Crack Paths 2012

aa-gfliexnrra

<4)

a.

Substituting Equation4 into Equationl leads to:

We)=2%(IQG—x)”i(é;é—x)dxi‘“l(é)

<5)

In order to simplify Equation5, the oath degree momentmatrix of function gflfif-x) is defined

by:

a.(:)= [(:—x)“i.(:;:—x)dx

(6)

Q

Then Equation5 will be rewritten in the form of Equation6:

MR (5) - Joana) + i “al? 17. (outta)

(7)

In order to exactly reproduce the nth order polynomial function, the following conditions need

to be satisfied;

a. (#1

{FAQ-0 (Jr-1,2,...,”

(8)

Or in summary:

"1(a) = 5,,

;

a = 0,1,2,...,n

(9)

If a correction function including n+1 unknowncoefficient is defined, n+1 Equations of 9 can

be satisfied simultaneously. The correction function is definedby EquationlO:

C(é,é—x)=ifla(é)(é—x)”

(10)

It can be also expressed in matrix form: C(§;§—x)=PT(§—X)fl(§)

(11)

where PT(ct - x) is a set of basic functions and including n+1 components and ,BK) is a set of

unknowncoefficient. Substituting Equationll into Equation9 and considering definition of

momentmatrix in Equation6 leads to Equation13: [gas-arr:art-“r:—xr*")i.<:—x>dx

(12)

m e

A ? ) =5“, ot=1,2,...,n

(13)

are

FromEquation13 the unknowncoefficient sets of £15) are obtained. Equation13 can also be

rewritten as Equation14.

are)

me) mare)

m...<:>> A?) =6..

(14)

are)

Or it can be shownin matrix form as Equations 15:

897