Node-based Smoothed Point Interpolation Method Research Articles

Particle node-based smoothed point interpolation method with stress regularisation for large deformation problems in geomechanics

In this study, a new numerical scheme is developed for modelling large deformation problems in geomechanics. The approach is based on employing the node based smoothed point interpolation methods (NSPIMs) in the framework of the particle finite element method (PFEM). The proposed technique is particle based in the sense that all variables are computed and stored at the nodes, hence constant transformation of the information between Gauss points and nodes, which is required in the particle finite element method (PFEM), is avoided. The accuracy of the proposed technique and its possible numerical instabilities, inherited through the use of NSPIMs in the method, are investigated in details. Accordingly, two remedial techniques, one based on stress regularisation by moving least square (MLS) and the other based on an improved formulation of NSPIMs, are presented to alleviate the instability problem and improve the accuracy of the numerical solutions. The proposed techniques and measures are thoroughly examined through three numerical examples, and conclusions are drawn.

An improved node-based smoothed point interpolation method for coupled hydro-mechanical problems in geomechanics

An improved version of the node-based smoothed point interpolation methods (NSPIMs) is proposed for coupled hydro-mechanical problems of geotechnical engineering. In the improved NSPIMs (I-NSPIMs), the polygonal nodal smoothing domains are estimated with area-equivalent circular domains. The smoothed strains over the circular nodal smoothing domains are then assumed to be linear, in contrast to the constant smoothed strain adopted in the original NSPIMs. The efficiency of the proposed method is investigated through five practical examples, all showing clear superiority of I-NSPIMs over the original formulation of NSPIMs in small deformation problems of geomechanics.

The forward and inversion analysis of high rock-fill dam during construction period using the node-based smoothed point interpolation method

PurposeThe meshfree node-based smoothed point interpolation method (NS-PIM) is extended to the forward and inversion analysis of a high gravelly soil core rock-fill dam during construction periods.Design/methodology/approachAs one member of the meshfree methods, the NS-PIM has the advantages of “softer” stiffness and adaptability to large deformations which is quite indispensable for the stability analysis of rock-fill dams. In this work, the present method contains a reconstruction procedure to deal with the existence or nonexistence of the construction layers. After verifying the validity of the NS-PIM method for nonlinear elastic model during construction period, the convergence features of the NS-PIM and FEM methods are further investigated with different mesh schemes. Furthermore, the NS-PIM and FEM methods are applied for the forward analysis of a high gravelly soil core rock-fill dam and the convergence features under complex stress conditions are also studied using the rock-fill dam model. Finally, the NS-PIM method is used to calculate the Duncan–Chang parameters of the deep overburden under the high gravelly soil core rock-fill dam based on the back-propagation neural network method.FindingsThe results show that: the NS-PIM solution for construction analysis still possesses the property of upper bound solution even under complex stress conditions and can provide comparatively more conservative results for safety evaluation. Furthermore, it can be used to evaluate the accuracy of results and mesh quality together with the FEM solution which has the property of lower bound solution; the inversion analysis in this work provides a set of material parameters for the deep overburden under high rock-fill dam during construction period and the calculated results show good agreement with the measured displacement values and it is feasible to apply the NS-PIM to the forward and inversion analysis of high rock-fill dams on deep overburden during construction periods.Research limitations/implicationsIn further study, the feasibility of three-dimensional problems, elastic–plastic problems, contact problems and multipoint inversion can still be probed in the NS-PIM solution for the forward and inversion analysis of high rock-fill dams on deep overburden.Practical implicationsThis paper introduced a method for the forward and inversion analysis of high rock-fill dams during construction period using the NS-PIM solution. The property of upper bound solution ensures that the NS-PIM can provide more conservative results for safety evaluation. The inversion analysis in this work provides a set of material parameters for the deep overburden under high rock-fill dam during construction periods.Originality/valueFirst, the analysis from forward to inversion for high rock-fill dams during construction period using the NS-PIM solution is accomplished in this work. A procedure dealing with the existence or nonexistence of the construction layers is also developed for the construction analysis. Second, it is confirmed in this work that the NS-PIM still possesses the property of upper bound solution even under complex stress conditions (the forward analysis of high rock-fill dams during construction period). Thus, more conservative results can be provided for safety evaluation. Furthermore, it can be used to evaluate the accuracy of results and mesh quality together with the FEM solution which has the property of lower bound solution. Third, the calculated material parameters of the deep overburden in this work can be used for further studies of the high rock-fill dam.

