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Plastic collapse analysis of cracked structures using extendedisogeometric elements and second-order cone programming H.. Led a Department of Mechanics, Faculty of Mathematics and Comput

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Plastic collapse analysis of cracked structures using extended

isogeometric elements and second-order cone programming

H Nguyen-Xuana,⇑, Loc V Tranb, Chien H Thaic, Canh V Led

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

b

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

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

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

expen-sive 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

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 with-out any meshing or the requirement of the element boundaries

to align the crack faces In addition, extended meshfree methods

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 neces-sary 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 struc-tures[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 inte-grate Computer Aided Design (CAD) and Computer Aided Engineer-ing (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 func-tions 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

⇑ Corresponding author.

E-mail address: nxhung@hcmus.edu.vn (H Nguyen-Xuan).

Contents lists available atScienceDirect

Theoretical and Applied Fracture Mechanics

j o u r n a l h o m e p a g e : w w w e l s e v i e r c o m / l o c a t e / t a f m e c

Please cite this article in press as: H Nguyen-Xuan et al., Plastic collapse analysis of cracked structures using extended isogeometric elements and

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second-high-order approximations with an arbitrary smoothness

Further-more, B-splines (or NURBS) provide a ﬂexible way to perform

reﬁne-ment 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

com-bination 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

func-tions up to Cp1 instead of C0continuity as in the standard FEM

[27] Benson et al.[45]recently combined the XFEM approach to

lin-ear 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

discontin-uous 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 Section2 Section3summarizes

an extended isogeometric approximation for limit analysis

prob-lem Solution procedure is given in Section4 Several numerical

examples are illustrated in Section5 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 X2 R2 with

boundary C of continuous and discontinuous parts such that

C¼Cu[Ct[Cc and Cu\Ct\Cc¼ 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

_

u ¼ _u½ v_Tbe 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]

Wexð _uÞ ¼

Z

X

fT_u dXþ

Z

C t

The internal work rate for sufﬁciently smooth stressesr and

velocity ﬁeld _u is given by the bilinear form

Winðr; _uÞ ¼

Z

X

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

where V denote a space of kinematically admissible velocity ﬁeld deﬁned as

where H1ðXÞ is a Hilbert space Furthermore, the stressesrmust 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

whereRis a space of symmetric stress tensor

If deﬁning C ¼ f _u 2 V j Wexð _uÞ ¼ 1g, the exact collapse multiplier kexactcan be determined by solving any of the following optimization problems[49]

kexact¼ maxfk j 9r2 B : Winðr; _uÞ ¼ kWexð _uÞ;8u 2 Vg_ ð6Þ

¼max

r 2Bmin

_

¼min

_ u2Cmax

¼min

_

where

Dð _uÞ ¼ max

in whichrare the admissible stresses contained within the convex yield surface

principles of limit analysis, respectively The limit load of both approaches converges to the exact solution Herein, a saddle point

ðr; _uÞ exists such that both the maximum of all lower bounds

kand the minimum of all upper bounds kþcoincide 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 isogeomet-ric approach

For a limit analysis problem, only plastic strains rate is interested in the associated ﬂow rule

_

e¼ _l@w

where _lis 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]

Dð_eÞ ¼r0

Z

X

ffiffiffiffiffiffiffiffiffiffiffiffi _

eTH _e

p

where

3

4 2 0

2 4 0

2 6

3

andr0is the yield stress

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second-2.2 A short introduction on NURBS functions

A knot vectorN¼ n1;n2; ;nnþpþ1

is deﬁned as a sequence of knot value ni2; 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 C1continuous inside a knot span and Cp1

contin-uous at a single knot The B-spline basis functions are constructed

by the following recursion formula

Ni;pðnÞ ¼ n ni

niþp ni

Ni;p1ðnÞ þ niþpþ1 n

niþpþ1 niþ1Niþ1;p1ðnÞ

with p = 0,

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 N¼ n 1;n2 ;nnþpþ1

and

H ¼ng1;g2 ;gmþqþ1o

as

Fig 1 illustrates the set of one-dimensional and

two-dimen-sional 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

RAðn;gÞ ¼PmnNAfA

A NAðn;gÞfA

ð17Þ

It is evident that the B-spline function is obtained when the

individual weight of the control points is constant

2.3 Extended isogeometric ﬁnite elements

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 approxi-mation 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

_uh

I2S

NIð Þ _qx IþX

J2S c

NJð Þ H xx ð Þ H x J _

aJ

K2S t

NKð ÞxX4

a ¼1

Fað Þ Fx aðxKÞ

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

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

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second-where NI;J;Kðn;gÞ are the NURBS basis functions and _qI¼ _u½ I v_ITis

the velocities of nodal displacements of _uhassociated with the set of

the control points S, additional nodal unknowns _aJ and _ba

K are associated with the set of the control points Scwhose support is

cut by the crack and Sf whose support contains the crack tip, as

shown inFig 2, respectively

We need to deﬁne two types of enrichments: the

Heaviside-type enrichment H xð Þ and the tip-enrichment functions Fað Þ.x

The Heaviside function is given by

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 3illustrates 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 N2;2;N3;2;N4;2 which are determined by intersection

between the basic function with the discontinuous position as

shown inFig 3b Thus, to model the discontinuity, just three shape

functions are used to multiply with Heaviside function as plotted

inFig 3c

The tip enrichments can be utilized as1

F r; hð Þ ¼ ffiffiffi

r p

ffiffiffi r p

2 sin hð Þ;

ffiffiffi r p

2 sin hð Þ

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

p þ 1

ð Þ p þ 1ð Þ (blue color) and p p (red color), respectively The compatible strain rates can be expressed through the approximate velocity ﬁeld as

