Edge-based Smoothed Point Interpolation Method Research Articles

An h-adaptive edge-based smoothed point interpolation method for elasto-plastic analysis of saturated porous media

This study presents the first application of the edge-based smoothed point interpolation method (ESPIM) for adaptive analysis of coupled elasto-plastic problems in saturated porous media. The original ESPIM is enhanced with an adaptive strategy involving an automated h-refinement scheme, a simple, yet effective, recovery-based error indicator, and an efficient data transfer scheme to analyse coupled elasto-plastic problems in geomechanics. The governing equations are presented and discretised in space and time to yield the fully coupled system of equations. The solution strategy is then discussed in terms of handling the material non-linearity, and accommodating the proposed adaptivity procedure in the numerical algorithm. Four numerical examples comprising two different bearing capacity problems, a two-dimensional consolidation problem, and a slope stability analysis are thoroughly investigated. Different adaptive analyses with a range of adaptivity control parameters are investigated in each example and the performances of different techniques are compared in terms of accuracy and the time of the analysis. The results show the ability of the proposed adaptive analyses in yielding accurate and efficient results. Conclusions are drawn in terms of the most efficient numerical set-up to solve non-linear problems in porous media using the proposed numerical scheme.

An Automatic Adaptive Edge-based Smoothed Point Interpolation Method for Coupled Flow-Deformation Analysis of Saturated Porous Media

An adaptive analysis based on an advanced computational method, referred to as edge-based smoothed point interpolation method (ESPIM), is proposed for numerical analysis of coupled problems in geomechanics. For the adaptive procedure, a simple error indicator derived based on the maximum difference of the strain energy among the three nodes of each background cell is adopted along with a simple automatic h-type local refinement strategy and automatic mesh generator based on the Delaunay triangulation. Novel activation and termination criteria are proposed for initiating and ending adaptivity analysis in each time step. The performance of the proposed adaptive procedure is investigated through three numerical examples. The results show that with the proposed adaptivity procedure, the adaptive ESPIMs have better computational efficiency compared to non-adaptive methods or the general uniform mesh refinement strategy. In particular, it is shown that the proposed termination criteria for the refinement steps ensures that the adaptivity procedure stops when the computational cost of the refinement overweights the resulting gain in the solution accuracy (determined by the user), hence unnecessary refinement steps are avoided in the analysis to yield optimal computational efficiency.

The hygro-thermo-electro-mechanical coupling edge-based smoothed point interpolation method for the response of functionally graded piezoelectric structure under hygrothermal environment

In this paper, we analyze the hygro-thermo-electro-mechanical (HTEM) coupling problem of functionally graded piezoelectric (FGPE) structure based on edge-based smoothed point interpolation method (ES-PIM). The basic equations for FGPE structure are derived in the multi-physical field. Triangular elements adopted in ES-PIM can be generated automatically and suitable for complicated structures. The responses for the HTEM coupling problem are calculated by ES-PIM and finite element method (FEM) under different hygrothermal conditions. Through the numerical examples, the factors related to the temperature and moisture fields are explored. The exponential factor and the hygrothermal effect on the responses of the FGPE structure are also studied. Besides, the advantages of proposed method for HTEM coupling problems are also validated by several numerical examples. These solutions also indicate that the increase of temperature and moisture concentration decreases the structural stiffness when empirical constants are set to negative numbers, leading to the decrease of frequency and variation of displacement. This article proposes an excellent numerical method and provides a reference for the design and application of piezoelectric structures.

A Fully Automatic h-Adaptive Analysis Procedure Using the Edge-Based Smoothed Point Interpolation Method

Within the framework of edge-based smoothed point interpolation method (ES-PIM), a fully automatic adaptive procedure has been proposed by introducing a novel local critical value definition. Owing to the softened stiffness, ES-PIM can generally provide much better results than the traditional finite element method using the simplest linear triangle elements. By comparing with most adaptive models which need to set a refinement rate manually, a novel local critical value considering the area-dependent average magnitude of error indicator for all the cells has been introduced in this work, which leads to an adaptive ES-PIM model without manual operation during the adaptive process. Numerical results have shown that the present adaptive model conducts the adaptive process dynamically, adds the new nodes properly and provides more accurate results with less field nodes than both the model with uniform refinement and the adaptive model using predefined refinement rate. Also considering linear triangular background elements used, the present method is supposed to have much potential for solving complicated engineering problems.

Code verification in computational geomechanics: Method of manufactured solutions (MMS)

Code verification is the process of ensuring that there are no coding mistakes or algorithm deficiencies in a scientific computing code. In this paper, code verification in computational geomechanics based on the method of manufactured solutions (MMS) is discussed. The application of this technique, which is the most rigorous method among all code verification techniques, is presented through verification of an in-house geomechanics code based on the Edge-based Smoothed Point Interpolation Method (ESPIM) for spatial, and three-point scheme for temporal discretisation of the governing equations. The MMS, as a generic approach to construct all-inclusive and complex test solutions for code verification, is explained. Two manufactured solutions along with an order of accuracy study are employed in several different setups to obtain the numerical and formal order of accuracies of the code. Results from the mesh and time step refinement studies confirm the verification of our code, and demonstrate the application of the MMS and order of convergence study in coupled problems in geomechanics.