Coupling Magneto-Electro-Elastic node-based smoothed radial point interpolation method for free vibration and transient analysis of Functionally Graded Magneto-Electro-Elastic structures

A Coupling Magneto-Electro-Elastic (MEE) Node-based Smoothed Radial Point Interpolation Method (CM-NS-RPIM) was proposed to solve the free vibration and transient responses of Functionally Graded Magneto-Electro-Elastic (FGMEE) structures. By introducing the modified Newmark method, the displacement, electrical potential and magnetic potential of the structures under transient mechanical loading were obtained. Based on G space theory and the weakened weak (W2) formulation, the equations of the multi-physics coupling problems were derived. Using triangular background elements, the free vibration and transient responses of three numerical examples were studied. Results proved that CM-NS-RPIM performed better than the standard FEM by reducing the overly-stiff of structures. Moreover, CM-NS-RPIM could reduce the number of nodes while guaranteeing the accuracy. Besides, triangular elements could be generated automatically even for complex geometries. Therefore, the effectiveness and validity of CM-NS-RPIM were demonstrated, which were valuable for the design of intelligence devices, such as energy harvesters and sensors.

Magneto-electro-elastic node-based smoothed point interpolation method for micromechanical analysis of natural frequencies of nanobeams

As an addition to the traditional finite element method (FEM), the magneto-electro-elastic node-based smoothed point interpolation method (MEE-NS-PIM) with asymptotic homogenization method (AHM) is presented to solve the micromechanical problems of MEE nanobeams, which overcomes the deficiency of FEM and improves the accuracy of the calculation results. Firstly, the basic equations of MEE medium are derived. Secondly, AHM is adopted to calculate the property parameters of MEE materials under microcosmic situations, and the AHM model is illustrated. Then, the relative formulations of the discretized system used to calculate the frequency of MEE nanostructures are deduced based on MEE-NS-PIM. Moreover, several numerical examples are calculated, and the results of MEE-NS-PIM are compared with those of FEM, which proves the convergence, precision, and effectiveness of MEE-NS-PIM. Therefore, MEE-NS-PIM combined with AHM can be used to analyze the microcosmic MEE coupling problems and obtain a more accurate and reliable solution for MEE micromechanics.

Meshfree method analysis of Biot’s consolidation using the node-based smoothed point interpolation method (NS-PIM)

PurposeThe purpose of the article is to extend the node-based smoothed point interpolation method (NS-PIM) for soil consolidation analysis based on the Biot’s theory.Design/methodology/approachThe shape functions for displacements and pore pressures are constructed using the PIM separately, leading to the Kronecker delta property and easy implementation of essential boundary conditions. Then, a benchmark problem of 2D consolidation under ramp load is solved to investigate the validity of this application. Meanwhile, convergence features of different solutions are studied. Furthermore, the incompressible and impermeable condition under instant load is investigated. The results calculated by the NS-PIM solution with different orders of shape functions are compared. Finally a 2D consolidation problem in construction period is solved. An error estimation method is applied to check the mesh quality.FindingsThe results of the NS-PIM solution show good agreement with those certified results. Useful convergence features are found when comparing the results of the NS-PIM and the FEM solutions. A simple method is introduced to estimate the errors of the model with rough grids. The convergence features and error estimation method can be applied to check the mesh quality and get accurate results. More stable results can be obtained using the NS-PIM solution with lower order of pore pressure shape functions under the incompressible and impermeable condition.Research limitations/implicationsIt cannot be denied that the calculation of NS-PIM solution takes more time than that of the FEM solution, and more work needs to be carried out to optimize the NS-PIM solution. Also, in further study, the feasibility of more complicated and practical engineering problems can still be probed in the NS-PIM solution.Practical implicationsThis paper introduced a method for the consolidation analysis on the situation of construction loads (“ramp load”) using the NS-PIM which is quite indispensable in many foundation problems. Also, more stable results can be obtained using the NS-PIM solution with lower order of pore pressure shape functions than that with same order of shape functions.Originality/valueThis study first focuses on the situation of construction loads (“ramp load”) in the NS-PIM consolidation analysis which is quite indispensable in many foundation problems. An error estimation method is introduced to evaluate the mesh quality and get accurate values based on the convergence features of the FEM and NS-PIM solutions. Then, the incompressible and impermeable condition under instant load is investigated, and the analysis show that the NS-PIM with lower order of pore pressure shape functions can get stable results in such conditions.

