Showing papers on "Smoothed finite element method published in 1991"

Update: Application of the Finite Element Method to Linear Elastic Fracture Mechanics

Abstract: Use of the finite element method to treat two and three-dimensional linear elastic fracture mechanics problems is becoming common place. In general, the behavior of the displacement field in ordinary elements is at most quadratic or cubic, so that the stress field is at most linear or quadratic. On the other hand, the stresses in the neighborhood of a crack tip in a linear elastic material have been shown to be square root singular. Hence, the problem begins by properly modeling the stresses in the region adjacent to the crack tip with finite elements. To this end, quarter-point, singular, isoparametric elements may be employed; these will be discussed in detail. After that difficulty has been overcome, the stress intensity factor must be extracted from either the stress or displacement field or by an energy based method. Three methods are described here: displacement extrapolation, the stiffness derivative and the area and volume J-integrals. Special attention will be given to the virtual crack extension which is employed by the latter two methods. A methodology for calculating stress intensity factors in two and three-dimensional bodies will be recommended.

A refined global‐local finite element analysis method

Abstract: A refined global-local method was proposed to improve the efficiency of finite element analysis The proposed method was based on the regular finite element method in conjunction with three basic step, ie the global analysis, the local analysis and the refined global analysis In the first two steps, a coarse finite element mesh was used to analyse the entire structure to obtain the nodal displacements which were subsequently used as displacement boundary conditions for local regions of interest These local regions with the prescribed boundary conditions were then analysed with refined meshes to obtain more accurate stresses In the third step, a new global displacement distribution based on the results of the previous two steps was assumed for the analysis, from which much improved solutions for both stresses and displacements were produced Numerical examples showed that the proposed method yielded accurate solutions with significant savings in computing time compared with the regular finite element method Further, this method is suitable for parallel computation

Stochastic Finite Element Method: Response Statistics

Abstract: An important objective of a stochastic finite element analysis of an engineering system should be the determination of a set of design criteria which can be implemented in a probabilistic context. In other words, it is important that any method that is used for the numerical treatment of stochastic systems be compatible with some rationale that permits a reliability analysis. Traditionally, this problem has been addressed by relying on the second order statistical moments of the response process. These moments offer useful statistical information regarding the response of the system and equations will be developed in an ensuing section for their determination. However, second order moments do not suffice for a complete reliability analysis. For this, higher order moments and related information are required. In this context, the response of any system to a certain excitation may be viewed as a point in the space defined by the parameters describing the system. In other words, for every set of such parameters, the response is uniquely determined as a function of the independent variables. When these parameters are regarded as random variables, the response function may be described as a point in the space spanned by these variables, defining a surface that corresponds to all possible realizations of the random parameters.

Finite Element Modeling of Ventricular Mechanics

Abstract: Computing the distributions of stress and strain in a body with the complex geometry, boundary conditions, and material properties of the heart is a difficult yet worthwhile endeavor. The most promising method for obtaining numerical solutions to this problem is the finite element method. In this chapter, we review the use of finite element analysis for modeling ventricular mechanics. And we conclude by presenting a new axisymmetric finite element model of the passive left ventricle with a realistic geometry and fibrous architecture, physiological boundary conditions, and a three-dimensional constitutive equation.

Numerical study of Fisher's equation by a Petrov-Galerkin finite element method

Abstract: Fisher's equation, which describes a balance between linear diffusion and nonlinear reaction or multiplication, is studied numerically by a Petrov-Galerkin finite element method. The results show that any local initial disturbance can propagate with a constant limiting speed when time becomes sufficiently large. Both the limiting wave fronts and the limiting speed are determined by the system itself and are independent of the initial values. Comparing with other studies, the numerical scheme used in this paper is satisfactory with regard to its accuracy and stability. It has the advantage of being much more concise.

Nonlinear Computational Mechanics: State of the Art

Abstract: The field of computational mechanics has grown very rapidly during the last decade. This is due to the fact that modern engineering design needs complex models which can only be analyzed and simulated on powerful computers and workstations using numerical methods such as finite element, boundary element or finite difference techniques. This volume presents an overview of current research areas representing the state-of-the-art in the field of nonlinear computational mechanics. The areas considered in more detail include the mathematical theory and numerical algorithms, nonlinear finite element procedures, boundary element techniques, beam, plate and shell formulations, inelastic constitutive models and contact formulations.

