Civil Engineering Reference
In-Depth Information
1
CHAPTER
Basic Concepts of the
Finite Element Method
1.1 INTRODUCTION
The finite element method (FEM), sometimes referred to as
finite element
analysis
(FEA), is a computational technique used to obtain approximate solu-
tions of boundary value problems in engineering. Simply stated, a boundary
value problem is a mathematical problem in which one or more dependent vari-
ables must satisfy a differential equation everywhere within a known domain of
independent variables and satisfy specific conditions on the boundary of the
domain. Boundary value problems are also sometimes called
field
problems. The
field is the domain of interest and most often represents a physical structure.
The
field variables
are the dependent variables of interest governed by the dif-
ferential equation. The
boundary conditions
are the specified values of the field
variables (or related variables such as derivatives) on the boundaries of the field.
Depending on the type of physical problem being analyzed, the field variables
may include physical displacement, temperature, heat flux, and fluid velocity to
name only a few.
1.2 HOW DOES THE FINITE ELEMENT
METHOD WORK?
The general techniques and terminology of finite element analysis will be intro-
duced with reference to Figure 1.1. The figure depicts a volume of some material
or materials having known physical properties. The volume represents the
domain of a boundary value problem to be solved. For simplicity, at this point,
we assume a two-dimensional case with a single field variable
(
x, y
) to be
determined at every point
P
(
x
,
y
) such that a known governing equation (or equa-
tions) is satisfied exactly at every such point. Note that this implies an exact
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