10 Finite Element Algorithms for Computational Biomechanics of the Brain
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the initial volume. In Eq. (10.8), the matrix of shape function derivatives B 0 and the
initial volume V 0 are constant and therefore can be precomputed (see Fig. 10.3).
The main benefits of the TLED algorithm in comparison to the explicit dynamics
algorithms using Updated Lagrangian formulation are:
• Allows precomputing of many variables involved (e.g. derivatives with respect
to spatial coordinates), Fig. 10.3.
• No accumulation of errors – increased stability for quasi-static solutions.
• Second Piola-Kirchhoff stress and Green strain are used – appropriate for
handling geometric non-linearities.
• Easy implementation of the material law for hyper-elastic materials using the
deformation gradient.
The fact that many quantities involved in the computation of nodal forces
can be precomputed leads to a significant decrease in the computational effort.
For instance, the TLED algorithm using eight-noded hexahedral under-integrated
elements requires approximately 35% fewer floating-point operations per element,
per time step than the Updated Lagrangian explicit algorithm using the same
elements [37].
10.4 Algorithms for Neurosurgery Modelling
As explained in Chap. 6, accurate warping of high-quality pre-operative radiographic images to the intra-operative (i.e. deformed) brain configuration in a process
known as nonrigid registration is a key element of image-guided neurosurgery. In
order to perform such warping, only the final (deformed during surgery) state of
the brain needs to be predicted. This requires algorithms for determining steady
state solution for the brain deformations. The deformations at the steady state must
be obtained within the real-time constraints of image-guided neurosurgery, which
practically means that the results should be available in 40–80 seconds.
As stated in Chap. 6, the neuroimage registration is a non-linear problem of
computational mechanics as it involves large deformations, non-linear material
properties and non-linear boundary conditions. However, it is a less demanding
problem than surgery simulation as only the steady state solution for deformations
is of interest, i.e. the time history of forces and deformations does not have
to be obtained. Therefore, for image registration, we advocate combining Total
Lagrangian (TL) formulation, discussed in Sect. 10.3 Algorithms for Surgery
Simulation, with Dynamic Relaxation (DR) which is an explicit iterative algorithm
that relies on introduction of an artificial mass-dependent damping term in the
equation of motion. The damping attenuates the oscillations in the transient
response, increasing the speed of convergence towards the steady state solution.
Because DR is an explicit algorithm, there is no need for solving large systems
of equations. All quantities can be treated as vectors, reducing the implementation
complexity and the memory requirements. Although the number of iterations to
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the initial volume. In Eq. (10.8), the matrix of shape function derivatives B 0 and the
initial volume V 0 are constant and therefore can be precomputed (see Fig. 10.3).
The main benefits of the TLED algorithm in comparison to the explicit dynamics
algorithms using Updated Lagrangian formulation are:
• Allows precomputing of many variables involved (e.g. derivatives with respect
to spatial coordinates), Fig. 10.3.
• No accumulation of errors – increased stability for quasi-static solutions.
• Second Piola-Kirchhoff stress and Green strain are used – appropriate for
handling geometric non-linearities.
• Easy implementation of the material law for hyper-elastic materials using the
deformation gradient.
The fact that many quantities involved in the computation of nodal forces
can be precomputed leads to a significant decrease in the computational effort.
For instance, the TLED algorithm using eight-noded hexahedral under-integrated
elements requires approximately 35% fewer floating-point operations per element,
per time step than the Updated Lagrangian explicit algorithm using the same
elements [37].
10.4 Algorithms for Neurosurgery Modelling
As explained in Chap. 6, accurate warping of high-quality pre-operative radiographic images to the intra-operative (i.e. deformed) brain configuration in a process
known as nonrigid registration is a key element of image-guided neurosurgery. In
order to perform such warping, only the final (deformed during surgery) state of
the brain needs to be predicted. This requires algorithms for determining steady
state solution for the brain deformations. The deformations at the steady state must
be obtained within the real-time constraints of image-guided neurosurgery, which
practically means that the results should be available in 40–80 seconds.
As stated in Chap. 6, the neuroimage registration is a non-linear problem of
computational mechanics as it involves large deformations, non-linear material
properties and non-linear boundary conditions. However, it is a less demanding
problem than surgery simulation as only the steady state solution for deformations
is of interest, i.e. the time history of forces and deformations does not have
to be obtained. Therefore, for image registration, we advocate combining Total
Lagrangian (TL) formulation, discussed in Sect. 10.3 Algorithms for Surgery
Simulation, with Dynamic Relaxation (DR) which is an explicit iterative algorithm
that relies on introduction of an artificial mass-dependent damping term in the
equation of motion. The damping attenuates the oscillations in the transient
response, increasing the speed of convergence towards the steady state solution.
Because DR is an explicit algorithm, there is no need for solving large systems
of equations. All quantities can be treated as vectors, reducing the implementation
complexity and the memory requirements. Although the number of iterations to