Showing papers on "Finite element method published in 1997"

Nonlinear Continuum Mechanics for Finite Element Analysis

Abstract: Designing engineering components that make optimal use of materials requires consideration of the nonlinear characteristics associated with both manufacturing and working environments. The modeling of these characteristics can only be done through numerical formulation and simulation, and this requires an understanding of both the theoretical background and associated computer solution techniques. By presenting both nonlinear continuum analysis and associated finite element techniques under one roof, Bonet and Wood provide, in this edition of this successful text, a complete, clear, and unified treatment of these important subjects. New chapters dealing with hyperelastic plastic behavior are included, and the authors have thoroughly updated the FLagSHyP program, freely accessible at www.flagshyp.com. Worked examples and exercises complete each chapter, making the text an essential resource for postgraduates studying nonlinear continuum mechanics. It is also ideal for those in industry requiring an appreciation of the way in which their computer simulation programs work.

Introduction to the Finite Element Method

Abstract: This chapter introduces the finite element method (FEM) as a tool for solution of classical electromagnetic problems. Although we discuss the main points in the application of the finite element method to electromagnetic design, including formulation and implementation, those who seek deeper understanding of the finite element method should consult some of the works listed in the bibliography section.

Mechanics of laminated composite plates : theory and analysis

Abstract: Introduction and Mathematical Preliminaries Fiber-Reinforced Composite Materials. Vectors and Tensors. Matrices. Transformation of Vector and Tensor Components. Integral Relations. Equations of Anisotropic Elasticity Classification of Equations. Kinematics. Kinetics. Constitutive Equations. Equations of Thermoelasticity and Electroelasticity. Summary. Virtual Work Principles and Variational Methods Virtual Work. The Variational Operator and Functionals. Extrema of Functionals. Virtual Work Principles. Variational Methods. Summary. Introduction to Composite Materials Basic Concepts and Terminology. Constitutive Equations of a Lamina. Transformation of Stresses and Strains. Plane Stress Constitutive Relations. Classical and First-Order Theories of Laminated Composite Plates Introduction. An Overview of ESL Laminate Theories. The Classical Laminated Plate Theory. The First-Order Laminated Plate Theory. Stiffness Characteristics for Selected Laminates. One-Dimensional Analysis of Laminated Plates Introduction. Analysis of Laminated Beams Using CLPT. Analysis of Laminated Beams Using FSDT. Cylindrical Bending Using CLPT. Cylindrical Bending Using FSDT. Closing Remarks. Analysis of Specially Orthotropic Plates Using CLPT Introduction. Bending of Simply Supported Plates. Bending of Plates with Two Opposite Edges Simply Supported. Bending of Rectangular Plates with Various Boundary Conditions. Buckling of Simply Supported Plates Under Compressive Loads. Buckling of Rectangular Plates Under Inplane Shear Load. Vibration of Simply Supported Plates. Buckling and Vibration of Plates with Two Parallel Edges Simply Supported. Closure. Analytical Solutions of Rectangular Laminates Using CLPT Governing Equations in Terms of Displacements. Admissible Boundary Conditions for the Navier Solutions. Navier Solutions of Antisymmetric Cross-Ply Laminates. The Navier Solutions of Antisymmetric Angle-Ply Laminates. The LTvy Solutions. Analysis of Midplane Symmetric Laminates. Transient Analysis. Summary. Analytical Solutions of Rectangular Laminates Using FSDT Introduction. Simply Supported Antisymmetric Cross-Ply Laminates. Simply Supported Antisymmetric Angle-Ply Laminates. Antisymmetric Cross-Ply Laminates with Two Opposite Edges Simply Supported. Antisymmetric Angle-Ply Laminates with Two Opposite Edges Simply Supported. Transient Solutions. Summary. Finite Element Analysis of Composite Laminates Introduction. Laminated Beams and Plate Strips by CLPT. Timoshenko Beam/Plate Theory. Numerical Results for Beams and Plate Strips. Finite Element Models of Laminated Plates (CLPT). Finite Element Models of Laminated Plates (FSDT). Summary. Refined Theories of Laminated Composite Plates Introduction. A Third-Order Plate Theory. Higher-Order Laminate Stiffness Characteristics. The Navier Solutions. LTvy Solutions of Cross-Ply Laminates. Displacement Finite Element Model. Layerwise Theories and Variable Kinematic Models In troduction. Development of the Theory. Finite Element Model. Variable Kinematic Formulations. Nonlinear Analysis of Composite Laminates Introduction. Nonlinear Stiffness Coefficients. Solution Methods for Nonlinear Algebraic Equations. Computational Aspects and Numerical Examples. Closure. Index Most chapters include Exercises and References for Additional Reading

