Computational Mechanicshttps://doi.org/10.1007/s00466-020-01821-5

ORIG INAL PAPER

Isogeometric analysis of thin Reissner–Mindlin shells: lockingphenomena and B-bar method

Qingyuan Hu1 · Yang Xia2 · Sundararajan Natarajan3 · Andreas Zilian4 · Ping Hu2 · Stéphane P. A. Bordas4

Received: 14 July 2019 / Accepted: 18 January 2020© Springer-Verlag GmbH Germany, part of Springer Nature 2020

AbstractWe propose a local type of B-bar formulation, addressing locking in degenerated Reissner–Mindlin shell formulation in thecontext of isogeometric analysis. Parasitic strain components are projected onto the physical space locally, i.e. at the elementlevel, using a least-squares approach. The formulation allows the flexible utilization of basis functions of different orders asthe projection bases. The introduced formulation is much cheaper computationally than the classical B method. We showthe numerical consistency of the scheme through numerical examples, moreover they show that the proposed formulationalleviates locking and yields good accuracy even for slenderness ratios of 105, and has the ability to capture deformations ofthin shells using relatively coarse meshes. In addition it can be opined that the proposed method is less sensitive to lockingwith irregular meshes.

Keywords Isogeometric · Reissner–Mindlin shell · Locking · B-bar method · Mesh distortion

1 Introduction

The conventional finite element method (FEM) employspolynomial basis functions to represent the geometry andthe unknown fields. The commonly employed approxima-tion functions are Lagrangian polynomials. However, theseLagrange polynomials are usually built upon amesh structurewhich needs to be generated, from a CAD (computer-aideddesign) model provided for the domain of interest. Thismesh generation leads to the loss of certain geometrical fea-tures: e.g. a circle becomes a polyhedral domain. Moreover,Lagrange polynomials lead to low order continuity betweenneighboring elements, which is inadequate in applicationsdescribed by high order partial differential equations.

B Qingyuan Huqingyuanhu@jiangnan.edu.cn

1 School of Science, Jiangnan University, Wuxi 214122,People’s Republic of China

2 School of Automotive Engineering, Dalian University ofTechnology, Dalian 116024, People’s Republic of China

3 Integrated Modelling and Simulation Lab, Department ofMechanical Engineering, Indian Institute of Technology,Madras, Chennai 600036, India

4 Institute of Computational Engineering Sciences, Faculty ofSciences, Technology and Medicine, University ofLuxembourg, Esch-sur-Alzette, Luxembourg

The introduction of isogeometric analysis (IGA) [1] pro-vides a general theoretical framework for the concept of“design-through-analysis” which has attracted considerableattention. The key idea of IGA consists in a direct linkbetween CAD and the simulation, by utilizing the same func-tions to approximate the unknown field variables as thoseused to describe the geometry of the domain under consid-eration, similar to the idea proposed in [2]. Moreover, it alsoprovides a systematic construction of high-order basis func-tions [3]. Note that, more recently, a generalization of theisogeometric concept was proposed, whereby the geome-try continues to be described by NURBS functions as inthe CAD, but the unknown field variables are allowed tolive in different (spline) spaces. This leads to the concept ofsub- and super-geometric analysis, also known as GeometryIndependent Field approximaTion (GIFT), described withina boundary element framework in [4] and proposed in [5,6]and later refined in [7]. Related ideas, aiming at the con-struction of tailored spline spaces for local refinement wereproposed recently in [8].

In the literature, the IGA has been applied to study theresponse of plate and shell structures, involving two maintheories, viz., the Kirchhoff–Love theory and the Reissner–Mindlin theory. Thanks to the C1-continuity of the adoptedNURBS basis functions, Kiendl et al. [9] developed anisogeometric shell element based on Kirchhoff–Love shell

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theory. The isogeometric Kirchhoff–Love shell element forlarge deformations was presented in [10]. The isogeometricReissner–Mindlin shell element was implemented in [11],including linear elastic and nonlinear elasto-plastic constitu-tive behavior. The blended shell formulation was proposed toglue a Kirchhoff–Love shell with a Reissner–Mindlin shellin [12]. In addition, the isogeometric Reissner–Mindlin shellformulation derived from the continuum theory was pre-sented in [13], in which the exact director vectors were usedto improve accuracy. The solid shell element was developedin [14], in this formulation the NURBS basis functions wereused to construct the mid-surface and a linear Lagrange basisfunction was used to interpolate the thickness field.

The Kirchhoff–Love type formulation is rotation-free, butit is only valid for thin structures. Moreover due to theabsence of rotational degrees of freedom (DoFs), specialtechniques are required to deal with the rotational boundaryconditions [9,15,16] and multi-patch connection [17]. Theo-retically, the Reissner–Mindlin theory is valid for both thickand thin shell structures. However it is observed from the lit-erature [11,18] that, when the shell kinematics is representedby Reissner–Mindlin theory, both FEM and IGA approachessuffer from locking for thin structures, especially for loworder elements with coarse meshes. This has been attractingengineers and mathematicians to develop robust elementsthat could alleviate this pathology. Adam et.al. proposed afamily of concise and effective selective reduced integration(SRI [19]) rules within the IGA framework for beams [20],plates/shells [21], and non-linear shells using T-splines [22].Elguedj et al. [23] presented B method and F method to han-dle nearly incompressible linear and non-linear problems.The B method has been successfully applied to shear lock-ing problems in curved beams [24], 2D solid shells [25], 3Dsolid shells [26] and later in nonlinear solid shell formula-tion [27]. Echter and Bischoff [18,28] proposed the discreteshear gap (DSG) method [29] to alleviate the shear lock-ing phenomena effectively. Other approaches include twistKirchhoff theory [30], virtual element method [31], collo-cation method [32,33], simple first order shear deformationtheory [34], and single variable method [35,36]. The aboveapproaches have been employed with Lagrangian elementsor IGA framework with varying orders of success.

The approaches proposed by Bouclier, Elguedj andCombescure [25–27] focus on solid-shells in IGA andachieve good un-locking performance with the B method,thus it is promising to implement and test this method fordegenerated Reissner–Mindlin shell elements in the contextof IGA. This paper is the extension of our previous paper [37]on Timoshenko beams. In this research, in order to alleviatethe locking phenomena, the locking strains are projected ontobi-dimensional low-order physical space by the least squaremethod. The novel idea behind the formulation is to use mul-tiple sets of basis functions to project the locking strains

locally i.e. element-wise, instead of projecting globally i.e. allover the patch, thereby reducing the computational effort sig-nificantly. More importantly, the local projection allows oneto use different orders of projecting basis functions, whichcould further improve the un-locking performance.

The outline of this paper is as follows: Sect. 2 gives anoverview of the degenerated Reissner–Mindlin shell formu-lation. In Sect. 3, we present the novel approach, the local Bmethod to alleviate the locking (both shear and membrane)problems encountered in thin structures whilst employingReissner–Mindlin formulation. The accuracy, robustness andthe convergence properties are demonstrated by some bench-mark examples in Sect. 4, followed by concluding remarksin the last Sect. 5. Some implementation details are providedin the “Appendix”.

2 Isogeometric formulation ofReissner–Mindlin shells

2.1 Reissner–Mindlin shell model

Figure 1 represents the mid-surface of a shell structure in theparametric and physical spaces. For simplicity, we considera shell of constant thickness h, and assume the linear elasticmaterial to be homogeneous and isotropic, which is governedby Young’s modulus E and Poisson’s ratio ν.

