a fully locking free isogeometric approach for plane linear elasticity problems a stream function...

8/13/2019 A Fully Locking Free Isogeometric Approach for Plane Linear Elasticity Problems a Stream Function Formulation

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A fully locking-free isogeometric approach for plane linearelasticity problems: A stream function formulation

F. Auricchio a,b , L. Beirao da Veiga d , A. Buffa b , C. Lovadina b,c, * , A. Reali a, G. Sangalli b,c

a Dipartimento di Meccanica Strutturale,Universita ` di Pavia, Italyb IMATI-CNR, Pavia, Italy

c Dipartimento di Matematica, Universita ` di Pavia, Via Ferrata 1 Pavia I-27100, Italyd Dipartimento di Matematica, Universita ` di Milano, Italy

Received 9 February 2007; accepted 11 July 2007Available online 25 July 2007

Abstract

In this paper, we study plane incompressible elastic problems by means of a stream-function formulation such that a divergence-freedisplacement eld can be computed from a scalar potential. The numerical scheme is constructed within the framework of NURBS-basedisogeometric analysis and we take advantage of the high continuity guaranteed by NURBS basis functions in order to obtain the dis-placement eld from the potential differentiation. As a consequence, the obtained numerical scheme is automatically locking-free inthe presence of the incompressibility constraint. A Discontinuous Galerkin approach is proposed to deal with multiple mapped, possiblymultiply connected, domains. Extensive numerical results are provided to show the method capabilities.

2007 Elsevier B.V. All rights reserved.

Keywords: Incompressible elasticity; Stream function formulation; Isogeometric analysis; NURBS

1. Introduction

In this work we focus on two-dimensional plane strainlinear elastic problems and we investigate the possibility of solving in an exact way the constraint induced by the incom-pressibility condition through the use of an isogeometricinterpolation based on Non-Uniform Rational B -Splines(NURBS), see [15]. In particular, we take advantage of thefact that high continuity across the elements is easily

achieved by NURBS functions.It is well known that it is possible to rewrite the incom-pressible elastic problem as an elliptic fourth-order problemin terms of a scalar potential (referred to as stream function ,see [12]) whose curl gives the displacement eld. That is, theproblem is rewritten such to have only a single scalar eld

unknown (implying less degrees of freedom than the origi-nal problem) and the displacements are obtained comput-ing the curl of such a scalar eld in post-processing. Theobtained solution is divergence-free by construction and,hence, the incompressibility constraint is automaticallyand exactly satised.

The trade-off of this methodology is the need to con-struct a solution characterized by higher continuity thanthe one required by the original problem. This aspect rep-

resents a severe drawback for standard numerical tech-niques such as nite elements, because rather complicatedmethods must be adopted (see, for instance, the elementsdetailed in [2]). In the present paper this difficulty is per-fectly circumvented and solved resorting to isogeometricanalysis [15], which may be seen as a generalization of classical nite element analysis. Indeed, the NURBS-basedisogeometric approach allows to use shape functionssmoother than the standard C 0-continuity of nite ele-ments. Another strength of that approach, which is of greatrelevance in all engineering applications, is that it makes

0045-7825/$ - see front matter 2007 Elsevier B.V. All rights reserved.

doi:10.1016/j.cma.2007.07.005

* Corresponding author. Address: Dipartimento di Matematica, Uni-versita di Pavia, Via Ferrata 1 Pavia I-27100, Italy. Tel.: +39 0382 985685;fax: +39 0382 985602.

E-mail address: [email protected] (C. Lovadina).

www.elsevier.com/locate/cma

Available online at www.sciencedirect.com

Comput. Methods Appl. Mech. Engrg. 197 (2007) 160172
mailto:[email protected]:[email protected]


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possible the exact geometric representation of CAD objectsand simplies mesh renement (see [15]).

Accordingly, combining the stream function reformula-tion and the NURBS-based technique we end up with adiscretization procedure for which volumetric lockingeffects cannot occur. Therefore, the proposed formulation

represents a highly accurate alternative to more classicalapproaches, such as mixed or enhanced strain nite ele-ment methods (see [9] or [21], for instance).

