XRPIM versus XFEM

2nd Reading

February 25, 2013 16:21 WSPC/0219-8762 196-IJCM 1340006

International Journal of Computational MethodsVol. 10, No. 1 (2013) 1340006 (12 pages)c© World Scientific Publishing CompanyDOI: 10.1142/S0219876213400069

XRPIM versus XFEM

S. LIU

Northeastern University, College of EngineeringBoston, MA 02115-5000, USA

[email protected]

Received 26 November 2011

Accepted 4 January 2012Published 28 February 2013

We present an extended radial point interpolation method (XRPIM) for modeling cracksand material interfaces in two-dimensional elasto-static problems. Therefore, partitionof unity enrichment is incorporated into RPIM. We employ both step enrichment andcrack tip enrichment for cracks. The studies are restricted to stationary cracks thoughthe method can be extended easily to moving boundaries. We compare the results to theextended finite element method to show the superiority of our method. We show for twoselected problems that the error is of magnitudes lower compared to XFEM simulations.

Keywords: Impact; SFEM; failure; large deformations.

1. Introduction

Partition of unity enrichment [Belytschko and Black (1999); Melenk and Babuska(1996)] has become popular to model a variety of problems; in this article we focus onapplications to stationary cracks. Partition of unity enrichment was originally usedin meshfree methods (MMs) [Fleming et al. (1997); Melenk and Babuska (1996)] toimprove approximation properties of MMs. Later, they were used to improve theaccuracy for fracture problems Fleming et al. (1997). In 1998, this approach wasfirst introduced in finite elements; the extended finite element method (XFEM)[Belytschko and Black (1999)] was born. Up to date, numerous publications onXFEM appeared.

In this article, we incorporate partition of unity enrichment into the radial pointinterpolation method (RPIM) [Wang and Liu (2002); Zhang et al. (2007)] for thefirst time. The RPIM is a very effective and highly accurate meshfree method[Belytschko et al. (1996); Nguyen et al. (2008)] developed by the group of Prof.G.R. Liu that was applied to a variety of problems [Cui et al. (2010a,b, 2011); Daiet al. (2006); Liu et al. (2007, 2008, 2011); Song et al. (2006); Wu et al. (2010); Zhaoet al. (2009)]. The Smoothed Finite Element Method (SFEM) [Cui et al. (2008);

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http://dx.doi.org/10.1142/S0219876213400069

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S. Liu

Le Canh et al. (2010); Liu and Dai (2007); Liu and Nguyen-Thoi (2010); Liu et al.(2007, 2009); Nguyen-Thanh et al. (2008, 2010, 2011); Nguyen-Thoi et al. (2009,a,b,2010); Nguyen-Xuan et al. (2008, 2010); Thai-Hoang et al. (2011); Tran et al. (2009)]was developed based on the RPIM. Recent efforts try to combine the advantagesof XFEM and SFEM [Baiz et al. (2011); Bordas et al. (2010, 2011); Rabczuk et al.(2008); Vu Bac et al. (2011)]. It is not the first time that the XFEM concept wasused in MMs [Le (2010a,b); Liu et al. (1995, 2008); Liu and Liu (2010); Lucy(1977); Moosavi et al. (2011); Rabczuk and Eibl (2003); Rabczuk et al. (2004)].Ventura et al. [2002] were the first who applied it in the element-free Galerkin(EFG) method [Belytschko et al. (1994)]. Later on, [Rabczuk and Areias (2006a);Rabczuk and Zi (2007)] extended this approach to nonlinear problems, shells andother applications [Bordas et al. (2008); Rabczuk and Belytschko (2007); Rabczuket al. (2007a,b, 2008, 2010); Zi et al. (2007)] and named the method XEFG. A sim-pler version with more efficient enrichment strategies was developed by Rabczukand Belytschko [2004,2006], Rabczuk and Areias [2006b], Rabczuk and Samaniego[2008], Rabczuk et al. [2007] and Rabczuk et al. [2009].

There are several advantages of using partition of unity enrichment in RPIM orin general in MMs:

• MMs are more accurate than finite elements.• Adaptivity can be incorporated in an easier fashion [Rabczuk and Belytschko

(2005)]. A certain refinement of the discretization is still needed in front of thecrack tip.

• Due to the lack of a mesh, MMs are more flexible than finite elements. Thisis particularly advantageous for representing the crack surface and for modelinglarge deformations, dynamic fracture and fragmentation.