The main differences between the Reissner–Mindlin shelltheory and theKirchhoff–Love shell theory lie in the assump-tions on the deformation behavior of the section and furtherin the resulting independent kinematic quantities attachedto the mid-surface. According to the Reissner–Mindlin the-ory, along the thickness direction a first order kinematicdescription is adopted for the transverse shear deformation.Assuming a Cartesian coordinate system, an arbitrary pointP belongs to the 3D shell structure can be found by

xP (ξ, η, ζ ) = x(ξ, η) + ζh

2n(ξ, η), (1)

where x stands for the shell mid-surface, modeled by a func-tion of (ξ, η), as shown in Fig. 1. Let parameter ζ ∈ [−1, 1]denotes the shell thickness. On mid-surface x, the normalvector is calculated by

n = x,ξ × x,η

||x,ξ × x,η||2 , (2)

and we denote

t1 := x,ξ

||x,ξ ||2 , t2 := n × t1. (3)

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Fig. 1 Mid-surface (blue) in theparameter space (left) andphysical space (right) for a shellmodel. The 3D model isrecovered by Eq. (1). (Colorfigure online)

ξ

η

n

ξ

η

hz

y

x

η

ξ

Note that (t1, t2, n) form a local frame on the mid-surface,and the components of these vectors are indicated by sub-script x, y, z respectively.

Assuming small deformations, the displacement of thepoint P is recovered by

uP (ξ, η, ζ ) = u(ξ, η) + ζh

2θ(ξ, η) × n(ξ, η), (4)

in which u and θ are the displacement vector and the rotationvector at the mid-surface point projected from P . Then wehave the linearized strain tensor

εεε = 1

2(uP,x + uTP,x). (5)

2.2 Isogeometric approach

Following the degenerated type formulation, we start oursimulation from the mid-surface geometry of a Reissner–Mindlin shell. In order to describe the shell mid-surface,bi-variate NURBS basis functions are employed. Let Ξ ={ξ1, . . . , ξn+p+1} and H = {η1, . . . , ηm+q+1} be open knotvectors, and wA be given weights, where p and q are theorders along the directions ξ and η respectively. DenotingA = {1, . . . , nm}, we construct the NURBS basis functionsRA(ξ, η). For more details about IGA, interested readers arereferred to [40] and references therein.

The geometry of the undeformedmid-surface is describedby

x(ξ, η) =nm∑

A=1

RA(ξ, η)xA, (6)

where xA define the locations of control points. The discretedisplacement field on the mid-surface is interpolated as

{u(ξ, η)

θ(ξ, η)=

nm∑

A=1

RA(ξ, η)

{uA

θ A, (7)

specifically uA = (u, v, w)T and θ A = (θx , θy, θz)T. Once

the mid-surface is modeled according to Eq. (6) and (7),an arbitrary point P in the shell body can be traced by thefollowing discrete forms

xP (ξ, η, ζ ) =nm∑

A=1

RA(ξ, η)xA + ζh

2n(ξ, η), (8)

and

uP (ξ, η, ζ ) =nm∑

A=1

RA(ξ, η)uA + ζh

2

nm∑

A=1

RA(ξ, η)θ A × n(ξ, η).

(9)

Regarding the exact normal vector field n, we constructan approximated field by collocated points, explained in“Appendix A”.

Upon employing the Galerkin framework and using thefollowing discrete spaces for the displacement field

S ={u ∈

[H1(Ω)

]d, u|Γu = ud

}, (10)

V ={v ∈

[H1(Ω)

]d, v|Γu = 0

}, (11)

the variational equation reads: find u ∈ S such that

b(u, u∗) = l(u∗) ∀u∗ ∈ v, (12)

in which the bi-linear term is

b(u, u∗) =∫

Ω

εεε(u∗)TDgεεε(u) dΩ, (13)

with Dg denoting the global constitutivematrix.More detailsare provided in “Appendix B”.

When the displacements and the rotations are approx-imated by polynomials of the same order, the discretized

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framework experiences locking (shear and membrane) whenthe shell thickness becomes very small. The numerical proce-dure fails to satisfy theKirchhoff limit as the shear strain doesnot vanish with the thickness approaching zero. One popularexplanation of shear locking is that variables involved are notcompatible [35], which is also known as field inconsistency.In the next section, we introduce the B method, trying toaddress the locking syndrome from the above point of view.

3 B-bar method for Reissner–Mindlin shells

In this section, we firstly illustrate the main idea behind theB method. After introducing the classical B method, in orderto reduce the computational cost of the formulation, a localform of the B formulation is proposed, followed by a novelprojecting scheme.

3.1 The idea behind the B-bar method

Assuming an unit length Timoshenko beam, which is dis-creted by one element with 2 nodes. The element lengthparameter x ∈ [0, 1]. By the help of shape functions

N1 = 1 − x, N2 = x, (14)

the deflection field w and the rotation field θ are constructedas

w = N1w1 + N2w2, θ = N1θ1 + N2θ2, (15)

in which wi and θi are nodal variables, i = 1, 2.For thin beams, the Timoshenko beam theory tends to

degenerate into the Euler beam theory, and the shear strain(in Timoshenko beam theory) tends to vanish:

γ = dw

dx− θ = 0, (16)

and in discrete form denoted by its superscript h:

γ h = (−w1 + w2) − [(1 − x)θ1 + xθ2

] = 0. (17)

The field inconsistency phenomenon occurs in the aboveequation, that is the order (wrt. x) of θi is higher than theorder of wi , which makes the shear strain difficult to vanish.

The B method modifies the field to be order-consistency.Firstly we build a pseudo shear strain as

γ h =∑

A

N Aγ Ah, (18)

where γ Ahis the corresponding pseudo DoF. Since the order

of the initial shape functions Ni is one, we define our one

order lower shape function(s) as N A = 1 with A = 1. Nextwe perform a least square (L2) projection and obtain a systemof linear equation(s):

∫ 1

0N1

(γ h − γ h

)dx = 0, (19)

specifically

∫ 1

0N1 N1 dx · γ 1h

−∫ 1

0N1

[(−w1 + w2) − (

(1 − x)θ1 + xθ2)]

dx = 0,

(20)

and then

γ 1h + w1 − w2 + 1

2θ1 + 1

2θ2 = 0. (21)

Solve the above equation for γ 1h

γ 1h = −w1 + w2 − 1

2θ1 − 1

2θ2, (22)

thus the pseudo shear strain we previously built becomes

γ h = N1γ1h = (−w1 + w2) −

(1

2θ1 + 1

2θ2

). (23)

Compare the pseudo shear strain in Eq. (23) with the originalone in Eq. (17), it is noticed that the shape functions for wi

remain unchanged, but the order of shape functions for θiis reduced to be the same as wi . The pseudo shear strain isorder consistency, it is believed that this strain could achievegood un-locking performance.

From the strainmatrix perspective, the Bmethodmodifiesthe original strain matrix

B = [ dN1dx

dN2dx −N1 −N2

](24)

into

B = N1

(∫ 1

0N1 N1 dx

)−1 ∫ 1

0N1

[dN1dx

dN2dx −N1 −N2

]dx,

(25)

which explains the name of this method.

