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 edge-based smoothed tetrahedron finite element method (ES-T-FEM) for 3D static and dynamic problems

Strain smoothing operation has been recently adopted to soften the stiffness of the model created using tetrahedron mesh, such as the Face-based Smoothed Finite Element Method (FS-FEM), with the aim to improve solution accuracy and the applicability of low order tetrahedral elements. In this paper, a new method with strain smoothing operation based on the edge of four-node tetrahedron mesh is proposed, and the edge-based smoothing domain of tetrahedron mesh is serving as the assembly unit for computing the 3D stiffness matrix. Numerical results demonstrate that the proposed method possesses a close-to-exact stiffness of the continuous system and gives better results than both the FEM and FS-FEM using tetrahedron mesh or even the FEM using hexahedral mesh in the static and dynamic analysis. In addition, a novel domain-based selective scheme is proposed leading to a combined ES-T-/NS-FEM model that is immune from volumetric locking and hence works well for nearly incompressible materials. The proposed method is an innovative and unique numerical method with its distinct features, which possesses strong potentials in the successful applications for static and dynamics problems.     

An edge-based smoothed finite element method for visco-elastoplastic analyses of 2D solids using triangular mesh

An edge-based smoothed finite element method (ES-FEM) using triangular elements was recently proposed to improve the accuracy and convergence rate of the existing standard finite element method (FEM) for the elastic solid mechanics problems. In this paper, the ES-FEM is extended to more complicated visco-elastoplastic analyses using the von-Mises yield function and the Prandtl–Reuss flow rule. The material behavior includes perfect visco-elastoplasticity and visco-elastoplasticity with isotropic and linear kinematic hardening. The formulation shows that the bandwidth of stiffness matrix of the ES-FEM is larger than that of the FEM, and hence the computational cost of the ES-FEM in numerical examples is larger than that of the FEM for the same mesh. However, when the efficiency of computation (computation time for the same accuracy) in terms of a posteriori error estimation is considered, the ES-FEM is more efficient than the FEM.     

光滑有限元的声学研究：时域和频域分析

摘　要：在使用有限元进行声场的数值模拟中,存在着两个主要误差,一个是数值方法中常规的插值误差,另外一个是计算声学中所特有的耗散误差(dispersion error),后者则是影响声学模拟仿真置信度的最重要因素。产生耗散误差的本质原因是由于有限元的数值模型刚度“偏硬”造成的。为了控制耗散误差,最重要的是使数值模型更好的反映真实模型。本文采用了一种基于边光滑的有限元方法(ES-FEM)来对声场的时域和频域进行数值模拟研究。该方法只采用对复杂问题域适应性很强的三角形网格,通过引进基于边的广义梯度光滑技术,能够使得有限元系统得到适当的“软化”。关于时域和频域的算例表明了在使用同样网格的情况下,本方法在声学模拟中的精度都要比有限元模型的高。     

An adaptive procedure for eigenvalue problems using the hierarchical finite element method

An adaptive procedure for the solution of the generalized linear eigenvalue problem within the hierarchical finite element method is described. The problem of finding for a given discretization, an upper limit eigenvalue that is accurate within a prescribed tolerance is especially studied. An error estimator is presented and a recomputational scheme for improved solutions is proposed. A numerical example is included.     

声学数值计算的分区光滑径向点插值无网格法

针对标准的有限元法分析声学问题时由于数值色散导致高波数计算结果不可靠问题,将分区光滑径向点插值法(cell-based smoothed radial point interpolation method, CS-RPIM)应用到二维声学分析中,推导了分区光滑径向点插值法分析二维声学问题的原理公式。该方法将问题域划分为三角形背景单元,每个单元进一步分成若干个光滑域,对每个光滑域进行声压梯度光滑处理,运用光滑Galerkin弱形式构造系统方程,并按有限元中方法施加必要的边界条件。CS-RPIM提供了合适的模型硬度,能有效降低色散效应,提高计算精度。对管道和二维轿车声学问题的数值分析结果表明,与标准有限元法相比,CS-RPIM具有更高的精度和准确度,在高波数计算时这种优势特别明显。     

