Acta Mech Sin (2010) 26:247–255 DOI 10.1007/s10409-010-0338-3

RESEARCH PAPER

Incompatible numerical manifold method for fracture problems Gaofeng Wei · Kaitai Li · Haihui Jiang

Received: 28 April 2008 / Revised: 10 June 2009 / Accepted: 11 November 2009 / Published online: 6 March 2010 © The Chinese Society of Theoretical and Applied Mechanics and Springer-Verlag GmbH 2010

Abstract The incompatible numerical manifold method (INMM) is based on the finite cover approximation theory, which provides a unified framework for problems dealing with continuum and discontinuities. The incompatible numerical manifold method employs two cover systems as follows. The mathematical cover system provides the nodes for forming finite covers of the solution domain and the weighted functions, and the physical cover system describes geometry of the domain and the discontinuous surfaces therein. In INMM, the mathematical finite cover approximation theory is used to model cracks that lead to interior discontinuities in the process of displacement. Therefore, the discontinuity is treated mathematically instead of empirically by the existing methods. However, one cover of a node is divided into two irregular sub-covers when the INMM is used to model the discontinuity. As a result, the method sometimes causes numerical errors at the tip of a crack. To The project was supported by the Natural Science Foundation of Shandong Province for Excellent Young and Middle-aged Scientist (2007BS04045 and 2008BS04009) and the Natural Science Foundation of Shandong Province (Y2006B24 and Y2008A11). G. Wei (B) · K. Li Institute of Science, Xi’an Jiaotong University, Xi’an 710049, China e-mail: [email protected] K. Li e-mail: [email protected] G. Wei Institute of Mechanical Engineering, Shandong Institute of Light Industry, Jinan 250353, China H. Jiang Department of Science and Technology, Shandong Institute of Light Industry, Jinan 250353, China e-mail: [email protected]

improve the precision of the INMM, the analytical solution is used at the tip of a crack, and thus the cover displacement functions are extended with higher precision and computational efficiency. Some numerical examples are given. Keywords Incompatible numerical manifold method · Finite cover approximation theory · Fracture · Stress intensity factors · Crack tip field

1 Introduction Numerical manifold method (NMM) is a new numerical simulation method proposed by Shi [1,2] based on topological manifold and unifies both the finite element method (FEM) and the discontinuous deformation analysis (DDA), in which two sets of covers are employed; one is called mathematical cover and the other physical cover. The mathematical cover is arbitrary and is used to define the approximations of the domain, and the physical cover is portraits of the geometry and is used to define the integrations field. The nodes and elements used in the computational model are obtained from these two sets of covers. Based on the two-cover procedure, this method can be used to solve problems dealing with both continuum and discontinuities in a unified framework. The methods related to NMM have been successfully applied to the prediction of crack growth, the failure simulation of slope, the linear and non-linear analysis for heterogeneous solids [3–5], while the efforts for developing numerical manifold element (NME) are still going on. Corresponding works can be found in Refs. [6–12]. One of the difficulties in its practical applications is the choice of suitable cover functions and weight functions. A simple and efficient way is the use of finite element covers, i.e. the finite element meshes are adopted to define the finite

123

248

covers, where the weight functions of the covers are nothing but the shape functions of FEM. Shi [13] gave details of the triangular finite element covers while Shyu and Salami[14] considered quadrilateral isoparametric element covers in NMM. The quadrilateral element is generally more efficient than the triangular element and is sufficient for most cases. However, the precision and efficiency of the quadrilateral isoparametric element is sometimes not adequate for bending problem. Wilson [15] suggested the use of incompatible isoparametric element in FEM and he demonstrated that incompatible element could be effective in practical applications, and it was showed in Refs.[16–18] that the adoption of incompatible elements in NMM (INMM) could be useful in obtaining more accurate results with simple covers. The choice of cover function is also important for proper NMM analysis, in particular when the size of each block is quite large. When the INMM is used for a crack problem, a cover is divided into two irregular sub-covers at the discontinuity. As a result, the method sometimes causes errors at the tip of a crack. To improve the precision of INMM, analytical solution near the tip of a crack is used in the present paper. Similar methods were also adopted in other papers, for example, Li [19] used analytical solution to present enriched meshless manifold for two-crack modeling, and Cheng [20] provided a complex variable meshless method for fracture problems. The enrichment near the tip of a crack for linear elastic problems is achieved by the expansion of the basis functions with singular functions. In the present paper, corresponding approximation functions for INMM are extended. From the minimum potential energy principle, an incompatible numerical manifold method is presented for fracture problems, and corresponding formulae are obtained. The INMM for fracture problems has a higher precision and greater computational efficiency than the conventional numerical manifold method.