3.2 Classical B-bar method in IGA

The novel idea behind the B method [23,24] is to use aprojected strain instead of the original one to formulate thebi-linear term

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b(u, u∗) =∫

Ω

εεε∗TDgεεεdΩ. (26)

The projected strain εεε and the original strain εεε are equivalentin the sense of the L2 projection. Denoting the approxi-mation space for displacement field by Qp,q , a commonway is using the projection space of one order lower, i.e.Q p,q = Qp−1,q−1 (see Fig. 2a). Lower order B-spline basisfunctions N A are built from lower order knot vectors, withall the weights given as w A = 1, A = {1, . . . , nm}. The L2

projection process is performed on the physical domain as

∫

Ω

NB

(εεεh − εεεh

)dΩ = 0,

B = 1, . . . , nm, (27)

in which the discretized form of the projected strain is

εεεh =nm∑

A=1

N Aε Ah, (28)

where ε Ahmeans the projection of εεεh onto N A. Finally, we

have

εεεh =nm∑

A,B=1

N AM−1A B

∫

Ω

NBεεεhdΩ, (29)

in which M A B is the inner product matrix

M A B =∫

Ω

N A NBdΩ := (N A, NB)Ω. (30)

The Gram matrix M is inverseable since the involvedbases are obviously linear-independent. In addition, the Hu-Washizu principle could be utilized to prove the variationalconsistency of the B method [23–25,41], and [24,26] provedthat the B method is equivalent to a mixed formulation.

3.3 Local B-bar formulation for Reissner–Mindlinshells

The motivation of the local B concept comes from the obser-vation that one needs to calculate the inverse of the matrixM in Eq. (29), which could be computationally expensive ifthe projection is applied patch-wise. Thus, from a practicalpoint of view, it is highly recommended to project the strainslocally [26,37], i.e. element-wise on Ωe:

εεεhe =( p+1)(q+1)∑

A,B=1

N AM−1A B

∫

Ωe

NBεεεhe dΩ, (31)

with

M A B = (N A, NB)Ωe . (32)

This kind of treatment can be also found from the work oflocal least square method [38,39]. It is worthy pointing outthat Eq. 31 implies the element strain matrix is modified as

B =( p+1)(q+1)∑

A,B=1

N AM−1A B

∫

Ωe

NBB dΩ (33)

comparing with the original one B in Appendix Eq. B.8.Moreover, with the assumption that projecting the orig-

inal strains onto a lower order space can help to reducelocking phenomena, it is expected that using the possiblylowest order for each element maximizes the un-lockingperformance. Therefore, instead of projecting the strainssimply onto Qp−1,q−1, different sets of projection spacesare adopted in this work. The projecting spaces need to bechosen carefully to avoid ill-condition or rand deficiency,readers interested in the corresponding mathematical the-ory is recommended to see [42], which is about volumelocking (nearly-incompressible) problems. Here, the pro-jection scheme in our previous work [37] is extended tobi-dimensional cases. Specifically, for space Q2,2, we adoptQ1,1 for corner elements, Q0,1 and Q1,0 for boundary ele-ments, and Q0,0 for inner elements, as shown in Fig. 2b. Thisgeneralized strategy is similarwith theSRI strategy inFig. 2c,however the SRI strategy strongly depends on continuity con-ditions across elements [21]. The shape of the projectingfunctions are plotted in Fig. 3 for an intuitive understanding.

The degenerated shell model is represented by its 2Dmid-surface, while the strain formula is expressed in 3D involvingthe thickness parameter ζ , in this case if the whole strain isprojected, rank deficiency appears and the formulation yieldsinaccurate results. Following the similar approach as outlinedin solid-shell element [25,26], in this work only the averagestrain through the thickness is projected:

MID(εεεhe ) =( p+1)(q+1)∑

A,B=1

N AM−1A B

∫

Ωe

MID(εεεhe )NB dΩ,

(34)

where MID(εεεhe ) is defined to be the average strain throughthe thickness within a single element

MID(εεεhe ) = 1

h

∫ h2

− h2

εεεhe dz. (35)

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Fig. 2 Projection strategies inbi-dimensional parameter space.Projecting spaces for lockingstrains are denoted as Qp,q . Forlocal B method (a), one orderlower B-spline basis functionsare employed for all elements.The proposed local B method(b) uses basis functions ofdifferent orders, similar to theSRI method (c), but the C1continuity of elements are notrestricted as in (c)

Q1,1 Q1,1 Q1,1 Q1,1

Q1,1 Q1,1 Q1,1 Q1,1

Q1,1 Q1,1 Q1,1 Q1,1

Q1,1 Q1,1 Q1,1 Q1,1

ξ, p = 2

η,q

=2

(a) Local B

Q1,1 Q0,1 Q0,1 Q1,1

Q1,0 Q0,0 Q0,0 Q1,0

Q1,0 Q0,0 Q0,0 Q1,0

Q1,1 Q0,1 Q0,1 Q1,1

ξ, p = 2

η,q

=2

(b) Present

ξ, p = 2, C1

η,q

=2,C1

(c) SRI [21]

0.75

0

1

0.25

0.5

0.75

0.25

0.5

0

1

0.250.5

0.75

0

1

0.250.5

0.75

0

1

B projection scheme

0.75

0

1

0.25

0.5

0.75

0.25

0.5

0

1

0.250.5

0.75

0

1

0.250.5

0.75

0

1

0.25

0.5

0.75

0

1value

(a) Local (b) Present local B projection scheme

Fig. 3 Bi-dimensional basis functions for projecting locking strains for each element, in the case of Qp,q = Q2,2

The modified bi-linear form is written as

b(U,U∗) =∫

Ω

εεε∗TDεεε

−MID(εεε∗)TDMID(εεε)︸ ︷︷ ︸originalaveragevirtualwork

+ MID(εεε)TDMID(εεε)︸ ︷︷ ︸

projectedaveragevirtualwork

dΩ.

(36)

The above calculation process is illustrated in Algorithm 1.A primary comparison of the computational efforts is per-

formed in Table 1, in which the meaning of the Step No.can be found in Algorithm 1. For classical IGA, the processis rather simple but it suffers from locking. In the classicalB method, the calculation of the inverse of matrix M =(N A, NB)Ω requires large amounts of memory. Reviewingthe literature of the B method since its appearance [43],one valuable contribution is the introduction of the local Bconcept [26,37], which saves lots of computational effort

with acceptable accuracy loss. Compared with the global Bmethod, the local B method shows promising advantagesin terms of computational efficiency. Based on the local Bmethod, in the present research we choose different ordersof basis functions as projection bases in order to achieve animproved un-locking performance.

4 Numerical examples

In this section, we demonstrate the performance of the pro-posed local B method for Reissner–Mindlin plates/shells bysolving several standard benchmark problems. The proposedformulation is implementedwithin the open sourceC++ IGAframework Gismo 1 [44]. The numerical examples include:(a) Square plate; (b) Scordelis-Lo roof; (c) Pinched cylinder

1 https://ricamsvn.ricam.oeaw.ac.at/trac/gismo/wiki/WikiStart.

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Computational Mechanics

Table 1 Computational effortchecklist for comparison

Step Classical IGA Classical B Local B Present

1 – Fig. 2a Fig. 2b Fig. 2b

2 – Qp−1,q−1 Qp−1,q−1 Q p,q

3 – (N A, NB)Ω (N A, NB)Ωe (N A, NB)Ωe

4 � � � �5 – � � �6 – On Ω On Ωe On Ωe

7∫Ωe

BTDgBdΩ � � �8 � � � �

Algorithm 1: Present local B method for Reissner–Mindlin shellInput: geometry information, basis functions informationOutput: stiffness matrix KStep1. Construct projecting basis functions map as in Fig. 2(b)foreach element e do

Step2. Determine projection bases N according to currentelement position (see Fig. 2(b))Step3. Coefficient matrix M A B = (N A, NB)Ωe and its inverseStep4. Strain matrix BStep5. Calculate average strain through the thickness

MID = 1

h

∫ h2

− h2

B dz

Step6. Least square projection

MID = N AM−1A B

∫

Ωe

MIDNB dΩ

Step7. Element stiffness matrix

K e =∫

Ωe

BTDgB − MIDTDgMID + MIDTDgMID dΩ

Step8. Assemble K = K + K eend

and (d) Pinched hemisphere with a hole. Unless otherwisementioned consistent units are employed in this study. In allthe numerical examples, the following knot vectors are cho-sen as the initial ones:

Ξ = {0, 0, 0, 1, 1, 1},H = {0, 0, 0, 1, 1, 1}. (37)

The following conventions are employed whilst discussingthe results:

– IGA 2: Classical IGA without any treatment of locking,approximation orders p = q = 2.

