13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- mi i mi i mi i i mi i d d d d d d d d w 0 , 2 cos2 2 2 ＋ x （4） where i id x x is the Euclidian distance between the point xand node i x , and is a small number to avoid the numerical difficulty resulting from the singularity at nodes. On each cover iC , a local approximation is first expressed as m k k k l i p a u 1 T ( ) ( ) ( ) ( ) x x P x a x （5） wheremis the number of terms in the basis, 1, , , ...... T x y xy p x are the nominal basis functions, ia is their coefficients. Here, the bilinear basis [1, , , ] T x y xy p x is used as an example to introduce the construction of the MSIM interpolation. 2.1 Cover interpolation for the nodes not located on the essential boundary For the typical nodei not located on the essential boundary, the local cover approximation is directly taken as a a x a y a xy u u i i i i i l i 4 3 2 1 ln ( ) ( ) x x （6） Let the cover function at the nodei satisfy the condition 0 4 3 2 1 ln ( ) i i i i i i i i i i iux a axayaxyu （7） Hence, i i i i i i i i i a u a x a y a x y 4 3 2 0 1 （8） Substituting Eq. (8) into Eq. (3) leads to ) ) ( ) ( ( ( ) 4 3 0 2 ln i i i i i i i i iu u axx ayy axyxy x i x iΨΤ （9） where ] [ 4 3 2 1 i i i i i x Ψ ] [1 i i i i x x y y xy x y （10） T i i i i i x u a a a ] [ 4 3 0 2 Τ （11） Similarly, the y-displacement interpolation ( ) ln i iv x is given by ) ) ( ) ( ( ( ) 4 3 0 2 ln i i i i i i i i i v bxx byy bxyxy v x i y i ΨΤ （12） where T i i i i i y v b b b ] [ 4 3 2 0 Τ （13） Eq. (9) and (12) can be rewritten in the form i i i i i v u ΨΤ x x u x ln ln ln （14） where i i i i i i i i i xy xy y y x x xy xy y y x x 0 0 0 1 0 0 0 0 1 0 Ψ （15） T i i i i i i i i i u v a b a b a b 0 0 2 2 3 3 4 4 Τ （16） ( , ) 0 0i iu v are the nodal displacement of node i , ( , , , , , ) 4 4 3 3 2 2 i i i i i i a b a b a b are the extra freedoms of cover iC . From Eq. (14), we observe that
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