G. Wei et al.

material volume, joints, blocks and the interfaces of different materials zones. It represents material condition that cannot be chosen arbitrarily. The mathematical mesh defines a fine or rough approximation of the unknown functions. It can be a mesh of some regular pattern, or a combination of some arbitrary figures. This mesh is chosen according to the problem geometry, solution accuracy requirements, and the physical property zoning. The mathematical mesh is used for building mathematical covers that represent small regions of the whole field and can be of any shape and sizes. They can overlap each other and need not coincide with the physical mesh. However, the whole mesh has to be large enough to cover every point of the physical mesh. Overlapping these two meshes provides a manifold description. The intersection of the mathematical cover and the physical mesh, or the common region of the two systems, defines the region of physical covers. A common area of the overlapped physical covers corresponds to an element in the manifold method. Piecing together all the common areas produces a complete cover of the whole field without overlapping. The mathematical mesh and the physical mesh are generally independent. The mathematical covers reflect the physical mesh through the application of weight functions. It is worth mentioning that the covers of the manifold method can span discontinuity boundaries. In addition, the manifold method does not require a mathematical mesh to conform to the physical boundary of a problem, and the mathematical cover can be partially out of the material volume. Therefore, the same size and shape to all the covers can always be used for the complicated geometric shapes of the material volumes and joint distributions. Further details of the geometrical aspect of manifolds can be found in the works of Shi [2] and Terada [21]. Based on the concept of covering systems, the approximation method by means of finite covers can be formulated as described in the next section. 2.2 Local cover weight functions in NMM

2 Incompatible numerical manifold method The core of the manifold method is a two-mesh description, from which the concept of node and element is generalized. Similar to the partition of unity method, it also invokes the finite covering of a problem domain in its basic construct. The construct of the method and its geometric interpretation are presented below. 2.1 The finite cover systems in NMM The finite cover systems used in the manifold method are referred to as the mathematical mesh and the physical mesh, respectively. The physical mesh is a unique portrait of the physical domain of a problem, and defines the integration fields. The physical mesh includes the boundary of the

123

In the manifold method, many individual folded domains are connected together to cover the entire material domain. The global displacement functions are the weighted averages of local independent cover functions on the common part of the associated covers. Determining the weight functions is very important for successful application of NMM. Assume area  is the common part of several overlapped covers i (i = 1, 2, . . . , m). The displacement functions (u, v) of area  can be expressed as u(x, y) =

m 

wi (x, y)u i (x, y),

i=1

v(x, y) =

m  i=1

(1) wi (x, y)vi (x, y),

Incompatible numerical manifold method for fracture problems

249

in which m is the number of covers that form the manifold element, wi (x, y) is called cover weight function and satisfies the following conditions wi (x, y) ≥ 0, (x, y) ∈ i ,

(2)

/ i , wi (x, y) = 0, (x, y) ∈ m  wi (x, y) = 1, (x, y) ∈ .

(3) (4)

i=1

The cover weight function is practically equivalent to the shape functions in the finite element analysis. u i (x, y) and vi (x, y) are the cover displacement functions, they can be constant basis functions, linear basis functions or series function. Suppose u i (x, y) and vi (x, y) are the linear basis functions expressed as ⎛ ⎞ di1   ⎜  ⎟ 1 0 x 0 y 0 ⎜ di2 ⎟ u i (x, y) (5) = ⎜ .. ⎟ = S · D i . 010x 0y ⎝ . ⎠ vi (x, y) di3 Theoretically, any shape of cover can be used in NMM. However, since the weight function, the common area and integration are all related to the cover shape, a reasonable choice of cover shape is very important for the actual implementation of NMM. It is generally convenient to use the triangular finite element covers and quadrilateral finite element covers. Suppose triangular domain  is the common part of three covers i, j, k, the weight functions of the triangular are ⎞ ⎡ ⎤⎛ ⎞ ⎛ f 11 f 12 f 13 1 we(1) (x, y) ⎝ we(2) (x, y) ⎠ = ⎣ f 21 f 22 f 23 ⎦ ⎝ x ⎠ , (6) f 31 f 32 f 33 we(3) (x, y) y where ⎡