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.

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...

A three-dimensional adaptive analysis using the meshfree node-based smoothed point interpolation method (NS-PIM)

In this paper, a three-dimensional (3-D) adaptive analysis procedure is proposed using the meshfree node-based smoothed point interpolation method (NS-PIM). Previous study has shown that the NS-PIM works well with the simplest four-node tetrahedral mesh, which is easy to be implemented for complicated geometry. In contrast to the displacement-based FEM providing lower bound solutions, the NS-PIM possesses the attractive property of providing upper bound solutions in strain energy norm. In the present adaptive procedure, a novel error indicator is devised for NS-PIM settings, which evaluates the maximum difference of strain energy values among four nodes in each of the tetrahedral cells. A simple h-type local refinement scheme is adopted and coupled with 3-D mesh automatic generator based on Delaunay technology. Numerical results indicate that the adaptive refinement procedure can effectively capture the stress concentration and solution singularities, and carry out local refinement automatically. The present adaptive procedure achieves much higher convergence in strain energy solution compared to the uniform refinement, and obtains upper bound solution in strain energy efficiently for force driven problems.

An efficient adaptive analysis procedure for node-based smoothed point interpolation method (NS-PIM)

This paper presents an efficient adaptive analysis procedure being able to operate in the framework of the node-based smoothed point interpolation method (NS-PIM). The NS-PIM uses three-node triangular cells and is very easy to be implemented, which make it an ideal candidate for adaptive analysis. In the present adaptive procedure, a new error indicator is devised for NS-PIM settings; two ways are proposed to calculate the local critical value; a simple h-type local refinement scheme is adopted and Delaunay technology is used for regenerating optimal new mesh. A number of typical numerical examples involving stress concentration and solution singularities have been tested. The results demonstrate that the present procedure achieves much higher convergence rate results compared to the uniform refinement, and can obtain upper bound solution in strain energy.

An edge-based smoothed point interpolation method (ES-PIM) for heat transfer analysis of rapid manufacturing system

This paper formulates an edge-based smoothed point interpolation method (ES-PIM) for analyzing 2D and 3D transient heat transfer problems with mixed boundary conditions and complicated geometries. In the ES-PIM, shape functions are constructed using the polynomial PIM with the Delta function property for easy treatment of essential boundary conditions. A generalized smoothing technique is used to reconstruct the temperature gradient field within the edge-based smoothing domains. The generalized smoothed Galerkin weak form is then used to establish the discretized system equations. Our results show that the ES-PIM can provide more close-to-exact stiffness compared with the “overly-stiff” finite element method (FEM) and the “overly-soft” node-based smoothed point interpolation method (NS-PIM). Owing to this important property, the present ES-PIM provides more accurate solutions than standard FEM using the same mesh. As an example, a practical cooling system of the rapid direct plasma deposition dieless manufacturing is studied in detail using the present ES-PIM, and a set of “optional” processing parameters of fluid velocity and temperature are found.

Certified solutions for hydraulic structures using the node‐based smoothed point interpolation method (NS‐PIM)

AbstractA meshfree node‐based smoothed point interpolation method (NS‐PIM), which has been recently developed for solid mechanics problems, is applied to obtain certified solutions with bounds for hydraulic structure designs. In this approach, shape functions for displacements are constructed using the point interpolation method (PIM), and the shape functions possess the Kronecker delta property and permit the straightforward enforcement of essential boundary conditions. The generalized smoothed Galerkin weak form is then applied to construct discretized system equations using the node‐based smoothed strains. As a very novel and important property, the approach can obtain the upper bound solution in energy norm for hydraulic structures. A 2D gravity dam problem and a 3D arch dam problem are solved, respectively, using the NS‐PIM and the simulation results of NS‐PIM are found to be the upper bounds. Together with standard fully compatible FEM results as a lower bound, we have successfully determined the solution bounds to certify the accuracy of numerical solutions. This confirms that the NS‐PIM is very useful for producing certified solutions for the analysis of huge hydraulic structures. Copyright © 2009 John Wiley & Sons, Ltd.

