Biomedical Engineering Reference
In-Depth Information
2 Methods
2.1 Meshless Total Lagrangian Explicit Dynamic Algorithm
The use of meshless methods is motivated by simple, automatic computational grid
generation for patient-specific simulations. Furthermore, compared with FEM, the
meshless method is more suitable for problems involving extremely large
deformations (where the mesh is easily destroyed) and boundary changes that occur
during neurosurgical procedures such as retractions, cuts, and tissue removal [
7
]. We
use the meshless total lagrangian explicit dynamic (MTLED) method described in [
8
].
Instead of elements, meshless method uses an unstructured cloud of nodes to
discretize the geometry. At each of the nodes in the problemdomain x
,weattach
a field variable that represents the displacement the node undergoes. Nodes are also
where mass exists and forces are calculated. To find the displacement of a point
x that is not a node, for example, integration point (where stresses and strains are
calculated for numerical integration), we must build shape functions based on the
geometry of nodes distribution and then perform approximation using the field
variables at nearby nodes.
To build shape functions from a cloud of unconnected nodes, we use support
domains and moving least squares [
9
] for their simplicity and robustness. For any
point x in the simulation domain, we consider a compact support domain
2
O
on x and
O
. The field variables u(x) is approximated with
u
h
find the n nodes within
ðxÞ
:
O
X
m
u
h
p
T
j
ðxÞa
j
ðxÞ
:
ðxÞ¼
(1)
j¼
1
Where p(x) is the basis function and a(x) is an m-vector of coefficient that
minimizes the energy function J. In this study, the quartic spline weight function
wdðÞ
is used.
X
n
2
wdðÞp
T
J ¼
xðÞaðxÞuxðÞ
:
(2)
i¼
1
k
x x
i
k
6
d
2
8
d
3
3
d
4
wdðÞ¼
1
i
þ
i
i
;
d
i
¼
(3)
:
r
A regular background grid is imposed over the geometry and integration is
performed in each grid cell following the general form of numerical integration
f(x) over a region V.
Z
X
fxðÞw
i
:
f ðxÞ
d
V
(4)
