An enriched radial point interpolation method based on weak-form and strong-form

In this article, an enriched radial point interpolation method (e-RPIM) is developed for computational mechanics. The conventional radial basis function (RBF) interpolation is novelly augmented by the suitable basis functions to reflect the natural properties of deformation. The performance of the enriched meshless RBF shape functions is first investigated using the surface fitting. The surface fitting results have proven that, compared with the conventional RBF, the enriched RBF interpolation has a much better accuracy to fit a complex surface than the conventional RBF interpolation. It has proven that the enriched RBF shape function will not only possess all advantages of the conventional RBF interpolation, but also can accurately reflect the deformation properties of problems. The system of equations for two-dimensional solids is then derived based on the enriched RBF shape function and both of the meshless strong-form and weak-form. A numerical example of a bar is presented to study the effectiveness and efficiency of e-RPIM. As an important application, the newly developed e-RPIM, which is augmented by selected trigonometric basis functions, is applied to crack problems. It has been demonstrated that the present e-RPIM is very accurate and stable for fracture mechanics problems.

A hybrid smoothed finite element method (H-SFEM) to solid mechanics problems

In this paper, a hybrid smoothed finite element method (H-SFEM) is developed for solid mechanics problems by combining techniques of finite element method (FEM) and Node-based smoothed finite element method (NS-FEM) using a triangular mesh. A parameter is equipped into H-SFEM, and the strain field is further assumed to be the weighted average between compatible stains from FEM and smoothed strains from NS-FEM. We prove theoretically that the strain energy obtained from the H-SFEM solution lies in between those from the compatible FEM solution and the NS-FEM solution, which guarantees the convergence of H-SFEM. Intensive numerical studies are conducted to verify these theoretical results and show that (1) the upper and lower bound solutions can always be obtained by adjusting ; (2) there exists a preferable at which the H-SFEM can produce the ultrasonic accurate solution.

A coupled SPH/FEM analysis of portable water filled barriers

Portable water filled barriers (PWFB) are semi-rigid roadside barriers which have the potential to display good crash attenuation characteristics at low and moderate impact speeds. The traditional mesh based numerical methods alone fail to simulate this type of impact with precision, stability and efficiency. This paper proposes to develop an advanced simulation model based on the combination of Smoothed Particles Hydrodynamics (SPH), a meshless method, and finite element method (FEM) for fluid-structure analysis using the commercially available software package LS-Dyna. The interaction between SPH particles and FEA elements is studied in this paper. Two methods of element setup at the element boundary were investigated. The response of the impacted barrier and fluid inside were analysed and compared. The system response and lagging were observed and reported in this paper. It was demonstrated that coupled SPH/FEM can be used in full scale PWFB modelling application. This will aid the research in determining the best initial setup to couple FEA and SPH in road safety barrier for impact response and safety analysis in the future.

Fluid-structure interaction analysis by coupled FE-SPH model based on a novel searching algorithm

Fluid–Structure Interaction (FSI) problem is significant in science and engineering, which leads to challenges for computational mechanics. The coupled model of Finite Element and Smoothed Particle Hydrodynamics (FE-SPH) is a robust technique for simulation of FSI problems. However, two important steps of neighbor searching and contact searching in the coupled FE-SPH model are extremely time-consuming. Point-In-Box (PIB) searching algorithm has been developed by Swegle to improve the efficiency of searching. However, it has a shortcoming that efficiency of searching can be significantly affected by the distribution of points (nodes in FEM and particles in SPH). In this paper, in order to improve the efficiency of searching, a novel Striped-PIB (S-PIB) searching algorithm is proposed to overcome the shortcoming of PIB algorithm that caused by points distribution, and the two time-consuming steps of neighbor searching and contact searching are integrated into one searching step. The accuracy and efficiency of the newly developed searching algorithm is studied on by efficiency test and FSI problems. It has been found that the newly developed model can significantly improve the computational efficiency and it is believed to be a powerful tool for the FSI analysis.