Specific drawbacks such as higher computational cost and difficulties in impos-ing boundary conditions hampered the applications of MMs. However, the RPIMdoes not suffer these drawbacks, so RPIM is an optimal candidate for partition ofunity enrichment. The paper is organized as follows: In the next section, we brieflyreview the RPIM. Then, partition of unity enrichment is addressed. Subsequently,we present two numerical examples involving cracks. We compare numerical resultsfrom XRPIM simulations to those of XFEM simulations and analytical solutions.The paper ends with conclusions.

2. Governing Equations

Consider a domain Ω bounded by the boundary Γ = Γu

⋃Γt

⋃Γc with Γu

⋂Γt = 0,

Γu

⋂Γc = 0 and Γc

⋂Γt = 0. We assume small strain linear elasticity in two

dimensions. The governing equation is the equilibrium equation:

σij,j + bi = 0 ∀ xi ∈ Ω, (1)

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XRPIM versus XFEM

where σij is the stress tensor and bi are the body forces. The governing equationsare complemented by the compatibility condition and the constitutive equations:

εij = usi,j = 0.5(ui,j + uj,i),

σij = Cijklεkl(2)

with the first order elasticity tensor Cijkl , the strain tensor εkl and the superimposeds denotes the symmetric part of a tensor. The boundary conditions are given by

ui = ui on Γu

σijnj = ti = ti on Γt (3)

tci = 0 on Γc,

where superimposed bar indicates the imposed boundary conditions. The last con-dition in Eq. (3) is the traction free boundary condition for cohesionless cracks. Thegoverning equation can be obtained in weak form by method of weighted residualsleading to the following equation:∫

Ω

δεij Cijklεkl︸︷︷︸σij

dΩ −∫

Ω

vibi dΩ −∫

Γt

viti dΓ = 0, (4)

where vi are arbitrary test functions.

3. Radial Point Interpolation Methods (RPIM)

RPIM shape functions [Cui et al. (2010a, 2011); Dai et al. (2006); Liu et al. (2007,2008, 2011); Wang and Liu (2002); Zhao et al. (2009)] are constructed based on a setof nodes located in a local domain of influence. The RPIM shape functions consist oftwo different basis functions, a polynomial basis and radial basis functions. Hence,an approximation of a function u(x) is given by

u(x) = upol(x) + uRBF(x) (5)

with

upol(x) =∑M

J=1 PJ(x)bJ ,

uRBF(x) =∑N

I=1 RI(x)aI ,(6)

or in matrix form:

upol(x) = PT (x)buRBF(x) = RT (x)a,

(7)

where P contains the terms of the polynomial basis, b are unknown coefficients, Ris the radial basis function — in this work we use the multi-quadrics RBF (MQ-RBF) — and a are unknown coefficients. The coefficients in Eq. (7) are obtainedso that the field functions are satisfied at N nodes in the domain of influence at

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x. This will lead to the following system of equations that needs to be solved withrespect to a and b:

D = RTa + PT b (8)

with D = [D1, . . . , Dn]. As there are N equations in (8) but N + M unknowns, anadditional constraint is required given by

N∑I=1

pJ(xK)aI = 0, J = 1, 2, . . . , M. (9)

This leads finally to the following system of equations:(D0

)=

[R PP 0

](ab

)= G

(ab

), (10)

that can easily be solved with respect to a and b. The final RPIM approximationreads

u(x) = NTD (11)

with the RPIM shape functions

N = [N1(x), . . . , NN(x)],

NK(x) =∑N

I=1 RIGI,K +∑M

J=1 pJGN+J,K .(12)

where GI,K is the element of matrix G−1. In contrast to many other MMs, the RPIMapproximation satisfies the Kronecker–Delta properties such that the imposition ofDirichlet boundary conditions is straightforward. RPIM also fulfills partition ofunity and polynomial completeness of any desired order. Here, we satisfy linearcompleteness and use linear basis functions. In summary, the RPIM approximationis well suited for partition of unity enrichment.

4. Partition of Unity Enrichment

The crack is modeled by partition of unity enrichment. Therefore the displacementfield is additively decomposed into a continuous part uC , a discontinuous part uD

and a singular part uS :

u = uC + uD + uS . (13)

For the discretization of the continuous displacement field, the standard RPIMformulation is used as described in the previous section. The discontinuous part isonly relevant for nodes adjacent to the crack surface that are sufficiently far awayfrom the crack tip (Fig. 1) and is given by:

uD = ND A, (14)

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XRPIM versus XFEM

Crack tipenrichmentdomain

Discontinuousenrichment domain

Crack

Fig. 1. Crack tip enrichment domain and discontinuous enrichment domain.

