Solid Mechanics Problems - Meshless Methods in Biomechanics - page 159

Table 5.4 NNRPIM acronyms
Basis functions Influence-cells

First degree Second degree

MQ-RBF V1P0 V2P0

MQ-RBF + constant unit basis V1P1 V2P1

MQ-RBF + linear polynomial basis V1P3 V2P3

MQ-RBF + quadratic polynomial basis V1P6 V2P6

First it is presented the optimization study of the MQ-RBF shape parameters
considering the NNRPIM formulation using first degree influence-cells. After-
wards the shape parameters optimization study is repeated for a NNRPIM for-
mulation assuming the second degree influence-cells. In the present subsection
only the nodal based basic integration scheme is considered, as shown in Section
' ' Basic Integration Scheme ''. To obtain the RPI shape functions four combinations
of basis functions are considered: (1) only the MQ-RBF; (2) MQ-RBF and an unit
constant basis; (3) MQ-RBF and a linear polynomial basis; (4) MQ-RBF and a
quadratic polynomial basis.
In order to identify the applied NNRPIM formulation, acronyms are used for a
better understanding. In each acronym the first two characters identifies the degree
of the influence-cell, first degree = ''V1'' and second degree = ''V2''. The
polynomial basis is identified by ''Pm'', where m is the number of monomials in
the polynomial basis. For example, the formulation using a second degree influ-
ence-cell and an unit constant basis is called: ''V2P1''. The complete list of
acronyms is presented in Table 5.4 .
In order to obtain the optimized MQ-RBF shape parameter c, the MQ-RBF
shape parameter p is fixed: p = 1.0001, and the parameter c 2

is varied between
c ¼½10 4 ; 10 1 . The solid domain of the unit square patch 1 1m 2 is discretized
in the two distinct nodal distributions presented in Fig. 5.1 . On the patch boundary
it is imposed the displacement field defined in Eq. ( 5.2 ). It is considered a unit
thickness and the material properties of the square patch are: E = 1 Pa and
t = 0.3. The problem is solved using the flow-chart of Table 5.2 and considering
the plane stress deformation theory.
The medium displacement error, E med , defined in Eq. ( 5.1 ), is determined and
the sum of the interior equivalent forces f tot is obtained with Eq. ( 5.3 ).
In Fig. 5.5 the medium displacement error E med is presented, together with f tot ,
as function of the shape parameter c, for the V1P0 NNRPIM formulation. The
results regarding the V1P1 NNRPIM formulation and the V1P3 NNRPIM for-
mulation are presented respectively in Figs. 5.6 and 5.7 .
It was observed that using the quadratic polynomial basis, V1P6 NNRPIM
formulation, generates a singular moment matrix, i.e., non invertible. Thus the
study of this polynomial basis applied to this formulation was abandoned. As
Fig. 5.5 indicates, for the V1P0 NNRPIM formulation the optimal values for the
shape

R

parameter c are c 2, however the stress field produced with this