Showing papers on "Smoothed finite element method published in 2011"

Mixed and Hybrid Finite Element Methods

Abstract: Variational Formulations and Finite Element Methods. Approximation of Saddle Point Problems. Function Spaces and Finite Element Approximations. Various Examples. Complements on Mixed Methods for Elliptic Problems. Incompressible Materials and Flow Problems. Other Applications.

Isogeometric finite element data structures based on Bézier extraction of T-splines

Abstract: We develop finite element data structures for T-splines based on Bezier extraction generalizing our previous work for NURBS. As in traditional finite element analysis, the extracted Bezier elements are defined in terms of a fixed set of polynomial basis functions, the so-called Bernstein basis. The Bezier elements may be processed in the same way as in a standard finite element computer program, utilizing exactly the same data processing arrays. In fact, only the shape function subroutine needs to be modified while all other aspects of a finite element program remain the same. A byproduct of the extraction process is the element extraction operator. This operator localizes the topological and global smoothness information to the element level, and represents a canonical treatment of T-junctions, referred to as ‘hanging nodes’ in finite element analysis and a fundamental feature of T-splines. A detailed example is presented to illustrate the ideas. Copyright © 2011 John Wiley & Sons, Ltd.

A Comparison of All Hexagonal and All Tetrahedral Finite Element Meshes for Elastic and Elasto-plastic Analysis

Abstract: This paper compares the accuracy of elastic and elastio-plastic solid continuum finite element analyses modeled with either all hexagonal or all tetrahedral meshes. Eigenvalues of element stiffness matrices, linear static displacements and stresses, dynamic modal frequencies, and plastic flow values in are computed and compared. Elements with both linear and quadratic displacement functions are evaluated. Linear incompressibility conditions are also investigated. A simple bar with a rectangular cross-section, fixed at one end, is modeled and results are compared to known analytical solutions wherever possible. The evaluation substantiates a strong preference for linear displacement hexagonal finite elements when compared solely to linear tetrahedral finite elements. The use of quadratic displacement formulated finite elements significantly improve the performance of the tetrahedral as well as the hexahedral elements. The nonlinear elastoplastic comparison indicates that linear hexagonal elements may be superior to even quadratic tetrahedrons when shear stress in dominant. Results of this work may serve as a guide in selecting appropriate finite element types to be used in three dimensional elastic and elastic plastic analysis. INTRODUCTION Consideration of the convergence characteristics of two dimensional solutions of elastic continuum problems, using both quadrilateral and triangular elements, has been covered in previous studies and some finite element textbooks[1,2]. Such studies conclude that the significant factors that effect convergence characteristics of finite element solutions include the element's basic shape, element distortion, polynomial order of the element, completeness of polynomial functions, integration techniques, and material incompressibility. It is generally accepted that simplex triangular elements are inferior when compared to bilinear quadrilaterals. For example, statements such as “... for reasons of better accuracy and efficiency, quadrilateral elements are preferred for two-dimensional meshes and hexahedral elements for three-dimensional meshes. This preference is clear in structural analysis and seems to also hold for other engineering disciplines.”[2] However, it is also generally accepted that triangular elements, with higher order displacement assumptions, provide acceptable accuracy and convergence characteristics. However, mesh locking due to material incompressibility as reported by Hughes[3], is a serious shortcoming of triangular elements. The current focus for developing rapidly converging finite element procedures is to incorporate h-p adaptive techniques.[4] Of particular note for this study is an article by Lo and Lee[5] which investigates the convergence of mixed element in h-p adaptive finite element analysis. A significant conclusion from this paper is, that by carefully controlling quality and grading, quadrilateral elements provide an increase in efficiency in h-p adaptivity over pure triangular elements. A few studies have been published comparing the convergence characteristics of hexahedral verse tetrahedral meshes. Cifuentes and Kalbag [6] conclude that the results obtained with quadratic tetrahedral elements, compared to bilinear hexahedral elements, were equivalent in terms of both accuracy and CPU time. Bussler and Ramesh [7] report more accuracy using the same order hexahedral elements over tetrahedrons. Weingarten [8] indicates that both quadratic tetrahedrons and hexahedrons were equivalent in accuracy and efficiency and recommends using p method tetrahedrons to achieve desired accuracy. No studies were found incorporating incompressibility or plasticity aspects relating to the convergence of hexagonal and tetrahedral elements. In this paper, stiffness matrix eigenvalues of a square geometrical volume, meshed with a single hexahedron is compared to the same geometrical volume meshed with five tetrahedrons. Next, results of a linear elastic, fixed end bar, meshed with either all hexahedrons or all tetrahedrons are compared. Both bending and torsional results are considered. The computed vibration modes of the fixed end bar problem are then evaluated. Finally, elasto-plastic calculations of the fixed end bar again meshed with both types of elements are evaluated. STIFFNESS MATRIX EIGENVALUES The evaluation of the eigenvalues and eigenvectors of a stiffness matrix is important when studying the convergence characteristics of any finite element.[9] Properly formulated elements have a zero valued eigenvalue associated with each rigid body motion. In addition, since the displacement based finite element technique overestimates the stiffness of a body, the smaller the eigenvalues for a particular deformation mode, the more effective is the element. Therefore, to provide an initial assessment of the effectiveness of simplex tetrahedrons compared with bilinear hexahedrons, the eigenvalues of equivalent models were computed. A regular unit cube volume, with a Young’s Modulus of 30,000,000 and Poisson’s Ratio of .3 was modeled with a single hexahedron and five tetrahedrons as shown in Figure 1. Note the configuration shown at the bottom of the figure 1 shows how the five tetrahedrons are positioned to fill the unit cube. The internal tetrahedron’s position results in some directional properties of the stiffness matrix. The eigenvalues of the hexahedron were computed from (1) the stiffness matrix generated