The paper starts with a presentation of the stream func-tion formulation for plane linear elasticity. Then, after abrief review of some basic concepts of Non-UniformRational B -Splines (NURBS), we introduce the isogeomet-ric approach that is employed both for describing the prob-lem geometry and for approximating the stream function.Moreover, we sketch an extension of our formulation todeal with 3D problems. We also outline a DiscontinuousGalerkin type approach to deal with the case of multi-patch domains, thus including multiply connected geome-tries. Finally, we present extensive numerical tests in orderto show the capabilities of the method on some signicantmodel problems. In particular, a rst comparison withsome mixed-enhanced nite elements (see [5,16]) shows asuperior behaviour of our NURBS-based discretization.As a conclusion, future directions of research on the pro-posed formulation are drawn.

2. The plane linear elasticity problem for incompressiblematerials

We consider the plane linear elasticity problem in the

framework of the innitesimal theory (cf., e.g. [14 or 20])for a homogeneous isotropic incompressible material.The elastic body occupies a regular region X in R 2 withboundary oX C D [ C N such that C D \ C N ; . We aretherefore led to solve the following boundary-valueproblem:

Find u; p such that

div 2l eu pd f in X;

div u 0 in X;

u 0 on C D;

2l eu pd n g on C N ;

8>>>>>>>>>>>>>>>>>>>:

1

where u u1; u2; X ! R 2 is the displacement eld,p; X ! R is the pressure-like eld, f f 1; f 2; X ! R 2and g g 1; g 2; C N ! R 2 are loading terms. Moreover,e is the usual symmetric gradient operator acting on vec-tor elds, while d is the second-order identity tensor. Final-ly, l is the Lame coefficient, for which we suppose that0 < l 0 6 l 6 l 1 < 1 .

Let H 1 DX be the Sobolev space of H 1 functions vanish-

ing on C D. We recall that a standard variational formula-tion of problem (1) consists in nding u; p 2 V Q H 1 DX

2 L2X which solves the system

2l Z X eu : ev Z X p div v Z X f v Z C N g v 8v 2 V;R X q div u 0 8q 2 Q:

8


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(CAD) and computer graphics (see [17,18]). A shortdescription of these functions follows.

3.1. B-Splines and NURBS

B -splines in the plane are piecewise polynomial curves

composed of linear combinations of B -spline basis func-tions. The coefficients are points in the plane, referred toas control points .

A knot vector is a set of non-decreasing real numbers rep-resenting coordinates in the parametric space of the curve

f n1 0; . . . ; nn p 1 1g; 8

where p is the order of the B -spline and n is the number of basis functions (and control points) necessary to describeit. A knot vector is said to be uniform if its knots are uni-formly-spaced and non-uniform otherwise. Moreover, aknot vector is said to be open if its rst and last knots are re-peated p + 1 times. In what follows, we always employ openknot vectors. Basis functions formed from open knot vec-tors are interpolatory at the ends of the parametric interval0; 1 but are not, in general, interpolatory at interior knots.

Given a knot vector, univariate B -spline basis functionsare dened recursively starting with p = 0 (piecewiseconstants)

N i;0n 1 if ni 6 n < ni 1

0 otherwise: 9

For p > 1

N i; p n n nini p ni

N i; p 1n ni p 1 nni p 1 ni 1

N i 1; p 1n:

10

In Fig. 1, we present an example consisting of n = 9cubic basis functions generated from the open knot vectorf 0; 0; 0; 0; 1=6; 1=3; 1=2; 2=3; 5=6; 1; 1; 1; 1g.

An important property of B -spline basis functions isthat they are C p 1-continuous, if internal knots are notrepeated. If a knot has multiplicity k , the basis is C p k -con-tinuous at that knot. In particular, when a knot has multi-plicity p, the basis is C 0 and interpolatory at that location.

