothed point interpolation method for elasto-plastic coupled hydro-mechanical analysis of saturated porous media

An edge-based smoothed point interpolation method is adopted for coupled hydro-mechanical analysis of saturated porous media with elasto-plastic behaviour. A novel approach for the evaluation of the coupling matrix of the porous media is adopted. Stress integration is performed using the substepping method, and the modified Newton-Raphson approach is utilised to address the nonlinearities arising from the elasto-plastic constitutive model used in the formulation. Numerical examples are studied and the results are compared with analytical solutions and those obtained from the conventional finite element method (FEM) to evaluate the performance of the proposed model.

Mathematical Basis of G Spaces

This paper represents some basic mathematic theories for G[Formula: see text] spaces of functions that can be used for weakened weak (W2) formulations, upon which the smoothed finite element methods (S-FEMs) and the smoothed point interpolation methods (S-PIMs) are based for solving mechanics problems. We first introduce and prove properties of G[Formula: see text] spaces, such as the lower boundedness and convergence of the norms, which are in contrast with H1spaces. We then prove the equivalence of the Gsnorms and its corresponding semi-norms. These mathematic theories are important and essential for the establishment of theoretical frame and the development of relevant numerical approaches. Finally, numerical examples are presented by using typical S-FEM models known as the NS-FEM and [Formula: see text]S-FEM to examine the properties of a smoothed method based on Gsspaces, in comparison with the standard FEM with weak formulation.

Meshfree Method Analysis of Biot's Consolidation Using Cell-Based Smoothed Point Interpolation Method

A set of cell-based smoothed point interpolation methods are proposed for the numerical analysis of Biot’s formulation. In the proposed methods, the problem domain is discretized using a triangular background mesh. Shape functions are constructed using either polynomial or radial point interpolation method (PIM), leading to the delta function property of shape functions and consequently, easy implementation of essential boundary conditions. The Biot’s equations are discretised in space and time. A variety of support domain selection schemes (T-schemes) are investigated. The accuracy and convergence rate of the proposed methods are examined by comparing the numerical results with the analytical solution for the benchmark problem of one dimensional consolidation.

Proofs of the stability and convergence of a weakened weak method using PIM shape functions

Recently, the smoothed point interpolation method (S-PIM) regarded as a weakened weak (W2) formulation method has been developed for solving engineering mechanics problems. It works well with distorted meshes. The G space theory offers the theoretical base for all the W2 methods that use smoothing operations. In this paper, we first prove mathematically that if a function is of Lipschitz continuity and its interpolated function is established using PIM shape functions, then the interpolated function belongs to a Gh,0s space. Our proofs work for smoothing operations that are the node-based, cell-based and a mixture of both smoothing domains. In addition, when mesh is refined under a given regularity condition, a sufficiently smooth target function can be approximated by its interpolated function with arbitrary accuracy, meaning that the interpolation error norm approaches to zero. Therefore, the stability and convergence of a W2 method using PIM shape functions and G space theory can be ensured.

A cell-based smoothed point interpolation method for flow-deformation analysis of saturated porous media

A group of cell-based smoothed point interpolation methods based on the generalised gradient smoothing technique are proposed for the numerical modelling of saturated porous media. In the methods proposed, the problem domain is first discretised with the use of a simple triangular background mesh. The purpose of the background mesh is twofold: (i) it is used to select the supporting nodes for each point of interest for the construction of nodal shape functions, and (ii) it provides cells to serve as the smoothing domains. Spatial discretisation of the coupled partial differential equations is derived by applying the weakened weak (W2) formulation referred to as the Generalised Smoothed Galerkin method. Both displacement and pressure fields are interpolated using the point interpolation shape functions (polynomial and radial). Shape function differentiations are effected through the use of the smoothed gradient technique, leading to smoothed strains and pressure gradients. Temporal discretisation is performed with a three-point time discretisation scheme with variable time steps. A host of node selection schemes, known as T-schemes, are adopted to guarantee the non-singularity of the moment matrices in creating shape functions. The proposed methods are thoroughly examined by simulation of a number of benchmark examples with analytical or semi-analytical solutions. The accuracy and convergence rate of the methods are investigated through comparison of the numerical results of the proposed methods with those obtained using analytical/semi-analytical solutions, point interpolation methods, and standard finite element methods.

A face-based smoothed point interpolation method (FS-PIM) for analysis of nonlinear heat conduction in multi-material bodies

A face-based smoothed point interpolation method (FS-PIM) is formulated to solve nonlinear heat transfer analysis of composite structures. For this method, the problem domain is first discretized using tetrahedral elements, and the face-based smoothing domains are further formed based on the faces of the tetrahedral meshes. Then, the field functions are approximated using PIM shape functions, which are constructed by different polynomial basis. Finally, the smoothed Galerkin weak form was utilized to obtain the discretized system equations. Numerical examples with different kinds of boundary conditions are investigated to verify the validity of the present method. The obtained results show that the presented FS-PIM schemes are very effective, when dealing with the nonlinear heat transfer analysis of composite structures.

