11 Meshless Algorithms for Computational Biomechanics of the Brain
289
The formula for determining the bounds for maximum eigenvalue λ I
max of the
stiffness matrix given in Eq. (11.24) is valid also for the finite element method,
as long as the same mass lumping scheme is used. Therefore, it applies also to
the eight-noded hexahedral element with a single integration point we discussed in
Chap. 10 as the element of choice for computational biomechanics of the brain.
11.6 Algorithm Verification
We follow the verification approach introduced in Chap. 10 where the results
obtained by means of new algorithms of non-linear computational mechanics
are compared with the solutions from the established algorithms. However, none
of the existing weak form meshless methods of computational mechanics has
been recognised by the research community as a gold standard yet. Therefore,
following Chap. 10, also in this chapter, we use the results obtained from the
established algorithms implemented in commercial non-linear finite element codes
as a reference solution.
In the following sections, we present verification results for some of the
algorithms described in this chapter: Meshless Total Lagrangian Explicit Dynamics
(MTLED) framework, Modified Moving Least Square (MMLS) shape function for
deformation interpolation and specialised visibility criterion for modelling surgical
dissection/tissues rupture.
11.6.1 Meshless Total Lagrangian Explicit Dynamics
(MTLED) Framework
The Meshless Total Lagrangian Explicit Dynamics (MTLED) framework has been
verified by comparing the results obtained using this framework with the established
finite element code (ABAQUS implicit dynamics non-linear solver [31]) that was
used when modelling semi-confined uniaxial compression and shear of a cylinder
made from a very soft (shear modulus of 1 kPa) hyperelastic (neo-Hookean)
material. In the meshless discretisation of the cylinder, almost arbitrary node
placement and hexahedral integration cells non-conforming to the geometry were
used (Fig. 11.12).
For 20% compression and shear of the cylinder, the difference in the total reaction
force on the displaced cylinder surface between MTLED framework and ABAQUS
implicit finite element solver [31] was no more than 5% (Fig. 11.13a). The force—
time histories obtained using the meshless framework—was qualitatively similar
to those from the finite element method. The maximum relative difference in the
computed deformations between the MTLED framework and ABAQUS was around
3.5% (it can be seen in Fig. 11.13b that some of the nodes in meshless discretisation
do not sit exactly on the deformed finite element boundary).
289
The formula for determining the bounds for maximum eigenvalue λ I
max of the
stiffness matrix given in Eq. (11.24) is valid also for the finite element method,
as long as the same mass lumping scheme is used. Therefore, it applies also to
the eight-noded hexahedral element with a single integration point we discussed in
Chap. 10 as the element of choice for computational biomechanics of the brain.
11.6 Algorithm Verification
We follow the verification approach introduced in Chap. 10 where the results
obtained by means of new algorithms of non-linear computational mechanics
are compared with the solutions from the established algorithms. However, none
of the existing weak form meshless methods of computational mechanics has
been recognised by the research community as a gold standard yet. Therefore,
following Chap. 10, also in this chapter, we use the results obtained from the
established algorithms implemented in commercial non-linear finite element codes
as a reference solution.
In the following sections, we present verification results for some of the
algorithms described in this chapter: Meshless Total Lagrangian Explicit Dynamics
(MTLED) framework, Modified Moving Least Square (MMLS) shape function for
deformation interpolation and specialised visibility criterion for modelling surgical
dissection/tissues rupture.
11.6.1 Meshless Total Lagrangian Explicit Dynamics
(MTLED) Framework
The Meshless Total Lagrangian Explicit Dynamics (MTLED) framework has been
verified by comparing the results obtained using this framework with the established
finite element code (ABAQUS implicit dynamics non-linear solver [31]) that was
used when modelling semi-confined uniaxial compression and shear of a cylinder
made from a very soft (shear modulus of 1 kPa) hyperelastic (neo-Hookean)
material. In the meshless discretisation of the cylinder, almost arbitrary node
placement and hexahedral integration cells non-conforming to the geometry were
used (Fig. 11.12).
For 20% compression and shear of the cylinder, the difference in the total reaction
force on the displaced cylinder surface between MTLED framework and ABAQUS
implicit finite element solver [31] was no more than 5% (Fig. 11.13a). The force—
time histories obtained using the meshless framework—was qualitatively similar
to those from the finite element method. The maximum relative difference in the
computed deformations between the MTLED framework and ABAQUS was around
3.5% (it can be seen in Fig. 11.13b that some of the nodes in meshless discretisation
do not sit exactly on the deformed finite element boundary).