By means of tensor products, a B -spline region can be

constructed starting from knot vectors fn1 0; . . . ;nn p 1 1g and fg1 0; . . . ; gm q 1 1g; and an n mnet of control points B i; j : Two-dimensional basis functions N i; p and M j;q (with i 1; . . . ; n and j 1; . . . ; m) of order pand q, respectively, are dened from the knot vectors, andthe B -spline region is the image of the mapS : 0; 1 0; 1 ! X given by

Sn; g Xn

i1 Xm

j1 N i; p n M j ;qgB i; j : 11

The two-dimensional parametric space is the domain0; 1 0; 1 . Observe that the two knot vectors fn1 0; . . . ; nn p 1 1g and fg1 0; . . . ; gm q 1 1g generatein a natural way a mesh of rectangular elements in theparametric space.

A rational B -spline in R 2 is the projection onto two-dimensional physical space of a polynomial B -splinedened in three-dimensional homogeneous coordinatespace. For a complete discussion of these space projections,see [11] and references therein. In this way, a great varietyof geometrical entities can be constructed and, in particu-lar, all conic sections can be obtained exactly. The projec-tive transformation of a B -spline curve yields a rationalpolynomial curve. Note that when we refer to the order

of a NURBS curve, we mean the order of the polynomialcurve from which the rational curve was generated.To obtain a NURBS curve in R 2, we start from a set Bwi

(i 1; . . . ; n) of control points (projective points) for aB -spline curve in R 3 with knot vector N. Then the controlpoints for the NURBS curve are

B i j Bwi j

wi; j 1; 2; 12

where B i j is the j th component of the vector Bi andwi Bwi 3 is referred to as the i th weight . The NURBS ba-sis functions of order p are then dened as

R p i n N i; p nwi

Pni1 N i; p nwi

: 13

The NURBS curve is dened by

Cn Xn

i1 R p i nB i: 14

Analogously to B -splines, NURBS basis functions onthe two-dimensional parametric space 0; 1 0; 1 aredened as

R p ;qi; j n; g N i; p n M j;qgwi; j

Pn

i1

Pm

j1 N i; p n M j ;qgwi; j; 15

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

N i , 3

Fig. 1. Cubic basis functions formed from the open knot vector

f 0; 0; 0; 0; 1=6; 1=3; 1=2; 2=3; 5=6; 1; 1; 1; 1g:

162 F. Auricchio et al. / Comput. Methods Appl. Mech. Engrg. 197 (2007) 160172


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where wi; j Bwi; j3. Observe that the continuity and sup-ports of NURBS basis functions are the same as for B -splines.

NURBS regions, similarly to B -spline regions, aredened in terms of the basis functions (15). In particularwe assume from now on that the physical domain X is a

NURBS region associated with the n m net of controlpoints Bi; j , and we introduce the geometrical mapF : 0; 1 0; 1 ! X given by

Fn; g Xn

i1 Xm

j1 R p ;qi; j n; gB i; j : 16

3.2. Isogeometric analysis

The image of the elements in the parametric spaces areelements in the physical spaces. The physical mesh istherefore

T h fFni; ni 1 g j ; g j 1 withi 1; . . . ; n p ; j 1; . . . ; m qg: 17

We denote by h the mesh-size, that is, the maximum diam-eter of the elements of T h.

Following the isoparametric approach, the space of NURBS functions on X is dened as the span of the push-forward of the basis functions (15)V h spanf R p ;qi; j F 1gi1;... ;n; j1;... ;m: 18

3.3. Numerical method

We introduce here the NURBS conforming discretiza-tion of thevariational problem (7).Let V h denote a NURBSspace as introduced in (18), of regularity C k , k P 1, anddegree p > k . We have V h H 2X, and then we can intro-duce the following conforming discretizations of U

Uh V h \ U: 19

The numerical method for (7) we propose is a plain Galer-kin formulation: the approximated displacement is denedas uh curl wh , where wh 2 Uh is the solution of theproblem

Find wh

2 Uh such that

2l Z X ecurl wh: ecurl u h Z X f curl u h Z C N g curl u h; 8u h 2 Uh:8


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F a : 0; 1 0; 1 ! Xa ; a 1; . . . ; N :

This occurs in the description of complex and/or multiplyconnected geometries, as depicted in Figs. 2 and 3 (notethat each patch is topologically equivalent to the paramet-ric domain 0; 1 0; 1 ).

