Solid Mechanics Fundamentals - Meshless Methods in Biomechanics - page 35

2.3.4 Force Vector

The virtual work of the middle terms in Eq. ( 2.69 ) can be expressed and developed
as,

2
4

3
5 ¼ f b

dL 2 ¼ d Z
X

b du dX ð 2 : 99 Þ

and

2 3

dL 3 ¼ d Z
C

4 5 ¼ f t

t du dC ð 2 : 100 Þ

being the total force vector f defined as,

f b þ f t ¼ f ð 2 : 101 Þ

Thus, the total force vector f can be developed in a matrix form,
f ¼ Z
X

H T b dX þ Z
C

H T t dC : ð 2 : 102 Þ

2.3.5 Essential Boundary Conditions Imposition

If the shape functions of the meshless method possess the Kronecker delta prop-
erty, then the boundary conditions can be imposed directly as in the FEM. The
continuum analysis involves two types of boundary conditions, the essential
boundary conditions (displacement related) and the natural boundary conditions
(force related). Neglecting dumping effects and assuming that the matricial form of
the equilibrium equations resulting from virtual work expression, Eq. ( 2.69 ), can
be presented as,

Ku þ Mu ¼ f ð 2 : 103 Þ

Such equation can be rewritten as,

u c
u d

þ M cc u c

u d

¼ f c
f d

K cc K cd M cd

ð 2 : 104 Þ

K dc K dd M dc M dd

where u c are the unknown displacements and u d the known, or prescribed, dis-
placements. The vectors f c and f d correspond respectively to the known applied