262 Computational Modelling in Hydraulic and Coastal Engineering
3. The Galerkin method and the approximated function
f
x
=
+
+
α α
0
1
α
α
2
2
3
3
x
x
+
4. The collocation method and the approximated function
f =
+
α 0
α
π
α
π
1
2
2
sin
s in( )
x
x
+
5. The Galerkin method and the approximated function
f =
+
α 0
α
π
α
π
1
2
2
sin
s in( )
x
x
+
9.3 FINITE ELEMENTS METHOD
The finite elements method is widely used in engineering and particularly in
the area of structural analysis, where the method is interpreted as the maximization of the potential energy of a structural system. However, the finite
elements method has also been extensively applied in computational fluid
dynamics (CFD) and other scientific and technical applications. The main
characteristic of the finite elements method is that it discretizes the solution
domain in segments (finite elements) and the solution is sought over each
element. The elements are connected to one another through their common
nodes. The resulting algebraic difference equations are similar to the ones
derived by the finite differences method but with some weighted factors.
The advantage of the finite element method is its ability to easily handle
even the most complex boundary shapes. The disadvantage of the method
is the somewhat laborious pre-treatment of the differential equations before
they are converted into difference equations. For that purpose, there are two
main approaches: the Rayleigh-Ritz method and the Galerkin method (that
has already been discussed as one of the weighted residual methods).
9.3.1 Rayleigh-Ritz method
The Rayleigh-Ritz method can be explained in terms of the theory of variational principles (Gelfand and Fomin 1963). In general, variational principles investigate the one-to-one correspondence and equivalence between
certain differential equations and their functionals. For those values of
the functions, f, which are solutions of the original differential equation
(known as the Euler’s equation), the functionals I[f] become stationary
(reaching a relative maximum or minimum value).
For example, for a second-order differential equation, a functional I[f]
is defined as
I f
F x f f f dx
D
[ ]
( , , , )
=
′ ′′
∫
(9.32)
3. The Galerkin method and the approximated function
f
x
=
+
+
α α
0
1
α
α
2
2
3
3
x
x
+
4. The collocation method and the approximated function
f =
+
α 0
α
π
α
π
1
2
2
sin
s in( )
x
x
+
5. The Galerkin method and the approximated function
f =
+
α 0
α
π
α
π
1
2
2
sin
s in( )
x
x
+
9.3 FINITE ELEMENTS METHOD
The finite elements method is widely used in engineering and particularly in
the area of structural analysis, where the method is interpreted as the maximization of the potential energy of a structural system. However, the finite
elements method has also been extensively applied in computational fluid
dynamics (CFD) and other scientific and technical applications. The main
characteristic of the finite elements method is that it discretizes the solution
domain in segments (finite elements) and the solution is sought over each
element. The elements are connected to one another through their common
nodes. The resulting algebraic difference equations are similar to the ones
derived by the finite differences method but with some weighted factors.
The advantage of the finite element method is its ability to easily handle
even the most complex boundary shapes. The disadvantage of the method
is the somewhat laborious pre-treatment of the differential equations before
they are converted into difference equations. For that purpose, there are two
main approaches: the Rayleigh-Ritz method and the Galerkin method (that
has already been discussed as one of the weighted residual methods).
9.3.1 Rayleigh-Ritz method
The Rayleigh-Ritz method can be explained in terms of the theory of variational principles (Gelfand and Fomin 1963). In general, variational principles investigate the one-to-one correspondence and equivalence between
certain differential equations and their functionals. For those values of
the functions, f, which are solutions of the original differential equation
(known as the Euler’s equation), the functionals I[f] become stationary
(reaching a relative maximum or minimum value).
For example, for a second-order differential equation, a functional I[f]
is defined as
I f
F x f f f dx
D
[ ]
( , , , )
=
′ ′′
∫
(9.32)