Plastic collapse analysis of cracked structures using extended isogeometric elements and second-order cone programming

Theoretical and Applied Fracture Mechanics xxx (2014) xxx–xxx

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Plastic collapse analysis of cracked structures using extended isogeometric elements and second-order cone programming H. Nguyen-Xuan a,⇑, Loc V. Tran b, Chien H. Thai c, Canh V. Le d a

Department of Mechanics, Faculty of Mathematics and Computer Science, University of Science, VNU-HCMC, 227 Nguyen Van Cu Street, Ho Chi Minh City, Viet Nam Department of Architectural Engineering, Sejong University, 98 Kunja Dong, Kwangjin Ku, Seoul 143-747, South Korea c Division of Computational Mechanics, Ton Duc Thang University, Viet Nam d Department of Civil Engineering, International University, VNU-HCMC, Viet Nam b

a r t i c l e

i n f o

Article history: Available online xxxx Keywords: Rigid-perfect plasticity Cracked structure Limit analysis Isogeometric analysis Second-order cone programming

a b s t r a c t We investigate a numerical procedure based on extended isogeometric elements in combination with second-order cone programming (SOCP) for assessing collapse limit loads of cracked structures. We exploit alternative basis functions, namely B-splines or non-uniform rational B-splines (NURBS) in the context of limit analysis. The optimization formulation of limit analysis is rewritten in the form of second-order cone programming (SOCP) such that interior-point solvers can be exploited efficiently. Numerical examples are given to demonstrate reliability and effectiveness of the present method. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction Accurate prediction of the load bearing capacity of structures plays an important role in many practical engineering problems. Traditional elastic designs cannot evaluate the actual load carrying capacity of structures and incremental elasto-plastic analyses can become cumbersome and present convergence issues for large scale structures. Therefore, various limit analysis approaches have been devised to investigate the behavior of structures in the plastic regime. Nowadays, limit analysis has become a well-known tool for assessing the safety load factor of engineering structures as an efficient direct method [1–7]. Limit analysis has emerged as an efficient approach to evaluate elastic–plastic fracture toughness and safety of fracture failure [8]. The earlier research on such a load bearing capacity of cracked structures was reported in [9]. Several analytical approaches can be found in Ewing and Richards [10] and Miller [11]. Numerical methods for assessing the safety factor of cracked structures have also been studied [8,12]. The standard finite element method enhanced with special singular elements [13] around the crack tips was proposed to accurately capture the singularity. This is well known in the literature due to its simplicity, but can lead to expensive computational cost, especially for complex cracked structures. As an alternative approach, the extended finite element method (XFEM) [14] is recently opening a new pathway for predicting ⇑ Corresponding author. E-mail address: [email protected] (H. Nguyen-Xuan).

plastic limit load of cracked structures. XFEM utilizes the Lagrange polynomials into approximation the enriched displacement field in order to capture the local discontinuous and singular fields without any meshing or the requirement of the element boundaries to align the crack faces. In addition, extended meshfree methods [15–18], phantom node method [17,19,20], and node-based smoothed extended finite element method (NS-XFEM) [21] are also potential candidates for the aforementioned issue. The other key interest in numerical assessment of limit analysis problem is mathematical programming. Discrete upper bound limit analysis results in a minimization problem involving linear or non-linear programming. Linear programming problems can be applied for piecewise linearization of yield criteria, but a necessary number of additional variables is often required. However, most of the yield criteria can be formed as an intersection of cones for which the limit analysis problem can be solved efficiently by the primal–dual interior point method [22,23] implemented in the MOSEK software package [24]. This algorithm was proved to be a very effective optimization tool for the limit analysis of structures [6,25,26], and therefore it will be used in our study. In the effort to advanced computational methodologies, Hughes et al. [27] introduced IsoGeometric Analysis (IGA) in order to integrate Computer Aided Design (CAD) and Computer Aided Engineering (CAE). The basic idea is to use same CAD basis functions as in the context of numerical analysis. While the finite element method (FEM) is most popular in CAE, the most common CAD basis functions are NURBS. One of the advantages of IGA is ability to represent exactly domains being conic sections and to handle easily

http://dx.doi.org/10.1016/j.tafmec.2014.07.008 0167-8442/Ó 2014 Elsevier Ltd. All rights reserved.

Please cite this article in press as: H. Nguyen-Xuan et al., Plastic collapse analysis of cracked structures using extended isogeometric elements and secondorder cone programming, Theor. Appl. Fract. Mech. (2014), http://dx.doi.org/10.1016/j.tafmec.2014.07.008