Fully coupled elastoplastic hydro‐mechanical analysis of unsaturated porous media using a meshfree method

SummaryThis paper presents the first application of an advanced meshfree method, ie, the edge‐based smoothed point interpolation method (ESPIM), in simulation of the coupled hydro‐mechanical behaviour of unsaturated porous media. In the proposed technique, the problem domain is spatially discretised using a triangular background mesh, and the polynomial point interpolation method combined with a simple node selection scheme is adopted for creating nodal shape functions. Smoothing domains are formed on top of the background mesh, and a constant smoothed strain, created by applying the smoothing operation over the smoothing domains, is assigned to each smoothing domain. The deformation and flow models are developed based on the equilibrium equation of the mixture, and linear momentum and mass balance equations of the fluid phases, respectively. The effective stress approach is followed to account for the coupling between the flow and deformation models. Further coupling among the phases is captured through a hysteretic soil water retention model that evolves with changes in void ratio. An advanced elastoplastic constitutive model within the context of the bounding surface plasticity theory is employed for predicting the nonlinear behaviour of soil skeleton. Time discretisation is performed by adopting a three‐point discretisation method with growing time steps to avoid temporal instabilities. A modified Newton‐Raphson framework is designed for dealing with nonlinearities of the discretised system of equations. The performance of the numerical model is examined through a number of numerical examples. The state‐of‐the‐art computational scheme developed is useful for simulation of geotechnical engineering problems involving unsaturated soils.

ES-PIM applied to buckling of variable angle tow laminates

The increasing need for automatic mesh generation has led to the development of efficient triangulation algorithms that are able to discretize any 2D or 3D domain. Modern finite element formulations based on strain smoothing techniques (SFEM) provide enhanced convergence properties, preventing yet the stiffening behavior of triangular meshes. Recent research has shown that meshless methods based on triangular mapping of the integration domain can be used to produce even better convergence properties than SFEM. The present study explores the Edge-based Smoothed Point Interpolation Method (ES-PIM) as a meshless solution to investigate linear buckling on variable angle tow (VAT) laminates. Such advanced composite structures show a heterogeneous distribution of constitutive properties and thickness, presenting additional challenges to the numerical solution. Important aspects related to the transverse shear correction herein adopted are investigated, leading to interesting conclusions regarding the possibility to use the ES-PIM for conservative estimates of the critical buckling load of VAT laminates.

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.

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.

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.

DESIGN OF ASYMMETRIC GEAR AND ACCURATE BENDING STRESS ANALYSIS USING THE ES-PIM WITH TRIANGULAR MESH

This paper proposes a novel design of an asymmetric gear that applies a larger pressure angle in the drive side and a standard pressure angle in the coast side via accurate stress analysis using the edge-based smoothed point interpolation method with triangular mesh (ES-PIM-(T3)). An asymmetric rack cutter is first developed. Then, the governing equations of the tooth profiles cut by the rack cutter are derived. Finally, five sets of asymmetric gears with the pressure angles of 20° /20°, 25°/20°, 30°/20°, 35°/20°, and 40°/20° are established to quantitatively check the stress distributions at the fillet of the drive side of these teeth. The best point for applying force in a static bending stress analysis and the best pressure angle in the drive side for a gear design are suggested.

ES-PIM with Cell Death and Birth Technique for Simulating Heat Transfer in Concrete Dam Construction Process

The meshfree edge-based smoothed point interpolation method (ES-PIM) is extended to the simulation of transient heat transfer problems for concrete dams in the construction process. The present ES-PIM consists of the following ingredients: (1) a novel cell death and birth technique is proposed to deal with the existence or nonexistence of the concrete layers, and hence the construction process of dams can be easily simulated; (2) a lumped diagonalization procedure for specific heat matrix is implemented to improve the stability and efficiency of transient computations; and (3) the triangular background cells are used to construct the nonoverlapping smoothing integral subdomains, and the cell-based T3- and T6/3-schemes are adopted for the node selection in the formation of PIM shape functions. On the basis of the weakened weak formulation, the ES-PIM is spatially stable and convergent to the exact solution. The ES-PIM can achieve better results in accuracy and convergence rate, in comparison with the finite element method using the same meshes.

An adaptive edge-based smoothed point interpolation method for mechanics problems

This paper develops a smoothing domain-based energy (SDE) error indicator and an efficient adaptive procedure using edge-based smoothed point interpolation methods (ES-PIM), in which the strain field is constructed via the generalized smoothing operation over smoothing domains associated with edges of three-node triangular background cells. Because the ES-PIM can produce a close-to-exact stiffness and achieve ‘super-convergence’ and ‘ultra-accurate’ solutions, it is an ideal candidate for adaptive analysis. A SDE error indicator is first devised to make use of the features of the ES-PIM. A local refinement technique based on the Delaunay algorithm is then implemented to achieve high efficiency. The refinement of nodal neighbourhood is accomplished simply by adjusting a scaling factor assigned to control local nodal density. Intensive numerical studies, including the problems with stress concentration and solution singularity, demonstrate that the proposed adaptive procedure is effective and efficient in producing solutions with desired accuracy.

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.

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.