Showing papers on "Smoothed finite element method published in 2013"

On Smoothed Finite Element Methods

Abstract: The paper presents an overview of the smoothed finite element methods (S-FEM) which are formulated by combining the existing standard FEM with the strain smoothing techniques used in the meshfree methods. The S-FEM family includes five models: CS-FEM, NS-FEM, ES-FEM, FS-FEM and α-FEM (a combination of NS-FEM and FEM). It was originally formulated for problems of linear elastic solid mechanics and found to have five major properties: (1) S-FEM models are always “softer” than the standard FEM, offering possibilities to overcome the so-called overly-stiff phenomenon encountered in the standard the FEM models; (2) S-FEM models give more freedom and convenience in constructing shape functions for special purposes or enrichments (e.g, various degree of singular field near the crack-tip, highly oscillating fields, etc.); (3) S-FEM models allow the use of distorted elements and general n-sided polygonal elements; (4) NS-FEM offers a simpler tool to estimate the bounds of solutions for many types of problems; (5) the αFEM can offer solutions of very high accuracy. With these properties, the S-FEM has rapidly attracted interests of many. Studies have been published on theoretical aspects of S-FEMs or modified S-FEMs or the related numerical methods. In addition, the applications of the S-FEM have been also extended to many different areas such as analyses of plate and shell structures, analyses of structures using new materials (piezo, composite, FGM), limit and shakedown analyses, geometrical nonlinear and material nonlinear analyses, acoustic analyses, analyses of singular problems (crack, fracture), and analyses of fluid-structure interaction problems.Copyright © 2013 by ASME

The Finite Element

Abstract: In this chapter we study the concept of a finite element in some more detail. We begin with the classical definition of a finite element as the triplet of a polygon, a polynomial space, and a set of functionals. We then show how to derive shape functions for the most common Lagrange elements. The isoparametric mapping is introduced as a tool to allow for elements with curved boundaries, and to simplify the computation of the element stiffness matrix and load vector. We finish by presenting some more exotic elements.

A Phantom-Node Method with Edge-Based Strain Smoothing for Linear Elastic Fracture Mechanics

Abstract: This paper presents a novel numerical procedure based on the combination of an edge-based smoothed finite element (ES-FEM) with a phantom-node method for 2D linear elastic fracture mechanics. In the standard phantom-node method, the cracks are formulated by adding phantom nodes, and the cracked element is replaced by two new superimposed elements. This approach is quite simple to implement into existing explicit finite element programs. The shape functions associated with discontinuous elements are similar to those of the standard finite elements, which leads to certain simplification with implementing in the existing codes. The phantom-node method allows modeling discontinuities at an arbitrary location in the mesh. The ES-FEM model owns a close-to-exact stiffness that is much softer than lower-order finite element methods (FEM). Taking advantage of both the ES-FEM and the phantom-node method, we introduce an edge-based strain smoothing technique for the phantom-node method. Numerical results show that the proposed method achieves high accuracy compared with the extended finite element method (XFEM) and other reference solutions.

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

Abstract: 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 ABAQUS toolbox for multiscale finite element computation

Abstract: In this paper, we propose to implement, in the framework of a commercial finite element software, a computational multilevel finite element method for the modeling of composite materials and structures. In the present approach, the unknown constitutive relationship at the macroscale is obtained by solving a local finite element problem at the microscale. The main advantages of the proposed computational approach are that it can greatly save computer memory and CPU time, and it has good accuracy at the same time while it allows to easily building nonlinear behavior for high order mechanical theories to deal with problems which cannot be handled by classical multiscale or homogenization theories. The linear and the non-linear cases are introduced and implemented in ABAQUS. A Python script and user-defined FORTRAN subroutines have been developed for this purpose. Finally numerical results show that the method presented in this paper is effective and reliable.