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high-order approximations with an arbitrary smoothness. Furthermore, B-splines (or NURBS) provide a flexible way to perform refinement and degree elevation [28]. Isogeometric analysis has been applied to a wide range of practical mechanics problems [29–44] and so on. In the framework of fracture mechanics, a so-called eXtended IsoGeometric Analysis (XIGA) is coined known as a combination of the enrichment functions through partition of unity method (PUM) with NURBS basic functions. Being different from XFEM, XIGA utilizes NURBS basis functions instead of the Lagrange polynomials. The present method inherits the following advantages of IGA: (1) retaining exact geometry at every meshing level; (2) being flexible in refinement, de-refinement, and degree elevation; and (3) archiving the continuous-order derivatives of shape functions up to C p1 instead of C 0 continuity as in the standard FEM [27]. Benson et al. [45] recently combined the XFEM approach to linear fracture analysis with higher-order NURBS basis functions which produce excellent accuracy for cracked solids. De Luycker et al. [46] proposed an isogeometric formulation using NURBS basis functions in combination with XFEM via incompatible meshes which produces high levels of accuracy with optimal convergence rates for linear fracture mechanics. Verhoosel et al. [47] applied IGA to modeling of cohesive cracks by inserting knots for discontinuous displacement field. Ghorashi et al. [48] proposed the XIGA formulation to deal with mixed-mode crack propagation problems which the results demonstrated the effectiveness and robustness of XIGA with an acceptable level of accuracy and convergence rate. In this paper, we further extend the extended isogeometric finite elements to upper bound limit analysis of cracked structures made of rigid-perfectly plastic materials. We investigate several higher-order isogeometric elements via NURBS basis functions. The resulting non-smooth optimization problem is formulated in the form of minimizing a sum of Euclidean norms, ensuring that the resulting optimization problem can be solved by an efficient second order cone programming algorithm. The reliability of the method is made for both uncracked and cracked structures. The paper is arranged as follows: a brief review of the B-spline and NURBS surface is described in Section 2. Section 3 summarizes an extended isogeometric approximation for limit analysis problem. Solution procedure is given in Section 4. Several numerical examples are illustrated in Section 5. Finally we close our paper with some concluding remarks. 2. Extended Isogeometric element for upper bound limit analysis 2.1. Kinematic formulation Consider a rigid-perfectly plastic body of area X 2 R2 with boundary C of continuous and discontinuous parts such that C ¼ Cu [ Ct [ Cc and Cu \ Ct \ Cc ¼ ø, where Cu ; Ct ; Cc are the Dirichlet and Neumann boundary and crack surface, respectively. The problem is subjected to body forces f in X and surface tractions g on Ct . The constrained boundary Cu is fixed. Let T u_ ¼ ½ u_ v_  be plastic velocity or flow fields that belong to a space V of kinematically admissible velocity fields, where u_ and v_ are the velocity components in x- and y-direction, respectively. The external work rate associated with a virtual plastic flow u_ is expressed in the linear form as [2]

_ ¼ W ex ðuÞ

Z

T f u_ dX þ

Z

gT u_ dC

ð1Þ

_ ¼ W ex ðuÞ; _ 8u_ 2 V W in ðr; uÞ

The internal work rate for sufficiently smooth stresses r and velocity field u_ is given by the bilinear form

Z X

_ dX rT eðuÞ

ð3Þ

where V denote a space of kinematically admissible velocity field defined as 2 _ on Cu g V ¼ fu_ 2 ðH1 ðXÞÞ ; u_ ¼ u

ð4Þ

where H1 ðXÞ is a Hilbert space. Furthermore, the stresses r must satisfy the yield condition for assumed material. This stress field belongs to a convex set, B, obtaining from the used field condition. For the von Mises criterion, one writes

B ¼ fr 2 R; j wðrÞ 6 0g

ð5Þ

where R is a space of symmetric stress tensor. _ ¼ 1g, the exact collapse If defining C ¼ fu_ 2 V j W ex ðuÞ multiplier kexact can be determined by solving any of the following optimization problems [49]

_ ¼ kW ex ðuÞ; _ 8u_ 2 Vg kexact ¼ maxfk j 9r 2 B : W in ðr; uÞ _ ¼maxmin W in ðr; uÞ _ r2B u2C

ð6Þ ð7Þ

_ ¼min maxW in ðr; uÞ

ð8Þ

_ ¼min DðuÞ

ð9Þ

r2B

_ u2C _ u2C

where

_ ¼ maxW in ðr; uÞ _ DðuÞ r2B

ð10Þ

in which r are the admissible stresses contained within the convex yield surface. Problems (6) and (9) are known as static and kinematic principles of limit analysis, respectively. The limit load of both approaches converges to the exact solution. Herein, a saddle point ðr ; u_  Þ exists such that both the maximum of all lower bounds k and the minimum of all upper bounds kþ coincide and are equal to the exact value kexact [49]. In our work, we concern on the kinematic formulation. Hence, problem (9) will be used to evaluate an upper-bound limit load factor using a NURBS-based isogeometric approach. For a limit analysis problem, only plastic strains rate is interested in the associated flow rule

e_ ¼ l_

@w @r

ð11Þ

where l_ is a non-negative plastic multiplier and the yield function wðrÞ is convex. The condition (11) serves as a kinematic constraint which enforces the vectors of admissible strain rates. In this work, the von Mises failure criterion is applied to plane stress problems. Hence, the plastic dissipation can be expressed as a function of strain as [1]

Dðe_ Þ ¼ r0

Z pffiffiffiffiffiffiffiffiffiffiffiffi e_ T H e_ dX

ð12Þ

X

where

Ct

X

_ ¼ W in ðr; uÞ

The equilibrium equation is then described in the form of virtual work rate as follows

2 H¼

4 2 0

3

16 7 42 4 05 3 0 0 1

ð13Þ

ð2Þ and

r0 is the yield stress.