– B-bar 2: B method, global projection, p = q = 2.– LB 2: Local B method [26], the projection is appliedelement by element as shown in Eq. (31) and Fig. 2a,p = q = 2.

– SRI 2: Selective reduced integration scheme introducedin [21], p = q = 2 and we restrict C1 continuity betweenelements.

– GLB 2: Local B method with present projection strategyas shown in Fig. 2b, p = q = 2.

4.1 Simply supported square plate

Consider a simply supported square plate with thickness hand edge length L = 1 m. Owing to symmetry, only onequarter of the plate, i.e, a = L/2 is modeled as shown inFig. 4. The plate is assumed to be made up of homogeneousisotropic material with Young’s modulus E = 200 GPa andPoisson’s ratio ν = 0.3, and subjected to uniform pressure p.The control points and the corresponding weights are givenin Table 2.

The analytical out-of-plane displacement field for a thinplate with simply supported edges is given by

w(x, y) = 16p

π6D

∞∑

m=1,3,5,...

∞∑

n=1,3,5,...

sin(mπx

L

)sin

( nπ yL

)

mn[(m

L

)2 + ( nL

)2]2,

(38)

where D = Eh3/(12(1− ν2)). The transverse displacementis constant when the applied load is proportional to h3, thusthe numerical solution is independent of the plate thickness.Due to the fact thatwA indicates the maximum displacementof the plate, only the errors of wA are studied instead of thefield error. The reference value iswA = −2.21804×10−6 m.

Figure 5 shows the normalized displacement as a functionof mesh refinement. For thickness h = 10−3 m, although theconventional IGAyields inaccurate results for coarsemeshes,the results tend to improve upon refinement. However, IGAsuffers from severe shear locking syndrome for h = 10−5 m.In this case the results remain almost identical when increas-ing the number of elements, indicating that the elements areextremely locked. For Bmethod slight rank deficiency occursfor coarse meshes. GLB gets more accurate results than LBwhen h = 10−5 m. It is inferred that the proposed formula-

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Computational Mechanics

Fig. 4 Mid-surface of therectangular plate in Sect. 4.1:geometry and boundaryconditions. Only one quarter ofthe plate is modeled. The redfilled squares are thecorresponding control points.(Color figure online)

A

L2

L2

simply supported

simply supportedsymmetry

symmetry

x

z

y

E = 200 × 109 Pa

ν = 0.3L = 1m

h = 10−3 m, p = 10−2 N/m2

or

h = 10−5 m, p = 10−8 N/m2

Table 2 Control points and weights of the rectangular plate (Fig. 4)

1 2 3 4 5 6 7 8 9

x 0.5 0.75 1 0.5 0.75 1 0.5 0.75 1

y 0 0 0 0.25 0.25 0.25 0.5 0.5 0.5

z 0 0 0 0 0 0 0 0 0

w 1 1 1 1 1 1 1 1 1

tion alleviates shear locking phenomena and yields accurateresults even for extremely thin plates of slenderness ratio 105.

Figure 6 compares the convergence behavior by a severelocking case, the plate thickness is fixed as h = 10−5 m.In this study, IGA of order 2 is locked for even more than1000 elements, the convergence rate is nearly zero. Elementsby IGA of order 3 start with smaller error, but suffer fromlocking until they are sufficiently refined. LB behaves thesame as GLB for mesh 2 × 2. GLB achieves a good conver-gence rate in general, but at the well-refined steps the errorsbecome unreasonably increased, which is also observed forIGA of order 3. The reason is that the converging resultsmismatch the reference value slightly, as pointed out in thepartial enlarged Fig. 6b.

In Fig. 7 we analysis the computational time of theglobal/local B methods and IGA. Here we use a ×64 com-puter with Inter CPU i5-4590 (3.30 GHz), 8GB RAM, thenrun the code only by a single thread. Overall the B methodsadd slightly computational effort to IGA, for instance about2.64% in case of 64×64 elements, thus for a given accuracythe B methods are more cost-effective than IGA. For presentmethod, the time is spend mostly on calculating the aver-age strain (Step No. 5) and the least square projection (StepNo. 6). For each element the dimension of the coefficientmatrix M is one or two, therefore the computational time isnot increased significantly. However in classical B method,inverting M is more time consuming as the dimension of Mis proportional to the number of elements (Fig. 7).

4.2 Scordelis–Lo roof

Here we consider the Scordelis–Lo roof problem to furtherdemonstrate the capability of the proposed formulation in

0 10 20 30 40 50 60 700.86

0.88

0.9

0.92

0.94

0.96

0.98

1

1.02

Number of surface elements per side

wA/w

ref

A IGA 2B-bar 2LB 2GLB 2

(a) h = 10−3

0 10 20 30 40 50 60 700.86

0.88

0.9

0.92

0.94

0.96

0.98

1

1.02

Number of surface elements per side

wA/w

ref

A IGA 2B-bar 2LB 2GLB 2

(b) h = 10−5

Fig. 5 Normalized central displacement wA with mesh refinement forthe rectangular plate (Fig. 4). h stands for thickness

the case of membrane locking. It features a cylindrical panelwith ends supported by rigid diaphragm. The roof is sub-jected to uniform pressure, pz = 6250 N/m2 and the verticaldisplacement of the mid-point of the side edge is monitored

123

Computational Mechanics

100 101 10210−6

10−5

10−4

10−3

10−2

10−1

100

Number of surface elements per side

|wA−

wref

A|/|

wref

A|

IGA 2B-bar 2LB 2GLB 2IGA 3

(a) Convergence

100 101 102

−2.26

−2.24

−2.22

−2.2

−2.18

·10−6

Number of surface elements per side

wA

B-bar 2LB 2GLB 2IGA 3wref

A

(b) Partial enlarged

Fig. 6 Convergence of deflection wA for the rectangular plate prob-lem (Fig. 4), thickness h = 10−5. IGA elements are locked until theelements are refined to a large number

to study the convergence behavior. The material propertiesare Young’s modulus E = 30 GPa and Poisson’s ratio ν = 0.Owing to symmetry only one quarter of the roof is modeledas shown in Fig. 8. The roof is modeled with control points(x, y, z) and weights w, given in Table 3. The analyticalsolution based on the deep shell theory iswref

B = −0.0361 m[45], however the results obtained by IGA of p = q = 4with mesh 100×100, wref

B = −0.0361776 m, is taken as thereference solution.

