Meshless Methods Introduction - Meshless Methods in Biomechanics - page 83

2
4

3
5 ¼ Z

Z

1 1

2 ð u u Þ T a ð u u Þ dC u 2 d ð u u Þ T a ð u u Þ dC u

d

Cu Cu

¼ Z

1
2 ð du du

Þ T a ð u u Þ dC u ð 3 : 68 Þ

|{z}
0

Cu

¼ Z

2 du T a u dC u Z
Cu

1 1

2 du T a u dC u

Cu

Recall that the virtual displacement approximation du of the integration point
q I 2 Q is obtained with Eq. ( 3.41 ) and the approximated displacement components
u I ¼ ð u x u y u z Þ of the integration point q I 2 Q are obtained with Eq. ( 3.1 ). Both
expressions Eqs. ( 3.41 ) and ( 3.1 ) permit to obtain local approximations which can
be extended to global approximations using Eq. ( 3.42 ) and following the process
described in Fig. 3.17 . Thus, using the global approximation presented in

Eq. ( 3.42 ) it is possible to develop Eq. ( 3.68 ),

2
4

3
5

2
4

3
5 dC u

T

Z

2 du T a u dC u Z
Cu

2 du T a u dC u ¼ Z
Cu

1 1 1

2

H I du ð x Þ

| {z }
½3N 1

a H I u ð x Þ

|{z}
½3N 1

|{z}
½3 3

|{z}
½3 3N

|{z}
½3 3N Cu

2
4

3
5

T

Z

1
2

H I
|{z}
½3 3N

du ð x Þ
| {z }
½3N 1

a u ð q I Þ

|{z}
½3 1

dC u

|{z}
½3 3

Cu

¼ Z
Cu

1

2 du ð x Þ T

H I

u ð x Þ
|{z}
½3N 1

a H I dC u

|{z}
½3 3

|{z}
½3 3N

|{z}
½1 3N

|{z}
½3N 3

Z

1

2 du ð x Þ T

H I

a u ð q I Þ dC u

|{z}
½3 3

| {z }
½1 3N

| {z }
½3 1

|{z}
½3N 3

Cu

2 3

Z

4 5

1
2 a
|{z}
½3 3

¼ du ð x Þ T
|{z}
½1 3N

H I

H I
|{z}
½3 3N

dC u u ð x Þ

|{z}
½3N 1

|{z}
½3N 3

Cu

|{z}
K a

2 3

Z

4 5

1
2 a
|{z}
½3 3

du ð x Þ T
|{z}
½1 3N

H I

u ð q I Þ
|{z}
½3 1

dC u

|{z}
½3N 3

Cu

|{z}
f a

ð 3 : 69 Þ