290
A. Wittek et al.
Fig. 11.12 Meshless model of a cylinder used in verification of the MTLED algorithm by Horton
et al. [7]. The nodes are indicated as (.) and integration points as (+). Note almost arbitrary node
placement. The integration points do not conform to geometry. The boundary conditions are shown
in the right-hand-side figure: the nodes on the top boundary were constrained, and the prescribed
displacement was applied to the nodes on the bottom boundary. (Adapted from Horton et al. [7])
The MTLED framework produces stable results even for very large deformations
as indicated by the energy—time histories obtained when modelling the cylinder
compressed to 20% of its original height (nominal compressive strain of 0.8)
(Fig. 11.14). For such large compression, no verification against the ABAQUS
finite solver could be done as the finite element solution became unstable. This
is also demonstrated by recent results obtained when comparing the performance
of the MTLED framework with the non-linear finite procedures available in
ABAQUS solver in modelling of indentation of cylindrical samples made from soft
incompressible material (Sylgard 527 silicone gel by Dow Corning) with properties
similar to the brain tissue (Fig. 11.15a) [68]. The MTLED framework facilitates the
solution for the indentation depth of 59% of the sample initial height, while the finite
element procedures become unstable (the solution ‘fails’) for the indentation depth
of 24% of the sample initial height (Fig. 11.15b, c).
11.6.2 Modified Moving Least Square (MMLS) Shape
Functions for Computing Soft Tissue Deformation
We verified the performance of the MTLED framework with Modified Moving
Least Square (MMLS) shape functions through application in modelling of an
extension of a 2-D rectangular specimen (dimensions 10 cm × 4 cm) (Fig. 11.16)
[27]. The results obtained using the MTLED framework were compared with wellestablished non-linear finite element code (ABAQUS static non-linear solver [31]
with default configuration was used). Detailed description is provided in Joldes
A. Wittek et al.
Fig. 11.12 Meshless model of a cylinder used in verification of the MTLED algorithm by Horton
et al. [7]. The nodes are indicated as (.) and integration points as (+). Note almost arbitrary node
placement. The integration points do not conform to geometry. The boundary conditions are shown
in the right-hand-side figure: the nodes on the top boundary were constrained, and the prescribed
displacement was applied to the nodes on the bottom boundary. (Adapted from Horton et al. [7])
The MTLED framework produces stable results even for very large deformations
as indicated by the energy—time histories obtained when modelling the cylinder
compressed to 20% of its original height (nominal compressive strain of 0.8)
(Fig. 11.14). For such large compression, no verification against the ABAQUS
finite solver could be done as the finite element solution became unstable. This
is also demonstrated by recent results obtained when comparing the performance
of the MTLED framework with the non-linear finite procedures available in
ABAQUS solver in modelling of indentation of cylindrical samples made from soft
incompressible material (Sylgard 527 silicone gel by Dow Corning) with properties
similar to the brain tissue (Fig. 11.15a) [68]. The MTLED framework facilitates the
solution for the indentation depth of 59% of the sample initial height, while the finite
element procedures become unstable (the solution ‘fails’) for the indentation depth
of 24% of the sample initial height (Fig. 11.15b, c).
11.6.2 Modified Moving Least Square (MMLS) Shape
Functions for Computing Soft Tissue Deformation
We verified the performance of the MTLED framework with Modified Moving
Least Square (MMLS) shape functions through application in modelling of an
extension of a 2-D rectangular specimen (dimensions 10 cm × 4 cm) (Fig. 11.16)
[27]. The results obtained using the MTLED framework were compared with wellestablished non-linear finite element code (ABAQUS static non-linear solver [31]
with default configuration was used). Detailed description is provided in Joldes