For the geometry considered here, R/h = 100 andL/h = 200, the structure experiences membrane lockingas the transverse shear strain is negligible. The roof is domi-nated bymembrane and bending deformations. The results ofthe normalized vertical displacementwithmesh refinement isshown in Fig. 9. The results from the proposed formulation is

10 20 30 40 50 60 700

200

400

600

800

1,000

1,200

1,400

1,600

Number of surface elements per side

Tim

e(s

)

IGA 2B-bar 2GLB 2

(a) Computational time

0 200 400 600 800 1,000 1,200 1,400 1,6000.86

0.88

0.9

0.92

0.94

0.96

0.98

1

1.02

Computational time (s)

wA/w

ref

A

IGA 2B-bar 2GLB 2

(b) Accuracy (h = 10−5) versus computational time

10 20 30 40 50 60 700

10

20

30

40

Number of surface elements per side

Tim

edi

ffere

nce

(s)

B-bar 2GLB 2

(c) Time difference from bi-quadratic IGA

Fig. 7 Computational time comparison for the plate problem (Fig. 4)

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Computational Mechanics

x

y

zL2

free

symmetry

symmetry

R

pz

pz

pz

pz

rigid diaphragm

E = 30 × 109 Pa

ν = 0L = 6m

R = 3m

h = 0.03m

pz = 6250N/m2

B

Fig. 8 Scordelis–Lo roof problem in Sect. 4.2: geometry and boundaryconditions. The red filled squares are the corresponding control points.The mid-surface of the cylindrical panel is modeled and the roof issubjected to a uniform pressure. (Color figure online)

compared with selective reduced integration technique [21].It can be seen that accurate results are achieved by the presentmethod. For coarse meshes the B methods, including classi-cal B method and local B method as well as present method,lead to rank insufficient matrices, however for other prob-lems slightly rank deficiency is also possible as recorded inthe strain smoothingmethod by a single subcell [46]. In addi-tion, the contour plot of the deflection wB by IGA and GLBare given in Fig. 10. It is can be noted that GLB captures thedeformation quite well even for coarse meshes.

4.3 Pinched cylinder

The examples in Sects. 4.1 and 4.2 show that the proposedformulation yields accurate results when the structure expe-rience either shear locking or membrane locking. We presenthere the pinched cylinder problem with the aim of demon-strating the robustness of the proposed formulation whenthe structure experiences both shear and membrane locking.Again, due to symmetry only one quarter of the cylinderis modeled as shown in Fig. 11. The corresponding con-trol points and weights are given in Table 4. The cylinderconsists of of homogeneous isotropic material with Young’smodulus, E = 30 GPa and Poisson’s ratio, ν = 0.3. Theconcentrated load acting on the cylinder is P = 0.25 N.The reference value of the vertical displacement is take aswrefC = −1.85942−7 m, which is obtained by bi-quartic

IGA with mesh 100 × 100. This example serves as a test

0 10 20 30 40 50 60 700

0.2

0.4

0.6

0.8

1

1.2

1.4

Number of surface elements per side

wB

/w

ref

B

IGA 2SRI 2, C1

B-bar 2LB 2GLB 2

(a) Normalized

0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

1.2

Number of surface elements per side

wB

/wref

B

IGA 2SRI 2, C1

B-bar 2LB 2GLB 2

(b) Partial enlarged

Fig. 9 Results of wB for the Scordelis–Lo roof problem (Fig. 8)

case to evaluate the performance of the method when thestructure is dominated by bending behaviour. The conver-gence of the vertical displacement with mesh refinement isshown in Fig. 12 and it is evident that the proposed for-mulation yields more accurate results than the conventionalbi-quadratic IGA. The contour plot ofwC in Fig. 13 indicates

Table 3 Control points and weights of the Scordelis–Lo roof problem (Fig. 8)

1 2 3 4 5 6 7 8 9

x 0 1.091910703 1.928362829 0 1.091910703 1.928362829 0 1.091910703 1.928362829

y 0 0 0 1.5 1.5 1.5 3 3 3

z 3 3 2.298133329 3 3 2.298133329 3 3 2.298133329

w 1 0.9396926208 1 1 0.9396926208 1 1 0.9396926208 1

123

Computational Mechanics

Fig. 10 Deflection fieldw × 102 for the Scordelis–Loroof (Fig. 8) by IGA of order 2(a, b), and by the proposed localB method (c, d)

× 4 IGA × 8 IGA

× 4 GLB

(a) 4 (b) 8

(c) 4 (d) 8 × 8 GLB

rigiddiaphragm

sym

met

ry

symmetry

symmetr

y

x

y

z

R

L2

CP

E = 30 × 109 Pa

ν = 0.3L = 6m

R = 3m

h = 0.03m

P = 1 × 14

N

Fig. 11 Themid-surface of a fourth of the pinched cylinder in Sect. 4.3.The red filled squares are the corresponding control points. (Color figureonline)

that the elements modified by GLB exhibit higher values ofdisplacements compared to IGA.

4.4 Pinched hemisphere with hole

Consider a pinched hemisphere with 18◦ hole subjected toequal and opposite concentrated forces applied at the fourcardinal points. The hemisphere is modeled with Young’smodulus E = 68.25 MPa, Poisson’s ratio ν = 0.3 andconcentrated force, P = 1 N. As before, owing to sym-metry, only one quadrant of the hemisphere is modeled asshown in Fig. 14. The control points and weights employedto model the hemisphere are given in Table 5. This struc-ture experiences severe membrane and shear locking, andthe discretization experiences severe mesh distortion. Themesh distortion further enhances the locking phenomenon.To evaluate the convergence properties, the horizontal dis-placement urefD = 0.0940 m [21] is taken as the referencesolution. The results form the proposed formulation are plot-

Table 4 Control points andweights of the pinched cylinderproblem (Fig. 11)

1 2 3 4 5 6 7 8 9

x 0 3 3 0 3 3 0 3 3

y 0 0 0 1.5 1.5 1.5 3 3 3

z 3 3 0 3 3 0 3 3 0

w 1 0.7071067812 1 1 0.7071067812 1 1 0.7071067812 1

123

Computational Mechanics

0 10 20 30 40 50 60 700

0.2

0.4

0.6

0.8

1

Number of surface elements per side

wC

/wref

C

IGA 2SRI 2, C1

B-bar 2LB 2GLB 2

(a) Normalized

0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

Number of surface elements per side

wC

/wref

C

IGA 2SRI 2, C1

B-bar 2LB 2GLB 2

(b) Partial enlarged

Fig. 12 Results of wC of the pinched cylinder (Fig. 11)

ted in Fig. 15. As the elements are refined, GLB behavessimilarly to SRI, but the results obtained by GLB is slightlyaccurate. Moreover, the field of displacement ux is plotted inFig. 16, it seems that the proposed method allows to capturethe correct structural response.

4.5 Square plate with irregular meshes

Throughout Sect. 4.1, it is pointed out that in the cases ofplates with high slenderness ratio, e.g. 103 or 105, the classi-cal IGA elements suffer from locking, and present B method

has still achieved satisfactory results. However, the stud-ies are performed in somewhat ideal conditions by usingstructured meshes. In order to explore the generality andapplicability of the present method, we focus on the influ-ence of two kinds of unstructured meshes, i.e. control meshdistortion and nonuniform knots.

The influence of the control mesh distortion on the per-formance of the proposed formulation is investigated byconsidering two different kinds of distortions: control meshexpansion and rotation, as shown in Fig. 17. Thicknessh = 10−5 m is considered here as the case when both lockingand mesh distortion appear, and h = 10−1 m is chosen asthe reference case when only mesh distortion appears. Theinfluence of mesh distortion on the normalized center dis-placement is pictured in Fig. 18. When locking and meshdistortion occur at the same time, accuracy of IGA decreasessubstantially, while GLB keeps a good degree of accuracyeven for severe distortions. It is inferred that when lockinghappens, the results with classical IGA deteriorates with con-trol mesh distortion, while the proposed formulation is lesssensitive to control mesh distortion.