A NOVEL SCHEME OF STRAIN-CONSTRUCTED POINT INTERPOLATION METHOD FOR STATIC AND DYNAMIC MECHANICS PROBLEMS

This paper presents a new scheme of strain-constructed point interpolation method (SC-PIM) for static, free and forced vibration analysis of solids and structures using triangular cells. In the present scheme, displacement fields are assumed using shape functions created via the point interpolation method (PIM), which possess the Kronecker delta property facilitating the straightforward enforcement of displacement boundary conditions. Using the generalized gradient smoothing technique, the "smoothed" strains at the middle points of the cells edges are first obtained using the corresponding edge-based smoothing domains and the assumed displacement field. In each triangular background cell, the strains at the vertices are assigned using these smoothed strains in a proper manner, and then piecewisely linear strain fields are constructed by the linear interpolation for each sub-triangular cell using the edge-based "smoothed" strains. With the assumed displacements and constructed linear strain fields, the discretized system equations are created using the Strain Constructed Galerkin (SC-Galerkin) weak form. A number of benchmark numerical examples, including the standard patch test, static, free and forced vibration problems, have been studied and intensive numerical results have demonstrated that the present method possesses the following properties: (1) it works well with the simplest triangular mesh, no additional degrees of freedom and parameters are introduced and very easy to implement; (2) it is at least linearly conforming; (3) it possesses a close-to-exact stiffness: it is much stiffer than the "overly-soft" node-based smoothed point interpolation method (NS-PIM) and much softer than the "overly-stiff" FEM model; (4) the results of the present method are of superconvergence and ultra-accuracy: about one order of magnitude more accurate than those of the linear FEM; (5) there are no spurious non-zeros energy modes found and it is also temporally stable, hence the present method works well for dynamic problems.

A point interpolation method with least square strain field (PIM-LSS) for solution bounds and ultra-accurate solutions using triangular mesh

A point interpolation method with least square strain field (PIM-LSS) is developed for solid mechanics problems using triangular background mesh. In the PIM-LSS, PIM shape functions are used for displacement field construction that may or may not be compatible, and a least square fitting technique is adopted to construct the strain field. A strain constructed Galerkin (SC-Galerkin) weak formulation is then proposed for establishing discretized PIM-LSS models that have a number of special properties. We proved theoretically (1) the PIM-LSS provides a “softening” effect to the FEM model, and a “stiffing” effect to the node-based smoothed point interpolation method (NS-PIM) model; (2) the exact solution is bounded from both by PIM-LSS solutions with strain field of zero-order fitting and that with higher order fitting; (3) There exists a preferred order of fitting for the strain field such that ultra-accurate (one order higher accuracy) solution can be obtained using the PIM-LSS. These theorems and properties have been confirmed in numerical examples.

EDGE-BASED SMOOTHED POINT INTERPOLATION METHODS

This paper formulates an edge-based smoothed point interpolation method (ES-PIM) for solid mechanics using three-node triangular meshes. In the ES-PIM, displacement fields are construed using the point interpolation method (polynomial PIM or radial PIM), and hence the shape functions possess the Kronecker delta property, facilitates the enforcement of Dirichlet boundary conditions. Strains are obtained through smoothing operation over each smoothing domain associated with edges of the triangular background cells. The generalized smoothed Galerkin weak form is then used to create the discretized system equations and the formation is weakened weak formulation. Four schemes of selecting nodes for interpolation using the PIM have been introduced in detail and ES-PIM models using these four schemes have been developed. Numerical studies have demonstrated that the ES-PIM possesses the following good properties: (1) the ES-PIM models have a close-to-exact stiffness, which is much softer than for the overly-stiff FEM model and much stiffer than for the overly-soft node-based smoothed point interpolation method (NS-PIM) model; (2) results of ES-PIMs are generally of superconvergence and "ultra-accurate"; (3) no additional degrees of freedom are introduced, the implementation of the method is straightforward, and the method can achieve much better efficiency than the FEM using the same set of triangular meshes.

