Showing papers on "Smoothed finite element method published in 2010"

Finite element approximation of eigenvalue problems

Abstract: We discuss the finite element approximation of eigenvalue problems associated with compact operators. While the main emphasis is on symmetric problems, some comments are present for non-self-adjoint operators as well. The topics covered include standard Galerkin approximations, non-conforming approximations, and approximation of eigenvalue problems in mixed form. Some applications of the theory are presented and, in particular, the approximation of the Maxwell eigenvalue problem is discussed in detail. The final part tries to introduce the reader to the fascinating setting of differential forms and homological techniques with the description of the Hodge–Laplace eigenvalue problem and its mixed equivalent formulations. Several examples and numerical computations complete the paper, ranging from very basic exercises to more significant applications of the developed theory.

Computational Contact and Impact Mechanics: Fundamentals of Modeling Interfacial Phenomena in Nonlinear Finite Element Analysis

Abstract: 1 Introduction.- 2 Finite Element Formulations in Nonlinear Solid Mechanics.- 3 The Kinematically Linear Contact Problem.- 4 Continuum Mechanics of Large Deformation Contact.- 5 Finite Element Implementation of Contact Interaction.- 6 Tribological Complexity in Interface Constitutive Models.- 7 Energy-Momentum Approaches to Impact Mechanics.- 8 Emerging Paradigms for Contact Surface Discretization.- References.

Smoothed Finite Element Methods

Abstract: Introduction Physical Problems in Engineering Numerical Techniques: Practical Solution Tools Why S-FEM? The Idea of S-FEM Key Techniques Used in S-FEM S-FEM Models and Properties Some Historical Notes Outline of the Book Basic Equations for Solid Mechanics Equilibrium Equation: In Stresses Constitutive Equation Compatibility Equation Equilibrium Equation: In Displacements Equations in Matrix Form Boundary Conditions Some Standard Default Conventions and Notations The Finite Element Method General Procedure of FEM Proper Spaces Weak Formulation and Properties of the Solution Domain Discretization: Creation of Finite-Dimensional Space Creation of Shape Functions Displacement Function Creation Strain Evaluation Formulation of the Discretized System of Equations FEM Solution: Existence, Uniqueness, Error, and Convergence Some Other Properties of the FEM Solution Linear Triangular Element (T3) Four-Node Quadrilateral Element (Q4) Four-Node Tetrahedral Element (T4) Eight-Node Hexahedral Element (H8) Gauss Integration Fundamental Theories for S-FEM General Procedure for S-FEM Models Domain Discretization with Polygonal Elements Creating a Displacement Field: Shape Function Construction Evaluation of the Compatible Strain Field Modify/Construct the Strain Field Minimum Number of Smoothing Domains: Essential to Stability Smoothed Galerkin Weak Form Discretized Linear Algebraic System of Equations Solve the Algebraic System of Equations Error Assessment in S-FEM and FEM Models Implementation Procedure for S-FEM Models General Properties of S-FEM Models Cell-Based Smoothed FEM Cell-Based Smoothing Domain Discretized System of Equations Shape Function Evaluation Some Properties of CS-FEM Stability of CS-FEM and nCS-FEM Standard Patch Test: Accuracy Selective CS-FEM: Volumetric Locking Free Numerical Examples Node-Based Smoothed FEM Introduction Creation of Node-Based Smoothing Domains Formulation of NS-FEM Evaluation of Shape Function Values Properties of NS-FEM An Adaptive NS-FEM Using Triangular Elements Numerical Examples Edge-Based Smoothed FEM Introduction Creation of Edge-Based Smoothing Domains Formulation of the ES-FEM Evaluation of the Shape Function Values in the ES-FEM A Smoothing-Domain-Based Selective ES/NS-FEM Properties of the ES-FEM Numerical Examples Face-Based Smoothed FEM Introduction Face-Based Smoothing Domain Creation Formulation of FS-FEM-T4 A Smoothing-Domain-Based Selective FS/NS-FEM-T4 Model Stability, Accuracy, and Mesh Sensitivity Numerical Examples The alphaFEM Introduction Idea of alphaFEM-T3 and alphaFEM-T4 alphaFEM-T3 and alphaFEM-T4 for Nonlinear Problems Implementation and Patch Tests Numerical Examples S-FEM for Fracture Mechanics Introduction Singular Stress Field Creation at the Crack-Tip Possible sS-FEM Methods sNS-FEM Models sES-FEM Models Stiffness Matrix Evaluation J-Integral and SIF Evaluation Interaction Integral Method for Mixed Mode Numerical Examples Solved Using sES-FEM-T3 Numerical Examples Solved Using sNS-FEM-T3 S-FEM for Viscoelastoplasticity Introduction Strong Formulation for Viscoelastoplasticity FEM for Viscoelastoplasticity: A Dual Formulation S-FEM for Viscoelastoplasticity: A Dual Formulation A Posteriori Error Estimation Numerical Examples ES-FEM for Plates Introduction Weak Form for the Reissner-Mindlin Plate FEM Formulation for the Reissner-Mindlin Plate ES-FEM-DSG3 for the Reissner-Mindlin Plate Numerical Examples: Patch Test Numerical Examples: Static Analysis Numerical Examples: Free Vibration of Plates Numerical Examples: Buckling of Plates S-FEM for Piezoelectric Structures Introduction Galerkin Weak Form for Piezoelectrics Finite Element Formulation for the Piezoelectric Problem S-FEM for the Piezoelectric Problem Numerical Results S-FEM for Heat Transfer Problems Introduction Strong-Form Equations for Heat Transfer Problems Boundary Conditions Weak Forms for Heat Transfer Problems FEM Equations S-FEM Equations Evaluation of the Smoothed Gradient Matrix Numerical Example Bioheat Transfer Problems S-FEM for Acoustics Problems Introduction Mathematical Model of Acoustics Problems Weak Forms for Acoustics Problems FEM Equations S-FEM Equations Error in a Numerical Model Numerical Examples Index References appear at the end of each chapter.

