Biomedical Engineering Reference
In-Depth Information
3.3.2 Domain Representation
As depicted in Fig.
3.26
, the problem domain is discretized by finite elements. The
elements must be chosen, such that they characterize the governing differential
equations of the physical problem. Element shape (triangles, quadrilaterals etc.)
and element type (linear or quadratic) must represent the geometry of the domain
with desired accuracy. Finite element mesh density should be increased at domain
regions where large gradients of the solution are likely to occur.
3.3.3 Weak Form
The partial differential equations of the problem must be formulated in an integral
or weak form to develop the discrete finite element equations which lead to large
matrices that can be solved by computers.
Based on the local balance of linear momentum with respect to current configu-
ration, cf. (
3.120
), in this Chapter (
Sect. 3.2.5.2
), the balance equation is given by
r
r
S
þ
k
q v
¼
0
:
ð
3
:
347
Þ
With D
IRICHLET
and
VON
N
EUMANN
boundary conditions, i.e. displacement or
essential boundary conditions and stress or natural boundary conditions, respec-
tively, the strong form of the boundary value problem is obtained to
0
¼r
r
S
þ
k
q v
u
¼
u
on
oX
u
ð
3
:
348
Þ
t
¼
t
¼
n
S
on
oX
S
where u is the prescribed displacement field and t is the surface traction.
Introducing an arbitrary vector-valued weighting or test function w and mul-
tiplying (
3.347
) with w and integrating over the whole body domain X enforces the
integral term to vanish in a weighted integral sense (
3.349
),
Z
w
ðr
r
S
þ
k
qv
Þ
dV
¼
Z
w
ðr
r
S
Þ
dV
þ
Z
w
k dV
Z
w
v q dV
¼
0
:
X
X
X
X
ð
3
:
349
Þ
Employing the partial integration rule to express the first integral term on the
RHS of (
3.349
) and making use of symmetry of the C
AUCHY
stress tensor, deriving
from the balance of angular momentum (
3.123
), yields