On each patch Xa , we dene a space of potentials

Ua fu 2 H 2Xa=R ; curl u 0 on C D \ o Xa g; 26

whose curls give the space of divergence-free displacementson the patch

Ka fvjXa ; with v 2 Kg f curl u ; u 2 Uag: 27

The space of globally-dened divergence-free displace-ments K, dened in (3), can be obtained by glueing theKa s, i.e. by imposing continuity of the traces across theinterior interfaces. More precisely, it holds

v 2 K ()vjXa 2 Ka 8a 1; . . . ; N ;vjXa vjXb on oXa \ o Xb 8a ; b 1; . . . ; N :( 28

For the corresponding (patch-wise) potential u , the glueingcondition is therefore

curl u jXa curl u jXb on oXa \ o Xb 8a ; b 1; . . . ; N :

29

Hence, we here dene the space U as

U fu 2 L2X : u jXa 2 Ua ; a 1; . . . ; N ;u fulfills Eq: 29g: 30

Remark 1. The space U dened in (30) is different from itsanalogous introduced in the single-patch approach (cf. (6)).Indeed, conditions (29) only imply that the jump of u isconstant across the interior interfaces, and, therefore, u isnot necessarily a continuous function. This is in accordanceto what happens for multiply connected regions, where theexistence of a globally continuous potential u is notguaranteed, as it is well-known (see [12], for instance).

Having introduced the space U, the variational formula-tion of the problem on X for the unknown w is still (7)

Find w 2 U such that

2l

Z X

ecurl w: ecurl u

Z X

f curl u

Z C N

g curl u ; 8u 2 U8>>>>>>>>>>>>>>>>>>: 32Here, ff gg and s t denote the average and jump operatorson the interface C (see [4], for example), while C S is a dimen-sionless stabilizing constant. The three integrals on C dealwith the non-conformity of Uh, that is, with the lack of con-tinuity of discrete displacements across the interface C.

Remark 2. We remark that formulation (32) is a stronglyconsistent discretization of problem (31), as typical for DG

0 1

1

1

2

3

F

F

F3

2

1

Fig. 2. Simply connected X, split into three domains X1, X2 and X3.

1

2

1

2

0 1

1

F

F

Fig. 3. Multiply connected X, split into two simply connected domains X1and X2.



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methods. A rigorous analysis of the NURBS schemesbased on (32) can be developed by using the results of [6] incombination with the techniques of [7,8,13].

5. Numerical tests

In this section we present some numerical experimentsperformed to show the capabilities of the numericalmethod previously introduced. For all tests, we select anincompressible material with l = 40. Moreover, we remarkthat here and in the following we express forces and lengthsas KN and m, respectively.

5.1. Square block under a uniform body load with twoconstrained sides

We start considering a well-established problem takenfrom standard engineering literature. This test consists of a square block, subjected to a uniform body load f , withtwo consecutive constrained sides ( Fig. 5), for which weset L = 1 and f 0; 80.

We then consider a reference solution obtained from the(locking-free) quadrilateral mixed-enhanced formulationproposed in [16], that we have coded within the research-oriented nite element code FEAP (see [19]). A very nemesh of 101 101 nodes (with two degrees of freedomper node) has been employed (see Fig. 6). We check the

solution in three points referred to as A; B and C , whosecoordinates are, respectively, 1=2; 1; 1; 1 and 1; 1=2(cf. Fig. 5). The reference solution in correspondence of A, B and C is as follows:

u Aref 0:0782; 0:1575; u Bref 0:0915; 0:1624; uC ref 0:1658; 0:1176.

In Figs. 7 and 8, we report contour plots of the referencesolution for the two displacement components over thedeformed shape.

In Table 1 , we report the solutions obtained with themethod proposed in the present paper for different choicesof approximation orders and grids, normalized with respectto the reference solutions (i.e., in terms of ~ui ui=uref i).

0 1

1

21

F 1

F 2

Fig. 4. Rectangular X, split into two domains X1 and X2.