⎤ ⎡ f 11 f 12 f 13 xe( j) ye(k) − xe(k) ye( j) 1 ⎣ f 21 f 22 f 23 ⎦ = ⎣ ye( j) − ye(k)  f 31 f 32 f 33 xe(k) − xe( j)

and displacement function of the manifold element for quadrilateral element covers can be written as      ui u N1 0 N2 0 N3 0 N4 0 u= = vi 0 N1 0 N2 0 N3 0 N4 v =

4 

N i (SS · D i ) = T · D ,

(11)

i=1

where T and D are, respectively, displacement matrix and degree of freedom vector, expressed as   N1 S 0 N2 S 0 N3 S 0 N4 S 0 , (12) T = 0 N1 S 0 N2 S 0 N3 S 0 N4 S and D 1 , D 2 , D 3 , D 4 )T D = (D = (u i1 , vi1 , u i2 , vi2 , u i3 , vi3 , u i4 , vi4 )T .

(13)

2.3 Incompatible numerical manifold method

In INMM, with the additional displacement N¯ λ e added to the displacement functions T D D, the displacement functions can be expressed as ⎤ xe(k) ye(i) − xe(i) ye(k) xe(i) ye( j) − xe( j) ye(i) ⎦, ye(k) − ye(i) ye(i) − ye( j) (7) xe(i) − xe(k) xe( j) − xe(i)

and

   1 xe(i) ye(i)     =  1 xe( j) ye( j)  ,  1 xe(k) ye(k) 

It is well known that Eq. (9) satisfies Eqs. (2), (3) and (4). We can take wi (x, y) = Ni (ξ, η) as the cover weight functions of quadrilateral covers in NMM. In considering Eq. (9), the cover displacement functions (5) can be expressed as ⎞ ⎛ di1   ⎜  ⎟ 1 0 ξ 0 η 0 ⎜ di2 ⎟ u i (ξ, η) = (10) ⎜ .. ⎟ = S · D i , vi (ξ, η) 010ξ 0η ⎝ . ⎠ di6

u h = T D + N¯ λ e , (8)

(14)

where T D is compatible displacement function and N¯ λ e is incompatible displacement function, λ e is internal degree of freedom. Substituting Eq. (14) into strain equation, we get the strains in an element

where xe(i) , xe( j) , xe(k) , ye(i) , ye( j) , ye(k) are coordinate values at three stars, respectively. So we can take the weight functions in NMM as the shape functions in FEM. In quadrilateral element covers, the shape functions of isoparametric element are

ε = ε e + ε λ = B D + B¯ λ e ,

1 (1 + ξ0 )(1 + η0 ), (i = 1, 2, 3, 4), 4 where ξ0 = ξi ξ , η0 = ηi η.

where ε e and ε λ are strains for compatible part and incompatible part, respectively, and B and B¯ are strain matrix for compatible part and incompatible part, respectively.

Ni =

(9)

(15)

123

250

G. Wei et al.

Displacement function (14) in an manifold element must satisfy the conditions of patch test  ε λ dV = 0.

(16)

Ve

The incompatible displacement functions can be obtained from Eq. (16). For quadrilateral element mathematical covers, supposing xi , yi are coordinate values at four nodes, we can define 1 xi ξi , 4 4

a1 = a2 = a3 =

1 4 1 4

i=1 4  i=1 4 

⎡

⎤ 0 ⎥   ∂ ⎥ ⎥ uλ ⎥ ∂ y ⎥ vλ ∂ ⎦ ∂y ∂x ⎤ ⎡ ∂ ∂ p11 + p12 0 ⎥  ⎢ ∂ξ ∂η ∂ ∂ ⎥ 1 ⎢ ⎥ uλ ⎢ 0 p21 + p22 = . ⎥ ⎢ ∂ξ ∂η ⎥ vλ |JJ | ⎢ ⎦ ⎣ ∂ ∂ ∂ ∂ p21 + p22 p11 + p12 ∂ξ ∂η ∂ξ ∂η (23)

∂ ⎢ ∂x ⎢ ⎢ ελ = ⎢ 0 ⎢ ⎣ ∂

Usually we suppose that u λ and vλ have the same form, so Eq. (18) can be equaled as

xi ηi ,

(17)

1 1  p11 −1 −1

xi ξi ηi .

