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Finite Element Method: 1 - Introduction

The finite element method (FEM) is a numerical technique used to find approximate solutions to boundary value problems. It involves discretizing a continuous domain into small finite pieces called elements and approximating solutions within each element. While an analytical solution is exact, FEM provides approximate solutions by interpolating between discrete points. FEM has been used since the 1950s with early computers and has since been applied to various fields like structural analysis, heat transfer, fluid flow, and electromagnetism by following general procedures of pre-processing, solving, and post-processing.

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79 views10 pages

Finite Element Method: 1 - Introduction

The finite element method (FEM) is a numerical technique used to find approximate solutions to boundary value problems. It involves discretizing a continuous domain into small finite pieces called elements and approximating solutions within each element. While an analytical solution is exact, FEM provides approximate solutions by interpolating between discrete points. FEM has been used since the 1950s with early computers and has since been applied to various fields like structural analysis, heat transfer, fluid flow, and electromagnetism by following general procedures of pre-processing, solving, and post-processing.

Uploaded by

IñigoAlberdi
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© © All Rights Reserved
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FINITE ELEMENT METHOD

1_INTRODUCTION
1_INTRODUCTION

Contents
A bit of history
FEM objective
Comparison of FEM and exact solutions
Basic definitions
Applications
A BIT OF HISTORY
Basic behavioural laws had been established well before the FE era
Linear elastic stress-strain relations
Prandtl-Reuss plasticity equations
Theory of elasticity
1909: Rayleigh-Ritz method
Attempt to circumvent the restrictions of the analytical approach by replacing the
differential equations by algebraic equations
Their solution required numerical methods (prior computers!)
1943: Courant
Attempt at dividing a structure into triangular sub-regions
Principle of minimization of potential energy and piecewise polynomial variations to
solve St Venant torsion problem
1950: Computers
Able to use the numerical analysis approach to solving the basic equations of physics
Early computers: limited, costly and difficult to program. Only available to aerospace
and nuclear industries
Matrix methods
Articles
1956: Turner, Clough, Martin and Topp: included a description of the CST
Argyris and Kelsey: subject of energy theorems and structural analysis, considered
numerical methods with discrete elements
1960: Clough: first time finite element term have been used

MONDRAGON UNIBERTSITATEA 3
A BIT OF HISTORY
1960: research in the aero-
engine industry in the UK
Isoparametric element
Development of the front
solution
1970: first general purpose FE
codes
Mainly static elasticity and
dynamics
Since 1970: Abaqus in the 1980s
Research and development: CFD
and other mathematical physics
1983: NAFEMS, UK (National
Agency for Finite Element
Methods and Standards)
Nowadays: just have a look in
google! New publications every
day

MONDRAGON UNIBERTSITATEA 4
FEM OBJECTIVE
To obtain an approximate solution of a boundary value problem in
engineering
Boundary value problem = field problem
The field is the domain of interest
Field variables: displacement, temperature, heat flux, fluid velocity
Normally: problems governed by differential equations

Analytical solution vs FEM solution


Analytical FEM
Exact solution Approximate solution
Easy problems Any problem

Example. Determination of the stress field in a structure


stress field = stress at hundred or even million points!
Analytically. Sometimes, even for 1 point is complex
FEM. Possible but approximate. Interpolation between points

MONDRAGON UNIBERTSITATEA 5
Comparison of FEM and exact solutions
FEM. Many approximations:
Adaptation of the real geometry
Interpolation within the element

Adaptation of a real geometry a)


41 elements; b) 192 elements

MONDRAGON UNIBERTSITATEA 6
Comparison of FEM and exact solutions

a) Structure; b) 1 truss element;


c) 2 truss elements; d) 4 truss elements

MONDRAGON UNIBERTSITATEA 7
BASIC DEFINITIONS
FEM: numerical method of
solving systems of
equations after
discretizing the geometry
with finite elements and
finding a solution after
applying boundary
conditions to the
assembled system

General procedure for FEM


1) pre-processing
2) processing (solution)
3) post-processing

MONDRAGON UNIBERTSITATEA 8
APPLICATIONS

MONDRAGON UNIBERTSITATEA 9
APPLICATIONS

MONDRAGON UNIBERTSITATEA 10

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