Book Introduction To The Finite Elements Method.pdf

INTRODUCTION TO THE

FINITE ELEMENT METHOD

Evgeny Barkanov

Institute of Materials and Structures

Faculty of Civil Engineering

Riga Technical University

Riga, 2001

Preface

Today the finite element method (FEM) is considered as one of the well established

and convenient technique for the computer solution of complex problems in different fields

of engineering: civil engineering, mechanical engineering, nuclear engineering, biomedical

engineering, hydrodynamics, heat conduction, geo-mechanics, etc. From other side, FEM

can be examined as a powerful tool for the approximate solution of differential equations

describing different physical processes.

The success of FEM is based largely on the basic finite element procedures used: the

formulation of the problem in variational form, the finite element dicretization of this

formulation and the effective solution of the resulting finite element equations. These basic

steps are the same whichever problem is considered and together with the use of the digital

computer present a quite natural approach to engineering analysis.

The objective of this course is to present briefly each of the above aspects of the

finite element analysis and thus to provide a basis for the understanding of the complete

solution process. According to three basic areas in which knowledge is required, the course

is divided into three parts. The first part of the course comprises the formulation of FEM

and the numerical procedures used to evaluate the element matrices and the matrices of the

complete element assemblage. In the second part, methods for the efficient solution of the

finite element equilibrium equations in static and dynamic analyses will be discussed. In

the third part of the course, some modelling aspects and general features of some Finite

Element Programs (ANSYS, NISA, LS-DYNA) will be briefly examined.

To acquaint more closely with the finite element method, some excellent books, like

[1-4], can be used.

Evgeny Barkanov

Riga, 2001

Contents

PREFACE……………………………………………………………………………….…2

PART I THE FINITE ELEMENT METHOD……………………………………..…5

Chapter 1 Introduction………………………………………………………………...…5

1.1 Historicalbackground………………………………………………………………...5

1.2 Comparison of FEM with other methods………………………………………….....5

1.3 Problem statement on the example of “shaft under tensile load”…………………….6

1.4 Variational formulation of the problem……………………………………………....9

1.5 Ritzmethod………………………………………………………………………….10

1.6 Solution of differential equation (analytical solution)………………………………12

1.7 FEM……………………………………………………………………………...….13

Chapter 2 Finite element of bending beam…………………………………………….20

Chapter 3 Quadrilateral finite element under plane stress………………………...…23

PART II SOLUTION OF FINITE ELEMENT EQUILIBRIUM EQUATIONS….30

Chapter 4 Solution of equilibrium equations in static analysis……………………….30

4.1 Introduction…………...……………………………………………………………..30

4.2 Gaussianeliminationmethod……………………………………………………..…31

4.3 Generalisation of Gauss method………………………………………………….…31

4.4 Simple vector iterations…………………………………………………….……….33

4.5 Introduction to nonlinear analyses……………………………………………….….34

4.6 Convergencecriteria………………………………………………………………...37

Chapter 5 Solution of eigenproblems…………………………………………………...39

5.1 Introduction…………………………………………………………………………39

5.2 Transformationmethods…………………………………………………………….40

5.3 Jacobimethod……………………………………………………………………….41

5.4 Vector iteration methods…………………………………………………………….42

5.5 Subspace iteration method…………………………………………………………..43

Chapter 6 Solution of equilibrium equations in dynamic analysis…………………...45

6.1 Introduction………………………………………………………………………….45

6.2 Directintegrationmethods…………………………………………………………..45

6.3 The Newmark method………………………………………………………………46

6.4 Modesuperposition………………………………………………………………….47

6.5 Change of basis to modal generalised displacements……………………………….48

6.6 Analysis with damping neglected…………………………………………………...49

6.7 Analysis with damping included…………………………………………………….50

PART III EMPLOYMENT OF THE FINITE ELEMENT METHOD……………...53

Chapter 7 Some modelling considerations…………………………………………..…53

7.1 Introduction………………………………………………………………………….53

7.2 Type of elements…………………………………………………………………….53

7.3 Size of elements……………………………………………………………………..55

7.4 Locationofnodes…………………………………………………………………....56

7.5 Number of elements…………………………………………………………………56

7.6 Simplifications afforded by the physical configuration of the body………………..58

7.7 Finite representation of infinite body………………………………………………..58

7.8 Node numbering scheme……………………………………………………………59

7.9 Automaticmeshgeneration…………………………………………………………59

Chapter 8 Finite element program packages…………………………………………..60

8.1 Introduction………………………………………………………………………….60

8.2 Build the model……………………………………………………………………...60

8.3 Apply loads and obtain the solution………………………………………………...61

8.4 Review the results…………………………………………………………………...62

LITERATURE……………………………………………………………………………63

APPENDIX

A typical ANSYS static analysis……………………………………...64

PART I THE FINITE ELEMENT METHOD

Chapter 1 Introduction

1.1 Historicalbackground

In 1909 Ritz developed an effective method [5] for the approximate solution of

problems in the mechanics of deformable solids. It includes an approximation of energy

functional by the known functions with unknown coefficients. Minimisation of functional

in relation to each unknown leads to the system of equations from which the unknown

coefficients may be determined. One from the main restrictions in the Ritz method is that

functions used should satisfy to the boundary conditions of the problem.

In 1943 Courant considerably increased possibilities of the Ritz method by

introduction of the special linear functions defined over triangular regions and applied the

method for the solution of torsion problems [6]. As unknowns, the values of functions in

the node points of triangular regions were chosen. Thus, the main restriction of the Ritz

functions – a satisfaction to the boundary conditions was eliminated. The Ritz method

together with the Courant modification is similar with FEM proposed independently by

Clough many years later introducing for the first time in 1960 the term “finite element” in

the paper “The finite element method in plane stress analysis” [7]. The main reason of

wide spreading of FEM in 1960 is the possibility to use computers for the big volume of

computations required by FEM. However, Courant did not have such possibility in 1943.

An important contribution was brought into FEM development by the papers of

Argyris [8], Turner [9], Martin [9], Hrennikov [10] and many others. The first book on

FEM, which can be examined as textbook, was published in 1967 by Zienkiewicz and

Cheung [11] and called “The finite element method in structural and continuum

mechanics”. This book presents the broad interpretation of the method and its applicability

to any general field problems. Although the method has been extensively used previously

in the field of structural mechanics, it has been successfully applied now for the solution of

several other types of engineering problems like heat conduction, fluid dynamics, electric

and magnetic fields, and others.

1.2 Comparison of FEM with other methods

The common methods available for the solution of general field problems, like

elasticity, fluid flow, heat transfer problems, etc., can be classified as presented in Fig. 1.1.

Below FEM will be compared with analytical solution of differential equation and Ritz

method considering the shaft under tensile load (Fig. 1.2).

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