Please cite this article in press as: H. Nguyen-Xuan et al., Plastic collapse analysis of cracked structures using extended isogeometric elements and secondorder cone programming, Theor. Appl. Fract. Mech. (2014), http://dx.doi.org/10.1016/j.tafmec.2014.07.008

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2.2. A short introduction on NURBS functions

2.3. Extended isogeometric finite elements

  A knot vector N ¼ n1 ; n2 ; . . . ; nnþpþ1 is defined as a sequence of knot value ni 2; i ¼ 1; . . . ; n þ p. An open knot, i.e, the first and the last knots are repeated p + 1 times, is used. A B-spline basis function forms C 1 continuous inside a knot span and C p1 continuous at a single knot. The B-spline basis functions are constructed by the following recursion formula

Ni;p ðnÞ ¼

niþpþ1  n n  ni Ni;p1 ðnÞ þ Niþ1;p1 ðnÞ niþp  ni niþpþ1  niþ1

with p > 0

u_ h ðxÞ ¼ ð14Þ

with p = 0,

Ni;0 ðnÞ ¼



The idea of XFEM is to introduce physical functions with a priori knowledge of the problem field to the approximation [14]. The basic difference between XFEM and FEM is that the former involves the solution of the additional parameters blended to the approximation by the partition of unity. Similar to the enrichment functions used in XFEM, the XIGA velocity field of the cracked solids can be expressed as

X X   NI ðxÞq_ I þ NJ ðxÞ HðxÞ  H xJ a_J

þ 1 if ni 6 n < niþ1 0

otherwise

J2Sc

I2S

X

NK ðxÞ

K2St

4 X

ðF a ðxÞ  F a ðxK ÞÞb_ aK

ð18Þ

a¼1

ð15Þ

Two-dimensional B-spline basis functions are defined by the tensor product of basis functions in two parametric dimensions n   and g with two knot vectors N ¼ n1 ; n2 . . . ; nnþpþ1 and n o H ¼ g1 ; g2 . . . ; gmþqþ1 as

NA ðn; gÞ ¼ Ni;p ðnÞM j;q ðgÞ

ð16Þ

Fig. 1 illustrates the set of one-dimensional and two-dimensional B-spline basis functions. To model exactly curved geometries (e.g. circles, cylinders, spheres, etc.), each control point A has additional value called an individual weight fA . We denote Non-uniform Rational B-splines (NURBS) functions which are expressed as

NA fA RA ðn; gÞ ¼ Pmn NA ðn; gÞfA A

ð17Þ

It is evident that the B-spline function is obtained when the individual weight of the control points is constant.

Fig. 2. Illustration of enriched control points for a quadratic NURBS net.

Fig. 1. 1D and 2D B-spline basis functions.

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Fig. 3. Illustrates a one-dimensional example of enrichment function for the elements cut by the crack: (a) The Heaviside function. (b) The quadratic B-spline basis functions with an open and uniform knot vector N ¼ f0; 0; 0; 13 ; 23 ; 1; 1; 1g. (c) The multiplication of Heaviside function and B-spline basis functions NH for the element at the discontinuous position n ¼ 0:45. T where N I;J;K ðn; gÞ are the NURBS basis functions and q_ I ¼ ½u_ I v_ I  is h the velocities of nodal displacements of u_ associated with the set of the control points S, additional nodal unknowns a_ J and b_ aK are associated with the set of the control points Sc whose support is cut by the crack and Sf whose support contains the crack tip, as shown in Fig. 2, respectively. We need to define two types of enrichments: the Heavisidetype enrichment HðxÞ and the tip-enrichment functions F a ðxÞ. The Heaviside function is given by

HðxÞ ¼



þ1 if ðx  x Þ  n > 0 1 otherwise

ð19Þ

where x is the projection of point x on the crack; n is normal vectors of the crack alignment in point x . Fig. 3 illustrates 1D example of the enrichment function for the elements cut by the crack. It is observed that the center element containing discontinuity at position n ¼ 0:45 is supported by shape functions N 2;2 ; N 3;2 ; N 4;2 which are determined by intersection between the basic function with the discontinuous position as shown in Fig. 3b. Thus, to model the discontinuity, just three shape functions are used to multiply with Heaviside function as plotted in Fig. 3c. The tip enrichments can be utilized as1

Fðr; hÞ ¼









 pffiffiffi pffiffiffi h pffiffiffi h pffiffiffi h h r sin ; r cos ; r sin sin ðhÞ; r cos sin ðhÞ 2 2 2 2 ð20Þ

which is defined in the polar coordinate ðr; hÞ at a crack tip. Fig. 4 presents the Gauss points distribution in the cracked structure with three types of elements. For the crack tip and slip elements that are intersected with the crack, the sub-triangulation technique as same as the XFEM is used with 7 Gauss points in each sub-triangle (black and green colors), while the Gauss points of the neighbor elements at crack tip and normal elements are ðp þ 1Þ  ðp þ 1Þ (blue color) and p  p (red color), respectively. The compatible strain rates can be expressed through the approximate velocity field as

e_ ¼

X BI d_ I

ð21Þ

I

where the strain matrix B is given by

h BI ¼ Bstd I

Benr I

i

ð22Þ

in which Bstd and Benr are the standard and enriched part of matrix B defined by

2

Bstd I

NI;x 6 ¼4 0 NI;y

3 0 7 NI;y 5; NI;x

2

6 Benr ¼4 I

NI;x wI þ NI wI;x 0 NI;y wI þ NI wI;y

0

3

NI;y wI þ NI wI;y 7 5 NI;x wI þ NI wI;x

ð23Þ 1

Such tip enrichments may not reflect sufficiently physical features of plasticity collapse zones [50] of cracked structures, but they can help to capture stability and accuracy of solutions in our approach. Future research will be necessary to improve the solution by using a plastic enrichment basis.

in which wI may represent either the Heaviside function H or the branch functions F a and d_ I is nodal velocities vector including the standard and enriched velocity unknowns.