where ND is the matrix that contains enriched shape functions

ND = N H. (15)

H being the shifted Heavisde function and A is the vector containing additionalnodal degrees of freedom. Shifting is standard procedure in partition of unityenriched methods for cracks and removes problems due to blending. The singu-lar part uS is called the crack-tip enrichment and is only applied to nodes aroundthe crack tip. It is well known that it is important to keep a fixed size of the domainin front of the crack tip in order to maintain optimal convergence rates when thediscretization is refined. For linear elastic fracture mechanics (LEFM), the nearcrack tip solution is known and is commonly included as enrichment function Binto the approximation:

B =[√

r sinθ

2,√

r cosθ

2,√

r sinθ

2sin θ,

√r cos

θ

2sin θ

], (16)

where r and θ are defined in Fig. 2. The approximation of the singular displacementfield is given by

uS = NS G. (17)

G being the vector containing additional nodal parameters and NS is the matrixcontaining the tip-enriched shape functions:

NS = N B (18)

In contrast to the discretization of uD, a simple shifting cannot remove problems dueto blending as the enrichment function B is not constant. Many efficient blendingtechniques have been developed and we exploit a technique that has been proposed

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virtual crackextension

Cracktta

r

Fig. 2. Definition of the variables r and θ for the crack tip enrichment.

by Fries et al. Another issue is integration of the nonpolynomial and singular terms.Laborde et al. have shown that several thousand Gauss points are needed for anaccurate integration. We adopt approach that was proposed by Bordas et al. [2010];it is based in transforming the domain integral into a boundary integral and requiresonly few Gauss points along the boundaries of an element.

5. Numerical Results

Let us consider the edge-crack problem as shown in Fig. 3; this problem was studiedby several researchers in order to test numerical methods. We consider mode 1 andmode 2-fracture and assume plane stress conditions. The analytical solution for thepure mode 1 problem is given by Ewalds and Wanhill [1989]:

ux =K1

2G

√r

2Πcos

θ

2

(κ − 1 + 2 sin2 θ

2

),

L

La

Fig. 3. Edge-crack problem.

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XRPIM versus XFEM

uy =K1

2G

√r

2Πsin

θ

2

(κ + 1 − 2 cos2

θ

2

),

σxx =K1

2Πrcos

θ

2

(1 − sin

θ

2sin

3θ

2

),

σxy =K1

2Πrcos

θ

2

(1 + sin

θ

2sin

3θ

2

),

τxy =K1

2Πrcos

θ

2sin

θ

2cos

3θ

2,

(19)

and for mode 2-fracture:

ux =K2

2G

√r

2Πsin

θ

2

(κ + 1 + 2 cos2

θ

2

),

uy = −K2

2G

√r

2Πcos

θ

2

(κ − 1 − 2 sin2 θ

2

),

σxx = − K2

2Πrsin

θ

2

(2 + cos

θ

2cos

3θ

2

),

σxy =K2

2Πrcos

θ

2cos

θ

2sin

θ

2cos

3θ

2,

τxy =K2

2Πrcos

θ

2

(1 − sin

θ

2sin

3θ

2

),

(20)

where G is shear modulus, K1 and K2 are the stress intensity factors (SIF) formode 1 and mode 2 fracture, respectively and r and θ are explained in Fig. 2; κ isthe Kosolov constant that depends on Poisson’s ratio ν. For plane stress:

κ =3 − ν

1 + ν. (21)

We assume linear elastic material behavior with Young’s modulus E = 10,000 andPoisson’s ratio ν = 0.3. The specimen is loaded in pure mode 1, so that K1 = 1 andK2 = 0 and pure mode 2 so that K1 = 0 and K2 = 1. For those two cases, we willstudy and compare numerical results of XFEM to XRPIM.

The numerical results are presented in Fig. 4. Figure 4(a) shows the error in theL2 displacement norm of XFEM compared to XRPIM; Fig. 4(b) illustrates error inenergy norm and Fig. 4(c) the error in the SIFs. The results for mode 1 and mode 2are included in the figures; mode 2 results are slightly more accurate for XFEM andXRPIM. Moreover, for XFEM and XRPIM optimal convergence rate is obtainedand in terms of convergence rate, XFEM and XRPIM are almost indistinguishable.However, XRPIM is around one order of magnitude more accurate than XFEMwhen the same mesh density h and therefore, the same number of DOFs are used.

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(a) Error in L2 displacement norm (b) Error in energy norm

(c) Error in SIFs

Fig. 4. Comparison of XFEM and XRPIM.

