Application of NEM in seepage analysis with a free surface

As one kind of meshless methods, the natural element method (NEM) constructs shape functions based on the Voronoi diagrams, and it has advantages of both the conventional meshless method and the finite element method. Since the nodes are independent of the integral mesh, it is more suitable for the analysis of seepage with a free surface than the finite element method. In addition, its shape functions satisfy the Kronecker δ conditions, therefore, its boundary conditions can be dealt with much easily than those of such meshless method as element-free Galerkin method (EFGM). In this paper, the NEM was used in the seepage analysis of dams. The initial free surface was assumed first in the calculations, and the location of the free surface was adjusted according to the calculation results. The examples showed that the natural element method lead to satisfactory results.     

A truly meshless pre- and post-processor for meshless analysis methods

In recent years, meshless methods have been developed to eliminate the known drawbacks in finite element methods. Generating the input file for a meshless method and interpreting the output obtained can be difficult without graphical pre-processing and post-processing support. Unfortunately, most existing pre- and post-processing techniques are based on using an underlying finite element mesh or finite difference grid. Since meshless methods have neither, new approaches are required for providing this support for meshless methods. In this paper, a pre-processor and a post-processor are presented for the meshless method using node-based and pixel-based approaches as opposed to an element-based approach. Pre-processing supports for automated generation of nodes, support domains, and sub-domains along with local refining are also included. An extensive example is presented to demonstrate the effectiveness of the given pre-processor and post-processor.     

A Structural Nonlinear Analysis Workspace (SNAW) based on meshless methods

In this paper, a Structural Nonlinear Analysis Workspace (SNAW) based on meshless methods is presented. Using the meshless methods in SNAW, the modeling of structures for nonlinear analysis is accomplished by a set of particles, and the use of an explicit structured mesh as in the conventional finite element methods is unnecessary. Due to the use of smooth shape functions in the meshless methods, SNAW effectively deals with large structural deformations that usually lead to mesh distortion and solution divergence in finite elements. The meshless formulation in SNAW is applicable to hyperelastic and elasto-plastic materials, explicit and implicit time integrations, and rigid-to-flexible-body frictional contact conditions. This SNAW development interfaces meshless analysis code with MSC/PATRAN for pre- and post-processing using graphical menu-driven functionalities. A number of example problems, including engine mount, rubber seal, metal bar impact and metal-forming simulations, are presented to demonstrate the effectiveness of SNAW.     

Recent advances in the meshless simulation of aluminium extrusion and other related forming processes

Summary In this paper we review some recent results in the field of numerical simulation of extrusion and other forming processes obtained by the authors by using a meshless approach, together with a wide review of the existing bibliography on the topic. Three main alternatives exist in the literature, namely (updated) Lagrangian, Eulerian and arbitrary Lagrangian-Eulerian (ALE) methods. A review of the most important characteristics of each of these three approaches is here presented and their possible advantages are pointed out. Finally, an updated Lagrangian approach over a meshless approximation, based on a class of methods globally coined as natural element methods (also as natural neighbour Galerkin methods) is analysed and its relative advantages studied. Some numerical examples are included that clearly show the potential capabilities of the proposed method.     

A generalized finite-difference method based on minimizing global residual

An improved generalized finite-difference method is proposed in this paper, as an alternative meshless method to solve differential equations. The method establishes discrete equations by minimizing a global residual. A general frame for constructing difference schemes is first described. As one choice the moving least square method is used in this paper. Compared with other generalized finite-difference methods, the improved method yields a set of discrete equations having the favorable properties such as symmetric, positive definite and well conditioned. Compared with meshless methods based on a variational principle or a weak form, the method described in this paper does not need a numerical integration and thus provides an alternative way to avoid the difficulties in implementing a numerical integration. In the proposed method there is no such inconvenience in applying essential boundary conditions as commonly encountered in other meshless methods. Numerical examples show that the improved method has a high convergence rate and can produce accurate results even with a coarse mesh.     

移动最小二乘法研究进展与述评

为使移动最小二乘法能更好地应用到无网格方法中,详细阐述移动最小二乘逼近法、移动最小二乘插值法、MUKHERJEE改进的移动最小二乘法以及程玉民等提出的改进的移动最小二乘法和复变量移动最小二乘法等的研究进展,述评各种移动最小二乘法的优缺点,并概述各种移动最小二乘法形成的无网格方法的研究进展.     

Galerkin based smoothed particle hydrodynamics

In this paper, we propose a Galerkin based smoothed particle hydrodynamics (SPH) formulation with moving least-squares meshless approximation, applied to solid mechanics and large deformation. Our method is truly meshless and based on Lagrangian kernel formulation and stabilized nodal integration. The performance of the methodology proposed is tested through various simulations, demonstrating the attractive ability of particle methods to handle severe distortions and complex phenomena.     