A Runge-Kutta discontinuous finite element method for high speed flows

Abstract: A Runge-Kutta discontinuous finite element method is developed for hyperbolic systems of conservation laws in two space variables. The discontinuous Galerkin spatial approximation to the conservation laws results in a system of ordinary differential equations which are marched in time using Runge-Kutta methods. Numerical results for the two-dimensional Burger's equation show that the method is (p+1)-order accurate in time and space, where p is the degree of the polynomial approximation of the solution within an element and is capable of capturing shocks over a single element without oscillations. Results for this problem also show that the accuracy of the solution in smooth regions is unaffected by the local projection and that the accuracy in smooth regions increases as p increases. Numerical results for the Euler equations show that the method captures shocks without oscillations and with higher resolution than a first-order scheme.

New finite element method for multidimensional phase change heat transfer problems

Abstract: A new method, called the exact specific heat method, is introduced for finite element analysis of transient heat transfer with phase change. The method is ideally suited for metals and alloys that exhibit discontinuous specific heat functions across the phase transition temperature. The method uses an exact representation of the specific heat function and exploits its mathematical properties in obtaining the latent heat matrix. A derivation of the finite element formulation is presented. Numerical results are presented for several two-dimensional test cases and compared with analytical solutions and other finite element and finite difference numerical results. Excellent agreement was found.

On the accuracy of shape sensitivity

Abstract: The calculation of sensitivity of the response of a structure modeled by finite elements to shape variation is known to be subject to numerical difficulties. The accuracy of a given method is typically measured against the yard stick of finite-difference sensitivity calculation. The present paper demonstrates with a simple example that this approach may be flawed because of discretization errors associated with the finite element mesh. Seven methods for calculating sensitivity derivatives are compared for a two-material beam problem with a moving interface. It is found that as the mesh is refined, displacement sensitivity derivatives converge more slowly than the displacements. Six of the methods agree fairly well, but the adjoint variational surface method provides substantially different results. However, the difference is found to reflect convergence from another direction to the same answer rather than reduced accuracy. Additionally, it is observed that small derivatives are particularly prone to accuracy problems.

Application of the hybrid-trefftz finite element model to thin shell analysis

Abstract: The paper presents the results of a preliminary study on thin shallow shell element based on the hybrid-Trefftz (HT) model. This model adopts an assumed nonconforming displacement field which satisfies a priori the governing differential equations. The interelement continuity and the boundary conditions are enforced by frame fields defined in terms of the conventional nodal freedoms. In the p-extension, the frame functions involve an optional number of hierarchic displacement modes. Numerical results present the capability of the new shell element which can be implemented in existing finite element codes.

Volume visualization of 3D finite element method results

Abstract: This paper describes a method for visualizing the output data set of a 3D finite element method result. A linear tetrahedral element is used as a primitive for the visualization processing, and a 3D finite element model is subdivided into a set of these primitives, which are generated at every s< îd element. With these primitives, isosurfaces are visualized semltransparentiy from scalar data at each node point. Two methods are developed for the visualization of isosurfaces with and without inteimediate geometries. The methods are applied to output data sets from some simulation results of a semiconductor chip. These are visualized, and the effectiveness of the method is discussed.

A hierarchical approach to automatic finite element mesh generation

Abstract: A point-based two-stage hierarchical method for automatic finite element mesh generation from a solid model is presented. Given the solid model of a component and the required nodal density distribution, nodes are generated according to the hierarchy—vertex, edge, face and solid. At the vertices, nodes are established naturally. Nodes on the edges, faces and inside the solid model are generated by recursive subdivision. The nodes are then connected to form a valid and well-conditioned finite element mesh of tetrahedron elements using modified Delaunay Triangulation. Checks are conducted to ensure the compatibility of geometry and topology between the solid model and the mesh.

