FINITE ELEMENT METHOD

Some Thoughts

TARUN KANT IIT BOMBAY

Real Life Systems

are continuum problems

which imply,

infinite number of elements

infinite degrees of freedom

Problem is mathematically defined by differential equations subjected to BCs and ICs.

Ω1

Ω2

1

2

Field Problems

That is, problems which involve the solution of partial

differential equation with appropriate boundary

conditions (BVPs).

Such problems occur in a number of important areas

of engineering science, including stress analysis,

fluid and thermal flow, diffusion and

electromagnetism.

LAPLACE EQUATION

POISSON EQUATION

x y zk = k = k = k

2k = f (x, y, z)

f = 0

2 = 0

Heat Equation

Ω

Y X

Z

in Ω

BCs

x y zk k k (x,y,z)fx x y y z z

1 = (x, y, z) on

x x y y z z 1k n k n k n (x,y,z) (x,y,z) 0 on Γg hx y z

Solution

Methods

Analytical

Methods

Numerical

Methods

The idealized problems that the theory could

handle generally turn out to be too simple to

represent the true problem.

Finite Element Method

In many situations an adequate simplified model of

the real continuum is obtained using a finite number

of well defined interconnected elements.

Such a model is called as discrete.

Discretization

Various methods of discretization have been

proposed both by engineers and mathematicians.

All involve an approximation which is of a kind that it

approaches as closely as desired, the true

continuum solution as the number of discrete

variables increases.

Finite Elements

The Finite Element Method represents one of the most significant developments in the history of computational methods.

The use of modern Finite Element Technology has transformed much of the theoretical mechanics and abstract science into practical and essential tools for multitude of technological developments which affect many facets of our life.

FEM – engineer’s view point

The finite element method is a numerical analysis technique for obtaining approximate solutions to a wide variety of engineering problems.

The method was originally developed to study the stresses in complex airframe structures.

Its usage has now extended to the broad fields of

Continuum Mechanics

Electromagnetics

Many related areas

Finite Element Method

versatile and has physical appeal and has advantages over other numerical analysis

techniques.

Applicable to any field problems – heat transfer, stress

analysis, magnetic fields, etc.

No geometric restriction

No loading and boundary conditions restriction

No restriction on material properties

Can deal with multifield problems

Approximation is improved by mesh grading.

THE EARLY HISTORICAL

DEVELOPMENTS

Finite Element Method

M.J. Turner, R.W. Clough, H.C. Martin and L.J. Topp, Stiffness and deflection analysis of complex structures, J. Aero. Sci., 23, 805-823 (1956).

showed that small portions or elements in a continuum behave in a simplified manner.

The formal presentation of the finite element method together with the direct stiffness method for assembling elements is attributed to Turner et al. (1956), who employed the equations of classical elasticity to obtain properties of a triangular element for use in the analysis of plane stress problems.

Finite Element Method

R.W. Clough,

The finite element in plane stress analysis,

Proc. 2nd ASCE Conf. on Electronic Computation, PA,

Sept.1960

It was Clough (1960) who first used the term finite

elements in this paper devoted to plane elasticity

problems.

Finite Element Method

R. Courant,

Variational methods for the solution of problems of equilibrium and vibration, Bull. Amer. Math. Soc.,49,1-23 (1943).

Presented an approximate solution to the St. Venant torsion problem in which he approximated the warping function linearly in each of an assemblage of triangular elements and proceeded to formulate the problem using the principle of minimum potential energy.

Courant’s piece-wise application of the Ritz method involves all the basic concepts of the procedure now known as the finite element method.

Finite Element Method

Pioneering works on FEM started in early 1960’s at:

University of California, Berkeley, U.S.A.

University of Wales, Swansea, U.K.

Important works were also carried out at:

MIT, Cambridge, Mass, U.S.A.

University of Stuttgart, Germany.

Type of Problems

Equilibrium problems

Time independent problems, BVPs.

Eigenvalue problems

Steady state problems whose solution often requires the determination of natural frequencies and modes of vibration of solid and fluids.

These are special class of BVPs where solution exists for only certain ‘particular’ or ‘characteristic’ value of the parameter.

Propagation problems

Time dependent problems, IVPs.

Results when the time dimension is added to the problems of the first two categories.

Finite Elements

The finite element method basically consist of the following procedures:

First, a given physical or mathematical problem is modeled by dividing it into small or fundamental parts called “elements.”

Next, an analysis of the physics or mathematics of the problem is made on these elements.

Finally, the elements are reassembled into the whole with the solution to the original problem obtained through this assembly procedure.

