Meshless Methods Introduction - Meshless Methods in Biomechanics - page 70

This simple procedure can be applied to impose the essential boundary con-
ditions in any other solid mechanics formulations, assuming other dimension
orders and degrees of freedom. Consider now the general case in which each field
node x 2

d possesses m degrees of freedom. The field node x I 2 X C presents a
displacement constrain u on the Jth degree of freedom. Thus, the essential
boundary condition can be imposed by,

R

8
<

0 if i ¼ I ^ n ¼ J

1 if i ¼ j ¼ I ^ n ¼ k ¼ J

K ð m i ð m n ÞÞð m j ð m k ÞÞ ¼

:

K ð m i ð m n ÞÞð m j ð m k ÞÞ if i 6 ¼ I _ð i ¼ I ^ n 6 ¼ J Þ

ð 3 : 33 Þ

and

u if i ¼ I ^ n ¼ J

ð 3 : 34 Þ

f ð m i ð m n ÞÞ ¼

f ð m i ð m n ÞÞ if i 6 ¼ I _ð i ¼ I ^ n 6 ¼ J Þ

being f i ; j g ¼ f 1 ; 2 ; ... ; N g and f n ; k g ¼ f 1 ; 2 ; ... ; m g . If the meshless shape
function possesses the Kronecker delta property, then the direct imposition method
is able to enforce exactly the essential boundary conditions.
Since the RPIM and the NNRPIM are interpolating meshless methods, the
numerical examples solved in Chaps. 5 , 7 use the direct imposition method to
enforce the essential boundary conditions.

Penalty Method

The penalty method is another simple and efficient technique to enforce the
essential boundary conditions. In the penalty method the diagonal stiffness com-
ponents corresponding to the constrained degree of freedom are multiplied by a
penalty coefficient a, which is a scalar value much larger than the biggest com-
ponent of the stiffness matrix, a max ð K ij Þ . In the FEM the penalty coefficient
usually varies between 10 4 and 10 8 [ 34 , 48 ], nevertheless in meshless methods the
same magnitude range is acceptable [ 47 ]. A detail description of the penalty
method can be found in the classic FEM literature [ 34 , 48 ].
Consider again the field node x i 2 X C ^ x i 2

3 presenting the same dis-
placement constrains referred in '' Direct imposition method '': free along ox;
constrained by u y along oy; and constrained by u z along oz.
Using the penalty technique, the referred displacements constrains are enforced

R

by multiplying the respective diagonal components of the stiffness matrix,

K ð 3i 1 Þð 3i 1 Þ and K ð 3i 0 Þð 3i 0 Þ , with the penalty number a,