Crack Paths 2012

The 4th International Conference on “Crack Paths”

Where uR(ξ) is the approximation function, ῼ is the domain of intrest, ϕa(ξ-x) is a kernel

function, and a is the smoothing parameter. This method is not accurate on the boundary

condition or when few particles are considered on the domain unless the lumped volume is

carefully selected which is a very hard and time consuming work. R K P Mis an alternative

method to formulate the discrete consistency that is lacking in SPHmethod. The foundation of

R K P Mproposed by Liu et al. [1] in 1993 and was first applied to computational mechanics.

R K P Mmodifies the kernel function by introducing a correction function C(ξ;ξ-x). Adding the

correction function in the kernel approximation significantly enhances the solution accuracy in

comparison to SPHmethod. The method of using corrected kernel approximation in reproducing

a function is called Reproducing Kernel Particle Method. The reproduced kernel function of u(x)

can be written as:

[ [ I

uR

³ : d x x x u ; ) (

[

(3)

Where ϕa(ξ;ξ-x) is the modified kernel function on domain ῼ that is expressed by:

)4(

x C x x I [ [ [ [ I [

[ I

ax

¨©§

x a [ I ) 1 ( i a

i

(5)

¸¹·

Where ϕa(ξ-x) is window function, C(ξ;ξ-x) is correction function, and a is the dilation parameter

of the kernel function. Dilation parameter is defined in order to make more flexibility for the

window function and this parameter will control the expansion of the window function on the

domain. The correction function C(ξ;ξ-x) proposed by Liu et al. [1] is shown by a linear

combination of polynomial including some unknown coefficients. These unknown coefficients

will be computed after imposing the boundary conditions. Consider the following Taylor series

expansion in order to get the equations for reproducing an arbitrary function:

D D

D

f

1

u x

x u

[

¦

[

D

)(

!

(6)

D

0

Substituting Equation 6 into Equation 3 leads to:

x d x u D I [ [

D

D

D

·

: §

f

1

;

0

!

R

a

D ¦ ³ ¨

[

[

u

x

¸

(7)

©

¹

In order to simplify the Equation 7, the αth degree moment matrix of function ϕa(ξ;ξ-x) is

defined by:

m a [ [ I [ [ D

d x x x

³ :

;

D

(8)

Then the Equation 7 will be rewritten in the following form:

887