A cell-based smoothed radial point interpolation method with virtual nodes for three-dimensional mid-frequency acoustic problems

It is well known that the finite element method (FEM) encounters dispersion errors in coping with mid-frequency acoustic problems due to its “overly stiff” nature. By introducing the generalized gradient smoothing technique and the idea of condensed shape functions with virtual nodes, a cell-based smoothed radial point interpolation method is proposed to solve the Helmholtz equation for the purpose of reducing dispersion errors. With the properly selected virtual nodes, the proposed method can provide a close-to-exact stiffness of continuum, leading to a conspicuous decrease in dispersion errors and a significant improvement in accuracy. Numerical examples are examined using the present method by comparing with both the traditional FEM using four-node tetrahedral elements (FEM-T4) and the FEM model using eight-node hexahedral elements with modified integration rules (MIR-H8). The present cell-based smoothed radial point interpolation method has been demonstrated to possess a number of superiorities, including the automatically generated tetrahedral background mesh, high computational efficiency, and insensitivity to mesh distortion, which make the method a good potential for practical analysis of acoustic problems.     

A singular edge-based smoothed finite element method (ES-FEM) for bimaterial interface cracks

The recently developed edge-based smoothed finite element method (ES-FEM) is extended to the mix-mode interface cracks between two dissimilar isotropic materials. The present ES-FEM method uses triangular elements that can be generated automatically for problems even with complicated geometry, and strains are smoothed over the smoothing domains associated with the edges of elements. Considering the stress singularity in the vicinity of the bimaterial interface crack tip is of the inverse square root type together with oscillatory nature, a five-node singular crack tip element is devised within the framework of ES-FEM to construct singular shape functions. Such a singular element can be easily implemented since the derivatives of the singular shape term ${(1/\sqrt r)}$ are not needed. The mix-mode stress intensity factors can also be easily evaluated by an appropriate treatment during the domain form of the interaction integral. The effectiveness of the present singular ES-FEM is demonstrated via benchmark examples for a wide range of material combinations and boundary conditions.     

A cell-based smoothed radial point interpolation method (CS-RPIM) for static and free vibration of solids

A cell-based smoothed radial point interpolation method (CS-RPIM) based on the generalized gradient smoothing operation is proposed for static and free vibration analysis of solids. In present method, the problem domain is first discretized using triangular background cells, and each cell is further divided into several smoothing cells. The displacement field function is approximated using RPIM shape functions which have Kronecker delta function property. Supporting node selection for shape function construction uses the efficient T2L-scheme associated with edges of the background cells. The system equations are derived using the generalized smoothed Galerkin (GS-Galerkin) weak form, and the essential boundary conditions are imposed directly as in the finite element method (FEM). The effects of the number of divisions smoothing cells on the solution properties of the CS-RPIM are investigated in detail, and preferable numbers of smoothing cells is recommended. To verify the accuracy and stability of the present formulation, a number of numerical examples are studied to demonstrate numerically the efficiency of the present CS-RPIM.     

A scaled boundary radial point interpolation method for 2‐D elasticity problems

The scaled boundary radial point interpolation method (SBRPIM), a new semi‐analytical technique, is introduced and applied to the analysis of the stress–strain problems. The proposed method combines the advantages of the scaled boundary finite element method and the boundary radial point interpolation method. In this method, no mesh is required, nodes are scattered only on the boundary of the domain, no fundamental solution is required, and as the shape functions constructed based on the radial point interpolation method possess the Kronecker delta function property, the boundary conditions of problems can be imposed accurately without additional efforts. The main ideas of the SBRPIM are introducing a new method based on boundary scattered nodes without the need to element connectivity information, satisfying Kronecker delta function property, and being capable to handle singular problems. The equations of the SBRPIM in stress–strain fields are outlined in this paper. Several benchmark examples of 2‐D elastostatic are analyzed to validate the accuracy and efficiency of the proposed method. It is found that the SBRPIM is very easy to implement and the obtained results of the proposed method show a very good agreement with the analytical solution. Copyright © 2017 John Wiley & Sons, Ltd.     

