ARUP; ISBN: 978-0-9562121-5-3 - CMBBE 2012 - Cardiff University

In this study, we apply a meshless algorithm in which the computational grid is built
using only nodes (Fig. 1). As such meshless discretisation eliminates the need for mesh
of the inter-connected elements, the problem of computational grid distortion under
large strains disappears. Discontinuities (cracks) due to fragmentation are introduced by
appropriate reconstruction of nodal domains of influence. We provide verification of the
algorithm in terms of its ability to model discontinuities due to fragmentation in the
deforming 2-D continuum and evaluate the algorithm’s performance through modelling
of elongation and fragmentation of a sample of pia-arachnoid-complex tissue.
3. METHODS
Fig. 1 Schematic representation of
meshless discretisation facilitating
arbitrary distribution of nodes.
3.1 Meshless algorithm for modelling of soft tissue fragmentation
Basic implementation of the Meshless Total Lagrangian Explicit Dynamics (MTLED)
algorithm for computing soft tissue deformations we used in this study has been
described in [8]. Its key features are: 1) Moving Least-Squares (MLS) shape functions
for interpolation of field variables (displacements) between the nodes; 2) A regular
independent grid of integration cells for spatial integration; 3) Total Lagrangian
formulation of continuum mechanics [9]; and 4) Explicit integration in time domain
using central difference method.
Basic implementation of the MTLED algorithm by Horton et al. [8] does not facilitate
modelling of tissue fragmentation. In this study, the discontinuity in the deformation
field caused by the fragmentation was introduced by modifying the shape functions to
reflect the fact that nodes and integration points located on the opposite sides of the
crack cannot influence each other. The alterations to the shape functions were
implemented using the visibility criterion [7] in which the crack/fragmentation line (or
surface in 3-D) is considered opaque when constructing the nodal domains of influence.
To efficiently implement the Visibility Criterion, we used a level set function method
[10, 11]. Details of our implementation of a level set method for 2-D fragmentation
problems have been described in Jin et al. [12]. Therefore, only a summary is presented
in Fig. 2.
Fig. 2 The concept of level set function method applied in this
study for implementation of visibility criterion for modelling of
fragmentation induced discontinuities: The analysed domain is
divided into four subdomains (A, B, C and D) according to the
sign of two level set functions φ and ψ. Division into these four
subdomains is used to determine the position of a point and
supporting node in relation to the fragmentation (crack, notch)
line L. L1 is extension of the fragmentation line. L2 is the line
normal to the fragmentation line at the crack tip.