10 Finite Element Algorithms for Computational Biomechanics of the Brain
249
Four-noded tetrahedral elements with linear shape functions and one Gauss
point do not suffer from hourglassing. However, for incompressible (or nearly
incompressible) continua, such as soft tissues, four-noded tetrahedral elements
exhibit artificial stiffening known as volumetric locking [35]. A more detailed
discussion on algorithms for hourglass control and volumetric locking reduction
is provided in Sect. 10.3 Algorithms for Surgery Simulation below.
10.3 Algorithms for Surgery Simulation
As discussed in Chap. 6, surgical simulation systems are used to provide visual and
haptic feedback to a surgeon or trainee. Such systems must provide time accurate
prediction of the deformation field within an organ and interaction force between the
surgical tool and the tissue at frequencies of at least 500 Hz. From the perspective of
continuum mechanics, such prediction requires solving the problem involving large
deformations, non-linear constitutive properties and non-linear boundary conditions
within very strict time constraints of haptic feedback.
For relatively slowly varying loads, such as those that occur due to interactions
between the tissue and surgical tool, non-linear finite element algorithms utilising
implicit time stepping are traditionally recommended in the literature for solving
non-linear problems of solid mechanics [4]. Such procedures rely on solving
systems of algebraic equations and require computationally expensive iterations.
In contrast, the algorithms for injury simulation lead to an explicit formula for
unknown nodal displacements (Eq. 10.5). For such algorithms, the number of
operations per time step is typically three orders of magnitude smaller than for
the algorithms relying on implicit time stepping [23]. Despite the fact that surgical
simulations involve phenomena of duration of orders of magnitude longer than those
that are of interest in injury simulation, the restrictions on the time step size (Courant
criterion) required for solution stability (Eq. 10.6) do not compromise efficiency
of explicit time stepping in such simulations. This is because the acoustic wave
speed is proportional to square root of the analysed continuum Young’s modulus
(Eq. 10.7) which is very low (under 10 4 Pa) for the brain tissue. For instance, in
the simulation of needle insertion into the brain conducted using non-linear explicit
dynamics finite element procedures reported in Wittek et al. [21], the time step was
over 1.5·10 −2 ms. In contrast, in typical engineering applications, such as metal
forming, the integration steps are of an order of 10 −5 ms [36].
In commercial finite element codes utilising explicit time stepping, the calculated
variables (such as displacement, strain and stress) are incremented by referring
them to the current configuration of the analysed continuum, which is known as
an Updated Lagrangian formulation [4]. However, we advocate a Total Lagrangian
(TL) formulation of computational mechanics in which all variables are referred
to the original configuration of the system [37]. The decisive advantage of this
formulation is that all derivatives with respect to spatial coordinates are calculated
with respect to the original configuration and therefore can be precomputed
249
Four-noded tetrahedral elements with linear shape functions and one Gauss
point do not suffer from hourglassing. However, for incompressible (or nearly
incompressible) continua, such as soft tissues, four-noded tetrahedral elements
exhibit artificial stiffening known as volumetric locking [35]. A more detailed
discussion on algorithms for hourglass control and volumetric locking reduction
is provided in Sect. 10.3 Algorithms for Surgery Simulation below.
10.3 Algorithms for Surgery Simulation
As discussed in Chap. 6, surgical simulation systems are used to provide visual and
haptic feedback to a surgeon or trainee. Such systems must provide time accurate
prediction of the deformation field within an organ and interaction force between the
surgical tool and the tissue at frequencies of at least 500 Hz. From the perspective of
continuum mechanics, such prediction requires solving the problem involving large
deformations, non-linear constitutive properties and non-linear boundary conditions
within very strict time constraints of haptic feedback.
For relatively slowly varying loads, such as those that occur due to interactions
between the tissue and surgical tool, non-linear finite element algorithms utilising
implicit time stepping are traditionally recommended in the literature for solving
non-linear problems of solid mechanics [4]. Such procedures rely on solving
systems of algebraic equations and require computationally expensive iterations.
In contrast, the algorithms for injury simulation lead to an explicit formula for
unknown nodal displacements (Eq. 10.5). For such algorithms, the number of
operations per time step is typically three orders of magnitude smaller than for
the algorithms relying on implicit time stepping [23]. Despite the fact that surgical
simulations involve phenomena of duration of orders of magnitude longer than those
that are of interest in injury simulation, the restrictions on the time step size (Courant
criterion) required for solution stability (Eq. 10.6) do not compromise efficiency
of explicit time stepping in such simulations. This is because the acoustic wave
speed is proportional to square root of the analysed continuum Young’s modulus
(Eq. 10.7) which is very low (under 10 4 Pa) for the brain tissue. For instance, in
the simulation of needle insertion into the brain conducted using non-linear explicit
dynamics finite element procedures reported in Wittek et al. [21], the time step was
over 1.5·10 −2 ms. In contrast, in typical engineering applications, such as metal
forming, the integration steps are of an order of 10 −5 ms [36].
In commercial finite element codes utilising explicit time stepping, the calculated
variables (such as displacement, strain and stress) are incremented by referring
them to the current configuration of the analysed continuum, which is known as
an Updated Lagrangian formulation [4]. However, we advocate a Total Lagrangian
(TL) formulation of computational mechanics in which all variables are referred
to the original configuration of the system [37]. The decisive advantage of this
formulation is that all derivatives with respect to spatial coordinates are calculated
with respect to the original configuration and therefore can be precomputed