Generalized Galerkin Methods for Convection Dominated Transport Phenomena

Abstract: A brief survey is made of recent advances in the development of finite element methods for convection dominated transport phenomena. Because of the nonsymmetric character of convection operators, the standard Galerkin formulation of the method of weighted residuals does not possess optimal approximation properties in application to problems in this class. As a result, numerical solutions are often corrupted by spurious node-to-node oscillations. For steady problems describing convection and diffusion, spurious oscillations can be precluded by the use of upwind-type finite element approximations that are constructed through a proper Petrov-Galerkin weighted residual formulation. Various upwind finite element formulations are reviewed in this paper, with a special emphasis on the major breakthroughs represented by the so-called streamline upwind Petrov-Galerkin and Galerkin least-squares methods. The second part of the paper is devoted to a review of time-accurate finite element methods recently developed for the solution of unsteady problems governed by first-order hyperbolic equations. This includes Petrov-Galerkin, Taylor-Galerkin, least-squares, and various characteristic Galerkin methods. The extension of these methods to deal with unsteady convection-diffusion problems is also considered.

Differentiation of finite element approximations to harmonic functions (EM field computation)

Abstract: Derivatives of finite-element solutions are essential for most postprocessing operations, but numerical differentiation is an error-prone process. High-order derivatives of harmonic functions can be computed accurately by a technique based on Green's second identity, even where the finite element solution itself has insufficient continuity to possess the desired derivatives. Data are presented on the sensitivity of this method to solution error as well as to the numerical quadratures used. The procedure is illustrated by application to finding second and third derivatives of a first-order finite-element solution. >

A three dimensional adaptive finite element method for magnetostatic problems

Abstract: A three dimensional adaptive finite element method for magnetostatic field problems is presented. Local errors are estimated from the discontinuity of the tangential components of the magnetic field intensity vector at the interfaces between elements in the finite element method, and mesh refinement is carried out by using a three-dimensional bisection method. The elements are refined according to the amount of error estimated. It is shown that the proposed method is easily used and acceptable for three-dimensional magnetostatic field problems. >

Experimental and finite element simulation methods for rate‐dependent metal forming processes

Abstract: An experimental procedure and a finite element simulation method for rate-dependent metal forming processes are developed. The development includes the formulation of a tangential stiffness matrix for an axisymmetric solid finite element with four node, eight degree of freedom, quadrilateral cross-section. The formulation includes the effects of elasticity, viscoplasticity, temperature, strain rate and large strains. The solution procedure is based on a Newton-Raphson incremental-iterative method which solves the non-linear equilibrium equations and gives temperatures and incremental stresses and strains. Three examples are studied. In example 1, finite element simulation for the upsetting of a cylindrical workpiece between two perfectly rough dies is performed and the results are compared with alternative finite element solutions. In examples 2 and 3, both experimental and finite element studies are performed for the upsetting of a cylindrical billet and the forging of a ball, respectively. Annealed aluminium 1100 workpieces are used in both examples. For the finite element analysis, uniaxial compression tests are first performed to provide the material properties. The tests generate elastic moduli and two sets of stress-strain curves (quasi-static and constant strain rate), which are used to establish a rate-dependent material model for input. For both examples 2 and 3, comparisons between the experimental and finite element simulation results for the forming force vs. die displacement relations and also for the deformed configurations show good agreement. The versatility of finite element methods allows for displaying detailed knowledge of the metal forming process, such as the distributions of temperature rise, yield stress, effective stress, plastic strain, plastic strain rate, forming forces and deformed configurations, etc. at any instance during the forming process.

Efficient Solution Of Three-Dimensional Finite Element Models For Defibrillation And Pacing Applications

Abstract: a larger amount of main memory and disk space To handle the large number of nodes and elements, typically on the order of 450,000 unknowns, we developed our own finite element solver METHODS An algorithm for solving three-dimensional finite element models has been developed and is used to solve models of the thorax generated from X-ray Computed Tomography scans or Magnetic Resonance images Using this finite element analysis method we can build and solve thorax models that include detailed anatomical structures and contain as many as 450,000 unknowns Our solution method is compared with a commercial finite element analysis software package and the performance of the algorithm and the necessary computer requirements are described

Finite Element Programs for Structural Vibrations

Abstract: The finite element method problem solving in structural vibrations the modular approach in finite element programming vibration of plane pin-jointed trusses vibrations of continous beams vibrations of rigid-jointed plane frames vibration of grillages.