An enriched radial point interpolation method (e-RPIM) for analysis of crack tip fields

In this paper, an enriched radial point interpolation method (e-RPIM) is developed for the determination of crack tip fields. In e-RPIM, the conventional RBF interpolation is novelly augmented by the suitable trigonometric basis functions to reflect the properties of stresses for the crack tip fields. The performance of the enriched RBF meshfree shape functions is firstly investigated to fit different surfaces. The surface fitting results have proven that, comparing with the conventional RBF shape function, the enriched RBF shape function has: (1) a similar accuracy to fit a polynomial surface; (2) a much better accuracy to fit a trigonometric surface; and (3) a similar interpolation stability without increase of the condition number of the RBF interpolation matrix. Therefore, it has proven that the enriched RBF shape function will not only possess all advantages of the conventional RBF shape function, but also can accurately reflect the properties of stresses for the crack tip fields. The system of equations for the crack analysis is then derived based on the enriched RBF meshfree shape function and the meshfree weak-form. Several problems of linear fracture mechanics are simulated using this newly developed e-RPIM method. It has demonstrated that the present e-RPIM is very accurate and stable, and it has a good potential to develop a practical simulation tool for fracture mechanics problems.     

An efficient point estimate method for probabilistic analysis

A new and efficient point estimate method is developed to calculate the statistical moments of a random quantity, Z, that is a function of n random variables, X. The method is an extension of Rosenblueth's two-point concentration method. The method uses m × n concentrations matching up to the first m × n non-crossed moments of each random variable and crossed second order moments of the random variables. The kth moment of Z is calculated by weighting the value of Z to the power of k evaluated at n × m locations. Simple to use formulas are provided for two special cases of the method, i.e. 2n-concentration scheme and 2n + 1-concentration scheme. This 2n-concentration scheme considers the skewness of probability density function. The 2n + 1-concentration scheme considers the skewness and kurtosis of probabilility density function. The correlations between the random variables are considered by using a rotational transformation based on the eigenvector of covariance matrix. Illustrative examples are presented.     

An automatic adaptive refinement finite element procedure for 2D elastostatic analysis

An automatic adaptive refinement procedure for finite element analysis is presented. The procedure is applied to two-dimensional elastostatic problems to obtain solutions of prescribed accuracy. Through the combined use of new mesh generator using contour developed by Lo¹ and the concept of strain energy concentration, high-quality graded finite element meshes are generated. The whole process is fully automatic and no user intervention is required during the successive cycles of the mesh refinements. The Zienkiewicz and Zhu² error estimator is found to be effective and has been adopted for the present implementation. In the numerical examples tested, the error estimator gives an accurate error norm estimation and the effectivity index of the estimator converges to a value close to unity.     

An edge‐based smoothed finite element method for 3D analysis of solid mechanics problems

The edge‐based smoothed finite element method (ES‐FEM) was proposed recently in Liu, Nguyen‐Thoi, and Lam to improve the accuracy of the FEM for 2D problems. This method belongs to the wider family of the smoothed FEM for which smoothing cells are defined to perform the numerical integration over the domain. Later, the face‐based smoothed FEM (FS‐FEM) was proposed to generalize the ES‐FEM to 3D problems. According to this method, the smoothing cells are centered along the faces of the tetrahedrons of the mesh. In the present paper, an alternative method for the extension of the ES‐FEM to 3D is investigated. This method is based on an underlying mesh composed of tetrahedrons, and the approximation of the field variables is associated with the tetrahedral elements; however, in contrast to the FS‐FEM, the smoothing cells of the proposed ES‐FEM are centered along the edges of the tetrahedrons of the mesh. From selected numerical benchmark problems, it is observed that the ES‐FEM is characterized by a higher accuracy and improved computational efficiency as compared with linear tetrahedral elements and to the FS‐FEM for a given number of degrees of freedom. Copyright © 2013 John Wiley & Sons, Ltd.     

A singular edge-based smoothed finite element method (ES-FEM) for crack analyses in anisotropic media

The recently developed edge-based smoothed finite element method (ES-FEM) is extended to fracture problems in anisotropic media using a specially designed five-node singular crack-tip (T5) element. In the formulation of singular ES-FEM, only the assumed displacement values (not the derivatives) on the boundaries of the smoothing domains are needed. Thus, a layer of T5 crack-tip element is devised to construct “singular” shape functions via a simple point interpolation with a fractional order basis, without mapping procedure. The effectiveness of the present singular ES-FEM is demonstrated by intensive examples for a wide range of degrees of anisotropy.     