1 1 

i=1

b j ( j = 1, 2, 3) has the same definition, the difference is just replacing xi by yi . Introducing isoparametric coordinate into Eq. (16), we have 1 1 ε λ |JJ |dξ dη = 0,

(18)

−1 −1

−1 −1

 J =



(19)

T  K eDλ = K eλD = 

J11 = a1 + a3 η, J12 = b1 + b3 η, J21 = a2 + a3 ξ, J22 = b2 + b3 ξ.

K eλλ =

J

P eD =

J

−1

1 = |JJ |



 p11 p12 , p21 p22



ε λ can be given as

123

(22)

B T E B¯ dV,

Ve

(26b)

(26c) 

N T f dV +

P eλ =

For convenience, Eq. (21) can be written in the simplified form as

(26a)

T B¯ E B¯ dV,

Ve

(21)

dξ dη = 0.

Ve



(20)



1 J22 −J12 . = |JJ | −J21 J11



Ve

So we can get Jacobian inverse matrix as 

(24)



B TE B BdV, Ve

J11 J12 , J21 J22

dξ dη = 0,



K eD D =

where

−1

∂u λ ∂u λ p21 + p22 ∂ξ ∂η



From Eq. (24) we can get u λ and vλ . Substituting Eqs. (14) and (15) into minimum potential energy principle, we obtain the following discrete form    e   e D PD K D D K eDλ = , (25) K eλD K eλλ P eλ λe where

where J is Jacobian matrix, and can be expressed as

∂u λ ∂u λ + p12 ∂ξ ∂η

N T T dS,

(26d)

T N¯ T dS,

(26e)

Sσe T N¯ f dV +



Sσe

in which E is elastic matrix, f and T are the body force and the prescribed tractions, respectively. From the second term in Eq. (25), we get −1  e   P λ − K eλD D . λe = K eλλ (27) Substituting Eq. (27) into Eq. (25), we can eliminate internal degree of freedom λ e , and obtain the system of equations for incompatible NMM as

Incompatible numerical manifold method for fracture problems

K e D = P e,

251

(28)

where K = e

K eD D

−

K eDλ



−1 K eλλ

K eλD ,

 −1 e P e = P eD − K eDλ K eλλ P λ.

(29a) (29b)

The above expressions are the explicit forms containing internal displacement terms.

3 Incompatible numerical manifold method for fracture problems

1 Q 11 (x, y) = 2G 1 Q 12 (x, y) = 2G 1 Q 21 (x, y) = 2G







1 Q 22 (x, y) = − 2G

  θ r 2 θ cos κ − 1 + 2 sin , 2π 2 2

(32a)

  r θ θ κ + 1 − 2 cos2 , sin 2π 2 2

(32b)

  θ r 2 θ sin κ + 1 + 2 cos , 2π 2 2

(32c)



  r θ θ κ − 1 − 2 sin2 , (32d) cos 2π 2 2

where r is the distance from the point to the tip of the crack, θ is the angle from the tangent to the crack path at the crack tip shown in Fig. 1, and G is the shear modulus ⎧ ⎨ 3 − 4ν, (plane strain), (33) κ = 3−ν ⎩ , (plane stress), 1+ν

An incompatible numerical manifold formulation for fracture problems is proposed in this section. There are many methods for improving its accuracy: for example, intrinsic enrichment and extrinsic enrichment etc., and in this paper we will use the enriched trial function method.

in which κ is the Kolosov constant.

3.1 The displacement field near the tip of a crack

3.2 The enriched displacement function in the incompatible NMM for fracture problems

For two-dimensional fracture problems, the displacement field near the tip of a mixed mode crack is 

  r θ θ cos κ − 1 + 2 sin2 2π 2 2    θ θ K II r sin κ + 1 + 2 cos2 , + 2G 2π 2 2    KI θ r 2 θ v(x, y) = sin κ + 1 − 2 cos 2G 2π 2 2    θ K II r 2 θ cos κ − 1 − 2 sin . − 2G 2π 2 2

u(x, y) =

KI 2G

(30)

(31)

The above terms can be described as Q 1α (x, y) and Q 2α (x, y), (α = 1, 2)