A unified approach to mimetic finite difference, hybrid finite volume and mixed finite volume methods

Abstract: We investigate the connections between several recent methods for the discretization of anisotropic heterogeneous diffusion operators on general grids. We prove that the Mimetic Finite Difference scheme, the Hybrid Finite Volume scheme and the Mixed Finite Volume scheme are in fact identical up to some slight generalizations. As a consequence, some of the mathematical results obtained for each of the methods (such as convergence properties or error estimates) may be extended to the unified common framework. We then focus on the relationships between this unified method and nonconforming Finite Element schemes or Mixed Finite Element schemes. We also show that for isotropic operators, on particular meshes such as triangular meshes with acute angles, the unified method boils down to the well-known efficient two-point flux Finite Volume scheme.

The Finite Element Method

Abstract: This chapter contains sections titled: Introduction to the Finite Element Method Finite Element Analysis of Scalar Fields Finite Element Analysis of Vector Fields Finite Element Analysis in the Time Domain Absorbing Boundary Conditions Some Numerical Aspects Summary References Problems

A node-based smoothed finite element method with stabilized discrete shear gap technique for analysis of Reissner-Mindlin plates

Abstract: In this paper, a node-based smoothed finite element method (NS-FEM) using 3-node triangular elements is formulated for static, free vibration and buckling analyses of Reissner–Mindlin plates. The discrete weak form of the NS-FEM is obtained based on the strain smoothing technique over smoothing domains associated with the nodes of the elements. The discrete shear gap (DSG) method together with a stabilization technique is incorporated into the NS-FEM to eliminate transverse shear locking and to maintain stability of the present formulation. A so-called node-based smoothed stabilized discrete shear gap method (NS-DSG) is then proposed. Several numerical examples are used to illustrate the accuracy and effectiveness of the present method.

A theoretical study on the smoothed FEM (S‐FEM) models: Properties, accuracy and convergence rates

Abstract: Incorporating the strain smoothing technique of meshfree methods into the standard finite element method (FEM), Liu et al. have recently proposed a series of smoothed finite element methods (S-FEM) for solid mechanics problems. In these S-FEM models, the compatible strain fields are smoothed based on smoothing domains associated with entities of elements such as elements, nodes, edges or faces, and the smoothed Galerkin weak form based on these smoothing domains is then applied to compute the system stiffness matrix. We present in this paper a general and rigorous theoretical framework to show properties, accuracy and convergence rates of the S-FEM models. First, an assumed strain field derived from the Hellinger–Reissner variational principle is shown to be identical to the smoothed strain field used in the S-FEM models. We then define a smoothing projection operator to modify the compatible strain field and show a set of properties. We next establish a general error bound of the S-FEM models. Some numerical examples are given to verify the theoretical properties established. Copyright © 2010 John Wiley & Sons, Ltd.

An edge‐based smoothed finite element method for primal–dual shakedown analysis of structures

Abstract: An edge-based smoothed finite element method (ES-FEM) using three-node linear triangular elements was recently proposed to significantly improve the accuracy and convergence rate of the standard finite element formulation for static, free and forced vibration analyses of solids. In this paper, ES-FEM is further extended for limit and shakedown analyses of structures. A primal-dual algorithm based upon the von Mises yield criterion and a non-linear optimization procedure is used to compute both the upper and lower bounds of the plastic collapse limit and the shakedown limit. In the ES-FEM, compatible strains are smoothed over the smoothing domains associated with edges of elements. Using constant smoothing function, only one Gaussian point is required for each smoothing domain ensuring that the total number of variables in the resulting optimization problem is kept to a minimum compared with standard finite element formulation. Three benchmark problems are presented to show the stability and accuracy of solutions obtained by the present method.

