Meshless Methods Introduction - Meshless Methods in Biomechanics - page 71

2
4

3
5

K 11 K 1 ð 3i 2 Þ K 1 ð 3i 1 Þ K 1 ð 3i 0 Þ K 1 ð 3N 0 Þ

. . . . .

K ð 3i 2 Þ 1 K ð 3i 2 Þð 3i 2 Þ K ð 3i 2 Þð 3i 1 Þ K ð 3i 2 Þð 3i 0 Þ K ð 3i 2 Þð 3N 0 Þ

K ¼

K ð 3i 1 Þ 1 K ð 3i 1 Þð 3i 2 Þ a K ð 3i 1 Þð 3i 1 Þ K ð 3i 1 Þð 3i 0 Þ K ð 3i 1 Þð 3N 0 Þ

K ð 3i 0 Þ 1 K ð 3i 0 Þð 3i 2 Þ K ð 3i 0 Þð 3i 1 Þ a K ð 3i 0 Þð 3i 0 Þ K ð 3i 0 Þð 3N 0 Þ

. . . . .

K ð 3N 0 Þ 1 K ð 3N 0 Þð 3i 2 Þ K ð 3N 0 Þð 3i 1 Þ K ð 3N 0 Þð 3i 0 Þ K ð 3N 0 Þð 3N 0 Þ

ð 3 : 35 Þ

Notice that in the penalty method, in opposition to the direct imposition
method, only the diagonal components are modified. Additionally, it is necessary
to substitute the components f ð 3i 1 Þ and f ð 3i 0 Þ of the global force vector by a
K ð 3i 1 Þð 3i 1 Þ u y and a K ð 3i 0 Þð 3i 0 Þ u z ,

T

f ¼ f 1

f ð 3i 2 Þ a K ð 3i 1 Þð 3i 1 Þ u y a K ð 3i 0 Þð 3i 0 Þ u z f ð 3N 0 Þ

ð 3 : 36 Þ

The penalty method does not permit to satisfy exactly the essential boundary
conditions. The accuracy of the solution depends on the relative magnitude of the
penalty coefficient a [ 34 , 48 ]. Therefore, the displacement constrains u y and u z are
approximately satisfied solving the discrete system of equation Ku ¼ f considering
the modified stiffness matrix and global force vector.
Due to the simplicity of the penalty method, several authors prefer this meth-
odology to numerically enforce the essential boundary conditions, regardless the
studied solid mechanics formulation of the problem, the solid dimension or the
number of degrees of freedom per node.
As in previous section, consider now that each field node x 2

d possesses m
degrees of freedom and that a generic field node x I 2 X C has a displacement
constrain u on the Jth degree of freedom. Using the penalty method the essential
boundary condition can be imposed by substituting only the stiffness matrix
diagonal component K ð m I ð m J ÞÞð m I ð m J ÞÞ by a K ð m I ð m J ÞÞð m I ð m J ÞÞ and by
replacing only the global force vector component f ð m I ð m J ÞÞ by
a K ð m I ð m J ÞÞð m I ð m J ÞÞ u. All the other stiffness matrix and global force vector
components maintain the original value.

R

3.4.5.2 Approximating Meshless Methods

Meshless methods using approximation shape functions lacking the Kronecker
delta function property, such as the MLS shape functions, cannot impose the
essential boundary conditions using the same methodologies as the ones used in