Meshless Methods Introduction - Meshless Methods in Biomechanics - page 72

interpolating meshless methods. Additionally, the shape functions obtained with
the MLS approximation are not polynomials. Thus, the enforcement of the
essential boundary conditions in the EFG methods is not straightforward as it is in
interpolating meshless methods.
Several approaches have been studied for enforcing the essential boundary
conditions in the EFG method [ 49 ]. In the first EFGM papers [ 3 , 50 ] the essential
boundary conditions were imposed using the Lagrange multipliers method.
Afterwards, other approaches were investigated, such as using tractions as
Lagrange multipliers [ 51 , 52 ] or using the weak form of the essential boundary
conditions [ 53 ]. Another popular approach is to combine the FEM with the EFG
[ 54 , 55 ]. Other authors were able to enforce the essential boundary conditions
using the penalty method [ 56 ].
In this topic two distinct methodologies are presented: the ''Lagrange multi-
pliers'' and the ''penalty method''.

Lagrange Multipliers

The Lagrange multiplier method is a very popular technique to impose the
essential boundary conditions in meshless methods using MLS shape functions.
The essential boundary conditionu u ¼ 0 is represented by a functional, which
can be written as an integral using the Lagrange multiplier k,
Z

k T ð u u Þ dC u ð 3 : 37 Þ

Cu

Being C u 2 C a boundary on the solid domain with displacements constrains.
The integral form indicates that somehow it will be require to numerically inte-
grate the subdomain C u . Therefore, it is necessary to discretize C u with integration
points. The procedure is quite similar with the numeric integration of the traction
forces, Sect. 3.4.4 .
The three-dimensional example presented in Fig. 3.15 shows two types of
essential boundaries: an essential boundary curve C u ð C Þ 2 C and an essential
boundary surface C u ð S Þ 2 C. In a first step the field nodes on the essential
boundaries C u ð C Þ 2 C and C u ð S Þ 2 C must be identified separately: X Cu ð C Þ ¼
x 1 ; x 2 ; ... ; x N f g 2 C u ð C Þ and X Cu ð S Þ ¼ x 1 ; x 2 ; ... ; x N f g 2 C u ð S Þ , being NC and NS
the number of field nodes along the boundary curve C u ð C Þ and the boundary surface
C u ð S Þ respectively. Additionally, both sets X Cu ð C Þ and X Cu ð S Þ respect the following
conditions: X Cu ð C Þ X and X Cu ð S Þ X, being X the complete nodal distribution
discretizing the problem domain X 2

3 . Then, new integration points have to be
determined on the boundary curve C u ð C Þ and on the boundary surface C u ð S Þ .

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