2.2 Computational Tools
43
Next, for a more compact representation, the shape functions and their gradients are
arranged into element-wise (local) row-matrices
N e :=
N 1 , . . . , N k , . . . , N n en
, B e :=
N
1 , . . . , N
k , . . . , N
n en
.
(2.60)
Likewise, the local dofs of the trial and test functions are assembled into element-wise
(local) column-matrices
u e :=
u| 1 , . . . , u| k , . . . , u| n en
t , δu e :=
δu| 1 , . . . , δu| k , . . . , δu| n en
t . (2.61)
Then the trial and test functions along with their gradients read in compact matrix
notation as
u h (ξ )| L e = N e u e and h (ξ )| L e = B e u e
(2.62)
and
δu h (ξ )| L e = N e δu e and δδ h (ξ )| L e = B e δu e .
(2.63)
Consequently, the discretized weak form takes the compact representation
n el
e=1
δu
t
e
B e
B
t
e σ h dx =
n el
e=1
δu
t
e
B e
N
t
e b dx +
∂B σ
e
N
t
e ¯
σ d∂ x
(2.64)
that may be further abbreviated by introducing element-wise (local) column-matrices
of internal and external forces
n el
e=1
δu
t
e s e =
n el
e=1
δu
t
e f e .
(2.65)
Here, the element-wise (local) column-matrices of internal and external forces are
defined as
s e :=
B e
B
t
e σ h dx and f e :=
B e
N
t
e b dx +
∂B σ
e
N
t
e ¯
σ d∂ x.
(2.66)
Note that σ h is determined from the algorithmic stress update as outlined in the
sequel of the Catalogue in much detail for each computational material model. The
element-wise (local) dofs are formally assigned to n np global dofs by Boolean matrices (containing mostly zeros and only a few ones)