Meshless Methods Introduction - Meshless Methods in Biomechanics - page 68

If the meshless shape function possess the Kronecker delta property the
essential boundary conditions can be imposed using the same simple techniques
used in the finite element method. However, if the meshless shape functions is an
approximation functions, i.e., it does not possess the Kronecker delta property, the
imposition of the essential boundary conditions requires complex methodologies.
To simplify the exposition, only the imposition of essential boundary conditions
on elastostatic problems is addressed,

Ku ¼ f

ð 3 : 28 Þ

In the literature [ 34 ] it is possible to find detailed descriptions on the imposition
of essential boundary conditions for elastodynamic problems.

3.4.5.1 Interpolating Meshless Methods

If the meshless shape functions used in the construction of the discretized system
of equations are interpolating shape functions, such as the RPI shape functions,
then the ''direct imposition method'' or the ''penalty method'' can be used to
enforce the essential boundary conditions.
Considering the three-dimensional problem presented in Fig. 3.15 , first it is
necessary to identify the field nodes with displacement constrains. Figure 3.15
indicates that the solid displacements are constrained on the curve C u ð C Þ 2 C and
on the surface C u ð S Þ 2 C. Therefore, all field nodes belonging to C u ð C Þ or to C u ð S Þ
have to be selected: X C ¼ x 1 ; x 2 ; ... ; x NC

, being X C X and
NC the total number of field nodes on the essential boundaries. Afterwards, the
displacement constrains in each field node belonging to X C must be determined. If
each field node discretizing the solid domain possess m degrees of freedom then a
field node x i 2 X C can have from just one displacement constrain up to m dis-
placement constrains.

f g 2 C u ð C Þ [ C u ð S Þ

Direct Imposition Method

3 presents the following dis-
placement constrains: along the direction of the ox axis can move freely; along the
direction of the oy axis the displacement component is prescribed by u y ; and along
the direction of the oz axis the displacement component is prescribed by u z .
These essential boundary conditions can be enforced directly on the discrete
system of equations, Ku ¼ f , by modifying directly the stiffness matrix and the
global force vector. Being a three-dimensional problem, in which each field node
possess three degrees of freedom, the initial stiffness matrix presents the following
disposition,

It is consider that the field node x i 2 X C ^ x i 2 R