258
A. Wittek et al.
10.5.2 Stability of Under-integrated Hexahedral Elements:
Hourglassing
As stated in Chap. 6, low-order hexahedral elements with one Gauss point (also
referred to as linear under-integrated hexahedral elements) are the preferred choice
for explicit dynamics-type algorithms from the perspective of computational efficiency. However, such elements exhibit unphysical zero energy deformation modes
(hourglass). The hourglass modes can be controlled by calculating hourglass forces
that oppose the hourglass deformation modes. We have shown in Joldes et al. [55]
that the hourglass control forces for each element can be computed (in matrix form)
as:
t
0 F
Hg
= k· 0 Y· 0 Y
T
·
t
0 u,
(10.35)
where k is a constant that depends on the element geometry and material properties,
0 Y is the matrix of hourglass shape vectors and u is the matrix of current
displacements. As we use Total Lagrangian (TL) formulation, all quantities except
u in Eq. (10.35) are constant and can be precomputed. This makes the hourglass
control mechanism very efficient.
10.6 Modelling of the Brain-Skull Interactions for
Image-Guided Neurosurgery: Efficient Finite Sliding
Contact Algorithm
Modelling of interactions between continua (e.g. soft organs) undergoing deformations is a challenging task. To facilitate such modelling, many sophisticated contact
algorithms have been proposed in the literature (e.g. [56–59].) and implemented
in commercial finite element codes such as ABAQUS [8, 60] and LS-DYNA [5].
Application of such algorithms tends to consume significant computing resources,
which substantially increases the solution time.
When computing the brain deformation for neuroimage registration, we are interested in the interactions between the brain and rigid skull that provide constraints
for the brain tissue deformation and brain rigid body motion. Accurate modelling
of such interactions can be done using a very efficient algorithm that treats these
interactions as a finite sliding, frictionless contact between a deformable object (the
brain) and a rigid surface (the skull) [61]. The main parts of such contact algorithm
(for detailed description see [61]) are detection of nodes on the brain surface (also
called the slave surface) which have penetrated the skull surface (master surface)
and the repositioning of each slave node that has penetrated the master surface to
the closest point on the master surface.
Efficient penetration detection algorithm can be formulated based on the closest
master node (nearest neighbour) approach [5]. As the surfaces of the anatomical
A. Wittek et al.
10.5.2 Stability of Under-integrated Hexahedral Elements:
Hourglassing
As stated in Chap. 6, low-order hexahedral elements with one Gauss point (also
referred to as linear under-integrated hexahedral elements) are the preferred choice
for explicit dynamics-type algorithms from the perspective of computational efficiency. However, such elements exhibit unphysical zero energy deformation modes
(hourglass). The hourglass modes can be controlled by calculating hourglass forces
that oppose the hourglass deformation modes. We have shown in Joldes et al. [55]
that the hourglass control forces for each element can be computed (in matrix form)
as:
t
0 F
Hg
= k· 0 Y· 0 Y
T
·
t
0 u,
(10.35)
where k is a constant that depends on the element geometry and material properties,
0 Y is the matrix of hourglass shape vectors and u is the matrix of current
displacements. As we use Total Lagrangian (TL) formulation, all quantities except
u in Eq. (10.35) are constant and can be precomputed. This makes the hourglass
control mechanism very efficient.
10.6 Modelling of the Brain-Skull Interactions for
Image-Guided Neurosurgery: Efficient Finite Sliding
Contact Algorithm
Modelling of interactions between continua (e.g. soft organs) undergoing deformations is a challenging task. To facilitate such modelling, many sophisticated contact
algorithms have been proposed in the literature (e.g. [56–59].) and implemented
in commercial finite element codes such as ABAQUS [8, 60] and LS-DYNA [5].
Application of such algorithms tends to consume significant computing resources,
which substantially increases the solution time.
When computing the brain deformation for neuroimage registration, we are interested in the interactions between the brain and rigid skull that provide constraints
for the brain tissue deformation and brain rigid body motion. Accurate modelling
of such interactions can be done using a very efficient algorithm that treats these
interactions as a finite sliding, frictionless contact between a deformable object (the
brain) and a rigid surface (the skull) [61]. The main parts of such contact algorithm
(for detailed description see [61]) are detection of nodes on the brain surface (also
called the slave surface) which have penetrated the skull surface (master surface)
and the repositioning of each slave node that has penetrated the master surface to
the closest point on the master surface.
Efficient penetration detection algorithm can be formulated based on the closest
master node (nearest neighbour) approach [5]. As the surfaces of the anatomical