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H. Nguyen-Xuan et al. / Theoretical and Applied Fracture Mechanics xxx (2014) xxx–xxx

Crack

5

Enriched element

Split element

Normal element Neighbour element

Fig. 4. The Gauss points distribution around the crack. The number of Gauss points of the crack tip and slip elements are 7 (black and green colors), while the Gauss points of the neighbor elements at crack tip are ðp þ 1Þ  ðp þ 1Þ (blue color) and normal elements are p  p Gauss points (red color). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

The plastic dissipation of the rigid-perfectly plastic body is computed by

Dh ðu_ h Þ ¼

Z

pffiffiffiffiffiffiffiffiffiffiffiffi

r0 e_ T H e_ dX ¼ r0

X

nel Z X e¼1

X

e

pffiffiffiffiffiffiffiffiffiffiffiffi e_ T H e_ dX

ð24Þ

The strains rate e_ is now evaluated at Gauss points over patch

Xe . Eq. (24) can hence be rewritten as qffiffiffiffiffiffiffiffiffiffiffiffiffi NG X  i jJi j e_ Ti H e_ i D ð u Þ ’ r0 w h

_h

ð25Þ

where NG ¼ nel  nG is the total number of Gauss points of the  i is problem, nG is the number of Gauss points in each element, w the weight value at the Gauss point i and jJi j is the determinant of the Jacobian matrix computed at the Gauss point i. The optimization problem (9) associated with XIGA can now be rewritten as

r0 (

s:t

3. Solution procedure 3.1. Second-order cone programming (SOCP)

i¼1

kþ ¼ min

Because the present approach uses the compatible strains rate, an upper bound solution that is derived from the problem (26) on the collapse multiplier of the original continuous problem is produced when a sufficient number of Gauss points is required.

qffiffiffiffiffiffiffiffiffiffiffiffiffi NG X  i jJi j e_ Ti H e_ i w

The above limit analysis problem is a non-linear optimization problem with equality constraints. It can be solved using a general non-linear optimization solver such as a sequential quadratic programming (SQP) algorithm (which is generalization of Newton’s method for unconstrained optimization) or a direct iterative algorithm [1]. In particular, the optimization problem can be reduced to the problem of minimizing a sum of norms by Andersen et al. [51]. In fact a problem of this sort can be reformed as a SOCP problem. The general form of a SOCP problem with N sets of constraints has the following form

i¼1

_ u_ ¼ u on Cu _ ¼1 W ex ðuÞ

ð26Þ

min

NG X ci t i i¼1

s:t: kHi t þ v i k 6 yTi t þ zi for i ¼ 1; . . . ; N

ð27Þ

Please cite this article in press as: H. Nguyen-Xuan et al., Plastic collapse analysis of cracked structures using extended isogeometric elements and secondorder cone programming, Theor. Appl. Fract. Mech. (2014), http://dx.doi.org/10.1016/j.tafmec.2014.07.008

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(a)

(b)

Fig. 5. Square plate with a central circular hole; (a) Full model subjected to biaxial uniform loads and (b) its quarter model with symmetric conditions imposed on the left and bottom edges.

programming problem. In the framework of limit analysis problems, the second-order cones are the quadratic cone

Table 1 Control points and its weights for a plate with a circular hole. I 1 2

P I;1

P I;2

P I;3

(0, 3) (0.75, 3)

(0, 5) (5, 5)

3

(0, 1)

pffiffiffi  2  1; 1

pffiffiffi  1; 2  1

(3, 0.75)

4

(1, 0)

(3, 0)

wI;1

wI;2

1

(

wI;3

1 1

1 1

(5, 5)

.pffiffiffi 1þ1 2 =2

.pffiffiffi 1þ1 2 =2

1

1

(5, 0)

1

1

1

where t i 2 R; i ¼ 1; NG or t 2 RNG are the optimization variables, and the coefficients are ci 2 R; Hi 2 RmNG ; v i 2 Rm ; yi 2 RNG , and zi 2 R. For optimization problems in 2D or 3D Euclidean space, m ¼ 2 or m ¼ 3. When m ¼ 1 the SOCP problem reduces to a linear

Cq ¼

t2R

NG

) rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi XNG 2 j t1 P t ¼ kt2!NG k i¼2 i

ð28Þ

3.2. Solution procedure using second-order cone programming The limit analysis problem (26) is a non-linear optimization problem with equality constraints. Furthermore, because H is a positive definite matrix in plane stress problems (see in Eq. (13)), the plastic dissipation function in (26) can be rewritten straightforwardly in the well-known form involving a sum of norms as

(a)

(b)

(c) Fig. 6. Coarse mesh and control net of a square plate with a circular hole: (a) Quadratic. (b) Cubic. (c) Quartic.

