Meshless Methods Introduction - Meshless Methods in Biomechanics - page 69

K K K K K K K K K

11 12 13 1(3 i 2) 1(3 i 1) 1(3 i 0) 1(3 N 2) 1(3 N 1) 1(3 N 0)

K K K K K K K K K 21 22 23 2(3 i 2) 2(3 i 1) 2(3 i 0) 2(3 N 2) 2(3 N 1) 2(3 N 0)

K K K K K K K K K 31 32 33 3(3 i 2) 3(3 i 1) 3(3 i 0) 3(3 N 2) 3(3 N 1) 3(3 N 0)

K K K K K K K K K (3 i 2)1 (3 i 2)2 (3 i 2)3 (3 i 2)(3 i 2) (3 i 2)(3 i 1) (3 i 2)(3 i 0) (3 i 2)(3 N 2) (3 i 2)(3 N 1) (3 i 2)(3 N 0)

K K K K K K K K K

K (31)1 i (31)2 i (31)3 i (31 (3 2) i i (31 (31) i i (31 (3 0) i i (31 (3 i N 2) (31 (3 1) i N (31 (3 i N 0)

K 0)1 K K K K K K K K

(3 i (3 i 0) 2 (3 i 0)3 (3 i 0)(3 i 2) (3 i 0)(3 i 1) (3 i 0)(3 i 0) (3 i N i N i N 0 )(3 2) (3 0)(3 1) (3 0)(3 0)

K K K K K K K K K

(3 )1 N (3 )2 N (3 )3 N (3 )(3 ) N i (3 )(3 ) N i (3 )(3 ) N i ( 3 )(3 ) N N (3 )(3 ) N N (3 NN 2)(3 0)

K K K K K K K K K

(3 N 1)1 (3 N 1) 2 (3 N 1)3 (3 N 1)(3 i 2) (3 N 1)(3 i 1) (3 N 1)(3 i 0) (3 N 1)(3 N 2) (3 N 1)(3 N 1) (3 N 1)(3 N 0)

K K K K K K K K K

(3 N 0)1 (3 N 0)2 (3 N 0)3 (3 N 0)(3 i 2) (3 N 0)(3 i 1) (3 N 0)(3 i 0) (3 N 0)(3 N 2) (3 N 0)(3 N 1) (3 N 0)(3 N 0)

ð 3 : 29 Þ

In order to impose the mentioned displacements constrains the following two
matrix lines ð 3i 1 Þ and ð 3i 0 Þ are modified,

K K K K K K K K K 11 12 13 1(3 i 2) 1(3 i 1) 1(3 i 0) 1(3 N 2) 1(3 N 1) 1(3 N 0)

K K K K K K K K K

21 22 23 2(3 i 2) 2(3 i 1) 2(3 i 0) 2(3 N 2) 2(3 N 1) 2(3 N 0)

K K K K K K K K K

31 32 33 3(3 i 2) 3(3 i 1) 3(3 i 0) 3(3 N 2) 3(3 N 1) 3(3 N 0)

K K K K K K K K K

(3 i 2)1 (3 i 2)2 (3 i 2)3 (3 i 2)(3 i 2) (3 i 2)(3 i 1) (3 i 2)(3 i 0) (3 i 2)(3 N 2) (3 i 2)(3 N 1) (3 i 2)(3 N 0)

K 0 0 0 0 1 0 0 0 0

0 0 0 0 0 1 0 0 0

K K K K K K K K K

(3 N 2)1 (3 N 2) 2 (3 N 2)3 (3 N 2)(3 i 2) (3 N 2)(3 i 1) (3 N 2)(3 i 0) ( 3 N 2)(3 N 2 ) (3 NN NN 2)(3 1) (3 2)(3 0)

K K K K K K K K K

(3 N 1)1 (3 N 1) 2 (3 N 1)3 (3 N 1)(3 i 2) (3 N 1)(3 i 1) (3 N 1)(3 i 0) (3 N 1)(3 N 2) (3 N 1)(3 N 1) (3 N 1)(3 N 0)

K K K K K K K K K

(3 N 0)1 (3 N 0) 2 (3 N 0)3 (3 N 0)(3 i 2) (3 N 0)(3 i 1) (3 N 0)(3 i 0) ( 3 N 0)(3 N 2) (3 N 0)(3 N 1) (3 N 0)(3 N 0)

ð 3 : 30 Þ

The initial global force vector presents the following components,

ð 3 : 31 Þ

The imposition of the essential boundary conditions require that the compo-
nents ð 3i 1 Þ and ð 3i 0 Þ of the global force vector are substituted respectively
by u y and u z ,

ð 3 : 32 Þ

Using the modified stiffness matrix and global force vector the discrete system
of equation Ku ¼ f is solved and the displacement constrains u y and u z are exactly