Fig. 5. Square block under a uniform body load with two constrained

sides.

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

x/L

y / L

Fig. 6. Square block under a uniform body load with two constrainedsides: 101 101 node mesh employed to compute the reference solution.

We recall that there are two degrees of freedom per node.

Fig. 7. Square block under a uniform body load with two constrainedsides: reference solution for the rst displacement component (over the

deformed shape).

F. Auricchio et al. / Comput. Methods Appl. Mech. Engrg. 197 (2007) 160172 165


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In the last column of the table, we also report the numberof degrees of freedom effectively employed for thecomputations.

We highlight that, in order to impose the boundary con-ditions, we have to prescribe zero displacements (i.e. zerogradient components for the stream function) on a partof the boundary (the left and the bottom side) that we indi-cate with C. To do that, we have just to impose zero valuesfor all of the control points constituting C and for those

adjacent to it, as in this way the stream function is forcedto have zero normal and tangential derivative on C.Moreover, in Fig. 9, we report the control net and the

mesh obtained using p q 4 and 31 31 control points,while in Figs. 1012 we report contour plots of the solu-tion, obtained using such a mesh, in terms respectively of the stream function and of the displacement components.The latter are plotted over the deformed shape and theyshow the same behavior as in Figs. 7 and 8.

As it is possible to observe from the numerical results,good solutions are obtained even using low-order NURBSinterpolations over coarse grids.

5.2. Fully constrained square block with a trigonometric bodyload

The second test we consider consists of a fully con-strained square block occupying the region X L; L

Fig. 8. Square block under a uniform body load with two constrainedsides: reference solution for the second displacement component (over the

deformed shape).

Table 1Square block under a uniform body load with two constrained sides: normalized solutions at points A; B and C

Order c.p. grid ~u A x ~u A y ~u B x ~u B y ~uC x ~uC y # of d.o.f.s

2 2 11 11 0.941 0.968 0.969 0.966 0.976 0.956 8121 21 0.974 0.988 0.986 0.986 0.991 0.981 36131 31 0.984 0.993 0.991 0.992 0.995 0.989 841

11 11 0.965 0.985 0.979 0.983 0.988 0.974 813 3 21 21 0.988 0.996 0.993 0.994 0.996 0.992 361

31 31 0.994 0.997 0.997 0.998 0.998 0.997 84111 11 0.972 0.989 0.988 0.990 0.991 0.979 81

4 4 21 21 0.992 0.997 0.997 0.997 0.998 0.997 36131 31 0.996 0.999 0.999 0.999 0.999 0.999 841

Fig. 9. Square block under a uniform body load with two constrained sides: control net and mesh obtained using p q 4 and 31 31 control points for

the NURBS-based method. We recall that there is only one degree of freedom per control point.



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L; L. We choose a particular body load f for which thecorresponding analytical solution u is available (cf. [5]).

For this problem we set

L p=2; f 1 l cos y sin y 1 4cos2 x 2 xy cos x2 y ; f 2 l cos x sin x1 4cos2 y x2 cos x2 y ;

such that the analytical solution in terms of displacementcomponents is

u1 cos2 x cos y sin y

2 ;

u2 cos2 y cos xsin x

2 ;

while the solution in terms of the stream function w readsas

w cos2 x cos2 y

4 :

Analogously to the previous test, in Figs. 1315, we reportcontour plots of the solution, obtained using p q 4 and31 31 control points, in terms of the stream function andof the displacement components. We remark that in thiscase we have to impose zero displacement conditions onthe whole boundary oX; this is obtained by prescribingthe stream function to have zero values for all of theboundary control points and for those adjacent to oX.

In Figs. 16 and 17, we also report the convergence plots

for the two components of the displacement eld when

Fig. 10. Square block under a uniform body load with two constrainedsides: stream function obtained using p q 4 and 31 31 control

points.

Fig. 11. Square block under a uniform body load with two constrainedsides: rst displacement component (over the deformed shape) obtainedusing p q 4 and 31 31 control points.