A three dimensional immersed smoothed finite element method (3D IS-FEM) for fluid---structure interaction problems

Abstract: A three-dimensional immersed smoothed finite element method (3D IS-FEM) using four-node tetrahedral element is proposed to solve 3D fluid---structure interaction (FSI) problems. The 3D IS-FEM is able to determine accurately the physical deformation of the nonlinear solids placed within the incompressible viscous fluid governed by Navier-Stokes equations. The method employs the semi-implicit characteristic-based split scheme to solve the fluid flows and smoothed finite element methods to calculate the transient dynamics responses of the nonlinear solids based on explicit time integration. To impose the FSI conditions, a novel, effective and sufficiently general technique via simple linear interpolation is presented based on Lagrangian fictitious fluid meshes coinciding with the moving and deforming solid meshes. In the comparisons to the referenced works including experiments, it is clear that the proposed 3D IS-FEM ensures stability of the scheme with the second order spatial convergence property; and the IS-FEM is fairly independent of a wide range of mesh size ratio.

Discrete Empirical Interpolation Method for Finite Element Structural Dynamics

Abstract: Model Order Reduction (MOR) in nonlinear structural analysis problems in usually carried out by a Galerkin projection of the primary variables on a sensibly smaller space. However, the cost of computing the nonlinear terms is still of the order of the full system. The Discrete Empirical Interpolation Method is an effective algorithm to reduce the computational of the nonlinear term. However, its efficiency is diminished when applied to a Finite Element (FE) framework. We present here an alternative formulation of the DEIM that suits an FE formulation and preserves the efficiency of the method.

Adaptive boundary element methods with convergence rates

Abstract: This paper presents adaptive boundary element methods for positive, negative, as well as zero order operator equations, together with proofs that they converge at certain rates. The convergence rates are quasi-optimal in a certain sense under mild assumptions that are analogous to what is typically assumed in the theory of adaptive finite element methods. In particular, no saturation-type assumption is used. The main ingredients of the proof that constitute new findings are some results on a posteriori error estimates for boundary element methods, and an inverse-type inequality involving boundary integral operators on locally refined finite element spaces.

Finite element approximation of steady flows of incompressible fluids with implicit power-law-like rheology

Abstract: We develop the analysis of finite element approximations of implicit power-law-like models for viscous incompressible fluids. The Cauchy stress and the symmetric part of the velocity gradient in the class of models under consideration are related by a, possibly multivalued, maximal monotone $r$-graph with $1

An edge-based smoothed finite element method for analysis of laminated composite plates

Abstract: This paper promotes a novel numerical approach to static, free vibration and buckling analyses of laminated composite plates by an edge-based smoothed finite method (ES-FEM). In the present ES-FEM formulation, the system stiffness matrix is established by using the strain smoothing technique over the smoothing domains associated with the edges of the triangular elements. A discrete shear gap (DSG3) technique without shear locking is combined into the ES-FEM to give a so-called edge-based smoothed discrete shear gap method (ES-DSG3) for analysis of laminated composite plates. The present method uses only linear interpolations and its implementation into finite element programs is quite simple. Numerical results for analysis of laminated composite plates show that the ES-DSG3 performs quite well compared to several other published approaches in the literature.

Numerical Solution of the Fokker–Planck Equation by Finite Difference and Finite Element Methods—A Comparative Study

Abstract: Finite element and finite difference methods have been widely used, among other methods, to numerically solve the Fokker–Planck equation for investigating the time history of the probability density function of linear and nonlinear 2d and 3d problems; also the application to 4d problems has been addressed. However, due to the enormous increase in computational costs, different strategies are required for efficient application to problems of dimension ≥3. Recently, a stabilized multi-scale finite element method has been effectively applied to the Fokker–Planck equation. Also, the alternating directions implicit method shows good performance in terms of efficiency and accuracy. In this paper various finite difference and finite element methods are discussed, and the results are compared using various numerical examples.

Free and forced vibration analysis using the n-sided polygonal cell-based smoothed finite element method (nCS-FEM)

Abstract: A n-sided polygonal cell-based smoothed finite element method (nCS-FEM) was recently proposed to analyze the elastic solid mechanics problems, in which the problem domain can be discretized by a set of polygons with an arbitrary number of sides. In this paper, the nCS-FEM is further extended to the free and forced vibration analyses of two-dimensional (2D) dynamic problems. A simple lump mass matrix is proposed and hence the complicated integrations related to computing the consistent mass matrix can be avoided in the nCS-FEM. Several numerical examples are investigated and the results found of the nCS-FEM agree well with exact solutions and with those of others FEM.

