3 Modeling of Friction Stir Welding Processes
113
d
2
dx 2
E I
d
2
dx 2
− q = 0.
(3.76)
The differential equation can be completely solved by specifying and d/ dx at
both ends or d
2
/dx
2 and/or d
3
/dx
3 as boundary conditions. However, out of the
four boundary conditions, two must be as follows:
i. is specified at both ends.
ii. is specified at one end and d/ dx at any end.
Here, and d/ dx constitute essential boundary conditions and d
2
/dx
2 and
d
3
/dx
3 constitute natural boundary conditions.
The variables in FEM are generally approximated using Galerkin formulation and
Ritz formulation. In case of Galerkin formulation, the approximation of the primary
variable is by using a continuous function inside the element. A residue is obtained
depending on the approximating function when primary variable u
e is substituted in
Eq. (3.75), i.e.,
Lu
e
+ q = R
(3.77)
The residue should be zero everywhere; however, it is difficult to approximate the
true value. Therefore, the weighted residual is made equal to zero, i.e.,
D
w RdA = 0
(3.78)
where w is the weight function. In order to weaken the requirement on the differentiability of the approximating function, Eq. (3.78) is integrated by parts to redistribute
the order of derivative in w and R. The weight function is of the same form as
the approximating function in Galerkin method, which is usually some algebraic
function. The unknown coefficients of the function are replaced by unknown nodal
degrees of freedom as
u
e
= [N ]
u
ne
(3.79)
where {u
ne } represents nodal degrees of freedom, and [N] represents the shape
functions matrix.
In Ritz formulation, using calculus of variation, the differential equation Eq. (3.75)
is converted in its integral form. Equation (3.79) is substituted in the integral equation,
and it is partially differentiated with respect to {u
ne } in order to extremize the form.
Assembly of the elemental equations is performed in the next step. The equations
of each element are written in global form and added to each similar equation of
all the elements, i.e., all first equations of each element are added to obtain the
assembled first equation and so on. After the boundary conditions are applied, the
equations are solved by suitable solver. It is followed by post-processing [25].
113
d
2
dx 2
E I
d
2
dx 2
− q = 0.
(3.76)
The differential equation can be completely solved by specifying and d/ dx at
both ends or d
2
/dx
2 and/or d
3
/dx
3 as boundary conditions. However, out of the
four boundary conditions, two must be as follows:
i. is specified at both ends.
ii. is specified at one end and d/ dx at any end.
Here, and d/ dx constitute essential boundary conditions and d
2
/dx
2 and
d
3
/dx
3 constitute natural boundary conditions.
The variables in FEM are generally approximated using Galerkin formulation and
Ritz formulation. In case of Galerkin formulation, the approximation of the primary
variable is by using a continuous function inside the element. A residue is obtained
depending on the approximating function when primary variable u
e is substituted in
Eq. (3.75), i.e.,
Lu
e
+ q = R
(3.77)
The residue should be zero everywhere; however, it is difficult to approximate the
true value. Therefore, the weighted residual is made equal to zero, i.e.,
D
w RdA = 0
(3.78)
where w is the weight function. In order to weaken the requirement on the differentiability of the approximating function, Eq. (3.78) is integrated by parts to redistribute
the order of derivative in w and R. The weight function is of the same form as
the approximating function in Galerkin method, which is usually some algebraic
function. The unknown coefficients of the function are replaced by unknown nodal
degrees of freedom as
u
e
= [N ]
u
ne
(3.79)
where {u
ne } represents nodal degrees of freedom, and [N] represents the shape
functions matrix.
In Ritz formulation, using calculus of variation, the differential equation Eq. (3.75)
is converted in its integral form. Equation (3.79) is substituted in the integral equation,
and it is partially differentiated with respect to {u
ne } in order to extremize the form.
Assembly of the elemental equations is performed in the next step. The equations
of each element are written in global form and added to each similar equation of
all the elements, i.e., all first equations of each element are added to obtain the
assembled first equation and so on. After the boundary conditions are applied, the
equations are solved by suitable solver. It is followed by post-processing [25].