New hybrid Laplace transform/finite element method for three‐dimensional transient heat conduction problem

Abstract: The paper presents results obtained by the implementation of a new hybrid Laplace transform/finite element method developed by the authors. The present method removes the time derivatives from the governing differential equation using the Laplace transform and then solves the associated equation with the finite element method. Previously reported hybrid Laplace transform/finite element methods1 have been confined to one nodal solution at a time. When applied to many nodes it takes an excessive amount of computer time. By using a similarity transform method on the matrix of the complex number coefficients this restriction is removed and the reported new method provides a more useful tool for the solution of linear transient problems. Test examples are used to show that the basic accuracy is comparable to that obtainable by analytical, finite difference and finite element methods.

Development of A Three‐Dimensional Membrane Element for the Finite Element Analysis of Tires

Abstract: A three‐dimensional membrane element was developed for the finite element analysis of tires. In general, the three‐dimensional finite element analysis of tires uses a lot of computing time because of the complex nature of the problem. Major sources of complexity are, for example, nonlinearities in kinematics, material properties, boundary conditions, and the multilayer structure which is inherent to the tire. One of the ways to overcome this situation can be in the modeling strategy. This paper describes an approach where the cord‐rubber composite components of the tire are modeled by membrane elements. The number of nodes required in the tire model using this strategy is considerably reduced, without any loss of accuracy, compared with models in which only ordinary solid elements are used. The nonlinear finite element formulation, numerical examples, and a comparison of the results with those obtained from models using solid elements and experimental values are given in the paper.

Some finite-element and finite-difference methods for solving mathematical physics problems with non-smooth data in an n-dimensional cube. Part I

Abstract: For second-order hyperbolic equations a method based on polylinear finite-elements with the splitting operator is constructed. The absolute stability is proved and the error estimates (including the super convergence of the gradient of solution) in the classes of nonsmooth data are derived. Similar results are obtained also for two finite-difference methods with the splitting operator. Preliminary, a finite-difference method for elliptic equations (with a specific approximation of coefficients) is constructed. For this method optimal error estimates in the spaces W^ (Ω) and L2(Q) are derived and an inequality similar to an estimate of the acute angle between two operators is established. , The theory of finite-difference methods for solving second-order elliptic, parabolic and hyperbolic problems of mathematical physics was originally developed for the case of smooth solutions in the domain which is a η-dimensional cube [3,4,11]. The theory of finite-element methods for solving elliptic problems [1,10] naturally covers the case of nonsmooth solutions. In the last decade much attention has been paid to justification of finite-difference methods in the nonsmooth case [5,12]. The most advanced results in this direction were obtained for two-dimensional elliptic problems. In this paper we show how within a unified framework one can study both finite-difference and finite-element methods with splitting operators for solving time dependent problems with nonsmooth data. The analysis of these methods is essentially based on the results obtained for elliptic problems and for the case of finite-difference methods such results are presented in the paper. The paper is organized as follows. In Section 2 we construct a finite-element method with the splitting operator for solving hyperbolic equations; this method is closely related to the approximate factorization method [9]. We also establish its absolute stability and derive error estimates in the classes of nonsmooth data. Earlier similar results were obtained in [16, Section 7] for the method with the simplest type splitting , see [2, 4, 11] (in [16] an additional assumption that τ = 0(h) for η > 3 was used). In the last decade many papers were devoted to the analysis of superconvergence of the finite-element method, see the survey paper [6]. In Section 3 we establish (without a priori assumptions on the solution) the superconvergence of the gradient for the method from Section 2 in the classes of nonsmooth data. Therein we also study the closely related problem of the simplest choice of initial data of the method. Section 2 and 3 essentially exploit certain results from [13-16]. In Section 4 for elliptic equations we construct a finite-difference method on a nonuniform mesh. It is based on the method with polylinear elements and utilizes a specific approximation of the coefficients. Optimal error estimates in the spaces