A node-based smoothed point interpolation method (NS-PIM) for three-dimensional heat transfer problems

A node-based smoothed point interpolation method (NS-PIM) is formulated for three-dimensional (3D) heat transfer problems with complex geometries and complicated boundary conditions. Shape functions constructed here through PIM possess the delta function property and hence allow the straightforward enforcement of essential boundary conditions. The smoothed Galerkin weak form is employed to create discretized system equations, and the node-based smoothing domains are used to perform the smoothing operation and the numerical integration. The accuracy and efficiency of the NS-PIM solutions are studied through detailed analyses of actual 3D heat transfer problems. It is found that the NS-PIM can provide higher accuracy in temperature and its gradient than the reference approach does, in which very fine meshes are used in standard FEM code available with homogeneous essential boundary conditions. More importantly, the upper bound property of the NS-PIM is obtained using the same tetrahedral mesh. Together with the FEM, we now have a simple means to obtain both upper and lower bounds of the exact solution to heat transfer using the same type of mesh.

A Node-based Smoothed Point Interpolation Method (NS-PIM) for Three-dimensional Thermoelastic Problems

A node-based smoothed point interpolation method (NS-PIM) is formulated to analyze 3-D steady-state thermoelastic problems subjected to complicated thermal and mechanical loads. Gradient smoothing technique with node-based smoothing domains is utilized to modify the gradient fields and to perform the numerical integration required in the weak form formulation. Numerical results show that NS-PIM can achieve more accurate solutions even when the 4-node tetrahedral mesh is used compared to the finite-element method (FEM) using the same mesh, especially for strains and hence stresses. Most importantly, it can produce an upper bound solution of the exact solution in energy norm for both temperature and stress fields when a reasonably fine mesh is used. Together with FEM, we now for the first time have a simple means to obtain both upper and lower bounds of the exact solution to complex thermoelastic problems.

A node-based smoothed point interpolation method (NS-PIM) for thermoelastic problems with solution bounds

A node-based smoothed point interpolation method (NS-PIM) is formulated to analyze steady-state thermoelastic problems. In this approach, shape functions are constructed using the point interpolation method (PIM), which permits the straightforward enforcement of essential boundary conditions. The smoothed Galerkin weak form is then used to construct discretized system equations using smoothing domains constructed based on nodes. The bound property, accuracy and convergence of the present formulation are studied using 1D and 2D thermoelasticity problems. It is found that the computed temperature and its resulted gradient are in very good agreement with the analytical results or those obtained using the finite element method (FEM). Compared with the 3-node triangular FEM, the NS-PIM can achieve better accuracy and higher convergence in energy norm using the same linear triangular mesh. Together with the FEM, we now for the first time have a simple way to obtain both upper and lower bounds of the exact solution to thermoelasticity problems.

A GENERALIZED GRADIENT SMOOTHING TECHNIQUE AND THE SMOOTHED BILINEAR FORM FOR GALERKIN FORMULATION OF A WIDE CLASS OF COMPUTATIONAL METHODS

This paper presents a generalized gradient smoothing technique, the corresponding smoothed bilinear forms, and the smoothed Galerkin weakform that is applicable to create a wide class of efficient numerical methods with special properties including the upper bound properties. A generalized gradient smoothing technique is first presented for computing the smoothed strain fields of displacement functions with discontinuous line segments, by "rudely" enforcing the Green's theorem over the smoothing domain containing these discontinuous segments. A smoothed bilinear form is then introduced for Galerkin formulation using the generalized gradient smoothing technique and smoothing domains constructed in various ways. The numerical methods developed based on this smoothed bilinear form will be spatially stable and convergent and possess three major important properties: (1) it is variationally consistent, if the solution is sought in a Hilbert space; (2) the stiffness of the discretized model will be reduced compared to the model of the finite element method (FEM) and often the exact model, which allows us to obtain upper bound solutions with respect to both the FEM solution and the exact solution; (3) the solution of the numerical method developed using the smoothed bilinear form is less insensitive to the quality of the mesh, and triangular meshes can be used perfectly without any problems. These properties have been proved, examined, and confirmed by the numerical examples. The smoothed bilinear form establishes a unified theoretical foundation for a class of smoothed Galerkin methods to analyze solid mechanics problems for solutions of special and unique properties: the node-based smoothed point interpolation method (NS-PIM), smoothed finite element method (SFEM), node-based smoothed finite element method (N-SFEM), edge-based smoothed finite element method (E-SFEM), cell-based smoothed point interpolation method (CS-PIM), etc.