Radial point interpolation collocation method (RPICM) for the solution of nonlinear poisson problems

This paper applies radial point interpolation collocation method (RPICM) for solving nonlinear Poisson equations arising in computational chemistry and physics. Thin plate spline (TPS) Radial basis functions are used in the work. A series of test examples are numerically analysed using the present method, including 2D Liouville equation, Bratu problem and Poisson-Boltzmann equation, in order to test the accuracy and efficiency of the proposed schemes. Several aspects have been numerically investigated, namely the enforcement of additional polynomial terms; and the application of the Hermite-type interpolation which makes use of the normal gradient on Neumann boundary for the solution of PDEs with Neumann boundary conditions. Particular emphasis was on an efficient scheme, namely Hermite-type interpolation for dealing with Neumann boundary conditions. The numerical results demonstrate that a good accuracy can be obtained. The h-convergence rates are also studied for RPICM with coarse and fine discretization models.     

A 3D upper bound limit analysis using radial point interpolation meshless method and second‐order cone programming

This paper presents a 3D formulation for quasi‐kinematic limit analysis, which is based on a radial point interpolation meshless method and numerical optimization. The velocity field is interpolated using radial point interpolation shape functions, and the resulting optimization problem is cast as a standard second‐order cone programming problem. Because the essential boundary conditions can be only guaranteed at the position of the nodes when using radial point interpolation, the results obtained with the proposed approach are not rigorous upper bound solutions. This paper aims to improve the computing efficiency of 3D upper bound limit analysis and large problems, with tens of thousands of nodes, can be solved efficiently. Five numerical examples are given to confirm the effectiveness of the proposed approach with the von Mises yield criterion: an internally pressurized cylinder; a cantilever beam; a double‐notched tensile specimen; and strip, square and rectangular footings. Copyright © 2016 John Wiley & Sons, Ltd.     

An efficient memetic algorithm for 3D shape matching problems

Shape representation plays a vital role in any shape optimization exercise. The ability to identify a shape with good functional properties is dependent on the underlying shape representation scheme, the morphing mechanism and the efficiency of the optimization algorithm. This article presents a novel and efficient methodology for morphing 3D shapes via smart repair of control points. The repaired sequence of control points are subsequently used to define the 3D object using a B-spline surface representation. The control points are evolved within the framework of a memetic algorithm for greater efficiency. While the authors have already proposed an approach for 2D shape matching, this article extends it further to deal with 3D shape matching problems. Three 3D examples and a real customized 3D earplug design have been used as examples to illustrate the performance of the proposed approach and the effectiveness of the repair scheme. Complete details of the problems are presented for future work in this direction.     

A regularized least-squares radial point collocation method (RLS-RPCM) for adaptive analysis

This paper presents a stabilized meshfree method formulated based on the strong formulation and local approximation using radial basis functions (RBFs). The purpose of this paper is two folds. First, a regularization procedure is developed for stabilizing the solution of the radial point collocation method (RPCM). Second, an adaptive scheme using the stabilized RPCM and residual based error indicator is established. It has been shown in this paper that the features of the meshfree strong-form method can facilitated an easier implementation of adaptive analysis. A new error indicator based on the residual is devised and used in this work. As shown in the numerical examples, the new error indicator can reflect the quality of the local approximation and the global accuracy of the solution. A number of examples have been presented to demonstrate the effectiveness of the present method for adaptive analysis.     

Isotropic-BEM coupled with a local point interpolation method for the solution of 3D-anisotropic elasticity problems

This paper presents a solution procedure for the three-dimensional linear elastic problem with anisotropic properties. The approach uses the partition of the displacement field into complementary and particular parts. The former is the solution of a differential equation similar to that of an isotropic elastostatic and is obtained by the isotropic boundary element method. The particular integral is obtained by solving the corresponding strong form differential equations, using the local radial point interpolation method. This promising approach is simple to implement and leads to highly accurate solutions in some simple tested situations.