Smoothed Point Interpolation Method for Elastoplastic Analysis

This work formulates the node-based smoothed radial point interpolation method (NS-RPIM), a typical model of smoothed point interpolation method, for the elastoplastic analysis of two-dimensional solids with gradient-dependent plasticity. The NS-RPIM uses radial point interpolation shape functions for field approximation and node-based gradient smoothing for strain field construction. The formulation is based on the parametric variational principle (PVP) in the form of complementarity with the gradient-dependent plasticity being represented by means of the linearization of the yield criterion and the flow rule. Numerical study results have demonstrated the accuracy and stability of the proposed approach for elastoplastic analysis.

Meshfree Sequentially Linear Analysis of Concrete

AbstractA new, meshfree method employing the node-based, smoothed point interpolation method (NS-PIM) is presented as an alternative to the nonlinear finite-element approach for concrete members. T...

AN EFFICIENT SMOOTHED POINT INTERPOLATION METHOD FOR DYNAMIC ANALYSES

In this work, a new procedure to compute the mass matrix in the smoothed point interpolation method is discussed. Therefore, the smoothed subdomains are employed to evaluate the mass matrix, which have already been computed for the construction of the stiffness matrix, rendering a more efficient methodology. The procedure is discussed, taking into account the edge-based, cell-based, and node-based smoothed point interpolation methods, as well as different T-schemes for the construction of the support domain of the approximating shape function, which is here formulated based on the radial point interpolation method. Numerical results of different dynamic analyses are presented, illustrating the potentialities of the proposed methodology.

Time-domain electromagnetic wave propagation analysis by edge-based smoothed point interpolation methods

In this work, smoothed meshfree methods are firstly applied to numerically analyze two-dimensional electromagnetic wave propagation models. A weakened weak formulation based on the edges of triangular cells obtained by a Delaunay triangulation is considered here, framing the so-called edge-based smoothed domains. The meshfree shape functions are computed by the Radial Point Interpolation Method (RPIM) considering two schemes for the shape function support domains, namely the T3- and the T6-scheme. These schemes are based on the above mentioned triangular cells and they consider fixed numbers of nodes for the support domains. A new procedure to compute the transient related matrices is discussed here, aiming to be more consistent with the edge-based smoothed point interpolation method. In this new procedure, finer local integration techniques are managed to be considered at lower computational costs, rendering more accurate and efficient techniques. At the end of the paper, numerical examples are presented, illustrating the effectiveness and potentialities of the proposed methodologies.

An efficient adaptive analysis procedure using the edge-based smoothed point interpolation method (ES-PIM) for 2D and 3D problems

Abstract In this paper, an efficient adaptive analysis procedure is proposed using the newly developed edge-based smoothed point interpolation method (ES-PIM) for both two dimensional (2D) and three dimensional (3D) elasticity problems. The ES-PIM works well with three-node triangular and four-node tetrahedral meshes, is easy to be implemented for complicated geometry, and can obtain numerical results of much better accuracy and higher convergence rate than the standard finite element method (FEM) with the same set of meshes. All these important features make it an ideal candidate for adaptive analysis. In the present adaptive procedure, a novel error indicator is devised for ES-PIM settings, which evaluates the maximum difference of strain energy values among the vertexes of each background cell. A simple h -type local refinement scheme is adopted together with a mesh generator based on Delaunay technology. Intensive numerical studies of 2D and 3D examples indicate that the proposed adaptive procedure can effectively capture the stress concentration and solution singularities, carry out local refinement automatically, and hence achieve much higher convergence for the solutions in strain energy norm compared to the general uniform refinement.

An Edge-Based Smoothed Point Interpolation Method for Material Discontinuity

In this work, the edge-based smoothed point interpolation method (ES-PIM) is extended to treat material discontinuity problems. The PIM shape functions used in the ES-PIM possess the Delta function property, which allows straightforward imposition of point essential boundary conditions and, hence, no additional treatment needs to be applied along the interface to meet the interface condition. By performing the generalized strain smoothing operation within the edge-based smoothing domains, the ES-PIM models possess a number of good features compared with the standard FEM using the same set of linear triangular elements, such as better accuracy, higher convergence rate, and efficiency. Three typical numerical examples, including the benchmark sandwiches cantilever beam, a hollow cylinder with two different materials and an automotive part of complicated shape, have been studied using the present ES-PIM. Analytical solutions for the first two problems have also been derived and presented. Very good agreement between the numerical results and the given analytical reference solutions demonstrate the effectiveness of the present ES-PIM method.

English

This paper extends the edge-based smoothed point interpolation method (ES-PIM) to bending strength analysis of asymmetric gears with complex outlines. Five sets of asymmetric gears with pressure angles of 20°/20°, 25°/20°, 30°/20°, 35°/20° and 40°/20° were generated by a specially designed rack cutter. Four key factors, e.g. accuracy, convergence, the convergence rate and the computational efficiency of the present ES-PIM were checked in great detail on these five models, and the distributions of bending stresses at the fillet of the drive side were carefully investigated. The finite element method (FEM) was also used to calculate the abovementioned factors to stress the advantages of ES-PIM. The numerical results indicate that ES-PIM can provide more efficient and accurate solutions in the stress field than the FEM, and is very suitable for stress analysis of complicated asymmetric gears.