Untitled

76 Fernando Flores and Eugenio Oñate ˜
n= ϕ 0
’n
j
0i
t3 M
t 1
t 2
j
i
k
original
symmetry
plane
Fig. 2. Local cartesian system for the treatment of symmetry boundary conditions
will consider symmetry planes only. This restriction can be imposed through the
definition of the tangent plane at the boundary, including the normal to the plane
of symmetry ϕ 0 ′ n that does not change during the process.
The tangent plane at the boundary (mid-side point) is expressed in terms of two
orthogonal unit vectors referred to a local-to-the-boundary Cartesian system (see
Fig. 2) defined as
0
ϕ′n, ¯ϕ′ s
(28)
where vectorϕ 0 ′ n is fixed during the process while direction ¯ϕ′ s emerges from the
intersection of the symmetry plane with the plane defined by the central element
(M). The plane (gradient) defined by the central element in the selected original
convective Cartesian system (t1, t2) is
M
ϕ′1 , ϕ M
′ 2
(29)
the intersection line (side i) of this plane with the plane of symmetry can be written
in terms of the position of the nodes that define the side (j and k) and the original
length of the side l M i ,i.e.
ϕ i ′ s = 1
l M i
(ϕk − ϕj) (30)
That together with the outer normal to the side n i =[n1, n2] T =[n · t1, n · t2] T
(resolved in the selected original convective Cartesian system) leads to
iT
ϕ′1 ϕ iT
iT
n1 −n2 ϕ′n =
′ 2
n2 n1 ϕ iT
′ s
Z
X
Y
where, noting that λ is the determinant of the gradient, the normal component of
the gradient ϕ i ′ n can be approximated by
ϕ i ′ n = ϕ0 ′ n
λ|ϕ i ′ s |
ϕ ι =ϕ 0
’n
’n
i
ϕ ’s
i
t3 ϕ Μ
’2
ϕ Μ
’1
j
i
deformed
(31)
(32)