Mixed finite element methods: implementation with one unknown per element, local flux expressions, positivity, polygonal meshes, and relations to other methods

Abstract: In this paper, we study the mixed finite element method for linear diffusion problems. We focus on the lowest-order Raviart–Thomas case. For simplicial meshes, we propose several new approaches to reduce the original indefinite saddle point systems for the flux and potential unknowns to (positive definite) systems for one potential unknown per element. Our construction principle is closely related to that of the so-called multi-point flux-approximation method and leads to local flux expressions. We present a set of numerical examples illustrating the influence of the elimination process on the structure and on the condition number of the reduced matrix. We also discuss different versions of the discrete maximum principle in the lowest-order Raviart–Thomas method. Finally, we recall mixed finite element methods on general polygonal meshes and show that they are a special type of the mimetic finite difference, mixed finite volume, and hybrid finite volume family.

Energy estimates and numerical verification of the stabilized Domain Decomposition Finite Element/Finite Difference approach for time-dependent Maxwell’s system

Abstract: We rigorously derive energy estimates for the second order vector wave equation with gauge condition for the electric field with non-constant electric permittivity function. This equation is used in the stabilized Domain Decomposition Finite Element/Finite Difference approach for time-dependent Maxwell’s system. Our numerical experiments illustrate efficiency of the modified hybrid scheme in two and three space dimensions when the method is applied for generation of backscattering data in the reconstruction of the electric permittivity function.

Introduction to Finite Element Analysis

Abstract: In the previous two chapters we have formulated and analyzed the primal and dual variational formulations of the elastoplasticity problem. Later on, we will study various numerical methods to solve the variational problems. In all the numerical methods to be considered, we will use finite differences to approximate the time derivative and use the finite element method to discretize the spatial variables. The finite elemfent method is widely used for solving boundary value problems of partial differential equations arising in physics and engineering, especially solid mechanics. The method is derived from discretizing the weak formulation of a boundary value problem. The analysis of the finite element method is closely related to that of the weak formulation of the boundary value problem.

An application of the ES-FEM in solid domain for dynamic analysis of 2D fluid-solid interaction problems

Abstract: An edge-based smoothed finite element method (ES-FEM-T3) using triangular elements was recently proposed to improve the accuracy and convergence rate of the existing standard finite element method (FEM) for the solid mechanics analyses. In this paper, the ES-FEM-T3 is further extended to the dynamic analysis of 2D fluid–solid interaction problems based on the pressure-displacement formulation. In the present coupled method, both solid and fluid domain is discretized by triangular elements. In the fluid domain, the standard FEM is used, while in the solid domain, we use the ES-FEM-T3 in which the gradient smoothing technique based on the smoothing domains associated with the edges of triangles is used to smooth the gradient of displacement. This gradient smoothing technique can provide proper softening effect, and thus improve significantly the solution of coupled system. Some numerical examples have been presented to illustrate the effectiveness of the proposed coupled method compared with some existing methods for 2D fluid–solid interaction problems.

A wave finite element analysis of the passive cochlea.

Abstract: Current models of the cochlea can be characterized as being either based on the assumed propagation of a single slow wave, which provides good insight, or involve the solution of a numerical model, such as in the finite element method, which allows the incorporation of more detailed anatomical features In this paper it is shown how the wave finite element method can be used to decompose the results of a finite element calculation in terms of wave components, which allows the insight of the wave approach to be brought to bear on more complicated numerical models In order to illustrate the method, a simple box model is considered, of a passive, locally reacting, basilar membrane interacting via three-dimensional fluid coupling An analytic formulation of the dispersion equation is used initially to illustrate the types of wave one would expect in such a model The wave finite element is then used to calculate the wavenumbers of all the waves in the finite element model It is shown that only a single wave type dominates the response until this peaks at the best place in the cochlea, where an evanescent, higher order fluid wave can make a significant contribution