In the enriched displacement function method, the displacement function for fracture problems can be intrinsically enriched by incorporating the enrichment functions. For fracture problems, one can incorporated the asymptotic near-tip displacement field into Eq. (32), or its important ingredient, √ such as r , into Eq. (10). The choice of displacement functions depends on the coarse-mesh accuracy desired. There are two methods: (1) full enrichment, where the entire near-tip asymptotic displacement field is included in the displacement functions for higher accuracy; and (2) radial enrichment, √ where only the r function is included in the displacement functions for higher speed at some cost of accuracy. In the full enrichment of incompatible manifold approximations for fracture problems, it can be shown that all the functions in Eq. (33) are spanned by the cover displacement functions

    √ √ θ θ 1 0 ξ 0 η 0 0 0 r sin r cos ⎢ 2 2     S=⎢ ⎣ √ √ θ θ 0 010ξ 0η 0 r sin r cos 2 2 ⎤     √ √ θ θ sin(θ ) 0 sin(θ ) 0 r sin r cos ⎥ 2 2 ⎥.     ⎦ √ √ θ θ sin(θ ) 0 sin(θ ) 0 r sin r cos 2 2 ⎡

(34)

123

252

G. Wei et al.

where L is differential operator as follows ⎤ ⎡ ∂ 0 ⎥ ⎢ ∂x ⎥ ⎢ ⎢ 0 ∂ ⎥ L=⎢ ⎥, ∂y ⎥ ⎢ ⎣ ∂ ∂ ⎦

(41)

∂y ∂x and the stress-strain law is σ = Eε,

Fig. 1 Local coordinate system at crack tip

This cover displacement function can be used in Eq. (10) and leads to approximations of the same form as Eq. (11). We can also use the partial enriched displacement func√ tion method, and use the function r to expand the cover displacement functions, i.e.   √ 10ξ 0η0 r 0 √ , (35) S= r 010ξ 0η 0 where r is the radial distance from the crack tip. This enrichment is useful because the angular variation around the crack tip is smooth, but the radial variation is singular in the stress.

(42)

which can be used to write the variational form in Eq. (39) in terms of the displacements u h . The discrete form can be obtained by using Eq. (34) for full enrichment, or Eq. (35) for local enrichment. This leads to the system of Eq. (25). The essential boundary conditions are often enforced by the penalty method since the boundary of the problem domain does not necessarily coincide with the mathematical cover boundary in the incompatible manifold method. The term dWu (uu h ) in Eq. (39) is used to enforce the essential boundary conditions by the penalty method  h (43) δWu (uu ) = α δuu h (uu h − u¯ )dSu ,

3.3 The incompatible NMM for fracture problems

where α is a penalty parameters. In computations, the penalty parameter is taken in the range of α = 103 –107 .

Consider a two-dimensional domain  bounded by S. The equation of equilibrium is

4 Numerical examples

σi j, j + f i = 0, in , i = x, y,

j = x, y,

(36)

where σi j is the stress tensor and f i is the body force. The boundary conditions are

Two examples are used to illustrate the effectiveness of the enriched incompatible numerical manifold formulation for fracture problems.

σi j n j = t¯i , on Sσ ,

(37)

u i = u¯ i , on Su ,

4.1 Infinite plate with a hole

(38)

where u¯ i and t¯i denote, respectively, the prescribed displacements and tractions, n j is the unit outward normal to S, Sσ and Su are complementary parts of S where essential and nature boundary conditions are prescribed. The variational form for Eq. (36) can be written as   h h ∇ δuu σ (uu )d − δuu h · f d 

 −



δuu · t¯dS − δWu (uu h ) = 0, h

(39)

Sσ

∇ is the symmetric gradient operator. The term δW (uu h ) where∇ is required for enforcing the essential boundary conditions in the incompatible manifold method and will be discussed below. For linear elasticity, the strain-displacement equation is ε = Luu ,

123

(40)

