10 Finite Element Algorithms for Computational Biomechanics of the Brain
257
Therefore, the convergence criterion can be defined as:
ρ
1 − ρ
· u n+1 − u n ∞ ≤ ε,
(10.33)
where ε is the imposed absolute accuracy. This convergence criterion gives an
approximation of the absolute error based on the displacement variation norm from
the current iteration.
Because our parameter estimation procedures overestimate the maximum eigenvalue A m and underestimate the minimum eigenvalue A 0 , the value of the computed
spectral radius ρ c we can use in Eq. (10.33) is lower than the real value of the
spectral radius (see Eq. 10.20). This can lead to an early termination of the iteration
process. Therefore, in Eq. (10.33) we use a corrected value of the computed spectral
radius:
ρ ∞ = ρ c + ς · (1 − ρ c ) ,
(10.34)
where ζ is a correction parameter with values between 0 and 1, defining the
maximum underestimation error for the spectral radius ρ. In our simulations we
use ζ = 0.2.
10.5 Element Formulation for Finite Element Algorithms for
Surgery Simulation and Neurosurgery Modelling
10.5.1 Volumetric Locking
As stated in Chap. 6, due to stringent computation time requirements, the finite
element meshes for models applied in surgery simulation and image registration
must be constructed using low-order elements that are computationally inexpensive.
Mixed meshes consisting of tetrahedral and hexahedral element are most convenient
from the perspective of automation of simulation process. However, the standard
formulation of the tetrahedral element exhibits volumetric locking, especially in
case of soft tissues such as the brain, which are modelled as almost incompressible
materials [43–49]. There is a number of improved linear tetrahedral elements
already proposed by different authors [50–53]. The Averaged Nodal Pressure (ANP)
tetrahedral element proposed by Bonet and Burton in [50] is computationally
inexpensive and provides much better results for nearly incompressible materials
than the standard tetrahedral element. Nevertheless, one problem with the ANP
element and its implementation in a finite element code is the handling of interfaces
between different materials. In Joldes et al. [54], we extended the formulation of the
ANP element so that all elements in a mesh are treated in a similar way, requiring
no special handling of the interface elements.
257
Therefore, the convergence criterion can be defined as:
ρ
1 − ρ
· u n+1 − u n ∞ ≤ ε,
(10.33)
where ε is the imposed absolute accuracy. This convergence criterion gives an
approximation of the absolute error based on the displacement variation norm from
the current iteration.
Because our parameter estimation procedures overestimate the maximum eigenvalue A m and underestimate the minimum eigenvalue A 0 , the value of the computed
spectral radius ρ c we can use in Eq. (10.33) is lower than the real value of the
spectral radius (see Eq. 10.20). This can lead to an early termination of the iteration
process. Therefore, in Eq. (10.33) we use a corrected value of the computed spectral
radius:
ρ ∞ = ρ c + ς · (1 − ρ c ) ,
(10.34)
where ζ is a correction parameter with values between 0 and 1, defining the
maximum underestimation error for the spectral radius ρ. In our simulations we
use ζ = 0.2.
10.5 Element Formulation for Finite Element Algorithms for
Surgery Simulation and Neurosurgery Modelling
10.5.1 Volumetric Locking
As stated in Chap. 6, due to stringent computation time requirements, the finite
element meshes for models applied in surgery simulation and image registration
must be constructed using low-order elements that are computationally inexpensive.
Mixed meshes consisting of tetrahedral and hexahedral element are most convenient
from the perspective of automation of simulation process. However, the standard
formulation of the tetrahedral element exhibits volumetric locking, especially in
case of soft tissues such as the brain, which are modelled as almost incompressible
materials [43–49]. There is a number of improved linear tetrahedral elements
already proposed by different authors [50–53]. The Averaged Nodal Pressure (ANP)
tetrahedral element proposed by Bonet and Burton in [50] is computationally
inexpensive and provides much better results for nearly incompressible materials
than the standard tetrahedral element. Nevertheless, one problem with the ANP
element and its implementation in a finite element code is the handling of interfaces
between different materials. In Joldes et al. [54], we extended the formulation of the
ANP element so that all elements in a mesh are treated in a similar way, requiring
no special handling of the interface elements.