An n-sided polygonal edge-based smoothed finite element method (nES-FEM) for solid mechanics

Abstract: SUMMARY 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 further extended to a more general case, n-sided polygonal edge-based smoothed finite element method (nES-FEM), in which the problem domain can be discretized by a set of polygons, each with an arbitrary number of sides. The simple averaging point interpolation method is suggested to construct nES-FEM shape functions. In addition, a novel domain-based selective scheme of a combined nES/NS-FEM model is also proposed to avoid volumetric locking. Several numerical examples are investigated and the results of the nES-FEM are found to agree well with exact solutions and are much better than those of others existing methods. Copyright q 2010 John Wiley & Sons, Ltd.

A cell-based smoothed finite element method for kinematic limit analysis

Abstract: This paper presents a new numerical procedure for kinematic limit analysis problems, which incorporates the cell-based smoothed finite element method with second-order cone programming. The application of a strain smoothing technique to the standard displacement finite element both rules out volumetric locking and also results in an efficient method that can provide accurate solutions with minimal computational effort. The non-smooth optimization problem is formulated as a problem of minimizing a sum of Euclidean norms, ensuring that the resulting optimization problem can be solved by an efficient second-order cone programming algorithm. Plane stress and plane strain problems governed by the von Mises criterion are considered, but extensions to problems with other yield criteria having a similar conic quadratic form or 3D problems can be envisaged.

Analysis of plates and shells using an edge-based smoothed finite element method

Abstract: In this paper, an approach to the analysis of arbitrary thin to moderately thick plates and shells by the edge-based smoothed finite element method (ES-FEM) is presented. The formulation is based on the first order shear deformation theory, and Discrete Shear Gap (DSG) method is employed to mitigate the shear locking. Triangular meshes are used as they can be generated automatically for complicated geometries. The discretized system equations are obtained using the smoothed Galerkin weak form, and the numerical integration is applied based on the edge-based smoothing domains. The smoothing operation can provide a much needed softening effect to the FEM model to reduce the well-known “overly stiff” behavior caused by the fully compatible implementation of the displacement approach based on the Galerkin weakform, and hence improve significantly the solution accuracy. A number of benchmark problems have been studied and the results confirm that the present method can provide accurate results for both plate and shell using triangular mesh.

A novel singular node‐based smoothed finite element method (NS‐FEM) for upper bound solutions of fracture problems

Abstract: It is well known that the lower bound to exact solutions in linear fracture problems can be easily obtained by the displacement compatible finite element method (FEM) together with the singular crack tip elements. It is, however, much more difficult to obtain the upper bound solutions for these problems. This paper aims to formulate a novel singular node-based smoothed finite element method (NS-FEM) to obtain the upper bound solutions for fracture problems. In the present singular NS-FEM, the calculation of the system stiffness matrix is performed using the strain smoothing technique over the smoothing domains (SDs) associated with nodes, which leads to the line integrations using only the shape function values along the boundaries of the SDs. A five-node singular crack tip element is used within the framework of NS-FEM to construct singular shape functions via direct point interpolation with proper order of fractional basis. The mix-mode stress intensity factors are evaluated using the domain forms of the interaction integrals. The upper bound solutions of the present singular NS-FEM are demonstrated via benchmark examples for a wide range of material combinations and boundary conditions. Copyright © 2010 John Wiley & Sons, Ltd.

A New Class of High Order Finite Volume Methods for Second Order Elliptic Equations

Abstract: In the numerical simulation of many practical problems in physics and engineering, finite volume methods are an important and popular class of discretization methods due to the local conservation and the capability of discretizing domains with complex geometry. However, they are limited by low order approximation since most existing finite volume methods use piecewise constant or linear function space to approximate the solution. In this paper, a new class of high order finite volume methods for second order elliptic equations is developed by combining high order finite element methods and linear finite volume methods. Optimal convergence rate in $H^1$-norm for our new quadratic finite volume methods over two-dimensional triangular or rectangular grids is obtained.

Temporal stabilization of the node-based smoothed finite element method and solution bound of linear elastostatics and vibration problems

Abstract: A stabilization procedure for curing temporal instability of node-based smoothed finite element method (NS-FEM) is proposed for dynamic problems using linear triangular element. A stabilization term is added into the smoothed potential energy functional of the original NS-FEM, consisting of squared-residual of equilibrium equation. A gradient smoothing operation on second order derivatives is applied to relax the requirement of shape function, so that the squared-residual can be evaluated using linear elements. Numerical examples demonstrate that stabilization parameter can “tune” NS-FEM from being “overly soft” to “overly stiff”, so that eigenvalue solutions can be stabilized. Numerical tests provide an empirical value range of stabilization parameter, within which the stabilized NS-FEM can still produce upper bound solutions in strain energy to the exact solution of force-driven elastostatics problems, as well as lower bound natural frequencies for free vibration problems.