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

Abstract: 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.

Spectral Finite Element Method

Abstract: This chapter presents the procedures for the development of various types of spectral elements. The chapter begins with basic outline of spectral finite element formulation and illustrates its utility for wave propagation studies is complex structural components. Two variants of spectral formulations, namely the Fourier transform-based, and Wavelet transform-based spectral FEM are presented for both 1D and 2D waveguides. A number of examples are solved using the formulated elements to show the effectiveness of the spectral FEM approach to solve problems involving high frequency dynamic response.

Adaptive analysis using the node‐based smoothed finite element method (NS‐FEM)

Abstract: The paper presents an adaptive analysis within the framework of the node-based smoothed finite element method (NS-FEM) using triangular elements. An error indicator based on the recovery strain is used and shown to be asymptotically exact by an effectivity index and numerical results. A simple refinement strategy using the newest node bisection is briefly presented. The numerical results of some benchmark problems show that the present adaptive procedure can accurately catch the appearance of the steep gradient of stresses and the occurrence of refinement is concentrated properly. The energy error norms of adaptive models for both NS-FEM and FEM obtain higher convergence rate compared with the uniformly refined models, but the results of NS-FEM are better and achieve higher convergence rate than those of FEM. The effectivity index of NS-FEM is also closer and approaches to unity faster than that of FEM. The upper bound property in the strain energy of NS-FEM is always verified during the adaptive procedure. Copyright © 2009 John Wiley & Sons, Ltd.

A Node-Based Smoothed eXtended Finite Element Method (NS-XFEM) for Fracture Analysis

Abstract: This paper aims to incorporate the node-based smoothed finite element method (NS-FEM) into the extended finite element method (XFEM) to form a novel numerical method (NS-XFEM) for analyzing fracture problems of 2D elasticity. NS-FEM uses the strain smoothing technique over the smoothing domains associated with nodes to compute the system stiffness matrix, which leads to the line integrations using directly the shape function values along the boundaries of the smoothing domains. As a result, we avoid integration of the stress singularity at the crack tip. It is not necessary to divide elements cut by cracks when we replace interior integration by boundary integration, simplifying integration of the discontinuous approximation. The key advantage of the NS-XFEM is that it provides more accurate solutions compared to the XFEM-T3 element. We will show for two numerical examples that the NS-XFEM significantly improves the results in the energy norm and the stress intensity factors. For the examples studied, we obtain super-convergent results