Fig. 12. Square block under a uniform body load with two constrainedsides: second displacement component (over the deformed shape) obtained

using p q 4 and 31 31 control points.

Fig. 13. Fully constrained square block with a trigonometric body load:stream function obtained using p q 4 and 31 31 control points.



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using cubic and quartic approximations along each direc-

tion. We remark that, here and in the following, we com-pute errors using the L2-norm (denoted by k k0). It ispossible to observe (see also Table 2) that, even if forcoarse grids we get higher rates of convergence thanexpected, such rates tend to the expected theoretical values(i.e., 3 and 4 respectively) as the grid gets ner and ner.

Moreover, in Fig. 18, we study the convergence of thesolution as a function of the order of approximation over

a 21 21 control point grid. As expected, an exponentialrate of convergence with respect to the employed order isobtained.

Then, as we are interested in a comparison between theperformance of the proposed approach with standard niteelement techniques, in Fig. 19 we show the convergenceplots of Figs. 16 and 17 as compared with the ones pre-sented in [5,16] and obtained using some mixed-enhancednite element formulations. The plots are in terms of rela-tive L2-norm errors of the displacement modulus, consid-ered as a function of:

Fig. 14. Fully constrained square block with a trigonometric body load:rst displacement component obtained using p q 4 and 31 31

control points.

Fig. 15. Fully constrained square block with a trigonometric body load:second displacement component obtained using p q 4 and 31 31control points.

101.1

101.3

101.5

101.7

106

105

104

103

102

n (# of c.p. per direction)

| | u

h u

e x | | 0

1st component error2nd component errorreference (n 3 /5)

Fig. 16. Fully constrained square block with a trigonometric body load:solution convergence for p q 3.

101.1

101.3

101.5

101.7

108

107

106

105

104

103


| | u

h u

e x

| | 0


Fig. 17. Fully constrained square block with a trigonometric body load:solution convergence for p q 4.

Table 2Fully constrained square block with a trigonometric body load: slope sequence of the piece-wise linear curves in Fig. 16 ( p q 3) and Fig. 17 ( p q 4)

p q 3 4.113 3.702 3.505 3.394 3.323 3.274 3.237 3.210

p q 4 6.286 5.338 4.942 4.727 4.592 4.499 4.432 4.380



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(1) the number of control points per direction, for theNURBS discretizations of the stream functionformulation;

(2) the number of nodes per direction, for the nite ele-ments discretizations essentially based on formula-tion (2).

It is worth noticing that:

(1) control points in the adopted NURBS formulationand nodes in mixed-enhanced nite elements do nothave an equal weight in terms of degrees of freedom,since for the new approach the unknown eld is sca-

lar. Namely, a control point corresponds to only one

degree of freedom, while in the case of the mixed-enhanced nite elements each node corresponds tothree degrees of freedom (two for the displacementcomponents, one for the pressure).

(2) The stiffness matrix corresponding to the NURBSapproach is less sparse than the one arising fromnite element procedures, due to the larger supportof NURBS basis functions. In addition, a larger con-dition number is expected, as a result of the discreti-zation of a fourth-order problem.

We may reasonably assume that these two effects willeven up in terms of computational costs. Therefore, wethink that Fig. 19 represents a fair preliminary comparison.

Accordingly, the formulation proposed in this papershows a superior behavior with respect to all the nite ele-ment approaches studied in [5].

We nally solve this example also using a multipatchapproach, in order to test the formulation proposed in Sec-

tion 4.2.

2 3 4 5 610

8

107

106

105

104

103

102

p = q

| | u

h u

e x | | 0

1st component error2nd component errorreference (10 (p+2) )

Fig. 18. Fully constrained square block with a trigonometric body load:solution convergence as a function of the order of approximation over a21 21 control point grid.

101

102

106

104

102

100

n (# of nodes or c.p. per direction)

| | u

h u

e x

| | 0

/ | | u | |

0

PantusoBathetriang. mixenh 1triang. mixenh 2stream p=q=3stream p=q=4

Fig. 19. Fully constrained square block with a trigonometric body load:solution convergence for different mixed-enhanced formulations (cf. [5,16],3 d.o.f.s per node) and for the proposed approach using cubic and quarticapproximation (1 d.o.f. per control point).