An Analysis of a Broken $P_1$-Nonconforming Finite Element Method for Interface Problems

Abstract: We study some numerical methods for solving a second order elliptic problem with interface. We introduce an immersed finite element method based on the “broken” $P_1$-nonconforming piecewise linear polynomials on interface triangular elements having edge averages as degrees of freedom. These linear polynomials are broken to match the homogeneous jump condition along the interface which is allowed to cut through the element. We prove optimal orders of convergence in the $H^1$- and $L^2$-norm. Next we propose a mixed finite volume method in the context introduced in [S. H. Chou, D. Y. Kwak, and K. Y. Kim, Math. Comp., 72 (2003), pp. 525-539] using the Raviart-Thomas mixed finite element and this “broken” $P_1$-nonconforming element. The advantage of this mixed finite volume method is that once we solve the symmetric positive definite pressure equation (without Lagrangian multiplier), the velocity can be computed locally by a simple formula. This procedure avoids solving the saddle point problem. Furthermore, we show optimal error estimates of velocity and pressure in our mixed finite volume method. Numerical results show optimal orders of error in the $L^2$-norm and broken $H^1$-norm for the pressure and in the $H(\mathrm{div})$-norm for the velocity.

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

Abstract: 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.

Interpolation functions in the immersed boundary and finite element methods

Abstract: In this paper, we review the existing interpolation functions and introduce a finite element interpolation function to be used in the immersed boundary and finite element methods. This straightforward finite element interpolation function for unstructured grids enables us to obtain a sharper interface that yields more accurate interfacial solutions. The solution accuracy is compared with the existing interpolation functions such as the discretized Dirac delta function and the reproducing kernel interpolation function. The finite element shape function is easy to implement and it naturally satisfies the reproducing condition. They are interpolated through only one element layer instead of smearing to several elements. A pressure jump is clearly captured at the fluid–solid interface. Two example problems are studied and results are compared with other numerical methods. A convergence test is thoroughly conducted for the independent fluid and solid meshes in a fluid–structure interaction system. The required mesh size ratio between the fluid and solid domains is obtained.

An approach to combining 3D discrete and finite element methods based on penalty function method

Abstract: An algorithm combining three-dimensional (3D) discrete and finite element methods is proposed. This new approach is conducted by decomposing the calculation domain into a finite element (FE) calculation domain and a discrete element (DE) calculation domain; the interaction between the two sub-domains is processed by using a penalty function method. Following the established model that combines spherical DEs and FEs, the corresponding numerical code is developed. The vibration process of two cantilever beams under dynamic force is simulated. By comparing the results calculated with different penalty factors set and also with that calculated by the finite element code LS-DYNA, it is found that the calculated results are unanimous and the precision is almost the same as LS-DYNA, as long as the penalty factor is large enough. Moreover, the vibration processes of two plates under impact of rigid spheres are simulated and the accuracy of the model proposed in this paper is further proved in the field of contact mechanics by comparing the simulating results with that calculated by using LS-DYNA. Finally, the impact fracture behavior of a laminated glass plate is simulated, with the influence of model parameters taken into consideration. And the numerical experiments show that the combined model can be used to predict some macroscopical physical quantities, such as the impact force of impactor.

Analysis of an efficient finite element method for embedded interface problems

Abstract: A stabilized finite element method based on the Nitsche technique for enforcing constraints leads to an efficient computational procedure for embedded interface problems, in which the finite element mesh need not be aligned with the interface geometry. We consider cases in which the jump of a field across the interface is given, as well as cases in which the primary field on the interface is given. Optimal rates of convergence hold. Representative numerical examples demonstrate the effectiveness of the proposed methodology.

Finite Element Model

Abstract: Objectives: s Build an initial surface mesh that will be used as a pattern to create the final 1, 2 and 3D mesh. s Edit and smooth the mesh. s Build a finite element model by sweeping a node, 1D and 2D elements in a 30 o arc.

Dynamic Finite Element Analysis of Structures

Abstract: Because of the large number of computer runs required for dynamic analysis, it is very important that accurate and numerically efficient methods be used within computer programs. These methods have been described adequately. Equation of motion for various structures with seismic devices is given. Their functions are identified when various solution procedures are used. Some of the solution procedures popularly adopted by various resea rches are explained. A complete finite element procedure is given which can easily be linked to typical dynamic analysis chosen for a specific case study. These analyses are generally adopted by many well-known finite element computer packages. The analysis given has the flexibility to be adopted in any new package in the offing. Computer subroutines can be developed to link the main finite element analysis package with the performance of specific seismic devices. Many computerized finite element packages have such facilities. The purpose of this chapter is to give credit to the power of finite element in the use of earthquake-resistant structures.