Preface
The classical approximation methods known in continuum mechanics are the finite
difference method, the finite element method, the finite volume method, and the
boundary element method. Each method has its advantages and disadvantages and a
different spread in different areas of applied mechanics, e.g. solid or fluid
mechanics. The oldest approximation method to solve partial differential equations
is the finite difference method since the mathematical set of tools is basically limited
to the series expansion of derivatives. More precisely, truncated Taylor’s series is
used for the local expansions of the variables. This ‘simple’ derivation of the
method makes it quite attractive to introduce approximation methods in tertiary
engineering education, e.g. in mechanical or civil engineering. This allows
acquiring a general understanding of numerical approximation methods and the
involved common steps such as discretization, assembly of a global system of
equations, consideration of boundary and load conditions as well as the solution of
a linear or nonlinear system of equations.
This book is focused on the introduction of the finite difference method based on
the classical one-dimensional structural members, i.e. rods/bars and beams. It is the
goal to provide a first introduction to the manifold aspects of the finite difference
method and to enable the reader to get a methodical understanding of important
subject areas in structural mechanics. The reader learns to understand the
assumptions and derivations of different structural members. Furthermore, she/he
learns to critically evaluate the possibilities and limitations of the finite difference
method. Additional comprehensive mathematical descriptions, which solely result
from advanced illustrations for two- or three-dimensional problems, are omitted.
Hence, the mathematical description largely remains simple and clear.
Chapter 1 illustrates the derivation of the finite difference method in a general
way for one-dimensional problems, i.e. partial differential equations. Chapter 2
covers the simplest one-dimensional element type, i.e. the rod/bar element.
Approximate equations are provided for rods of constant and varying tensile
stiffnesses. Chapter 3 covers the simplest one-dimensional beam formulation
according to Euler-Bernoulli. This element is also called the thin beam. Again,
approximate equations are provided for thin beams of constant and varying bending
vii
The classical approximation methods known in continuum mechanics are the finite
difference method, the finite element method, the finite volume method, and the
boundary element method. Each method has its advantages and disadvantages and a
different spread in different areas of applied mechanics, e.g. solid or fluid
mechanics. The oldest approximation method to solve partial differential equations
is the finite difference method since the mathematical set of tools is basically limited
to the series expansion of derivatives. More precisely, truncated Taylor’s series is
used for the local expansions of the variables. This ‘simple’ derivation of the
method makes it quite attractive to introduce approximation methods in tertiary
engineering education, e.g. in mechanical or civil engineering. This allows
acquiring a general understanding of numerical approximation methods and the
involved common steps such as discretization, assembly of a global system of
equations, consideration of boundary and load conditions as well as the solution of
a linear or nonlinear system of equations.
This book is focused on the introduction of the finite difference method based on
the classical one-dimensional structural members, i.e. rods/bars and beams. It is the
goal to provide a first introduction to the manifold aspects of the finite difference
method and to enable the reader to get a methodical understanding of important
subject areas in structural mechanics. The reader learns to understand the
assumptions and derivations of different structural members. Furthermore, she/he
learns to critically evaluate the possibilities and limitations of the finite difference
method. Additional comprehensive mathematical descriptions, which solely result
from advanced illustrations for two- or three-dimensional problems, are omitted.
Hence, the mathematical description largely remains simple and clear.
Chapter 1 illustrates the derivation of the finite difference method in a general
way for one-dimensional problems, i.e. partial differential equations. Chapter 2
covers the simplest one-dimensional element type, i.e. the rod/bar element.
Approximate equations are provided for rods of constant and varying tensile
stiffnesses. Chapter 3 covers the simplest one-dimensional beam formulation
according to Euler-Bernoulli. This element is also called the thin beam. Again,
approximate equations are provided for thin beams of constant and varying bending
vii