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8
The Finite Difference Method
It has been emphasized in this treatise that most problems in the mechanics of solids, when
approached from an analytical viewpoint, can be represented locally by a system of differ-
ential equations or globally by an equivalent variational functional. Often the variational
functional is formed first and used to establish the governing equations. In any event, once
the mathematical (analytical) model has been established, solutions to particular prob-
lems are sought. Such solutions, which must satisfy the governing equations including the
boundary conditions, can be derived from the functional or from the differential equations.
As is immediately evident, contemporary mathematical methods can provide the exact so-
lution to only the simplest forms of the governing equations. Hence, advantage is taken of
the availability of the powerful digital computer by recasting problems in algebraic form. In
so doing, the mathematical
continuum model
, which requires an infinite number of degrees
of freedom (DOF) for its description, is replaced by a
discrete model
, which utilizes a finite
number of DOF. This transformation usually involves some sort of approximation.
The trial function approximations of Chapters 6 and 7 are popular methods for imple-
menting the necessary discretization. Another method, which is sometimes classified as a
subset of the finite element approach, is the
finite difference method.
This is the subject of this
chapter.
The finite difference method can be used to solve the governing differential equations
directly or can be applied to the variational functional to obtain a solution. Both approaches,
which are often characterized as mathematical discretizations in contrast to the more phys-
ically oriented discretization of the finite element method, will be discussed in this chapter.
8.1
Fundamentals
Consider a one-dimensional problem with governing differential equations supplemented
by boundary conditions. This is usually referred to as a boundary value problem. Suppose
we seek to determine a function
u
(
x
)
,
which satisfies the governing equation and boundary
conditions, over the interval 0
With the finite difference method, a
grid
or
mesh
of
points and corresponding intervals are established. Finite difference approximations to the
derivatives in the mathematical model are set up at these points which are usually equally
spaced. This results in a system of algebraic equations that can be solved for
u
at the grid
points of
x
≤
x
≤
L
.
.
495
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