Higher order meshless schemes applied to the finite element method in elliptic problems

This paper presents selected approximation techniques, typical for the meshless finite difference method (MFDM), although applied to the finite element method (FEM). Finite elements with standard or hierarchical shape functions are coupled with higher order meshless schemes, based upon the correction terms of a simple difference operator. Those terms consist of higher order derivatives, which are evaluated by means of the appropriate formulas composition as well as a numerical solution, which corresponds to the primary interpolation order, assigned to element shape functions. Correction terms modify the right-hand sides of algebraic FE equations only, yielding an iterative procedure. Therefore, neither re-generation of the stiffness matrix nor introduction of any additional nodes and/or degrees of freedom is required. Such improved FE-MFD solution approach allows for the optimal application of advantages of both methods, for instance, a high accuracy of the nodal FE solution and a derivatives’ super-convergence phenomenon at arbitrary domain points, typical for the meshless FDM. Existing and proposed higher order techniques, applied in the FEM, are compared with each other in terms of the solution accuracy, algorithm efficiency and computational complexity.In order to examine the considered algorithms, numerical results of several two-dimensional benchmark elliptic problems are presented. Both the accuracy of a solution and the solution’s derivatives as well as their convergence rates, evaluated on irregular and structured meshes as well as arbitrarily irregular adaptive clouds of nodes, are taken into account.     

Parallel EFG algorithm for heat transfer problems

At present, meshless element free Galerkin (EFG) method is being successfully applied in the areas such as solid mechanics, fracture mechanics and thermal. Being a meshless method, it has many advantages over finite element method. One big hurdle with the wide implementation of this method is its computational cost. Therefore, in this paper, a parallel algorithm is proposed for the EFG method. The parallel code has been written in FORTRAN language using MPI message passing library and executed on a four node (eight processors) MIMD type, distributed memory ‘PARAM 10000’ parallel computer. The total time, communication time, speedup and efficiency have been estimated for a three-dimensional heat transfer problem to validate the proposed algorithm. For eight processors, the speedup and efficiency are obtained to be 4.66 and 58.22%, respectively, for a data size of 1320 nodes.     

An improved pseudospectral meshless radial point interpolation (PSMRPI) method for 3D wave equation with variable coefficients

In this paper, a pseudospectral meshless radial point interpolation (PSMRPI) technique is applied to the three-dimensional wave equation with variable coefficients subject to given appropriate initial and Dirichlet boundary conditions. The present method is a kind of combination of meshless methods and spectral collocation techniques. The point interpolation method along with the radial basis functions is used to construct the shape functions as the basis functions in the frame of the spectral collocation methods. These basis functions will have Kronecker delta function property, as well as unitary possession. In the proposed method, operational matrices of higher order derivatives are constructed and then applied. The merit of this innovative method is that, it does not require any kind of integration locally or globally over sub-domains, as it is essential in meshless methods based on Galerkin weak forms, such as element-free Galerkin and meshless local Petrov–Galerkin methods. Therefore, computational cost of PSMRPI method is low. Further, it is proved that the procedure is stable with respect to the time variable over some conditions on the 3D wave model, and the convergence of the technique is revealed. These latest claims are also shown in the numerical examples, which demonstrate that PSMRPI provides excellent rate of convergence.

     

无网格方法中的结点分布算法

本文针对无网格法的特点,设计了一种适应于气固两相流直接模拟计算的结点分布算法．该算法在计算域内均匀分布结点,在颗粒周围以辐射状分布结点,并删除其中距离过近以及强烈影响当地结点均匀度的结点．计算表明,该算法在分布结点过程中不需要借助网格,可以直接得到比较理想的结点分布,从而为无网格法应用于气固两相流的直接模拟计算打下坚实的基础．     

Peridynamics‐Based Fracture Animation for Elastoplastic Solids

In this paper, we exploit the use of peridynamics theory for graphical animation of material deformation and fracture. We present a new meshless framework for elastoplastic constitutive modelling that contrasts with previous approaches in graphics. Our peridynamics‐based elastoplasticity model represents deformation behaviours of materials with high realism. We validate the model by varying the material properties and performing comparisons with finite element method (FEM) simulations. The integral‐based nature of peridynamics makes it trivial to model material discontinuities, which outweighs differential‐based methods in both accuracy and ease of implementation. We propose a simple strategy to model fracture in the setting of peridynamics discretization. We demonstrate that the fracture criterion combined with our elastoplasticity model could realistically produce ductile fracture as well as brittle fracture. Our work is the first application of peridynamics in graphics that could create a wide range of material phenomena including elasticity, plasticity, and fracture. The complete framework provides an attractive alternative to existing methods for producing modern visual effects.     

