42
2 Preliminaries
x ≈ x h with x h (ξ )| L e =
n en
k=1
x| k N k (ξ ).
(2.53)
An example for the (local) element-wise node and dof numbering for an element
with quadratic shape functions is depicted in Fig. 2.11.
Accordingly, the Jacobian of the isoparametric map computes element-wise
J e (ξ ) := ∂ ξ x h (ξ )| L e =
n en
k=1
x| k ∂ ξ N k (ξ ).
(2.54)
The Jacobian allows in particular to determine the total differential of the physical
coordinate
dx h | L e = J e dξ.
(2.55)
Thus, the gradient of the shape functions expands element-wise into
N
k = ∂ ξ N k J
−1
e .
(2.56)
Consequently, the gradients of the trial and test functions compute as
≈ h with h (ξ )| L e =
n en
k=1
u| k N
k (ξ )
(2.57)
and
δδ ≈ δδ h with δδ h (ξ )| L e =
n en
k=1
δu| k N
k (ξ ).
(2.58)
Accordingly, the discretized weak form expands eventually into
n el
e=1
n en
k=1
δu| k
B e
N
k σ dx =
n el
e=1
n en
k=1
δu| k
B e
N k b dx +
∂B σ
e
N k ¯
σ d∂ x
. (2.59)
Fig. 2.11 Local node and
dof numbering for an
element with quadratic shape
functions. The isoparametric
coordinate is ξ ∈ [−1, +1]
− 1
+1
x | 1
x | 2
x | 3
u | 1
u | 2
u | 3
ξ