This is consistently observed for error in displacement norm, energy norm and SIF.This shows clearly the advantage of meshfree RPIM over FEM. The computationalcost for RPIM is slightly higher but as we did not optimize our matlab code withrespect to computational time, we did not include efficiency plots.

6. Conclusions

We presented an extended meshfree method for modeling cracks by partition-of-unity enrichment. We named the method XRPIM (Extended Radial Point Inter-polation Method) as the meshfree method is based on RPIM. The method hasbeen developed for two-dimensional problems in linear elasticity and combines theadvantages of XFEM and MMs. We have studied two benchmark problems involvingmode 1 and 2 fracture and demonstrated the effectivity and accuracy of XRPIM.Therefore, we compared XRPIM to results of the XFEM. The results of XRPIM areone order more accurate as compared to XFEM when the same number of nodes(= degrees of freedom) are used. Therefore, XRPIM is a competitive alternative toXFEM.

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Nguyen-Thoi, T., Vu-Do, H. C., Rabczuk, T. and Nguyen-Xuan, H. [2010] “A node-based smoothed finite element method (ns-fem) for upper bound solution to visco-elastoplastic analysis of solids using triangular and tetrahedral meshes,” Comput.Method Appl. Mech. Eng. 199, 3005–2027.

Nguyen-Xuan, H., Rabczuk, T., Bordas, S. and Debongnie, J. F. [2008] “A smoothedfinite element method for plate analysis,” Comput. Method. Appl. Mech. Eng. 197,1184–1203.

Nguyen-Xuan, H., Liu, G. R., Thai-Hoang, C. and Nguyen-Thoi, T. [2010] “An edge-based smoothed finite element method (es-fem) with stabilized discrete shear gap tech-nique for analysis of reissner-mindlin plates,” Comput. Method. Appl. Mech. Eng. 199,471–489.

Rabczuk, T. and Areias, P. [2006a]“A meshfree thin shell for arbitrary evolving cracksbased on an extrinsic basis,” Comput. Model. Eng. Sci. 16(2), 115–130.

Rabczuk, T. and Areias, P. M. A. [2006b] “A new approach for modelling slip lines ingeological materials with cohesive models,” Int. J. Numer. Anal. Method Geomech.30(11), 1159–1172.

Rabczuk, T. and Belytschko, T. [2006] “Application of particle methods to static fractureof reinforced concrete structures,” Int. J. Fract. 137(1–4), 19–49.

Rabczuk, T. and Belytschko, T. [2004] “Cracking particles: A simplified meshfree methodfor arbitrary evolving cracks,” Int. J. Numer. Method Eng. 61(13), 2316–2343.

Rabczuk, T. and Belytschko, T. [2005] “Adaptivity for structured meshfree particle meth-ods in 2d and 3d,” Int. J. Numer. Method Eng. 63(11), 1559–1582.

Rabczuk, T. and Belytschko, T. [2007] “A three-dimensional large deformation mesh-free method for arbitrary evolving cracks,” Comput. Method Appl. Mech. Eng. 196,2777–2799.

Rabczuk, T. and Eibl, J. [2003] “Simulation of high velocity concrete fragmentation usingSPH/MLSPH,” Int. J. Numer. Method Eng. 56, 1421–1444.

Rabczuk, T. and Samaniego, E. [2008] “Discontinuous modelling of shear bands usingadaptive meshfree methods,” Comput. Method. Appl. Mech. Eng. 197 641–658.

Rabczuk, T. and Zi, G. [2007] “A meshfree method based on the local partition of unityfor cohesive cracks,” Comput. Mech. 39(6), 743–760.

Rabczuk, T., Belytschko, T. and Xiao, S. P. [2004] “Stable particle methods based onlagrangian kernels,” Comput. Method Appl. Mech. Eng. 193(12–14), 1035–1063.

Rabczuk, T., Eibl, J. and Stempniewski, L. [2004] “Numerical analysis of high speedconcrete fragmentation using a meshfree lagrangian method,” Eng. Fract. Mech. 71(4–6), 547–556.

Rabczuk, T., Areias, P. M. A. and Belytschko, T. [2007a] “A meshfree thin shell methodfor non-linear dynamic fracture,” Int. J. Numer. Method Eng. 72(5), 524–548.

Rabczuk, T. Bordas, S. and Zi, G. [2007] “A three-dimensional meshfree method for con-tinuous multiple crack initiation, nucleation and propagation in statics and dynamics,”Comput. Mech. 40(3), 473–495.