Tight and efficient surface bounds in meshless animation

This paper presents a fast approach for computing tight surface bounds in meshless animation, and its application to collision detection. Given a high-resolution surface animated by a comparatively small number of simulation nodes, we are able to compute tight bounding volumes with a cost linear in the number of simulation nodes. Our approach extends concepts about bounds of convex sets to the meshless deformation setting, and we introduce an efficient algorithm for finding extrema of these convex sets. The extrema can be used for efficiently updating bounding volumes such as AABBs or k-DOPs, as we show in our results. The choice of particular bounding volume may depend on the complexity of the contact configurations, but in all cases we can compute surface bound orders of magnitude faster and/or tighter than with previous methods.     

扩散过程数值仿真的径向基函数方法

扩散问题仿真的传统网格方法在求解中常面临网格剖分和近似精度较低的问题,因而将径向基函数方法引入进来,作为一类配点型无网格方法,它不再需要网格剖分,而且基函数光滑性好,近似精度高.详细阐述了径向基函数方法和差分法结合求解扩散方程的原理,给出方法的具体实施方案和离散求解模型,并以热传导和涡流模型为例进行仿真和分析,结果表明该方法在仿真扩散过程中不仅实施简单,而且计算精度较高,即使在较大的时间步距下也能得到较高的精度.     

Turing models in the biological pattern formation through spectral meshless radial point interpolation approach

In the present paper, the spectral meshless radial point interpolation (SMRPI) technique is applied to the solution of pattern formation in nonlinear reaction diffusion systems. Firstly, we obtain a time discrete scheme by approximating the time derivative via a finite difference formula, then we use the SMRPI approach to approximate the spatial derivatives. This method is based on a combination of meshless methods and spectral collocation techniques. The point interpolation method with the help of radial basis functions is used to construct shape functions which act as basis functions in the frame of SMRPI. In the current work, to eliminate the nonlinearity, a simple predictor–corrector (P–C) scheme is performed. The effect of parameters and conditions are studied by considering the well-known Schnakenberg model.

     

A remedy to gradient type constraint dilemma encountered in RKPM

A major disadvantage of conventional meshless methods as compared to finite element method (FEM) is their weak performance in dealing with constraints. To overcome this difficulty, the penalty and Lagrange multiplier methods have been proposed in the literature. In the penalty method, constraints cannot be enforced exactly. On the other hand, the method of Lagrange multiplier leads to an ill-conditioned matrix which is not positive definite. The aim of this paper is to boost the effectiveness of the conventional reproducing kernel particle method (RKPM) in handling those types of constraints which specify the field variable and its gradient(s) conveniently. Insertion of the gradient term(s), along with generalization of the corrected collocation method, provides a breakthrough remedy in dealing with such controversial constraints. This methodology which is based on these concepts is referred to as gradient RKPM (GRKPM). Since one can easily relate to such types of constraints in the context of beam-columns and plates, some pertinent boundary value problems are analyzed. It is seen that GRKPM, not only enforces constraints and boundary conditions conveniently, but also leads to enhanced accuracy and substantial improvement of the convergence rate.     

A practical framework for generating volumetric meshes of subject-specific soft tissue

Studying human motion using musculoskeletal models is a common practice in the field of biomechanics. By using such models, recorded subject’s motions can be analyzed in successive steps from kinematics and dynamics to muscle control. However simulating muscle deformation and interaction is not possible, but other methods such as a finite element (FE) simulation are very well suited to simulate deformation and interaction of objects. In this paper we present a practical framework for the automatic generation of FE ready meshes based on subject-specific segmented MRI data. The proposed method resolves several types of data inconsistencies: noise, an incomplete data set and self-intersections. This paper shows the different steps of the method, such as solving overlaps in the segmented surfaces, generating the volume mesh and the connection to a musculoskeletal simulation.     

Using spectral method as an approximation for solving hyperbolic PDEs

We demonstrate an application of the spectral method as a numerical approximation for solving hyperbolic PDEs. In this method a finite basis is used for approximating the solutions. In particular, we demonstrate a set of such solutions for cases which would be otherwise almost impossible to solve by the more routine methods such as the Finite Difference Method. Eigenvalue problems are included in the class of PDEs that are solvable by this method. Although any complete orthonormal basis can be used, we discuss two particularly interesting bases: the Fourier basis and the quantum oscillator eigenfunction basis. We compare and discuss the relative advantages of each of these two bases.     

Boundary value identification of inverse Cauchy problems in arbitrary plane domain through meshless radial point Hermite interpolation

This article presents an efficient method to solve elliptic partial differential equations which are the nucleus of several physical problems, especially in the electromagnetic and mechanics, such as the Poisson and Laplace equations, while the subject is to recover a harmonic data from the knowledge of Cauchy data on some part of the boundary of the arbitrary plane domain. This method is a local nodal meshless Hermite-type collocation technique. In this method, we use the radial-based functions to call out the shape functions that form the local base in the vicinity of the nodal points. We also take into account the Hermit interpolation technique for imposing the derivative conditions directly. The proposed technique called pseudospectral meshless radial point Hermit interpolation is applied on some illustrative examples by adding random noises on source function and reliable results are observed.