_

I

where the strain matrix B is given by

I BenrI

ð22Þ

in which Bstdand Benrare the standard and enriched part of matrix B deﬁned by

Bstd

I ¼

NI;y NI;x

2 6

3 7

NI;ywIþ NIwI;y NI;xwIþ NIwI;x

2 6

3 7 ð23Þ

in which wI may represent either the Heaviside function H or the branch functions Faand _dI is nodal velocities vector including the standard and enriched velocity unknowns

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 vectorN¼ f0; 0; 0; 1 ; 2 ;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.

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.

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second-The plastic dissipation of the rigid-perfectly plastic body is

com-puted by

Dhð _uhÞ ¼

Z

X

r0

ffiffiffiffiffiffiffiffiffiffiffiffi

_

eTH _e

p

Xnel e¼1

Z

X e

ffiffiffiffiffiffiffiffiffiffiffiffi _

eTH _e

p

The strains rate _eis now evaluated at Gauss points over patch

Xe Eq.(24)can hence be rewritten as

Dh

ð _uhÞ ’r0

XNG

i¼1

wijJij

ffiffiffiffiffiffiffiffiffiffiffiffiffi _

eT

iH _ei

q

ð25Þ

where NG ¼ nel nG is the total number of Gauss points of the

problem, nG is the number of Gauss points in each element, wiis

the weight value at the Gauss point i and Jj j is the determinant ofi

the Jacobian matrix computed at the Gauss point i

The optimization problem(9)associated with XIGA can now be

rewritten as

XNG

i¼1

wij jJi ffiffiffiffiffiffiffiffiffiffiffiffiffi

_

eT

iH _ei

q

Wexð _uÞ ¼ 1

(

ð26Þ

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

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

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

citi

s:t: kHit þvik 6 yT

Normal element

Enriched element Crack

Split 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.)

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second-where ti2 R; i ¼ 1; NG or t 2 RNGare the optimization variables, and

the coefﬁcients are ci2 R; Hi2 RmNG; vi2 Rm; yi2 RNG, and

zi2 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

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

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

i¼2t2 i

r

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 straightfor-wardly in the well-known form involving a sum of norms as

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.

Table 1

Control points and its weights for a plate with a circular hole.

2

p

1; 1

(0.75, 3) (5, 5) 1 þ 1 ffiffiffi

2 p

3 1; ffiffiffi

2

p

1

(3, 0.75) (5, 5) 1 þ 1 ffiffiffi

2 p

(c)

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

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second-Dhð _uhÞ ’ r0

XNG i¼1

where jj:jj denotes the Euclidean norm appearing in the plastic dis-sipation function, i.e, jjvjj ¼ ðvTvÞ1=2;C is the so-called Cholesky fac-tor of H

3 p

3 p 0

2 6

3

Table 2

Convergence of limit load factor for a square plate with a central circular hole.

16 8 24 12 32 16 40 20

Fig 7 Four meshes of square plate with circular hole.

Table 3

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

LB (lower bound); UB (upper bound).

a

Meshing 40 20.

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second-For convenience, a vector of additional variablesqiis introduced as

qi¼

q1

q2

q3

2 6

3 7

i

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

XNG i¼1

wij jtJi i

s:t

qi

_

Wexð _uhÞ ¼ 1

8

>

where the ﬁrst constraint in Eq.(32)represents the inequality con-straints of quadratic cones The total number of variables of the optimization problem is Nvar¼ 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]

Fig 8 Plastic dissipation distribution of a square plate with a circular hole.

Table 4

The limit load factor: P 2 ¼ 0 and P 1 ¼ry

R=L Heitzer [53] Tran et al [54] Quadratic Cubic Quartic

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.

Table 6 Weights for a grooved rectangular plate.

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second-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 Qua-dratic, cubic and quartic NURBS elements are studied for all numerical examples

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.

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second-4.1 Uncracked structures 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 P1;P2as 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 illus-trated inFig 7 A rational quadratic basis is used to describe exactly

a square plate with a central circular hole Knot vectorsN H of the coarse mesh with two quadratic elements are deﬁned as followsN¼ 0 0 0 0:5 1 1 1f g; H ¼ 0 0 0 1 1 1f g Control points and weights are given inTable 1 Coarse mesh and control net with

Table 8 Limit load factor for a grooved rectangular plate.

Authors Collapse multiplier Nature of solution Yield criterion

Casciaro and Cascini [65] 0.568 Numerical von Mises

Present (Quadratic)

Present (Cubic)

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

Fig 12 A central cracked plate.

Table 9

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

Method Nvar

XFEM 0.6179 0.5823 0.5738 0.5532 0.5454

(2880) (7568) (8640) (15,000) (21,344) a

XIGA (p = 2) 0.5566 0.5482 0.5376 0.5299 0.5252

(3328) (6524) (9416) (15,812) (22,184)

XIGA (p = 3) 0.5431 0.5351 0.5289 0.5227 0.5202

(6210) (12,712) (19,224) (32,296) (45,456)

XIGA (p = 4) 0.5358 0.5275 0.5225 – –

(9938) (21,172) (32,336) – –

a

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

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