Please cite this article in press as: H. Nguyen-Xuan et al., Plastic collapse analysis of cracked structures using extended isogeometric elements and secondorder cone programming, Theor. Appl. Fract. Mech. (2014), http://dx.doi.org/10.1016/j.tafmec.2014.07.008

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H. Nguyen-Xuan et al. / Theoretical and Applied Fracture Mechanics xxx (2014) xxx–xxx Table 2 Convergence of limit load factor for a square plate with a central circular hole. Methods

Quadratic Cubic Quartic Exact [52]

Dh ðu_ h Þ ’

r0

NG X  i jJi j jjCT e_ i jj w

ð29Þ

i¼1

Mesh 16  8

24  12

32  16

40  20

0.8046 0.8044 0.8043 –

0.8024 0.8023 0.8022 –

0.8017 0.8015 0.8014 –

0.8013 0.8011 0.8010 0.8

where jj:jj denotes the Euclidean norm appearing in the plastic dis1=2 sipation function, i.e, jjv jj ¼ ðv T v Þ ; C is the so-called Cholesky factor of H

2

3 2 0 0 p ffiffiffi 1 6 7 3 05 C ¼ pffiffiffi 4 1 3 0 0 1

(a) 16 × 8

(b) 24 × 12

(c) 32 × 16

(d) 40 × 20

ð30Þ

Fig. 7. Four meshes of square plate with circular hole.

Table 3 Comparisons of numerical results for a square plate with a central hole. Methods

Equilibrium FEM (LB) Equilibrium FEM (LB) Reduced basis technique (LB) Element-free Galerkin (LB) Linear programming approach (LB) Mixed model Boundary element method (LB) ES-FEM (Dual algorithm) NS-FEM (Dual algorithm) Analytical IGA (UB) IGA (UB) IGA (UB)

Authors

Belytschko [55] Nguyen and Palgen [56] Gross-Weege [57] Chen et al. [58] Corradi and Zavelani [59] Zouain et al. [60] Zhang et al. [61] Tran et al. [54] Nguyen-Xuan et al. [62] Gaydon and McCrum [52] Present (Quadratic)a Present (Cubic)a Present (Quartic)a

Load cases P2 ¼ P1

P 2 ¼ P 1 =2

P2 ¼ 0

– 0.704 0.882 0.874 0.767 0.894 0.889 0.896 0.894 0.894 0.8957 0.8956 0.8955

– – 0.891 0.899 – 0.911 0.898 0.912 0.911 – 0.9112 0.9112 0.9112

0.78 0.564 0.782 0.798 0.691 0.803 0.784 0.805 0.802 0.8 0.8013 0.8011 0.8010

LB (lower bound); UB (upper bound). a Meshing 40  20.

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H. Nguyen-Xuan et al. / Theoretical and Applied Fracture Mechanics xxx (2014) xxx–xxx Table 6 Weights for a grooved rectangular plate. I

P I;1

P I;2

P I;3

P I;4

P I;5

P I;6

1 2 3 4 5

1 0.8536 0.8536 0.8536 1

1 1 1 1 1

1 1 1 1 1

1 1 1 1 1

1 1 1 1 1

1 0.8536 0.8536 0.8536 1

For convenience, a vector of additional variables qi is introduced as

2

3

q1 7 T qi ¼ 6 4 q2 5 ¼ C e_ i q3 i Fig. 8. Plastic dissipation distribution of a square plate with a circular hole.

The optimization problem (26) becomes a problem of minimizing a sum of norms as

kþ ¼ min

Table 4 The limit load factor: P2 ¼ 0 and P1 ¼ ry .

ð31Þ

r0

NG X  i jJi jti w i¼1

R=L

Heitzer [53]

Tran et al. [54]

Quadratic

Cubic

Quartic

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.8951 0.7879 0.6910 0.5720 0.4409 0.2556 0.1378 0.0565 0.0193

0.8932 0.7967 0.6930 0.5760 0.4011 0.2429 0.1277 0.0521 0.0133

0.9006 0.8013 0.7016 0.5935 0.4174 0.2533 0.1316 0.0530 0.0124

0.9005 0.8011 0.7016 0.5934 0.4169 0.2532 0.1315 0.0529 0.0121

0.9005 0.8010 0.7016 0.5933 0.4168 0.2532 0.1315 0.0528 0.0119

8 > < kqi k 6 ti 8i ¼ 1; NG s:t u_h ¼ u _ on Cu > : h _ W ex ðu Þ ¼ 1

ð32Þ

where the first constraint in Eq. (32) represents the inequality constraints of quadratic cones. The total number of variables of the optimization problem is Nv ar ¼ NoDofs þ 4  NG where NoDofs is the total number of the degrees of freedom (DOFs) of the underlying problem. As a result, the optimization problem defined by Eq. (32) can be effectively solved by the Mosek optimization package [24].

(a)

(b)

Fig. 9. A grooved rectangular plate subjected to in-plane tension load: (a) full model and (b) its haft model with symmetric conditions.

Table 5 Control points for a grooved rectangular plate. I

P I;1

P I;2

P I;3

P I;4

P I;5

P I;6

1 2 3 4 5

(0, 1) (0.26, 0.99) (0.73, 0.73) (0.99, 0.26) (1.0, 0)

(0, 4) (0.32, 3.49) (0.91, 2.32) (1.44, 0.77) (1.5, 0)

(0, 4) (0.74, 3.45) (1.5, 2.22) (1.7, 0.74) (1.75, 0)

(4, 4) (3.25, 3.46) (2.5, 2.2) (2.25, 0.75) (2.25, 0)

(4, 4) (3.67, 3.49) (3.07, 2.31) (2.55, 0.78) (2.5, 0)

(4, 1) (3.73, 0.99) (3.26, 0.73) (3, 0.26) (3, 0)

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H. Nguyen-Xuan et al. / Theoretical and Applied Fracture Mechanics xxx (2014) xxx–xxx

(a)

9

(b)

Fig. 10. Coarse mesh and control points of a grooved rectangular plate: (a) Quadratic element. (b) Cubic element.