Analysis of fully-mixed finite element methods for the Stokes-Darcy coupled problem

Abstract: In this paper we analyze fully-mixed finite element methods for the coupling of fluid flow with porous media flow. Flows are governed by the Stokes and Darcy equations, respectively, and the corresponding transmission conditions are given by mass conservation, balance of normal forces, and the Beavers-Joseph-Saffman law. The fully-mixed concept employed here refers to the fact that we consider dual-mixed formulations in both the Stokes domain and the Darcy region, which means that the main unknowns are given by the pseudostress and the velocity in the fluid, together with the velocity and the pressure in the porous medium. In addition, the transmission conditions become essential, which leads to the introduction of the traces of the porous media pressure and the fluid velocity as the associated Lagrange multipliers. We apply the Fredholm and Babuska-Brezzi theories to derive sufficient conditions for the unique solvability of the resulting continuous formulation. Since the equations and unknowns can be ordered in several different ways, we choose the one yielding a doubly mixed structure for which the inf-sup conditions of the off-diagonal bilinear forms follow straightforwardly. Next, adapting to the discrete case the arguments of the continuous analysis, we are able to establish suitable hypotheses on the finite element subspaces ensuring that the associated Galerkin scheme becomes well posed. In addition, we show that the existence of uniformly bounded discrete liftings of the normal traces simplifies the derivation of the corresponding stability estimates. A feasible choice of subspaces is given by Raviart-Thomas elements of lowest order and piecewise constants for the velocities and pressures, respectively, in both domains, together with continuous piecewise linear elements for the Lagrange multipliers. This example confirms that with the present approach the Stokes and Darcy flows can be approximated with the same family of finite element subspaces without adding any stabilization term. Finally, several numerical results illustrating the good performance of the method with these discrete spaces, and confirming the theoretical rate of convergence, are provided.

The Mechanics of Solids and Structures - Hierarchical Modeling and the Finite Element Solution

Abstract: Mathematical Models in the Finite Element Solution.- Fundamental Steps in Structural Mechanics.- The Linear 3-D Elasticity Mathematical Model.- Mathematical Models used in Engineering Structural Analysis.- The Principle of Virtual Work.- The Finite Element Process of Solution.- Hierarchical Modeling Examples.- Modeling for Nonlinear Analysis.

Dispersion error reduction for acoustic problems using the edge‐based smoothed finite element method (ES‐FEM)

Abstract: The paper reports a detailed analysis on the numerical dispersion error in solving 2D acoustic problems governed by the Helmholtz equation using the edge-based smoothed finite element method (ES-FEM), in comparison with the standard FEM. It is found that the dispersion error of the standard FEM for solving acoustic problems is essentially caused by the 'overly stiff' feature of the discrete model. In such an 'overly stiff' FEM model, the wave propagates with an artificially higher 'numerical' speed, and hence the numerical wave-number becomes significantly smaller than the actual exact one. Owing to the proper softening effects provided naturally by the edge-based gradient smoothing operations, the ES-FEM model, however, behaves much softer than the standard FEM model, leading to the so-called very 'close-to-exact' stiffness. Therefore the ES-FEM can naturally and effectively reduce the dispersion error in the numerical solution in solving acoustic problems. Results of both theoretical and numerical studies will support these important findings. It is shown clearly that the ES-FEM suits ideally well for solving acoustic problems governed by the Helmholtz equations, because of the crucial effectiveness in reducing the dispersion error in the discrete numerical model.

A coupled ES-FEM/BEM method for fluid–structure interaction problems

Abstract: The edge-based smoothed finite element method (ES-FEM) developed recently shows some excellent features in solving solid mechanics problems using triangular mesh. In this paper, a coupled ES-FEM/BEM method is proposed to analyze acoustic fluid–structure interaction problems, where the ES-FEM is used to model the structure, while the acoustic fluid is represented by boundary element method (BEM). Three-node triangular elements are used to discretize the structural and acoustic fluid domains for the important adaptability to complicated geometries. The smoothed Galerkin weak form is adopted to formulate the discretized equations for the structure, and the gradient smoothing operation is applied over the edge-based smoothing domains. The global equations of acoustic fluid–structure interaction problems are then established by coupling the ES-FEM for the structure and the BEM for the fluid. The gradient smoothing technique applied in the structural domain can provide the important and right amount of softening effects to the “overly-stiff” FEM model and thus improve the accuracy of the solutions of coupled system. Numerical examples of acoustic fluid–structure interaction problems have been used to assess the present formulation, and the results show that the accuracy of present method is very good and even higher than those obtained using the coupled FEM/BEM with quadrilateral mesh.