A three-dimensional hybrid smoothed finite element method (H-SFEM) for nonlinear solid mechanics problems

This paper presents a novel three-dimensional hybrid smoothed finite element method (H-SFEM) for solid mechanics problems. In 3D H-SFEM, the strain field is assumed to be the weighted average between compatible strains from the finite element method (FEM) and smoothed strains from the node-based smoothed FEM with a parameter α equipped into H-SFEM. By adjusting α, the upper and lower bound solutions in the strain energy norm and eigenfrequencies can always be obtained. The optimized α value in 3D H-SFEM using a tetrahedron mesh possesses a close-to-exact stiffness of the continuous system, and produces ultra-accurate solutions in terms of displacement, strain energy and eigenfrequencies in the linear and nonlinear problems. The novel domain-based selective scheme is proposed leading to a combined selective H-SFEM model that is immune from volumetric locking and hence works well for nearly incompressible materials. The proposed 3D H-SFEM is an innovative and unique numerical method with its distinct features, which has great potential in the successful application for solid mechanics problems.

CFD tools in engineering design studies and medical sciences

In the recent time CFD tools have become increasingly useful in the engineering design studies especially in the area of aerospace vehicles. This is largely due to the advent of high speed computing platforms in addition to the development of new efficient algorithms. The algorithms based on kinetic schemes have been shown to be very robust and further meshless methods offer certain advantages over the other methods. Preliminary investigations of blood flow visualization through artery using CFD tool have shown encouraging results which further needs to be verified and validated.

On the performance of strain smoothing for quadratic and enriched finite element approximations (XFEM/GFEM/PUFEM)

By using the strain smoothing technique proposed by Chen et al. (Comput. Mech. 2000; 25: 137-156) for meshless methods in the context of the finite element method (FEM), Liu et al. (Comput. Mech. 2007; 39(6): 859-877) developed the Smoothed FEM (SFEM). Although the SFEM is not yet well understood mathematically, numerical experiments point to potentially useful features of this particularly simple modification of the FEM. To date, the SFEM has only been investigated for bilinear and Wachspress approximations and is limited to linear reproducing conditions. The goal of this paper is to extend the strain smoothing to higher order elements and to investigate numerically in which condition strain smoothing is beneficial to accuracy and convergence of enriched finite element approximations. We focus on three widely used enrichment schemes, namely: (a) weak discontinuities; (b) strong discontinuities; (c) near-tip linear elastic fracture mechanics functions. The main conclusion is that strain smoothing in enriched approximation is only beneficial when the enrichment functions are polynomial (cases (a) and (b)), but that non-polynomial enrichment of type (c) lead to inferior methods compared to the standard enriched FEM (e.g. XFEM). Copyright (C) 2011 John Wiley & Sons, Ltd.

Boundary knot method for 2D and 3D Helmholtz and convection-diffusion problems under complicated geometry

The boundary knot method (BKM) of very recent origin is an inherently meshless, integration-free, boundary-type, radial basis function collocation technique for the numerical discretization of general partial differential equation systems. Unlike the method of fundamental solutions, the use of non-singular general solution in the BKM avoids the unnecessary requirement of constructing a controversial artificial boundary outside the physical domain. The purpose of this paper is to extend the BKM to solve 2D Helmholtz and convection-diffusion problems under rather complicated irregular geometry. The method is also first applied to 3D problems. Numerical experiments validate that the BKM can produce highly accurate solutions using a relatively small number of knots. For inhomogeneous cases, some inner knots are found necessary to guarantee accuracy and stability. The stability and convergence of the BKM are numerically illustrated and the completeness issue is also discussed.

无网格法及其最新进展

无网格法具有许多独特的优点,因此有人认为无网格法将成为继有限元法之后新一代的数值方法.到目前为止,已提出了不下十几种无网格法,这些无网格法各有不同的优缺点.本文评述几种主要无网格法(着重应用于固体力学中的基于弱式的无网格法),将它们适当地分类,论述典型的无网格形状函数的构造方法,介绍无网格法的发展现状,评价已提出无网格法的优缺点,并比较典型无网格法的相同和不同点,以及讨论无网格法发展所面临的问题等.最后就无网格法的发展趋势进行了展望.