Fig. 11. Four meshes of a grooved rectangular plate.

Table 7 The convergence of limit load factor of a grooved rectangular plate using IGA. Method

Quadratic Cubic Quartic

Mesh 68

12  16

18  24

24  32

0.5783 0.5722 0.5670

0.5667 0.5633 0.5612

0.5629 0.5608 0.5596

0.5610 0.5596 0.5593

4. Numerical validation In this section, we examine the performance of the present method through a series of benchmark problems under plane stress assumption. Rigid-perfect plastic materials are used. Quadratic, cubic and quartic NURBS elements are studied for all numerical examples.

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H. Nguyen-Xuan et al. / Theoretical and Applied Fracture Mechanics xxx (2014) xxx–xxx Table 8 Limit load factor for a grooved rectangular plate. Authors

Collapse multiplier

Nature of solution

Yield criterion

Prager and Hodge [63] Casciaro and Cascini [65] Yan [64] Yan [64] Vu [12] Tran et al. [54] Nguyen-Xuan et al. [62] Present (Quadratic)  Present (Cubic)  Present (Quartic) 

0.5 0.568 0.5⁄–0.577  0.558 0.557 0.562 0.559 0.5610 0.5596 0.5593

Analytical Numerical Analytical Numerical Numerical Numerical Numerical Numerical Numerical Numerical

Tresca von Mises von Mises von Mises von Mises von Mises von Mises von Mises von Mises von Mises

⁄ And   are the lower bound and upper bound solutions, respectively.

(a) Problem model

(b) control net and enrichment

Fig. 12. A central cracked plate.

4.1. Uncracked structures

Table 9 Convergence of limit load factor of a central plate with a=b ¼ 0:5. Method

N v ar

XFEM

0.6179 (2880) 0.5566 (3328) 0.5431 (6210) 0.5358 (9938)

XIGA (p = 2) XIGA (p = 3) XIGA (p = 4)

0.5823 (7568) 0.5482 (6524) 0.5351 (12,712) 0.5275 (21,172)

0.5738 (8640) 0.5376 (9416) 0.5289 (19,224) 0.5225 (32,336)

0.5532 (15,000) 0.5299 (15,812) 0.5227 (32,296) – –

Exact a

The total number of variables N v ar is given in parentheses.

0.5454 (21,344)a 0.5252 (22,184) 0.5202 (45,456) – – 0.500

4.1.1. Square plate with a central circular hole subjected to biaxial uniform loads This example deals with a square plate with a central circular hole which is subjected to biaxial uniform loads P 1 ; P 2 as shown in Fig. 5. The ratio between the diameter of the hole and the side length of the plate is chosen to be 0.2 ðR=L ¼ 0:2Þ. This problem has been well known as the benchmark for various numerical approaches. Due to its symmetry, one fourth of the plate is modeled with 16  8; 24  12; 32  16 and 40  20 NURBS elements as illustrated in Fig. 7. A rational quadratic basis is used to describe exactly a square plate with a central circular hole. Knot vectors N  H of the coarse mesh with two quadratic elements are defined as follows N ¼ f 0 0 0 0:5 1 1 1 g; H ¼ f 0 0 0 1 1 1 g. Control points and weights are given in Table 1. Coarse mesh and control net with

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H. Nguyen-Xuan et al. / Theoretical and Applied Fracture Mechanics xxx (2014) xxx–xxx Table 10 The collapse limit factors of a central cracked plate via various ratios a=b. Approach

Crack length ratio a=b

Analytical solution XFEM (29  59) XIGA (p = 2)(19  39) XIGA (p = 3)(19  39)

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9 0.9452 0.9183 0.8917

0.8 0.8426 0.8239 0.8240

0.7 0.7443 0.7293 0.7236

0.6 0.6398 0.6303 0.6225

0.5 0.5419 0.5299 0.5227

0.4 0.4356 0.4295 0.4216

0.3 0.3381 0.3284 0.3222

0.2 0.2292 0.2266 0.2192

Fig. 13. A central cracked plate: (a) Plastic dissipation using XFEM. (b) Plastic dissipation using XIGA. (c) Collapse mechanism.

Table 11 The collapse limit factor of a grooved cracked plate with mesh of 20  25 elements. Approach

XIGA (p = 2) XIGA (p = 3) XIGA (p = 4)

Crack length ratio a=ðL  2RÞ 0

0.1

0.2

0.3

0.4

0.5

0.5610 0.5596 0.5593

0.4787 0.4749 0.4687

0.3746 0.3724 0.3540

0.2865 0.2777 0.2693

0.2006 0.1993 0.1877

0.1395 0.1312 0.1287

respect to quadratic, cubic and quartic elements are depicted in Fig. 6. For P2 ¼ 0, the analytical solution of limit load factor based on von Mises yield criterion provided by Gaydon and McCrum [52] is kexact ¼ 0:8P 1 =ry . The convergence of limit load factors of the present method in comparison with the analytical solution is provided in Table 2. We observe that strict upper bound solutions are obtained. All solutions converge to exact values when the mesh is refined. The results obtained using the IGA are compared with those of other numerical methods as listed in Table 3. A good agreement is expected. This confirms that the IGA can be regarded as the alternative approach together with other existing methods such as finite element method (FEM), boundary element method (BEM), smoothed finite element method (SFEM) and element-free Galerkin method (EFG). Fig. 8 shows high reproduction of plastic dissipation distribution of the plate with a circular hole. Next we study the geometric effect of a circular hole on the solution via various ratios R=L. The results obtained using 800 NURBS elements are given in Table 4. It is observed again that the present solutions are compared very well with other available ones [53,54].