Design of materials with extreme thermal expansion using a three-phase topology optimization method

Abstract: Composites with extremal or unusual thermal expansion coefficients are designed using a three-phase topology optimization method. The composites are made of two different material phases and a void phase. The topology optimization method consists in finding the distribution of material phases that optimizes an objective function (e.g. thermoelastic properties) subject to certain constraints, such as elastic symmetry or volume fractions of the constituent phases, within a periodic base cell. The effective properties of the material structures are found using the numerical homogenization method based on a finite-element discretization of the base cell. The optimization problem is solved using sequential linear programming. To benchmark the design method we first consider two-phase designs. Our optimal two-phase microstructures are in fine agreement with rigorous bounds and the so-called Vigdergauz microstructures that realize the bounds. For three phases, the optimal microstructures are also compared with new rigorous bounds and again it is shown that the method yields designed materials with thermoelastic properties that are close to the bounds. The three-phase design method is illustrated by designing materials having maximum directional thermal expansion (thermal actuators), zero isotropic thermal expansion, and negative isotropic thermal expansion. It is shown that materials with effective negative thermal expansion coefficients can be obtained by mixing two phases with positive thermal expansion coefficients and void.

Flexible Multibody Dynamics: Review of Past and Recent Developments

Abstract: In this paper, a review of past and recent developments in the dynamics of flexible multibody systems is presented. The objective is to review some of the basic approaches used in the computer aided kinematic and dynamic analysis of flexible mechanical systems, and to identify future directions in this research area. Among the formulations reviewed in this paper are the floating frame of reference formulation, the finite element incremental methods, large rotation vector formulations, the finite segment method, and the linear theory of elastodynamics. Linearization of the flexible multibody equations that results from the use of the incremental finite element formulations is discussed. Because of space limitations, it is impossible to list all the contributions made in this important area. The reader, however, can find more references by consulting the list of articles and books cited at the end of the paper. Furthermore, the numerical procedures used for solving the differential and algebraic equations of flexible multibody systems are not discussed in this paper since these procedures are similar to the techniques used in rigid body dynamics. More details about these numerical procedures as well as the roots and perspectives of multibody system dynamics are discussed in a companion review by Schiehlen [79]. Future research areas in flexible multibody dynamics are identified as establishing the relationship between different formulations, contact and impact dynamics, control-structure interaction, use of modal identification and experimental methods in flexible multibody simulations, application of flexible multibody techniques to computer graphics, numerical issues, and large deformation problem. Establishing the relationship between different flexible multibody formulations is an important issue since there is a need to clearly define the assumptions and approximations underlying each formulation. This will allow us to establish guidelines and criteria that define the limitations of each approach used in flexible multibody dynamics. This task can now be accomplished by using the “absolute nodal coordinate formulation” which was recently introduced for the large deformation analysis of flexible multibody systems.

Is the Pollution Effect of the FEM Avoidable for the Helmholtz Equation Considering High Wave Numbers

Abstract: The development of numerical methods for solving the Helmholtz equation, which behaves robustly with respect to the wave number, is a topic of vivid research. It was observed that the solution of the Galerkin finite element method (FEM) differs significantly from the best approximation with increasing wave number. Many attempts have been presented in the literature to eliminate this lack of robustness by various modifications of the classical Galerkin FEM. However, we will prove that, in two and more space dimensions, it is impossible to eliminate this so-called pollution effect. Furthermore, we will present a generalized FEM in one dimension which behaves robustly with respect to the wave number.

In-plane seismic response of brick masonry walls

Abstract: The paper addresses the problems of evaluation of strength, deformability, and energy dissipation capacity of unreinforced brick masonry walls, within the context of seismic assessment of existing buildings. Possible approaches to simplified strength evaluation are discussed on the basis of experimental and numerical data, and formulae for assessment are presented. The role of the shear ratio in the shear failure mechanisms is put in evidence and shear strength formulae are proposed accordingly. The most significative parameters regarding deformability under cyclic loading are highlighted and energy dissipation due to hysteretic behaviour is quantified for possible use in dynamic models. Experimental results show how ultimate drift seems to be a parameter with high regularity for walls failing in shear. Based on such result, a possible approach for seismic assessment is outlined. © 1997 John Wiley & Sons, Ltd.