101.1

101.2

101.3

101.4

101.5

101.6

106

105

104

103

102


| | u

n u

e x | | 0


Fig. 20. Fully constrained square block with a trigonometric body load:solution convergence for p q 3 for the two-patch case.

R 2

R 1

Fig. 21. Fully constrained quarter of an annulus with a polynomial bodyload: domain geometry.



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Fig. 22. Fully constrained quarter of an annulus with a polynomial body load: control net and mesh obtained using p q 2 and 4 4 control points.

Fig. 23. Fully constrained quarter of an annulus with a polynomial body load: control net and mesh obtained using p q 4 and 38 36 control points.

Fig. 24. Fully constrained quarter of an annulus with a polynomial bodyload: stream function obtained using p q 4 and 38 36 control

points.

Fig. 25. Fully constrained quarter of an annulus with a polynomial bodyload: rst displacement component obtained using p q 4 and 38 36

control points.



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The problem consists of dividing the square domainX L; L L; L into two subdomains X1 L; 0 L; L and X2 0; L L; L, sharing thecommon interface C f0g L; L (cf. Fig. 4). The for-mulation is then modied by the introduction of the inte-grals along C, as prescribed by (32), and choosing the

stabilizing constant C S 10.We conclude this example showing the convergenceplots for cubic approximation ( Fig. 20), where it is shownthat also for the multipatch solution the expected conver-gence rate is recovered.

5.3. Fully constrained quarter of an annulus witha polynomial body load

As a last numerical experiment, we focus on the behav-ior of the method when studying a problem on a domainthat cannot be represented exactly with standard nite ele-

ments. The selected domain X is the quarter of an annulusreported in Fig. 21, which is considered to be fully con-strained along its boundary oX. We remark that, again,boundary conditions are imposed by prescribing the streamfunction to have zero values for all of the boundary controlpoints and for those adjacent to oX.

Also for this example, we choose a particular body loadf for which the corresponding analytical solution u is avail-able. We set

R1 1; R2 4;

f 1 2 10 6 y 2l 91485 x4 y 2 2296 x6 y 4 2790 x4 y 6

7680 x2

645 x8

y 2

15470 x6

y 2

3808 y 4

2889 y 6

1280 y 2 36414 x4 y 4 107856 x2 y 4 13 y 10 374 y 8

1122 y 8 x2 22338 y 6 x2 16320 x4 73440 x2 y 2

9630 x6 1020 x8 30 x10;

f 2 8 10 6 y 3 xl 7704 y 4 1280 258 x6 y 2 492 x4 y 4

4641 x4 y 2 310 y 6 x2 5202 x2 y 4 25 x8 680 x6

4815 x4 51 y 8 1241 y 6 18297 x2 y 2 5440 x2

7344 y 2;

such that the analytical solution in terms of displacementcomponents is

u1 10 6 x2 y 4 x2 y 2 16 x2 y 2 1 5 x4 18 x2 y 2 85 x2 13 y 4 80 153 y 2;

u2 2 10 6 xy 5 x2 y 2 16 x2 y 2 1 5 x4 51 x2 6 x2 y 2 17 y 2 16 y 4;

while the solution in terms of the stream function w readsas

Fig. 26. Fully constrained quarter of an annulus with a polynomial bodyload: second displacement component obtained using p q 4 and

38 36 control points.

101

104

103

102

101

100

101

n (square root of total # of control points)

| | u

h u

e x

| | 0

1st

component error2nd component errorreference (50n 3 )

Fig. 27. Fully constrained quarter of an annulus with a polynomial bodyload: solution convergence for p q 3.

101

106

105

104

103

102

101

100

n (square root of total # of control points)

| | u

h u

e x

| | 2

1st component error

2nd component errorreference (100n 4 )

Fig. 28. Fully constrained quarter of an annulus with a polynomial body

load: solution convergence for p q 4.



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a fully locking free isogeometric approach for plane linear elasticity problems a stream function...

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