An infinitely long plate with a hole at the centre is subjected to a remote σ0 = 1.0 distributed traction in the longitudinal x-direction. The solution of this problem is given in Ref. [22] as   3a 4 a2 3 cos(2θ ) + cos(4θ ) + 4 cos(4θ ),(44) σx x = 1 − 2 r 2 2r   a2 1 3a 4 σ yy = − 2 cos(2θ ) − cos(4θ ) − 4 cos(4θ ), (45) r 2 2r   a2 1 3a 4 σx y = − 2 sin(2θ ) + sin(4θ ) + 4 sin(4θ ), (46) r 2 2r where a is the radius of the hole. A discrete model of the linear elastic problem is constructed by applying the exact tractions corresponding to Eqs. (44)–(46) on the boundaries, as shown in Figs. 2 and 3. Owing to symmetry only a quarter of the plate is modeled. The dimensions and parameters used are a = 1.0, square plate 10 × 10, Young’s modulus

Incompatible numerical manifold method for fracture problems

253

Fig. 2 The problem of infinite plate with a hole Fig. 5 σ y distribution

Fig. 3 Computational model using a quarter of the plate

Fig. 6 A rectangular plate with an edge crack

4.2 Near-tip crack problem

Fig. 4 σx distribution

E = 106 and Poisson’s ratio ν = 0.3. The enriched incompatible numerical manifold method is used to solve the plane strain problem. The discretization contained 441 nodes and 400 integration cells, and 4 × 4 Gaussian quadrature was used for a cell. The exact and numerical solutions are shown in Figs. 4 and 5. The results show that the enriched cover displacement functions significantly improve the accuracy of the solution compared to that using the non-enriched cover displacement functions.

A rectangular plate with an edge crack is shown in Fig. 6. The plate is loaded in tension at the top with σ = 0.2 GPa, and essential boundary conditions are applied at the bottom of the plate. The parameters used in the numerical simulations are L = 52 mm, D = 20 mm, a = 12 mm, E = 76 GPa, and ν = 0.286 while plane strain is assumed. The closed form solution for the crack is obtained by using the well-known near tip field in a domain around the crack tip and prescribing the displacements along the boundary of this field. The computed mode I stress intensity factors are compared with √ a finite geometry corrected value K I = Cσ aπ where the correction is given in Ref. [23] as C = 1.12 − 0.231(a/L) + 10.55(a/L)2 −21.72(a/L)3 + 30.39(a/L)4 .

(47)

This enriched method and the finite element method are used in the numerical simulations. For the finite element method, 1,722 regular elements are used to form the mesh

123

254

G. Wei et al.

Fig. 7 σx versus the distance from the crack tip

Fig. 8 σ y versus the distance from the crack tip

system, whereas the node system for the solution with enriched cover displacement functions can be given with a reduced node number of 860. √ The stress intensity factor K I is normalized by σ aπ , and the normalized K I value given by the finite element method and the enriched method are 1.32 and 1.42, respectively. The analytical solution is 1.44, and thus their relative errors are 8.3% and 1.4%, respectively. It indicated clearly that the enrichment of cover function results in a higher accuracy of solution and requires less number of nodes. Without enrichment, incompatible numerical manifold method requires considerable nodal refinement near the crack tip to capture the singular stress field to achieve sufficient accuracy. The singular stress cannot be adequately captured by the regular incompatible numerical manifold method without enriching the cover displacement function of the nodes around the crack tip. The stress field in front of the crack tip is shown in Figs. 7 and 8, where small fluctuation can be seen in the response curves. It can also be seen that the singularity at the crack tip is better modeled by the enriched method. Therefore, the enriched incompatible numerical manifold method can better capture the singularity and reduce fluctuations at the crack tip, even with a much reduced node number. The displacement field around the crack tip is shown in Fig. 9, where the two approximate solutions are also found in good agreement with the exact solution.

Fig. 9 Displacement distribution on the crack

5 Conclusions The enriched incompatible numerical manifold method for crack problem is presented in the present paper. Compared with the conventional finite element method, the finite cover approximation theory provides test functions that are not affected by the discontinuity in the solution domain. Therefore, this method can overcome difficulties of the conven-

123

tional finite element method for problems with a discontinuous domain. The enriched incompatible numerical manifold method has both the advantage of incompatible numerical manifold method and finite element method, and the ability in treating local problems. When problems of crack propagation are solved with the method, each node in the affected domain is separated into two or more nodes. All nodes that are not affected by the crack remain unchanged. As a result, arbitrary crack growth can be easily treated. The enrichment of cover displacement functions can also be used to solve problem with local singularity. The effectiveness of the proposed method in reducing computational degrees of freedom and increasing the numerical accuracy are demonstrated by numerical examples.