Immersed Interface Methods for Stokes Flow with Elastic Boundaries or Surface Tension

Abstract: A second-order accurate interface tracking method for the solution of incompressible Stokes flow problems with moving interfaces on a uniform Cartesian grid is presented. The interface may consist of an elastic boundary immersed in the fluid or an interface between two different fluids. The interface is represented by a cubic spline along which the singularly supported elastic or surface tension force can be computed. The Stokes equations are then discretized using the second-order accurate finite difference methods for elliptic equations with singular sources developed in our previous paper [SIAM J. Numer. Anal., 31(1994), pp. 1019--1044]. The resulting velocities are interpolated to the interface to determine the motion of the interface. An implicit quasi-Newton method is developed that allows reasonable time steps to be used.

Evaluation of Nonlinear Frame Finite-Element Models

Abstract: In recent years nonlinear dynamic analysis of three-dimensional structural models is used more and more in the assessment of existing structures in zones of high seismic risk and in the development...

Time-domain finite-element methods

Abstract: Various time-domain finite-element methods for the simulation of transient electromagnetic wave phenomena are discussed. Detailed descriptions of test/trial spaces, explicit and implicit formulations, nodal and edge/facet element basis functions are given, along with the numerical stability properties of the different methods. The advantages and disadvantages of mass lumping are examined. Finally, the various formulations are compared on the basis of their numerical dispersion performance.

Artificial Neural Networks for Solving Ordinary and Partial Differential Equations

Abstract: We present a method to solve initial and boundary value problems using artificial neural networks. A trial solution of the differential equation is written as a sum of two parts. The first part satisfies the boundary (or initial) conditions and contains no adjustable parameters. The second part is constructed so as not to affect the boundary conditions. This part involves a feedforward neural network, containing adjustable parameters (the weights). Hence by construction the boundary conditions are satisfied and the network is trained to satisfy the differential equation. The applicability of this approach ranges from single ODE's, to systems of coupled ODE's and also to PDE's. In this article we illustrate the method by solving a variety of model problems and present comparisons with finite elements for several cases of partial differential equations.

Analysis of the Inexact Uzawa Algorithm for Saddle Point Problems

Abstract: In this paper, we consider the so-called "inexact Uzawa" algorithm for iteratively solving linear block saddle point problems. Such saddle point problems arise, for example, in finite element and finite difference discretizations of Stokes equations, the equations of elasticity, and mixed finite element discretization of second-order problems. We consider both the linear and nonlinear variants of the inexact Uzawa iteration. We show that the linear method always converges as long as the preconditioners defining the algorithm are properly scaled. Bounds for the rate of convergence are provided in terms of the rate of convergence for the preconditioned Uzawa algorithm and the reduction factor corresponding to the preconditioner for the upper left-hand block. In the case of nonlinear iteration, the inexact Uzawa algorithm is shown to converge provided that the nonlinear process approximating the inverse of the upper left-hand block is of sufficient accuracy. Bounds for the nonlinear iteration are given in terms of this accuracy parameter and the rate of convergence of the preconditioned linear Uzawa algorithm. Applications to the Stokes equations and mixed finite element discretization of second-order elliptic problems are discussed and, finally, the results of numerical experiments involving the algorithms are presented.

Finite Element Solution of the Helmholtz Equation with High Wave Number Part II: The h - p Version of the FEM

Abstract: In this paper, which is part II in a series of two, the investigation of the Galerkin finite element solution to the Helmholtz equation is continued. While part I contained results on the $h$ version with piecewise linear approximation, the present part deals with approximation spaces of order $p \ge 1$. As in part I, the results are presented on a one-dimensional model problem with Dirichlet--Robin boundary conditions. In particular, there are proven stability estimates, both with respect to data of higher regularity and data that is bounded in lower norms. The estimates are shown both for the continuous and the discrete spaces under consideration. Further, there is proven a result on the phase difference between the exact and the Galerkin finite element solutions for arbitrary $p$ that had been previously conjectured from numerical experiments. These results and further preparatory statements are then employed to show error estimates for the Galerkin finite element method (FEM). It becomes evident that the error estimate for higher approximation can---with certain assumptions on the data---be written in the same form as the piecewise linear case, namely, as the sum of the error of best approximation plus a pollution term that is of the order of the phase difference. The paper is concluded with a numerical evaluation.

Application of the Strain Energy Damage Detection Method to Plate-like Structures

Abstract: In this paper the problem of using measured modal parameters to detect and locate damage in plate-like structures is investigated. Many methods exist for locating damage in a structure given the modal properties before and after damage. Unfortunately, many of these methods require a correlated finite element model or mass normalized mode shapes. If the modal properties are obtained using ambient excitation then the mode shapes will not be mass normalized. In this paper a method based on the changes in the strain energy of the structure will be discussed. This method has been successfully applied to beam-like structures, that is, structures characterized by one-dimensional curvature. In this paper the method will be generalized to plate-like structures that are characterized by two-dimensional curvature. This method only requires the mode shapes of the structure before and after damage. To evaluate the effectiveness of the method it will be applied to simulated data.