Fig. 14. The edge cracked plate.

4.1.2. Grooved rectangular plate subjected to axial uniform load Next example is a grooved rectangular plate subjected to in-plane tension load p as shown in Fig. 9. Limit analysis of this problem has been investigated by several authors such as the

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H. Nguyen-Xuan et al. / Theoretical and Applied Fracture Mechanics xxx (2014) xxx–xxx

(a)

(b) Fig. 16. The collapse mechanism and plastic dissipation distribution of the edge cracked plate with a=b ¼ 0:138 and H=b ¼ 2.

Fig. 15. The limit load factor via ratios a=b for an edge cracked plate. Fig. 17. The model of a grooved cracked plate.

analytical approach using Tresca yield criterion proposed by Prager and Hodge [63], the analytical approach using von Mises yield criterion studied by Yan [64] and several numerical methods using von Mises yield criterion reported in Casciaro and Cascini [65], Yan [64], Vu [12], Tran et al. [54] and Nguyen-Xuan et al. [62]. Knot vectors N  H of the coarse mesh with NURBS elements are defined as follows N ¼ f 0 0 0 1=3 2=3 1 1 1 g; H ¼ f 0 0 0 1=4 2=4 3=4 1 1 1 g. Data of control net and weights are listed in Tables 5 and 6. The structure is discretized into a coarse mesh and control net corresponding to quadratic and cubic elements as shown in Fig. 10. Meshes of 6  8; 12  16; 18  24 and 24  32 NURBS elements are described in Fig. 11. Table 7 confirms the convergence of the present method. A comparison between the obtained results and those of other analytical and numerical approaches are reported in Table 8. As expected, the present solutions shows high reliability. The present method produces the solutions belonging to the reliable interval of the analytical approach proposed by Yan [64]. 4.2. Cracked structures 4.2.1. Central cracked plate subjected to tension Let us consider the 2D plate with dimension b  H having an initial crack at center with length a as shown in Fig. 12a. Firstly,

the convergence of the limit load factor k ¼ rlim it =r in case of cracked length ratio a/b = 0.5 which a mesh is plotted in Fig. 12b is tabulated in Table 9 (see Table 10). The present results calculated from XIGA with p = 2, 3, 4 are compared to that of XFEM and analytical solution from [11]. It can be seen that, combining with SOCP, numerical methods (XFEM and XIGA) give the upper bound convergence. Herein, XIGA gains the better result although it uses coarser mesh with less variables compared to XFEM. Furthermore, as order of NURBS basis functions increases, the obtained result is more accuracy. However, higher order of NURBS increases significantly number of variables according to number of control points which is identified in Fig. 4. Therefore, in this study, we prefer to use quadratic and cubic NURBS functions. At the collapse state, the collapse mechanism and plastic dissipation distribution in the plate are plotted in Fig. 13. It is seen that XFEM cannot produce as smoothly plastic dissipation as XIGA although it uses extremely finer mesh. Because XIGA utilizes NURBS basis function with C p1 continuity through element boundaries. Table 11 summarizes the limit load factor of a center cracked plate via cracked length ratio a/b using XIGA in comparison with results obtained from XFEM and the analytical solution k ¼ 1  a=b calculated by [11]. Herein, using the mesh of 29  59 Q4 elements for XFEM and 19  39 cubic elements for XIGA, respectively, the same number of variables is

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H. Nguyen-Xuan et al. / Theoretical and Applied Fracture Mechanics xxx (2014) xxx–xxx

13

Fig. 19. Plastic dissipation in a grooved crack plate via length crack ratio.

of short – crack, numerical results are slightly lower the analytical solution [10]. The limit loads also depend strongly on the ratio a=b. The results obtained agree very well with the exact solution. Fig. 16 illustrates the plastic dissipation distribution and collapse mechanism of the rectangular plate at H=b=2.

Fig. 18. The grooved cracked plate.

gained approximately up to 32000 for both numerical methods. It is again seen that present results are more accurate than those of XFEM. 4.2.2. Edge cracked plate subjected to tension Next example is a plate of length H width b as shown in Fig. 14a and a single edge cracked length a subjected to tension is studied. The plate is discreted into 15  31 elements as shown in Fig. 14b. This benchmark problem was solved by the analytical method [10] and the limit load factor is given as

kexact

8 1  x  x2 if x ¼ a=b 6 0:146 > > < qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ðcx þ 0:5ðcx  1ÞÞ2 þ cð1  xÞ2  ðcx  0:5ðc  1ÞÞ; > > pffiffiffi : and c ¼ 2= 3 otherwise ð33Þ

Fig. 15a provides results of analytical and numerical approaches. Both XFEM and XIGA solutions mach well with the analytical one. Moreover, it is also seen that the results obtained from the XIGA using a fewer number of DOFs are slightly closer to the exact one than the XFEM. The limit load factors for various ratios a=b and H=b are exhibited in Fig. 15b. It seems that in case

4.2.3. Grooved cracked plate Finally, we consider a grooved plate with a single edge cracked length a subjected to in-plane tension load p as shown in Fig. 17. We believe that this investigation is useful to recognize a case study of limit load estimation of structures with defects [11]. The data are as same as subSection 5.1.2. The analytical solution was not available. The aim of this study is to estimate the load bearing capacity of structures involving holes and cracks. The full geometry of a grooved plate is modeled into a coarse mesh of 4  5 quadratic NURBS elements as shown in Fig. 18a. A fine mesh of 20  25 NURBS elements is then obtained from a coarse mesh as plotted in Fig. 18b. At the collapse state, the collapse mechanism in the plate is plotted in Fig. 18c. Table 11 shows that the presence of the crack affects very significantly the load bearing capacity of this structure. The same conclusion is obtained as p increases the limit load decreases. Finally, Fig. 19 shows the plastic dissipation in the plate through the change of the crack length.