A contact algorithm for 3D discrete and finite element contact problems based on penalty function method

Abstract: A contact algorithm in the context of the combined discrete element (DE) and finite element (FE) method is proposed. The algorithm, which is based on the node-to-surface method used in finite element method, treats each spherical discrete element as a slave node and the surfaces of the finite element domain as the master surfaces. The contact force on the contact interface is processed by using a penalty function method. Afterward, a modification of the combined DE/FE method is proposed. Following that, the corresponding numerical code is implemented into the in-house developed code. To test the accuracy of the proposed algorithm, the impact between two identical bars and the vibration process of a laminated glass plate under impact of elastic sphere are simulated in elastic range. By comparing the results with the analytical solution and/or that calculated by using LS-DYNA, it is found that they agree with each other very well. The accuracy of the algorithm proposed in this paper is proved.

Numerical dynamic analysis of uncertain mechanical structures based on interval fields

Abstract: This paper introduces the concept of interval fields for the dynamic analysis of uncertain mechanical structures in the context of finite element analysis. The theoretic background of the concept is explained, and it is shown how it can be applied to represent dependent uncertainty in the model definition phase and in the post-processing phase. Further, the paper concentrates on the calculation of interval fields resulting from a dynamic analysis. A procedure is introduced that enables the calculation of a joint representation of multiple output quantities of a single interval finite element problem while preserving the mutual dependence between the components of the output vector. The application of the approach is illustrated using a vibro-acoustic finite element analysis.

An optimal adaptive mixed finite element method

Abstract: Various applications in fluid dynamics and computational continuum mechanics motivate the development of reliable and efficient adaptive algorithms for mixed finite element methods. In order to save degrees of freedom, not all but just some selected set of finite element domains are refined. Hence the fundamental question of convergence as well as the question of optimality require new mathematical arguments. The presented adaptive algorithm for Raviart-Thomas mixed finite element methods solves the Poisson model problem, with optimal convergence rate. Chen, Holst, and Xu presented "convergence and optimality of adaptive mixed finite element methods" (2008) following arguments of Rob Stevenson for the conforming finite element method. Their algorithm reduces oscillations separately, before approximating the solution by some adaptive algorithm in the spirit of W. Dorfler (1996). The algorithm proposed here appears more natural in switching to either reduction of the edge-error estimator or of the oscillations.

Conforming Rectangular Mixed Finite Elements for Elasticity

Abstract: We present a new family of rectangular mixed finite elements for the stress-displacement system of the plane elasticity problem. Based on the theory of mixed finite element methods, we prove that they are stable and obtain error estimates for both the stress field and the displacement field. Using the finite element spaces in this family, an exact sequence is established as a discrete version of the elasticity complex in two dimensions. And the relationship between this discrete version and the original one is shown in a commuting diagram.

A smoothed Hermite radial point interpolation method for thin plate analysis

Abstract: A smoothed Hermite radial point interpolation method using gradient smoothing operation is formulated for thin plate analysis. The radial basis functions augmented with polynomial basis are used to construct the shape functions that have the important Delta function property. The smoothed Galerkin weakform is adopted to discretize the governing partial differential equations, and a curvature smoothed operation is developed to relax the continuity requirement and achieve accurate bending solutions. The approximation based on both deflection and rotation variables make the proposed method very effective in enforcing the essential boundary conditions. The effects of different numbers of sub-smoothing-domains created based on the triangular background cell are investigated in detail. A number of numerical examples have been studied and the results show that the present method is very stable and accurate even for extremely irregular background cells.