Mixed Finite Elements for Elliptic Problems with Tensor Coefficients as Cell-Centered Finite Differences

Abstract: We present an expanded mixed finite element approximation of second-order elliptic problems containing a tensor coefficient. The mixed method is expanded in the sense that three variables are explicitly approximated, namely, the scalar unknown, the negative of its gradient, and its flux (the tensor coefficient times the negative gradient). The resulting linear system is a saddle point problem. In the case of the lowest order Raviart--Thomas elements on rectangular parallelepipeds, we approximate this expanded mixed method by incorporating certain quadrature rules. This enables us to write the system as a simple, cell-centered finite difference method requiring the solution of a sparse, positive semidefinite linear system for the scalar unknown. For a general tensor coefficient, the sparsity pattern for the scalar unknown is a 9-point stencil in two dimensions and 19 points in three dimensions. Existing theory shows that the expanded mixed method gives optimal order approximations in the $L^2$- and $H^{-s}$-norms (and superconvergence is obtained between the $L^2$-projection of the scalar variable and its approximation). We show that these rates of convergence are retained for the finite difference method. If $h$ denotes the maximal mesh spacing, then the optimal rate is $O(h)$. The superconvergence rate $O(h^{2})$ is obtained for the scalar unknown and rate $O(h^{3/2})$ for its gradient and flux in certain discrete norms; moreover, the full $O(h^{2})$ is obtained in the strict interior of the domain. Computational results illustrate these theoretical results.

Shear deformable shell elements for large strains and rotations

Abstract: Well-known finite element concepts like the Assumed Natural Strain (ANS) and the Enhanced Assumed Strain (EAS) techniques are combined to derive efficient and reliable finite elements for continuum based shell formulations. In the present study two aspects are covered: The first aspect focuses on the classical 5-parameter shell formulation with Reissner–Mindlin kinematics. The above-mentioned combinations, already discussed by Andelfinger and Ramm for the linear case of a four-node shell element, are extended to geometrical non-linearities. In addition a nine-node quadrilateral variant is presented. A geometrically non-linear version of the EAS-approach is applied which is based on the enhancement of the Green–Lagrange strains instead of the displacement gradient as originally proposed by Simo and Armero. In the second part elements are derived in a similar way for a higher order, so-called 7-parameter non-linear shell formulation which includes the thickness stretch of the shell (Buchter and Ramm). In order to avoid artificial stiffening caused by the three dimensional displacement field and termed ‘thickness locking’, special provisions for the thickness stretch have to be introduced. © 1997 John Wiley & Sons, Ltd.

Variation simulation for deformable sheet metal assemblies using finite element methods

Abstract: Traditional variation analysis methods, such as Root Sum Square method and Monte Carlo simulation, are not applicable to sheet metal assemblies because of possible part deformation during the assembly process. This paper proposes the use of finite element methods (FEM) in developing mechanistic variation simulation models for deformable sheet metal parts with complex two or three dimensional free form surfaces. Mechanistic variation simulation provides improved analysis by combining engineering structure models and statistical analysis in predicting the assembly variation. Direct Monte Carlo simulation in FEM is very time consuming, because hundreds or thousands of FEM runs are required to obtain a realistic assembly distribution. An alternative method, based on the Method of Influence Coefficients, is developed to improve the computational efficiency, producing improvements by several orders of magnitude. Simulations from both methods yield almost identical results. An example illustrates the developed methods used for evaluating sheet metal assembly variation. The new approaches provide an improved understanding of sheet metal assembly processes.

Nonlinear dynamics of structures assembled by bolted joints

Abstract: The nonlinear transfer behaviour of an assembled structure such as a large lightweight space structure is caused by the nonlinear influence of structural connections. Bolted or riveted joints are the primary source of damping compared to material damping, if no special damping treatment is added to the structure. Simulation of this damping amount is very important in the design phase of a structure. Several well known lumped parameter joint models used in the past to describe the dynamic transfer behaviour of isolated joints by Coulomb friction elements are capable of describing global states of slip and stick only. The present paper investigates the influence of joints by a mixed experimental and numerical strategy. A detailed Finite Element model is established to provide understanding of different slip-stick mechanisms in the contact area. An advanced lumped parameter model is developed and identified by experimental investigations for an isolated bolted joint. This model is implemented in a Finite Element program for calculating the dynamic response of assembled structures incorporating the influence of micro- and macroslip of several bolted joints.