Rabczuk, T., Areias, P. M. A. and Belytschko, T. [2007b] “A simplified mesh-free methodfor shear bands with cohesive surfaces,” Int. J. Numer. Method Eng. 69(5), 993–1021.

Rabczuk, T., Zi, G., Bordas, S. and Nguyen-Xuan, H. [2008] “A geometrically non-linearthree dimensional cohesive crack method for reinforced concrete structures,” Eng.Fract. Mech. 75, 4740–4758.

Rabczuk, T., Zi, G., Gerstenberger, A. and Wall, W. A. [2008] “A new crack tip elementfor the phantom node method with arbitrary cohesive cracks,” Int. J. Numer. MethodEng. 75, 577–599.

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2nd Reading


S. Liu

Rabczuk, T. Song, J.-H. and Belytschko, T. [2009] “Simulations of instability in dynamicfracture by the cracking particles method,” Eng. Fract. Mech. 76, 730–741.

Rabczuk, T., Bordas, S. and Zi, G. [2010] “On three-dimensional modelling of crack growthusing partition of unity methods,” Comput. Struct. 88, 1391–1411.

Rabczuk, T., Gracie, R., Song, J.-H. and Belytschko, T. [2010] “Immersed particle methodfor fluid-structure interaction,” Int. J. Numer. Method Eng. 81, 48–71.

Rabczuk, T., Zi, G., Bordas, S. and Nguyen-Xuan, H. [2010] “A simple and robust three-dimensional cracking-particle method without enrichment,” Comput. Method. Appl.Mech. Eng. 199, 2437–2455.

Song, C. X., Liu, G. R., Li, H. and Han, X. [2006] “Simulation of an extruded quadrupo-lar dielectrophoretic trap using meshfree approach,” Eng. Anal. Bound. Elem. 30,994–1005.

Thai-Hoang, C., Nguyen-Thanh, N., Nguyen-Xuan, H. and Rabczuk, T. [2011] “An alter-native alpha finite element method with discrete shear gap technique for analysis oflaminated composite plates,” Appl. Math. Comput. 217(17), 7324–7348.

Thai-Hoang, C., Nguyen-Thanh, N., Nguyen-Xuan, H., Rabczuk, T. and Bordas, S. [2011]“A cell-based smoothed finite element method for free vibration and buckling analysisof shells, KSCE J. Civil Eng. 15(2), 347–361.

Tran, N. T., Liu, G.R., Nguyen-Xuan, H. and Nguyen-Thoi, T. [2009] “An edge-basedsmoothed finite element method for primal-dual shakedown analysis of structures,”Int. J. Numer. Method Eng. page 10.1002/nme.2804.

Ventura, G., Xu, J. and Belytschko, T. [2002] “A vector level set method and new discon-tinuity approximation for crack growth by efg,” Int. J. Numer. Method Eng. 54(6),923–944.

Nguyen, V. P., Rabczuk, T., Bordas, S. and Duflot, M. [2008] “Meshless methods: A reviewand computer implementation aspects,” Math. Comput. Simul. 79, 763–813.

Vu Bac, N., Nguyen-Xuan, H., Chen, L., Bordas, S., Kerfriden, P., Simpson, R. N., Liu, G.R. and Rabczuk, T. [2011] “A node-based smoothed extended finite element method(nsxfem) for fracture analysis,” CMES-Comput. Model. Eng. Sci. 1898(1), 1–25.

Wang, J. G. and Liu, G. R. [2002] “A point interpolation meshless method based on radialbasis functions,” Int. J. Numer. Method Eng. 54, 1623–1648

Wu, S.C., Liu, G. R., Cui, X. Y., Nguyen, T. T. and Zhang, G. Y. [2010] “An edge-based smoothed point interpolation method (es-pim) for heat transfer analysis of rapidmanufacturing system,” Int. J. Heat Mass Trans. 53, 1938–1950.

Zhang, G. Y., Liu, G. R., Wang, Y. Y., Huang, H. T., Zhong, Z. H., Li, G. Y. andHan, X. [2007] “A linearly conforming point interpolation method (lc-pim) for three-dimensional elasticity problems,” 72, 1524–1543.

Zhao, Z., Liu, G. R., Dai, K. Y., Zhong, Z. H., Li, G. Y. and Han, X. [2009] “Free-vibrationanalysis of shells via a linearly conforming radial point interpolation method (lc-rpim),”Finite Elem. Anal. Des. 45, 917–924.

Zi, G., Rabczuk, T. and Wall, W. [2007] “Extended meshfree methods without branchenrichment for cohesive cracks,” Comput. Mech. 40(2), 367–382.

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XRPIM versus XFEM

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