Finite Elements

Finite element technology has emerged as a new discipline combining theoretical mechanics and applied science with approximation theory, numerical analysis and computer science.

It draws from developments made in each of the individual disciplines and is nurtured and nourished by them, but it also stands as a viable branch of knowledge in its own right.

Today finite element technology can be used on any physical problem which can be stated in terms of variational, differential, integral or integro-differential equations.

Finite Elements

The process of subdividing all systems into their

individual components or parts or elements, whose

behavior is readily understood, and then rebuilding the

original system from such elements to study its behavior

is a natural way in which the engineer, the scientist, or

even the economist proceeds.

OCZ, 1977

Finite Element Method

The finite element method is a general method of

structural analysis in which a continuous structure is

replaced by a finite number of elements interconnected

at a finite number of nodal points.

Regardless of the approach used for element formulation, the

solution of a continuum problem by the FEM always follows

an orderly step-by-step process:

1. Discretization of the continuum.

2. Selection of interpolation functions.

3, Establishment of element properties.

4. Assembly of element properties to obtain the system

equations.

5. Solution of the system equations

K d = f

6. Additional computations, design modifications, etc.

Finite Element Method

Finite Element Method

The first book

O. C. Zienkiewicz and Y.K. Cheung,

The Finite Element Method In Structural and Continuum

Mechanics,

McGraw-Hill, London, 1967.

Finite Element Method

The second book

C.S. Desai and J. Abel,

Introduction to Finite Element Method,

Van Nostrand-Reinhold, New York, 1971.

Finite Element Method

A.K. Noor,

Bibliography of books and monographs on finite element

technology,

Appl. Mech. Rev., 44(8), 307-317, 1991.

This bibliography lists about 400 finite element books in English and

other languages.

Finite Element Method

Some statistics as on 1991

320 books

340 Conf. Proc.

3500 papers/year.

BIG BUSINESS!

Theoretical Mechanics

deals with fundamental laws and principles of mechanics studied for their scientific intrinsic value.

Applied Mechanics

transfers the theoretical knowledge to scientific and engineering applications especially as regards the construction of mathematical models of physical phenomena.

Computational Mechanics

solves specific problems by simulation through numerical methods implemented on digital computers.

Finite Element Method

Alternatively (A joke about mathematician)

Computational mechanician is a person who searches for solutions to a physical problem at hand.

Applied mechanician searches for problems that fit given solutions.

and a Theoretical mechanician tries to prove the existence of problems and solutions.

Finite Element Method

Finite Element Method (misuse of FEM)

A few fundamental ideas on which the direct stiffness method (of structural analysis) is built are important for proper understanding of the Finite Element Method (FEM).

This is because, initially, FEM was applied to real complex structural systems as an extrapolation of the ‘matrix displacement method’, by none other than engineers, and that too, civil engineers.

Continuum Elements

Basic approach in isoparametric formulation is to express both the element coordinates and the element displacements in the form of interpolations using natural coordinates.

The formulation is same for the 1D, 2D and 3D elements (C0).

1D 2D

3D

Kij is known as the stiffness (influence) coefficient. It is of great importance in the structural theory.

It is the most important equation in the displacement or stiffness method of analysis.

The key task lies in forming K and F.

The solution follows a standard numerical procedure,

1 2

n

3

F = K d

1d = K F

1

1

1

2

2

2

.

.

.

n

n

n

X

Y

Z

X

Y

ZF =

X

Y

Z

1

1

1

2

2

n

n

n

u

v

w

u

v

w2

.

.

.

u

v

w

d =

11 12 13 . . . . . . . . . . . 1n

21 22 33 . . . . . . . . . . . 2n

n1 n2 n3 . . . . . . . . . . . nn

K K K K

K K K K

.

K .

.

.

K K K K

General Purpose Software

FEM HAS BECOME COMMERCIAL

ANSYS

ASKA

MARC

NASTRAN

SAP

ABAQUS

User friendliness

Preprocessor

Postprocessor

Mainframe vs. PC versions

Approaches to formulation of element

properties

Direct approach

Variational approach

Virtual work

Principle of minimum potential energy

Weighted residual approach

Finite Element Method

R.K. Livesley,

Matrix Methods of Structural Analysis,

Pergamon Press, Oxford, 1964.

H.C. Martin,

Introduction to Matrix Methods of Structural

Analysis,

McGraw-Hill, New York, 1966.

Finite Element Method

J. Mackerle,

Linkoping Institute of Technolgy, S-581 83 Linkoping,

Sweden, http://ohio.ikp.liu.se/fe

This website lists about 600 finite element books published

between 1967 and 2005.

Thank you