The influence of the non-uniformly distributed knots onthe performance of the proposed formulation is investigatedby considering a chosen knot ξi moving from its left neighborξi−1 to its right neighbor ξi+1. In the following example webuild the plate geometry by

Ξ = {. . . , ξ4 = 0.25, ξ5 = 0.375, ξ6 = 0.5, . . .},H = {. . . , η4 = 0.25, η5 = 0.375, η6 = 0.5, . . .}, (39)

and let ξ5 move from ξ4 to ξ6, η5 move from η4 to η6 simul-taneously, as illustrated in Fig. 19, while keeping the controlmesh normally distributed. Figure 20 shows the results interms of knots distribution, once again the classical IGAleads to poor results for unstructured knots distribution, butthe proposed B formulation keeps good accuracy. It shouldbe noted that, when ξ5 = ξ4 and η5 = η4, or when ξ5 = ξ6and η5 = η6, the special cases of repeated knots (C0 continu-ity) are also included. For present B formulation, there is noneed to consider inter knot multiplicity explicitly since theprojection procedure is applied element-wise. Thus, lowerorder knot vectors can be used without inner repeated knotsfor projection, in spite of element continuity. A similar anddetailed conclusion is drawn in [37].

123

Computational Mechanics

Fig. 13 Field of w × 107 of thepinched cylinder (Fig. 11) byIGA of order 2 (a, b), and bypresent local B method (c, d)

× 4 IGA × 8 IGA

× 4 GLB

(a) 4 (b) 8

(c) 4 (d) 8 × 8 GLB

123

Computational Mechanics

free

freesy

mm

etry

symm

etry

x

y

z

R

D

F

F

E = 68.25 × 106 Pa

ν = 0.3R = 10m

h = 0.04m

F = 2 × 12

N

Fig. 14 The mid-surface of a fourth of the pinched hemisphere inSect. 4.4. The red filled squares are the corresponding control points

5 Conclusion remarks

The local B method is adopted to unlock the degener-ated Reissner–Mindlin shell elements within the frameworkof isogeometric analysis. The shell mid-surface and theunknown field are described with non-uniform rational B-splines. The proposed method uses multiple sets of lowerorder B-spline basis functions as projection bases, by whichthe locking strains are modified in the sense of L2 projection,in this way field-consistent strains are obtained.

The salient features of the proposed local B method are:(a) has less computational effort than the classical, globalB method; (b) yields better accuracy than IGA of order 2especially in cases of coarse meshes or mesh distortions;(c) suppresses both shear and membrane locking commonlyencountered when lower order elements are employed andsuitable for both thick and thin models.

Future work includes extending the approach to largedeformations, large deflections and large rotations as wellas investigating the behaviour of the stabilization techniquefor enriched approximations such as those encountered inpartition of unity methods [46–48].

0 10 20 30 40 50 60 700

0.5

1

1.5

2

Number of surface elements per side

wD

/wref

D

IGA 2SRI 2, C1

B-bar 2LB 2GLB 2

(a) Normalized

0 2 4 6 8 10 12 14 16 180

0.2

0.4

0.6

0.8

1

Number of surface elements per side

wD

/wref

D

IGA 2SRI 2, C1

B-bar 2LB 2GLB 2

(b) Partial enlarged

Fig. 15 Results of uD of the pinched hemisphere (Fig. 14)

Table 5 Control points and the corresponding weights of the pinched hemisphere problem (Fig. 14)

1 2 3 4 5 6 7 8 9

x 10 10 0 10 10 0 3.090169944 3.090169944 0

y 0 10 10 0 10 10 0 3.090169944 3.090169944

z 0 0 0 7.265425281 7.265425281 7.265425281 9.510565163 9.510565163 9.510565163

w 1 0.7071067810 1 0.8090169942 0.5720614025 0.8090169942 1 0.7071067810 1

123

Computational Mechanics

Fig. 16 Field of u × 102 of thepinched hemisphere (Fig. 14) byIGA of order 2 (a, b), and bypresent local B method (c, d)

× 4 IGA × 8 IGA

× 4 GLB

(a) 4 (b) 8

(c) 4 (d) 8 × 8 GLB

123

Computational Mechanics

e = 0-2-1

123

(a) Expansion

r = 012345

(b) Rotation

Fig. 17 Illustration of control mesh distortions. In (a), four selectedcontrol points are moved along the diagonal directions. In (b), fourselected control points are rotated around the center of the patch. Indexese and r are employed to indicate the stages of distortions

−2 −1 0 1 2 30

0.2

0.4

0.6

0.8

1

e

wA/w

ref

A

IGA 2, h = 10−1

IGA 2, h = 10−5

GLB 2, h = 10−1

GLB 2, h = 10−5

(a) Expansion e

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

r

wA/w

ref

A

IGA 2, h = 10−1

IGA 2, h = 10−5

GLB 2, h = 10−1

GLB 2, h = 10−5

(b) Rotation r

Fig. 18 Normalized results of wA of the rectangular plate (Fig. 4) withcontrol mesh distortions (Fig. 17). h stands for thickness.When lockingand mesh distortion occur at the same time, errors of IGA of order 2drop down quickly, while GLB keeps good accuracy even for severedistortions

0 0.25 0.5 0.75 1

0.25

0.5

0.75

1

Fig. 19 Nonuniformknots distribution for the rectangular plate (Fig. 4).Let ξ5 move from ξ4 to ξ6, η5 move from η4 to η6 simultaneously

0.25 0.3125 0.375 0.4375 0.50

0.2

0.4

0.6

0.8

1

ξ5 and η5

wA/w

ref

A

IGA 2, h = 10−1

IGA 2, h = 10−5

GLB 2, h = 10−1

GLB 2, h = 10−5

Fig. 20 Normalized results of wA of the rectangular plate (Fig. 4) withnonuniformly knots distribution (Fig. 19). h stands for thickness

Acknowledgements Q. Hu is thankful for Prof. Gengdong Cheng forthe valuable suggestions of this research subject. Y. Xia is fundedby National Natural Science Foundation of China (Nos. 11702056,61572021). Stéphane Bordas thanks partial funding for his timeprovided by the European Research Council Starting IndependentResearch Grant (ERC Stg grant agreement No. 279578) “RealTCutTowards real time multiscale simulation of cutting in non-linear mate-rials with applications to surgical simulation and computer guidedsurgery”. We also thank the funding from the Luxembourg NationalResearch Fund (INTER/MOBILITY/14/8813215/CBM/Bordas andINTER/FWO/15/10318764).