References 1. Shi, G.H.: Manifold method of material analysis. In: Transactions of the 9th Army Conference on Applied Mathematics and Computing. pp. 57–76. U.S. Army Research Office, Minneapolis MN (1991)

Incompatible numerical manifold method for fracture problems 2. Shi, G.H.: Numerical manifold method. In: Proceedings of the 2nd International Conference on Analysis of Discontinuous Deformation, pp. 1–35. Kyoto (1997) 3. Tsay, R.J., Chiou, Y.J., Chuang, W.L.: Crack growth prediction by manifold method. J. Eng. Mech. 125, 884–890 (1999) 4. Lin, J.S., Ku, C.Y.: Simulation of slope failure using a meshed based partition of unity method. In: 15th ASCE Engineering Mechanics Conference, New York (2002) 5. Li, S.C., Cheng, Y.M. : Meshless numerical manifold method based on unit partition. Acta Mech. Sin. 36, 496–500 (in Chinese) (2004) 6. Li, S.C., Cheng, Y.M., Wu, Y.F.: Numerical manifold method based on the method of weighted residuals. Comput. Mech. 35, 470– 480 (2005) 7. Jeen-Shang, L.: A mesh-based partition of unity method for discontinuity modeling. Comput. Methods Appl. Mech. Eng. 192, 1515– 1532 (2003) 8. Yaw-Jeng, C., Yu-Min, L., Ren-Jow, T.: Mixed mode fracture propagation by manifold method. Int. J. Fract. 114, 327–347 (2002) 9. Kenjiro, T., Asai, M., Yamagishi, M.: Finite cover method for linear and non-linear analyses of heterogeneous solids. Int. J. Numer. Methods Eng. 58, 1321–1346 (2003) 10. Zhang, H.W., Zhou, L.: Numerical manifold method for dynamic nonlinear analysis of saturated porous media. Int. J. Numer. Anal. Methods Geomech. 30, 927–951 (2006) 11. Chen, G.Q., Ohnishi, Y., Ito, T.: Development of high order manifold method. Int. J. Numer. Methods Eng. 43, 685–712 (1998) 12. Li, S.C., Cheng, Y.M. : Manifold method and its application to rock mechanical. Adv. Mech. 34(4), 446–454 (in Chinese) (2004) 13. Shi, G.H.: Manifold method. In: Proceedings of the First International Forum on Discontinuous Deformation Analysis (DDA) and Simulations of Discontinuous Medium, pp. 52–204. New Mexico, USA (1996)

255 14. Shyu, K., Salami, M.R.: Manifold with four-node isoparametric finite element method. In: Working Forum on the Manifold Method of Material Analysis, pp. 165–182. California, USA (1995) 15. Wilson, E.L., Taylor, R.L., Doherty, W., Ghaboussi, J.: Incompatible displacement models. In: Fenves, S.T. (ed.) Numerical and Computer Methods in Structural Mechanics, pp. 43–57. Academic Press, New York (1973) 16. Cheng, Y.M., Zhang, Y.H., Chen, W.S.: Wilson non-conforming element in numerical manifold method. Commun. Numer. Methods Eng. 18, 877–884 (2002) 17. Wei, G.F., Feng, W.: An incompatible numerical manifold method for elasticity problem. Chin. J. Theor. Appl. Mech. 38(1), 79–88 (in Chinese) (2006) 18. Wei, G.F., Feng, W. : Wilson incompatible numerical manifold method. Rock Soil Mech. 27(2), 189–192 (in Chinese) (2006) 19. Li, S.C., Li, S.C., Cheng, Y.M.: Enriched meshless manifold method for two-crack modeling. Theor. Appl. Fract. Mech. 44, 234–248 (2005) 20. Cheng, Y.M., Li, J.H.: A complex variable meshless method for fracture problems. Sci. China Ser. G 49, 46–59 (2006) 21. Terada, K., Asai, M., Yamagishi, M.: Finite cover method for linear and non-linear analyses of heterogeneous solids. Int. J. Numer. Methods Eng. 58, 1321–1346 (2003) 22. Timoshenko, S.P., Goodier, J.N.: Theory of Elasticity, 3rd edn. McGraw-Hill, New York (1970) 23. Sih, G.C.: Mechanics of Fracture Initiation and Propagation. Kluwer, Boston (1990)

123