5. Conclusions We have presented an efficient approach for plastic limit analysis of both uncracked and cracked structures. The method was based on the framework of isogeometric analysis including the enrichments of discontinuous and singularity fields of cracks. The underlying optimization problem based on the von Mises yield criterion was transformed into the compact form of second-order cone programming problem in order to exploit efficiently primal–dual interior-point solvers. Numerical illustration of the plastic collapse limit is here valid for plane stress cases. Through the

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H. Nguyen-Xuan et al. / Theoretical and Applied Fracture Mechanics xxx (2014) xxx–xxx

examples tested, some concluding remarks can be shown as follows:  The method is flexible with refinement and degree elevation relied on NURBS basic functions. It thus allows to produce the desired accuracy of approximate solutions by high-order discretizations.  The present solution is dramatically improved that of XFEM, even the coarse mesh used.  The results obtained are in good agreement with the analytical and reference solutions. However, the present formulation is still under studying for treating locking issue of incompressibility constraints in plane strain and 3D problems. In addition, we are facing the computational cost that is significant due to an excessive overhead of control points with increasing refinement. Adaptive local refinement [66] will be therefore very promising to enhance computational effect. Closely, a simple and efficient quadrature algorithm [67,68] for NURBS-based isogeometric analysis will also be worth to pursue for a future research. This is a work in progress and our findings will be devoted in a forthcoming paper. Acknowledgements The support of Vietnam National University HoChiMinh City (VNU-HCM) under Grant No. B2013-18-04 is gratefully acknowledged. The second author appreciates for the support from the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science, and Technology (2010-0019373 and 2012R1A2A1A01007405). The authors would like to thank the Mosek software package team for their free release policy for academic research. References [1] A. Capsoni, L. Corradi, A finite element formulation of the rigid-plastic limit analysis problem, Int. J. Numer. Methods Eng. 40 (1997) 2063–2086. [2] E. Christiansen, K.D. Andersen, Computation of collapse states with von Mises type yield condition, Int. J. Numer. Methods Eng. 46 (1999) 1185–1202. [3] M. Staat, M. Heitzer, LISA– a European project for FEM-based limit and shakedown analysis, Nucl. Eng. Des. 206 (2001) 151–166. [4] K. Krabbenhoft, L. Damkilde, A general nonlinear optimization algorithm for lower bound limit analysis, Int. J. Numer. Methods Eng. 56 (2003) 165–184. [5] M. Vicente da Silva, A.N. Antao, A non-linear programming method approach for upper bound limit analysis, Int. J. Numer. Methods Eng. 72 (2007) 1192– 1218. [6] H. Ciria, J. Peraire, J. Bonet, Mesh adaptive computation of upper and lower bounds in limit analysis, Int. J. Numer. Methods Eng. 75 (2008) 899–944. [7] J.J. Muñoz, A. Huerta, J. Bonet, J. Peraire, A note on upper bound formulations in limit analysis, Int. J. Numer. Methods Eng. 91 (2012) 896–908. [8] A.M. Yan, H. Nguyen-Dang, Limit analysis of cracked structures by mathematical programming and finite element technique, Comput. Mech. 24 (1999) 319–333. [9] R. Hill, On discontinuous plastic states, with special reference to localised necking in thin sheets, J. Mech. Phys. Solids 1 (1952) 19–30. [10] D.J.F. Ewing, C.E. Richards, The yield-point loading of singly-notched pin loaded tensile strips, J. Mech. Phys. Solids 22 (1974) 27–36. [11] A.G. Miller, Review of limit loading of structures containing defects, Int. J. Numer. Methods Eng. 32 (1988) 197–327. [12] D.K. Vu, Dual Limit and Shakedown Analysis of Structures, Université de Liège, Belgium, 2001. [13] R.S. Barsoum, Triangular quarter-point elements as elastic and perfectlyplastic crack tip elements, Int. J. Numer. Methods Eng. 11 (1977) 85–98. [14] N. Moes, J. Dolbow, T. Belytschko, A finite element method for crack growth without remeshing, Int. J. Numer. Methods Eng. 46 (1) (1999) 131–150. [15] T. Rabczuk, T. Belytschko, Cracking particles: a simplified meshfree method for arbitrary evolving cracks, Int. J. Numer. Methods Eng. 61 (13) (2004) 2316– 2343. [16] T. Rabczuk, G. Zi, A meshfree method based on the local partition of unity for cohesive cracks, Comput. Mech. 39 (6) (2007) 743–760. [17] T. Rabczuk, G. Zi, A. Gerstenberger, W.A. Wall, A new crack tip element for the phantom node method with arbitrary cohesive cracks, Int. J. Numer. Methods Eng. 75 (5) (2008) 577–599.

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