Appendix A: Using an approximated normalvector field

In Eqs. 8 and 9, the normal vector field n(ξ, η) should becommonly built as

∑RAnA, where nA needs to be specified

for each control point: it denotes the normal vector of a mid-

123

Computational Mechanics

surface point associated with the control point xA. Althoughno approximation error occurs, it is time-consuming to solvefor the mid-surface projecting points of xA. In this paper thenormal vectors at the Greville abscissae (ξ , η) are adoptedto construct a normal vector field approximately [21], thefield is re-written as

∑RA(ξ, η)nA. Following Eq. 2, these

normal vectors nA at the Greville abscissae points can beeasily calculated. Thus we have

xP (ξ, η, ζ ) =nm∑

A=1

RA

(xA + ζ

h

2nA

), (A.1)

and

uP (ξ, η, ζ ) =nm∑

A=1

RA

(uA + ζ

h

2θ A × nA

). (A.2)

Appendix B: Detailed formulation of degen-erated Reissner–Mindlin shell element

In the following we introduce some necessary aspects toobtain the element stiffness matrix:

(a) Using Voigt notation, the relation between the strain vec-tor and the stress vector is expressed as

σ = Dgεεε, (B.1)

where Dg is the global constitutive matrix, and

Dg = TTDlT , (B.2)

here Dl is a given local constitutive matrix. To fulfill theplane stress state σ33 = 0, the local constitutive matrix isgiven by

Dl = E

1 − ν2

⎡

⎢⎢⎢⎢⎢⎢⎣

1 ν 0 0 0 0ν 1 0 0 0 00 0 0 0 0 00 0 0 1−ν

2 0 00 0 0 0 κ 1−ν

2 00 0 0 0 0 κ 1−ν

2

⎤

⎥⎥⎥⎥⎥⎥⎦, (B.3)

in which κ is the shear correction factor and it is set to be 5/6in this research. To fit the physical shape of the mid-surface,a transformation is employed for Dl , specifically

T =

⎡

⎢⎢⎢⎢⎢⎢⎣

t1x t1x t1yt1y t1z t1z t1x t1y t1yt1z t1z t1xt2x t2x t2yt2y t2z t2z t2x t2y t2yt2z t2z t2xnxnx nyny nznz nxny nynz nznx2t1x t2x 2t1yt2y 2t1zt2z t1x t2y + t2x t1y t1yt2z + t2yt1z t1zt2x + t2zt1x2t2xnx 2t2yny 2t2znz t2xny + nx t2y t2ynz + nyt2z t2znx + nzt2x2t1xnx 2t1yny 2t1znz t1xny + nx t1y t1ynz + nyt1z t1znx + nzt1x

⎤

⎥⎥⎥⎥⎥⎥⎦. (B.4)

At each Gauss quadrature point, vectors in Eqs. 2 and 3 arecalculated, then the transformation in Eq. B.2 is performed.

(b) Based on Eq. A.1, the Jacobi matrix is calculated by

J =

⎡

⎢⎢⎣

∂xxP∂ξ

∂x yP∂ξ

∂xzP∂ξ

∂xxP∂η

∂x yP∂η

∂xzP∂η

∂xxP∂ζ

∂x yP∂ζ

∂xzP∂ζ

⎤

⎥⎥⎦ =⎡

⎢⎣

∑ ∂RA∂ξ

(xxA + ζ h2 n

xA)

∑ ∂RA∂ξ

(x yA + ζ h

2 nyA)

∑ ∂RA∂ξ

(xzA + ζ h2 n

zA)∑ ∂RA

∂η(xxA + ζ h

2 nxA)

∑ ∂RA∂η

(x yA + ζ h

2 nyA)

∑ ∂RA∂η

(xzA + ζ h2 n

zA)∑

RA(xxA + h2 n

xA)

∑RA(x y

A + h2 n

yA)

∑RA(xzA + h

2 nzA)

⎤

⎥⎦ , (B.5)

then the physical gradient of the shape functions are

⎡

⎢⎢⎣

∂RA∂x

∂RA∂ y

∂RA∂z

⎤

⎥⎥⎦ = J−1

⎡

⎢⎢⎣

∂RA∂ξ

∂RA∂η

∂RA∂ζ

⎤

⎥⎥⎦ = J−1

⎡

⎢⎢⎣

∂RA∂ξ

∂RA∂η

0

⎤

⎥⎥⎦ . (B.6)

Moreover for ζ h2 RA := RA, their physical gradients are

given by

⎡

⎢⎢⎢⎣

∂ RA∂x

∂ RA∂ y

∂ RA∂z

⎤

⎥⎥⎥⎦ = J−1

⎡

⎢⎢⎢⎣

∂ RA∂ξ

∂ RA∂η

∂ RA∂ζ

⎤

⎥⎥⎥⎦ = J−1

⎡

⎢⎢⎣

ζ h2

∂RA∂ξ

ζ h2

∂RA∂η

h2 RA

⎤

⎥⎥⎦ . (B.7)

123

Computational Mechanics

(c) Taking a look at the strain formula Eq. 5, we form thestrain matrix as

B =

⎡

⎢⎢⎢⎢⎢⎢⎢⎣

RA,x 0 0 0 RA,x nzA −RA,x n

yA

0 RA,y 0 −RA,y nzA 0 RA,y n

xA

0 0 RA,z RA,z nyA −RA,z n

xA 0

RA,y RA,x 0 −RA,x nzA RA,y n

zA RA,x n

xA − RA,y n

yA

0 RA,z RA,y RA,y nyA − RA,z n

zA −RA,y n

xA RA,z n

xA

RA,z 0 RA,x RA,x nyA RA,z n

zA − RA,x n

xA −RA,z n

yA

⎤

⎥⎥⎥⎥⎥⎥⎥⎦

. (B.8)

Finally the element stiffness matrix is calculated by

K e =∫

Ωe

BTDgBdΩ =∫ 1

−1

∫

ηe

∫

ξe

BTDgB|J |dξdηdζ.

(B.9)

References

1. Hughes TJ, Cottrell JA, Bazilevs Y (2005) Isogeometric analysis:CAD, finite elements, NURBS, exact geometry and mesh refine-ment. Comput Methods Appl Mech Eng 194(39):4135–4195

2. Kagan P, Fischer A, Bar-Yoseph PZ (1998) New B-spline finiteelement approach for geometrical design and mechanical analysis.Int J Numer Methods Eng 41(3):435–458

3. Lipton S, Evans JA, Bazilevs Y, Elguedj T, Hughes TJ (2010)Robustness of isogeometric structural discretizations under severemesh distortion.ComputMethodsApplMechEng199(5):357–373

4. Marussig B, Zechner J, Beer G, Fries T-P (2015) Fast isogeometricboundary element method based on independent field approxima-tion. Comput Methods Appl Mech Eng 284:458–488

5. Xu G, Atroshchenko E, Bordas S (2014) Geometry-independentfield approximation for spline-based finite element methods. In:Proceedings of the 11th world congress in computational mechan-ics

6. Xu G, Atroshchenko E, Bordas S (2014) Geometry-independentfield approximation for spline-based finite element methods. In:Proceedings of the 11th world congress in computational mechan-ics

7. AtroshchenkoE,XuG,TomarS,BordasSWeakening the tight cou-pling between geometry and simulation in isogeometric analysis:from sub-and super-geometric analysis to Geometry IndependentField approximaTion (GIFT), arXiv preprint arXiv:1706.06371

8. Toshniwal D, Speleers H, Hughes TJ (2017) Smooth cubicspline spaces on unstructured quadrilateral meshes with particularemphasis on extraordinary points geometric design and isogeo-metric analysis considerations. Comput Methods Appl Mech Eng327:411–458

9. Kiendl J, BletzingerK-U, Linhard J,Wüchner R (2009) Isogeomet-ric shell analysis with Kirchhoff–Love elements. Comput MethodsAppl Mech Eng 198(49):3902–3914

10. Benson D, Bazilevs Y, Hsu M-C, Hughes T (2011) A large defor-mation, rotation-free, isogeometric shell. Comput Methods ApplMech Eng 200(13):1367–1378

11. Benson D, Bazilevs Y, Hsu M-C, Hughes T (2010) Isogeometricshell analysis: the Reissner–Mindlin shell. Comput Methods ApplMech Eng 199(5):276–289

12. Benson D, Hartmann S, Bazilevs Y, Hsu M-C, Hughes T (2013)Blended isogeometric shells. Comput Methods Appl Mech Eng255:133–146

13. Dornisch W, Klinkel S, Simeon B (2013) Isogeometric Reissner–Mindlin shell analysis with exactly calculated director vectors.Comput Methods Appl Mech Eng 253:491–504

14. Hosseini S, Remmers JJ, Verhoosel CV, Borst R (2013) An isogeo-metric solid-like shell element for nonlinear analysis. Int J NumerMethods Eng 95(3):238–256

15. Cottrell JA, Reali A, Bazilevs Y, Hughes TJ (2006) Isogeometricanalysis of structural vibrations. Comput Methods Appl Mech Eng195(41):5257–5296

16. HuQ,ChoulyF,HuP,ChengG,BordasSP (2018)Skew-symmetricnitsche’s formulation in isogeometric analysis: Dirichlet and sym-metry conditions, patch coupling and frictionless contact. ComputMethods Appl Mech Eng 341:188–220

17. Kiendl J, Bazilevs Y, Hsu M-C, Wüchner R, Bletzinger K-U(2010) The bending strip method for isogeometric analysis ofKirchhoff-Love shell structures comprised of multiple patches.Comput Methods Appl Mech Eng 199(37):2403–2416

18. Echter R, Bischoff M (2010) Numerical efficiency, locking andunlocking ofNURBSfinite elements. ComputMethodsApplMechEng 199(5):374–382

19. Hughes TJ, Cohen M, Haroun M (1978) Reduced and selectiveintegration techniques in the finite element analysis of plates. NuclEng Des 46(1):203–222

20. Adam C, Bouabdallah S, Zarroug M, Maitournam H (2014)Improved numerical integration for locking treatment in isogeo-metric structural elements. Part I: Beams. Comput Methods ApplMech Eng 279:1–28

21. Adam C, Bouabdallah S, Zarroug M, Maitournam H (2015)Improved numerical integration for locking treatment in isoge-ometric structural elements. Part II: plates and shells. ComputMethods Appl Mech Eng 284:106–137

22. Adam C, Bouabdallah S, Zarroug M, Maitournam H (2015) Areduced integration for Reissner–Mindlin non-linear shell analysisusing T-splines. In: Isogeometric analysis and applications 2014,Springer, Berlin, pp 103–125

23. Elguedj T, Bazilevs Y, Calo VM, Hughes TJR (2008) B over-barand F over-bar projection methods for nearly incompressible linearand non-linear elasticity and plasticity using higher-order nurbselements. Comput Methods Appl Mech Eng 197(33–40):2732–2762

24. Bouclier R, Elguedj T, Combescure A (2012) Locking free isogeo-metric formulations of curved thick beams. Comput Methods ApplMech Eng 245:144–162

25. Bouclier R, Elguedj T, Combescure A (2013) On the developmentof NURBS-based isogeometric solid shell elements: 2D problemsand preliminary extension to 3D. Comput Mech 52(5):1085–1112

123

Computational Mechanics

26. Bouclier R, Elguedj T, Combescure A (2013) Efficient isogeo-metric NURBS-based solid-shell elements: mixed formulation andB-bar method. Comput Methods Appl Mech Eng 267:86–110

27. Bouclier R, Elguedj T, Combescure A (2015) An isogeomet-ric locking-free nurbs-based solid-shell element for geometricallynonlinear analysis. Int J Numer Methods Eng 101(10):774–808

28. Echter R, Oesterle B, Bischoff M (2013) A hierarchic family ofisogeometric shell finite elements. Comput Methods Appl MechEng 254:170–180

29. Bletzinger K-U, Bischoff M, Ramm E (2000) A unified approachfor shear-locking-free triangular and rectangular shell finite ele-ments. Comput Struct 75(3):321–334

30. Brezzi F, Evans J, Hughes T, Marini L (2011) New quadrilateralplate elements based on Twist–Kirchhoff theory. Comput MethodsAppl Mech Eng 200:2547–2561

31. Brezzi F,Marini LD (2013)Virtual elementmethods for plate bend-ing problems. Comput Methods Appl Mech Eng 253:455–462

32. da Veiga LB, Lovadina C, Reali A (2012) Avoiding shear lockingfor the Timoshenko beam problem via isogeometric collocationmethods. Comput Methods Appl Mech Eng 241:38–51

33. Auricchio F, da Veiga LB, Kiendl J, Lovadina C, Reali A (2013)Locking-free isogeometric collocation methods for spatial Timo-shenko rods. Comput Methods Appl Mech Eng 263:113–126

34. Yin S, Hale JS, Yu T, Bui TQ, Bordas SP (2014) Isogeometriclocking-free plate element: a simple first order shear deformationtheory for functionally graded plates. Compos Struct 118:121–138

35. Kiendl J, Auricchio F, Hughes T, Reali A (2015) Single-variableformulations and isogeometric discretizations for shear deformablebeams. Comput Methods Appl Mech Eng 284:988–1004

36. Beirão Da Veiga L, Hughes T, Kiendl J, Lovadina C, Niiranen J,Reali A, Speleers H (2015) A locking-free model for Reissner–Mindlin plates: analysis and isogeometric implementation viaNURBS and triangular NURPS. Math Models Methods Appl Sci25(08):1519–1551

37. Hu P, Hu Q, Xia Y (2016) Order reduction method for lockingfree isogeometric analysis ofTimoshenkobeams.ComputMethodsAppl Mech Eng 308:1–22

38. Mitchell TJ, Govindjee S, Taylor RL (2011) Amethod for enforce-ment of Dirichlet boundary conditions in isogeometric analysis.In: Mueller-Hoeppe D, Loehnert S, Reese S (eds) Recent devel-opments and innovative applications in computational mechanics.Springer, Berlin, pp 283–293

39. Govindjee S, Strain J, Mitchell TJ, Taylor RL (2012) Convergenceof an efficient local least-squares fitting method for bases withcompact support. Comput Methods Appl Mech Eng 213:84–92

40. Nguyen VP, Anitescu C, Bordas SP, Rabczuk T (2015) Isogeomet-ric analysis: an overview and computer implementation aspects.Math Comput Simul 117:89–116

41. Simo J,HughesT (1986)On the variational foundations of assumedstrain methods. J Appl Mech 53(1):51–54

42. Antolin P, Bressan A, Buffa A, Sangalli G (2017) An isogeometricmethod for linear nearly-incompressible elasticity with local stressprojection. Comput Methods Appl Mech Eng 316:694–719

43. Hughes TJR (1980) Generalization of selective integration proce-dures to anisotropic and nonlinear media. Int J Numer MethodsEng 15:1413–1418

44. Juettler B, Langer U, Mantzaflaris A, Moore S, ZulehnerW (2014)Geometry + simulation modules: implementing isogeometric anal-ysis. Proc Appl Math Mech 14(1):961–962

45. Macneal RH, Harder RL (1985) A proposed standard set of prob-lems to test finite element accuracy. FiniteElemAnalDes1(1):3–20

46. Bordas SP, Rabczuk T, Hung N-X, Nguyen VP, Natarajan S, BogT, Hiep NV et al (2010) Strain smoothing in FEM and XFEM.Comput Struct 88(23):1419–1443

47. ChenL,RabczukT,BordasSPA,LiuG,ZengK,KerfridenP (2012)Extended finite element method with edge-based strain smoothing(ESm-XFEM) for linear elastic crack growth. Comput MethodsAppl Mech Eng 209:250–265

48. Surendran M, Natarajan S, Bordas S, Palani G (2017) Lin-ear smoothed extended finite element method. arXiv preprintarXiv:1701.03997

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